import PhysicsLogic.Papers.AdamsEtAl2006.Fidelity import Lean.Elab.Command import Lean.Util.CollectAxioms import Mathlib.Probability.Distributions.Gaussian.Real import Mathlib.MeasureTheory.Group.Prod import Mathlib.MeasureTheory.Integral.Prod import Mathlib.Analysis.Calculus.FDeriv.Pow import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv import Mathlib.Analysis.SpecialFunctions.NonIntegrable import Mathlib.Tactic.NormNum.RealSqrt import Mathlib.LinearAlgebra.Matrix.Trace import PhysicsLogic.Papers.Adler1969.Fidelity import PhysicsLogic.Papers.Rosen1940.Fidelity import PhysicsLogic.Papers.CallanMaldacena1996.Fidelity import PhysicsLogic.Papers.DeLaOssaQuevedo1992.Fidelity import PhysicsLogic.Papers.RocekVerlinde1991.Fidelity import PhysicsLogic.Papers.SeibergDuality1994.Fidelity import PhysicsLogic.Papers.WittenBaryons1979.Fidelity import PhysicsLogic.Papers.WittenDynamicalBreaking1981.Fidelity import PhysicsLogic.Papers.GubserKlebanovPolyakov1998.Fidelity import PhysicsLogic.Papers.HorowitzStrominger1996.Fidelity import PhysicsLogic.Papers.ItzhakiMaldacenaSonnenscheinYankielowicz1998.Fidelity import PhysicsLogic.Papers.KutasovSchwimmer1995.Fidelity import PhysicsLogic.Papers.LewkowyczMaldacena2013.Fidelity import PhysicsLogic.Papers.Polchinski1995.Fidelity import PhysicsLogic.Papers.StromingerVafa1996.Fidelity import PhysicsLogic.Papers.TownsendSupermembrane1995.Fidelity import PhysicsLogic.Papers.WittenHolography1998.Fidelity import PhysicsLogic.Papers.WittenStringDynamics1995.Fidelity import PhysicsLogic.Papers.AbrahamsAndersonLicciardelloRamakrishnan1979.Fidelity import PhysicsLogic.Papers.MacKinnonKramer1981.Fidelity import PhysicsLogic.Papers.Young1979.Fidelity import PhysicsLogic.Papers.HawkingOpenUniverses1965.Fidelity import PhysicsLogic.Papers.Vishveshwara1970.Fidelity import PhysicsLogic.Papers.EnglertBrout1964.Fidelity import PhysicsLogic.Papers.LiuSinclairSrivastava2018.Fidelity import PhysicsLogic.Papers.PetersRegts2017.Fidelity import PhysicsLogic.Papers.AharonovBohm1959.Fidelity import PhysicsLogic.Papers.AlbrechtSteinhardt1982.Fidelity import PhysicsLogic.Papers.AMPS2012.Fidelity import PhysicsLogic.Papers.Anderson1963.Fidelity import PhysicsLogic.Papers.AndersonBergmann1951.Fidelity import PhysicsLogic.Papers.AndersonLocalization1958.Fidelity import PhysicsLogic.Papers.ArovasSchriefferWilczek1984.Fidelity import PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity import PhysicsLogic.Papers.BedaqueHammerVanKolck1998.Fidelity import PhysicsLogic.Papers.Bekenstein1973.Fidelity import PhysicsLogic.Papers.Bell1964.Fidelity import PhysicsLogic.Papers.Bell1966.Fidelity import PhysicsLogic.Papers.BernsteinGreeneKruskal1957.Fidelity import PhysicsLogic.Papers.Berry1984.Fidelity import PhysicsLogic.Papers.Bethe1949.Fidelity import PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.Fidelity import PhysicsLogic.Papers.Blasius1908.Fidelity import PhysicsLogic.Papers.BlattJackson1949.Fidelity import PhysicsLogic.Papers.Bohr1935.Fidelity import PhysicsLogic.Papers.Bohm1952 import PhysicsLogic.Papers.Bohm1952.Fidelity import PhysicsLogic.Papers.BognerFurnstahlPerry2007.Fidelity import PhysicsLogic.Papers.BogoliubovSuperfluidity1947.Fidelity import PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.Fidelity import PhysicsLogic.Papers.BornCollisions1926.Fidelity import PhysicsLogic.Papers.CalderbankShor1996.Fidelity import PhysicsLogic.Papers.CallenWelton1951.Fidelity import PhysicsLogic.Papers.Casimir1945.Fidelity import PhysicsLogic.Papers.Chandrasekhar1931.Fidelity import PhysicsLogic.Papers.CHSH1969.Fidelity import PhysicsLogic.Papers.Choptuik1993.Fidelity import PhysicsLogic.Papers.CoffmanKunduWootters2000.Fidelity import PhysicsLogic.Papers.Creutz1980.Fidelity import PhysicsLogic.Papers.Crooks1999.Fidelity import PhysicsLogic.Papers.Dieks1982.Fidelity import PhysicsLogic.Papers.Dirac1933.Fidelity import PhysicsLogic.Papers.Dirac1950.Fidelity import PhysicsLogic.Papers.DurrGoldsteinZanghi1992.Fidelity import PhysicsLogic.Papers.Eardley1975.Fidelity import PhysicsLogic.Papers.Einstein1911.Fidelity import PhysicsLogic.Papers.Einstein1916.Fidelity import PhysicsLogic.Papers.EinsteinBrownian1905.Fidelity import PhysicsLogic.Papers.EinsteinSpecialRelativity1905.Fidelity import PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.Fidelity import PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.Fidelity import PhysicsLogic.Papers.EPR1935.Fidelity import PhysicsLogic.Papers.EvansColeman1994.Fidelity import PhysicsLogic.Papers.EvansCohenMorriss1993 import PhysicsLogic.Papers.EvansCohenMorriss1993.Fidelity import PhysicsLogic.Papers.Everett1957 import PhysicsLogic.Papers.Everett1957.Fidelity import PhysicsLogic.Papers.FermiPastaUlam1955.Fidelity import PhysicsLogic.Papers.Feynman1948.Fidelity import PhysicsLogic.Papers.FeynmanCohen1956.Fidelity import PhysicsLogic.Papers.FeynmanTwoFluid1954.Fidelity import PhysicsLogic.Papers.FixsenEtAl1996.Fidelity import PhysicsLogic.Papers.FixsenEtAl1996 import PhysicsLogic.Papers.ForsterNelsonStephen1977.Fidelity import PhysicsLogic.Papers.Friedmann1922.Fidelity import PhysicsLogic.Papers.FuKane2009.Fidelity import PhysicsLogic.Papers.GoldstoneSalamWeinberg1962.Fidelity import PhysicsLogic.Papers.GallavottiCohen1995 import PhysicsLogic.Papers.GallavottiCohen1995.Fidelity import PhysicsLogic.Papers.Gamow1928.Fidelity import PhysicsLogic.Papers.GarwinLedermanWeinrich1957.Fidelity import PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.Fidelity import PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.Fidelity import PhysicsLogic.Papers.GlazekWilson1993.Fidelity import PhysicsLogic.Papers.GRW1986.Fidelity import PhysicsLogic.Papers.GurneyCondon1928.Fidelity import PhysicsLogic.Papers.Guth1981.Fidelity import PhysicsLogic.Papers.Haldane1983.Fidelity import PhysicsLogic.Papers.Haldane1988.Fidelity import PhysicsLogic.Papers.HalperinNelson1978.Fidelity import PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity import PhysicsLogic.Papers.Hannay1985.Fidelity import PhysicsLogic.Papers.HastingsKoma2006.Fidelity import PhysicsLogic.Papers.HaxelJensenSuess1949.Fidelity import PhysicsLogic.Papers.Hawking1975.Fidelity import PhysicsLogic.Papers.Heisenberg1927.Fidelity import PhysicsLogic.Papers.Higgs1964.Fidelity import PhysicsLogic.Papers.HohenbergKohn1964.Fidelity import PhysicsLogic.Papers.Hubble1929.Fidelity import PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.Fidelity import PhysicsLogic.Papers.JacobiPDEReduction1837.Fidelity import PhysicsLogic.Papers.Jarzynski1997.Fidelity import PhysicsLogic.Papers.Jaynes1957.Fidelity import PhysicsLogic.Papers.Kadanoff1966.Fidelity import PhysicsLogic.Papers.KaplanSavageWise1998.Fidelity import PhysicsLogic.Papers.Kennard1927.Fidelity import PhysicsLogic.Papers.Kerr1963.Fidelity import PhysicsLogic.Papers.Kibble1976.Fidelity import PhysicsLogic.Papers.Kitaev2001.Fidelity import PhysicsLogic.Papers.Knox1995.Fidelity import PhysicsLogic.Papers.KochenSpecker1967.Fidelity import PhysicsLogic.Papers.KohnSham1965.Fidelity import PhysicsLogic.Papers.KolmogorovLocalStructure1941.Fidelity import PhysicsLogic.Papers.KomargodskiSchwimmer2011.Fidelity import PhysicsLogic.Papers.KortewegDeVries1895.Fidelity import PhysicsLogic.Papers.KosterlitzThouless1973.Fidelity import PhysicsLogic.Papers.Kubo1957.Fidelity import PhysicsLogic.Papers.LandauDamping1946.Fidelity import PhysicsLogic.Papers.LandauSuperfluidity1941.Fidelity import PhysicsLogic.Papers.Laughlin1983.Fidelity import PhysicsLogic.Papers.LaughlinHall1981.Fidelity import PhysicsLogic.Papers.LeeYang1952.Fidelity import PhysicsLogic.Papers.Lemaitre1927.Fidelity import PhysicsLogic.Papers.LiebRobinson1972.Fidelity import PhysicsLogic.Papers.LindeLindeMezhlumian1993.Fidelity import PhysicsLogic.Papers.Lorenz1963.Fidelity import PhysicsLogic.Papers.Maldacena1997.Fidelity import PhysicsLogic.Papers.Mayer1949.Fidelity import PhysicsLogic.Papers.MerminWagner1966.Fidelity import PhysicsLogic.Papers.Metropolis1953.Fidelity import PhysicsLogic.Papers.Miller1964.Fidelity import PhysicsLogic.Papers.Minkowski1908.Fidelity import PhysicsLogic.Papers.Mori1965.Fidelity import PhysicsLogic.Papers.NachtergaeleSims2006.Fidelity import PhysicsLogic.Papers.Nakano1956.Fidelity import PhysicsLogic.Papers.Nambu1960.Fidelity import PhysicsLogic.Papers.NiuThoulessWu1985.Fidelity import PhysicsLogic.Papers.Noether1918.Fidelity import PhysicsLogic.Papers.Nordtvedt1968.Fidelity import PhysicsLogic.Papers.Nyquist1928.Fidelity import PhysicsLogic.Papers.Onsager1931.Fidelity import PhysicsLogic.Papers.OppenheimerVolkoff1939.Fidelity import PhysicsLogic.Papers.OrdonezRayVanKolck1995.Fidelity import PhysicsLogic.Papers.OregRefaelVonOppen2010.Fidelity import PhysicsLogic.Papers.Ozawa2003.Fidelity import PhysicsLogic.Papers.Parker1969.Fidelity import PhysicsLogic.Papers.Penrose1965.Fidelity import PhysicsLogic.Papers.PerlmutterEtAl1999.Fidelity import PhysicsLogic.Papers.PenziasWilson1965.Fidelity import PhysicsLogic.Papers.PortegiesZwartBoekholt2014.Fidelity import PhysicsLogic.Papers.PoundRebka1960.Fidelity import PhysicsLogic.Papers.Prandtl1904.Fidelity import PhysicsLogic.Papers.PressSchechter1974.Fidelity import PhysicsLogic.Papers.Rabi1939.Fidelity import PhysicsLogic.Papers.Ramsey1950.Fidelity import PhysicsLogic.Papers.Raychaudhuri1955.Fidelity import PhysicsLogic.Papers.ReggeWheeler1957.Fidelity import PhysicsLogic.Papers.RiessEtAl1998.Fidelity import PhysicsLogic.Papers.Robertson1929.Fidelity import PhysicsLogic.Papers.Rovelli1996 import PhysicsLogic.Papers.Rovelli1996.Fidelity import PhysicsLogic.Papers.RungeGross1984.Fidelity import PhysicsLogic.Papers.RyuTakayanagi2006.Fidelity import PhysicsLogic.Papers.SeibergWitten1994.Fidelity import PhysicsLogic.Papers.Schwarzschild1916.Fidelity import PhysicsLogic.Papers.ScottSrednickiWhite1993.Fidelity import PhysicsLogic.Papers.SmoluchowskiBrownian1906.Fidelity import PhysicsLogic.Papers.Stanley1968.Fidelity import PhysicsLogic.Papers.Steane1996.Fidelity import PhysicsLogic.Papers.Thomas1935.Fidelity import PhysicsLogic.Papers.ThoulessTransport1983.Fidelity import PhysicsLogic.Papers.THooft1974.Fidelity import PhysicsLogic.Papers.TKNN1982.Fidelity import PhysicsLogic.Papers.WegnerHoughton1973.Fidelity import PhysicsLogic.Papers.Weinberg1987.Fidelity import PhysicsLogic.Papers.Wetterich1993.Fidelity import PhysicsLogic.Papers.WilczekFractionalSpin1982.Fidelity import PhysicsLogic.Papers.Wilson1974.Fidelity import PhysicsLogic.Papers.WittenIndex1982.Fidelity import PhysicsLogic.Papers.WoottersZurek1982.Fidelity import PhysicsLogic.Papers.WuEtAl1957.Fidelity import PhysicsLogic.Papers.ZabuskyKruskal1965 import PhysicsLogic.Papers.ZabuskyKruskal1965.Fidelity import PhysicsLogic.Papers.Zamolodchikov1986.Fidelity import PhysicsLogic.Papers.Zeh1970.Fidelity import PhysicsLogic.Papers.Zerilli1970.Fidelity import PhysicsLogic.Papers.Zwanzig1973.Fidelity /-! # Independent Client Reuse Checks These examples live outside the active library. They use public paper and paradigm interfaces as a downstream consumer would, while preserving the original conditional premises and source-specific conclusion boundaries. -/ set_option autoImplicit false namespace DownstreamDependencyAudit open Lean Elab Command /-- Match the immediate edges in the pinned Lean/Util/CollectAxioms.lean (Leonardo de Moura, Microsoft, Apache-2.0), including checked opaque values, inductive constructors, and the collector's intentionally empty quotient case. -/ def directDependencies (environment : Environment) (name : Name) : Array Name := match environment.checked.get.find? name with | some (.axiomInfo value) => value.type.getUsedConstants | some (.defnInfo value) => value.type.getUsedConstants ++ value.value.getUsedConstants | some (.thmInfo value) => value.type.getUsedConstants ++ value.value.getUsedConstants | some (.opaqueInfo value) => value.type.getUsedConstants ++ value.value.getUsedConstants | some (.quotInfo _) => #[] | some (.ctorInfo value) => value.type.getUsedConstants | some (.recInfo value) => value.type.getUsedConstants | some (.inductInfo value) => value.type.getUsedConstants ++ value.ctors.toArray | none => #[] /-- Direct evidence avoids constructing the entire transitive closure. All other queries still use Lean's collector from a fresh state in the current environment. This is only an API-use query, never a replacement for a complete trust audit. -/ def usesConstant (environment : Environment) (declaration required : Name) : Bool := if declaration == required || (directDependencies environment declaration).contains required then true else let (_, dependencies) := ((Lean.CollectAxioms.collect declaration).run environment).run {} dependencies.visited.contains required namespace Examples def leftApi : Nat := 37 def rightApi : Nat := 41 def leftHelper : Nat := leftApi def leftClient : Nat := leftHelper def rightClient : Nat := rightApi abbrev TypeApi := Nat def typeClient : TypeApi := Nat.zero opaque opaqueClient : Nat := leftHelper theorem theoremClient : leftClient = 37 := rfl inductive Tree where | leaf | branch : Tree -> Tree -> Tree def treeClient : Tree := .branch .leaf .leaf def treeSize : Tree -> Nat | .leaf => 1 | .branch first second => treeSize first + treeSize second end Examples private def checkDependencyQueryExamples : CommandElabM Unit := do let environment <- getEnv let roots := #[``Examples.leftApi, ``Examples.leftHelper, ``Examples.leftClient, ``Examples.rightClient, ``Examples.typeClient, ``Examples.opaqueClient, ``Examples.theoremClient, ``Examples.Tree, ``Examples.Tree.branch, ``Examples.Tree.rec, ``Examples.treeClient, ``Examples.treeSize, ``Quot, ``Quot.lift, ``Quot.ind, ``propext, ``Classical.choice, ``Quot.sound, `MissingDependencyExample] let targets := #[``Examples.leftApi, ``Examples.rightApi, ``Examples.leftHelper, ``Examples.leftClient, ``Examples.TypeApi, ``Examples.Tree, ``Examples.Tree.branch, ``Examples.Tree.rec, ``Nat, ``Quot, ``Iff, ``propext, ``Classical.choice, ``Quot.sound, `MissingDependencyExample, `EntirelyUnreachable] for root in roots do let (_, reference) := ((Lean.CollectAxioms.collect root).run environment).run {} for target in targets do unless usesConstant environment root target == reference.visited.contains target do throwError "Dependency query disagrees with kernel collector: {root}, {target}" let some typeInfo := environment.checked.get.find? ``Examples.typeClient | throwError "Missing type-only regression declaration" unless typeInfo.type.getUsedConstants.contains ``Examples.TypeApi do throwError "Type-only regression does not retain its named type" if (typeInfo.value?.getD (.sort .zero)).getUsedConstants.contains ``Examples.TypeApi then throwError "Type-only regression also mentions its API in the value" if (directDependencies environment ``Examples.leftClient).contains ``Examples.leftApi then throwError "Transitive regression unexpectedly has a direct API edge" unless usesConstant environment ``Examples.leftClient ``Examples.leftApi do throwError "Dependency query lost a transitive value dependency" if usesConstant environment ``Examples.rightClient ``Examples.leftApi then throwError "Dependency query leaked another client's API evidence" unless (directDependencies environment ``Examples.opaqueClient).contains ``Examples.leftHelper do throwError "Dependency query omitted an opaque value" unless (directDependencies environment ``Examples.Tree).contains ``Examples.Tree.branch do throwError "Dependency query omitted an inductive constructor" unless (directDependencies environment ``Quot.lift).isEmpty do throwError "Dependency query expanded a quotient primitive" logInfo m!"Downstream dependency-query differential examples passed: {roots.size * targets.size} comparisons." run_cmd checkDependencyQueryExamples end DownstreamDependencyAudit namespace PhysicsLogic.IndependentReuse open PhysicsLogic.Paradigm.FluctuationsAndArrow open PhysicsLogic.Paradigm.QuantumNonseparability.Countermodels open PhysicsLogic.Framework.BipartiteEntanglementContext.Countermodels example : PhysicsLogic.Logic.PaperFidelity.independentClaimLedgersComplete PhysicsLogic.Papers.BardeenCooperSchrieffer1957.routedArgument.argument PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.claimDependencyLedger PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.claimEpistemicLedger = true := PhysicsLogic.Logic.PaperFidelity.static_checks_expose_complete_independent_claim_ledgers PhysicsLogic.Papers.BardeenCooperSchrieffer1957.routedArgument PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.claimDependencyLedger PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.claimEpistemicLedger PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.routeDependencyLedger PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.static_fidelity_checks_pass example : PhysicsLogic.Logic.PaperFidelity.independentRouteLedgerComplete PhysicsLogic.Papers.HamiltonGeneralMethod1834.routedArgument PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.routeDependencyLedger = true := PhysicsLogic.Logic.PaperFidelity.static_checks_expose_complete_independent_route_ledger PhysicsLogic.Papers.HamiltonGeneralMethod1834.routedArgument PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.claimDependencyLedger PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.claimEpistemicLedger PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.routeDependencyLedger PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.static_fidelity_checks_pass example : PhysicsLogic.Logic.PaperFidelity.independentClaimLedgersComplete PhysicsLogic.Papers.Jaynes1957.routedArgument.argument PhysicsLogic.Papers.Jaynes1957.Fidelity.claimDependencyLedger PhysicsLogic.Papers.Jaynes1957.Fidelity.claimEpistemicLedger = true := PhysicsLogic.Logic.PaperFidelity.static_checks_expose_complete_independent_claim_ledgers PhysicsLogic.Papers.Jaynes1957.routedArgument PhysicsLogic.Papers.Jaynes1957.Fidelity.claimDependencyLedger PhysicsLogic.Papers.Jaynes1957.Fidelity.claimEpistemicLedger PhysicsLogic.Papers.Jaynes1957.Fidelity.routeDependencyLedger PhysicsLogic.Papers.Jaynes1957.Fidelity.static_fidelity_checks_pass example {T : PhysicsLogic.Papers.BardeenCooperSchrieffer1957.BCSTypes} (context : PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SemanticContext T) (certificate : PhysicsLogic.Logic.PaperFidelity.Certificate PhysicsLogic.Papers.BardeenCooperSchrieffer1957.routedArgument context.interpretAtom PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.claimDependencyLedger PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.claimEpistemicLedger PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.routeDependencyLedger PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.countermodelMatrix) : PhysicsLogic.Logic.PaperFidelity.independentClaimLedgersComplete PhysicsLogic.Papers.BardeenCooperSchrieffer1957.routedArgument.argument PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.claimDependencyLedger PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.claimEpistemicLedger = true := PhysicsLogic.Logic.PaperFidelity.certificate_exposes_complete_independent_claim_ledgers PhysicsLogic.Papers.BardeenCooperSchrieffer1957.routedArgument context.interpretAtom PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.claimDependencyLedger PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.claimEpistemicLedger PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.routeDependencyLedger PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Fidelity.countermodelMatrix certificate example {T : PhysicsLogic.Papers.HamiltonGeneralMethod1834.HamiltonTypes} (context : PhysicsLogic.Papers.HamiltonGeneralMethod1834.SemanticContext T) (certificate : PhysicsLogic.Logic.PaperFidelity.Certificate PhysicsLogic.Papers.HamiltonGeneralMethod1834.routedArgument context.interpretAtom PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.claimDependencyLedger PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.claimEpistemicLedger PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.routeDependencyLedger PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.countermodelMatrix) : PhysicsLogic.Logic.PaperFidelity.independentRouteLedgerComplete PhysicsLogic.Papers.HamiltonGeneralMethod1834.routedArgument PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.routeDependencyLedger = true := PhysicsLogic.Logic.PaperFidelity.certificate_exposes_complete_independent_route_ledger PhysicsLogic.Papers.HamiltonGeneralMethod1834.routedArgument context.interpretAtom PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.claimDependencyLedger PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.claimEpistemicLedger PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.routeDependencyLedger PhysicsLogic.Papers.HamiltonGeneralMethod1834.Fidelity.countermodelMatrix certificate example : ∃ boundary, PhysicsLogic.Papers.FermiPastaUlam1955.Fidelity.countermodelMatrix boundary := PhysicsLogic.Logic.PaperFidelity.inhabited_countermodel_matrix_exhibits_realized_boundary PhysicsLogic.Papers.FermiPastaUlam1955.Fidelity.countermodelMatrix ⟨PhysicsLogic.Papers.FermiPastaUlam1955.Fidelity.BoundaryId.modalNearReturnNotExactFullStateRecurrence⟩ PhysicsLogic.Papers.FermiPastaUlam1955.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.WittenIndex1982.SpectralTraceCountermodels.logarithmicTower.index = 1 ∧ ¬PhysicsLogic.Papers.WittenIndex1982.SpectralTraceCountermodels.logarithmicTower.HeatTraceDefinedAt 1 := by have spectrum := PhysicsLogic.Papers.WittenIndex1982.SpectralTraceCountermodels.finite_energy_sublevels_and_positive_pairing_do_not_supply_every_heat_trace exact ⟨spectrum.2.2.1, spectrum.2.2.2⟩ example : let witness := PhysicsLogic.Papers.WittenIndex1982.PeriodicCartanGaugeReadout.unquotientedCoordinateWitness witness.HasSourceGaugeTranslation ∧ ¬witness.IdentifiesPhysicalGaugeOrbit := PhysicsLogic.Papers.WittenIndex1982.PeriodicCartanGaugeReadout.periodic_gauge_translation_does_not_identify_an_unquotiented_wavefunction example : let component := PhysicsLogic.Papers.WittenIndex1982.CartanFock.vOperator 1 (PhysicsLogic.Papers.WittenIndex1982.CartanFock.componentSpins 1 0) component ≠ 0 ∧ component * component = 0 := PhysicsLogic.Papers.WittenIndex1982.CartanFock.a_nonzero_rank_one_v_component_can_have_zero_square example : let orbit := PhysicsLogic.Papers.WittenIndex1982.TwistedGaugeOrbitReadout.twoColorOrbitWitness orbit.nthPowerWinding = 1 ∧ orbit.totalParallelSectorIndex = 2 := by constructor · rfl · exact PhysicsLogic.Papers.WittenIndex1982.TwistedGaugeOrbitReadout.source_magnetic_orbit_has_total_index_equal_to_color_count PhysicsLogic.Papers.WittenIndex1982.TwistedGaugeOrbitReadout.twoColorOrbitWitness 2 PhysicsLogic.Papers.WittenIndex1982.TwistedGaugeOrbitReadout.two_color_orbit_witness_realizes_source_magnetic_orbit example : PhysicsLogic.Papers.WittenIndex1982.DeformationCountermodels.PointwiseContinuousGradedFamilyAt 2 ∧ (PhysicsLogic.Papers.WittenIndex1982.DeformationCountermodels.quantumMechanics 2 (1 / 3)).gradedKernelIndex ≠ (PhysicsLogic.Papers.WittenIndex1982.DeformationCountermodels.quantumMechanics 2 0).gradedKernelIndex := by obtain ⟨family, _normalizable, _escapes, changes⟩ := PhysicsLogic.Papers.WittenIndex1982.DeformationCountermodels.pointwise_factorized_deformation_does_not_protect_the_normalizable_kernel_index 2 (by norm_num) exact ⟨family, changes (1 / 3) (by norm_num)⟩ example : PhysicsLogic.Framework.EquilibriumNoiseDissipationContext.Countermodels.missingEquilibriumNoiseCalculation.calculatedNoisePower ≠ PhysicsLogic.Framework.EquilibriumNoiseDissipationContext.Countermodels.missingEquilibriumNoiseCalculation.requiredEquilibriumNoisePower := by have noCalculation := PhysicsLogic.Framework.EquilibriumNoiseDissipationContext.Countermodels.equilibrium_dissipative_setup_does_not_supply_noise_calculation.2 exact noCalculation example : PhysicsLogic.Framework.ResonantTransitionSpectroscopyContext.Countermodels.driveWithoutCalculatedTransitionReadout.calculatedTransitionCount ≠ PhysicsLogic.Framework.ResonantTransitionSpectroscopyContext.Countermodels.driveWithoutCalculatedTransitionReadout.appliedDriveAmplitude := by have noCalculatedTransition := PhysicsLogic.Framework.ResonantTransitionSpectroscopyContext.Countermodels.controlled_drive_does_not_supply_transition_calculation.2 exact noCalculatedTransition example : PhysicsLogic.Framework.BoundaryMajoranaModeContext.Countermodels.localizedNonMajoranaReadout.localizationDistance < PhysicsLogic.Framework.BoundaryMajoranaModeContext.Countermodels.localizedNonMajoranaReadout.localizationLength ∧ PhysicsLogic.Framework.BoundaryMajoranaModeContext.Countermodels.localizedNonMajoranaReadout.conjugatedAmplitude ≠ PhysicsLogic.Framework.BoundaryMajoranaModeContext.Countermodels.localizedNonMajoranaReadout.originalAmplitude := by have witness := PhysicsLogic.Framework.BoundaryMajoranaModeContext.Countermodels.a_localized_zero_mode_need_not_be_a_majorana_mode exact ⟨witness.1, witness.2.2⟩ example : (PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.HigherFormCountermodels.defectExchangeReadout .commutingProductsOnly).firstFusionCharge = (PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.HigherFormCountermodels.defectExchangeReadout .commutingProductsOnly).secondFusionCharge ∧ (PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.HigherFormCountermodels.defectExchangeReadout .commutingProductsOnly).initialSupportPosition ≠ (PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.HigherFormCountermodels.defectExchangeReadout .commutingProductsOnly).continuousExchangeEndpoint := by exact PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.HigherFormCountermodels.commuting_ordered_products_do_not_supply_continuous_defect_exchange example : PhysicsLogic.Papers.Heisenberg1927.HeisenbergFramework.Countermodels.discontinuityWithoutBeamCorrection.DiscontinuousRecoilPresent ∧ ¬PhysicsLogic.Papers.Heisenberg1927.HeisenbergFramework.Countermodels.discontinuityWithoutBeamCorrection.CorrectedBeamDirectionMechanismPresent := PhysicsLogic.Papers.Heisenberg1927.HeisenbergFramework.Countermodels.discontinuous_recoil_alone_does_not_supply_corrected_microscope_mechanism namespace MeasurementPhysicalReuseAudit open PhysicsLogic.Philosophy.Measurement.CountermodelData open PhysicsLogic.Philosophy.Measurement.ValueIndicationContext.Countermodels open PhysicsLogic.Philosophy.Measurement.ReferenceCalibrationContext.Countermodels open PhysicsLogic.Philosophy.Measurement.StateChangeMeasurementContext.Countermodels example : noIndicatedRuns.performs .controlProcedure .controlDetector .idle .controlSystem .control .control .controlRecord .declared /\ noIndicatedRuns.recordIndicates .controlRecord .controlQuantity .actual .declared /\ Not (noIndicatedRuns.IndicatedRunAt .selectedProcedure .selectedQuantity .selectedSystem .declared) := control_run_exists_without_selected_value_indication example : biasedIndication.quantityHasValue .selectedQuantity .selectedSystem .prepared .actual /\ Not (biasedIndication.quantityHasValue .selectedQuantity .selectedSystem .prepared .reported) /\ biasedIndication.recordIndicates .selectedRecord .selectedQuantity .reported .declared /\ Not (biasedIndication.recordIndicates .selectedRecord .selectedQuantity .actual .declared) := biased_record_distinguishes_actual_and_indicated_values example : noReferenceRuns.prepares .referencePreparation .selectedSystem .prepared .declared /\ noReferenceRuns.performs .controlProcedure .controlDetector .idle .controlSystem .control .control .controlRecord .declared /\ Not (noReferenceRuns.CalibratedBy .selectedProcedure .selectedQuantity .referencePreparation .actual .selectedSystem .declared) := prepared_reference_and_control_run_do_not_create_selected_calibration example : changingRun.performs .selectedProcedure .calibratedDetector .probe .selectedSystem .prepared .disturbed .selectedRecord .declared /\ Not (changingRun.performs .selectedProcedure .calibratedDetector .idle .selectedSystem .prepared .disturbed .selectedRecord .declared) /\ Not (changingRun.stateEquivalent .prepared .disturbed .declared) := disturbance_distinguishes_intervention_states_and_detector_record end MeasurementPhysicalReuseAudit example : (PhysicsLogic.Papers.Heisenberg1927.HeisenbergFramework.Countermodels.microscopeWithoutSelectedGaussian.OperationalResultAvailable ∧ ¬PhysicsLogic.Papers.Heisenberg1927.HeisenbergFramework.Countermodels.microscopeWithoutSelectedGaussian.SelectedGaussianResultAvailable) ∧ (PhysicsLogic.Papers.Heisenberg1927.HeisenbergFramework.Countermodels.selectedGaussianWithoutMicroscope.SelectedGaussianResultAvailable ∧ ¬PhysicsLogic.Papers.Heisenberg1927.HeisenbergFramework.Countermodels.selectedGaussianWithoutMicroscope.OperationalResultAvailable) := PhysicsLogic.Papers.Heisenberg1927.HeisenbergFramework.Countermodels.microscope_and_selected_gaussian_results_are_logically_independent example {T : PhysicsLogic.Papers.Heisenberg1927.HeisenbergTypes} (framework : PhysicsLogic.Papers.Heisenberg1927.HeisenbergFramework T) (operational : framework.EstablishedOperationalUncertainty) (qualified : framework.EstablishedQualifiedScopeAndAddendum) (distinction : framework.EstablishedOperationalGaussianRoleDistinction) : framework.toOperationalErrorDisturbanceContext |>.EstablishedOperationalErrorDisturbanceAt framework.sharedOperationalErrorDisturbanceModel := framework.microscope_realizes_shared_operational_error_disturbance operational qualified distinction example : PhysicsLogic.Papers.Kennard1927.KennardFramework.Countermodels.gaussianWithoutGeneralFactorization.GaussianSaturatesBound ∧ ¬PhysicsLogic.Papers.Kennard1927.KennardFramework.Countermodels.gaussianWithoutGeneralFactorization.GeneralInequalityEstablished := PhysicsLogic.Papers.Kennard1927.KennardFramework.Countermodels.gaussian_equality_does_not_establish_general_inequality example : PhysicsLogic.Papers.Kennard1927.KennardFramework.Countermodels.dispersionWithoutDisturbanceAnalysis.HasStateDispersionBound ∧ ¬PhysicsLogic.Papers.Kennard1927.KennardFramework.Countermodels.dispersionWithoutDisturbanceAnalysis.HasObservedDisturbance := PhysicsLogic.Papers.Kennard1927.KennardFramework.Countermodels.a_state_dispersion_formula_does_not_supply_measurement_disturbance example {T : PhysicsLogic.Papers.Kennard1927.KennardTypes} (framework : PhysicsLogic.Papers.Kennard1927.KennardFramework T) (precision : framework.PrecisionSetup) (calculation : framework.GeneralPositivityCalculation) : framework.toStateDispersionUncertaintyContext |>.EstablishedStateDispersionUncertaintyAt framework.sharedStateDispersionModel := framework.kennard_general_inequality_realizes_shared_state_dispersion_uncertainty precision calculation example {T : PhysicsLogic.Papers.Kennard1927.KennardTypes} (framework : PhysicsLogic.Papers.Kennard1927.KennardFramework T) (alternative : framework.EstablishedAlternativeExactObservationInterpretation) : framework.toSharedAmplitudeOutcomeProbabilityContext |>.EstablishedAmplitudeOutcomeProbabilityAt framework.selectedAmplitudeOutcomeProbabilityModel := framework.kennard_probability_curve_realizes_shared_amplitude_outcome_probability alternative example {T : PhysicsLogic.Papers.Kennard1927.KennardTypes} (framework : PhysicsLogic.Papers.Kennard1927.KennardFramework T) (precision : framework.PrecisionSetup) (calculation : framework.GeneralPositivityCalculation) : framework.planckConstant / (2 * Real.pi) ≤ framework.precisionValueAt framework.coordinatePrecision * framework.precisionValueAt framework.momentumPrecision := framework.canonicalAmplitudePair.factorization_calculation_derives_precision_product_bound (framework.precision_setup_selects_actual_moments precision) (calculation precision) example {T : PhysicsLogic.Papers.Kennard1927.KennardTypes} (framework : PhysicsLogic.Papers.Kennard1927.KennardFramework T) (result : framework.EstablishedGeneralKennardInequality) : framework.canonicalAmplitudePair.factorCurrent = 0 ∧ framework.canonicalAmplitudePair.RemainderBalance := result.2.1.2 example {T : PhysicsLogic.Papers.Kennard1927.KennardTypes} (framework : PhysicsLogic.Papers.Kennard1927.KennardFramework T) (result : framework.EstablishedGeneralKennardInequality) (positiveRemainder : 0 < framework.canonicalAmplitudePair.remainder) : framework.planckConstant / (2 * Real.pi) < framework.precisionValueAt framework.coordinatePrecision * framework.precisionValueAt framework.momentumPrecision := framework.canonicalAmplitudePair.positive_remainder_makes_precision_product_strict (framework.precision_setup_selects_actual_moments result.1) result.2.1.2.2 positiveRemainder example {T : PhysicsLogic.Papers.Kennard1927.KennardTypes} (framework : PhysicsLogic.Papers.Kennard1927.KennardFramework T) (setup : framework.GaussianMinimumSetup) : ∀ q, framework.amplitudeValueAt framework.coordinateAmplitude q = framework.canonicalAmplitudePair.referenceGaussian q := framework.gaussian_minimum_setup_uses_reference_amplitude setup example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Heisenberg1927.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Heisenberg1927.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Kennard1927.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Kennard1927.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.Robertson1929.RobertsonFramework.Countermodels.boundedProductWithWrongCommutator.HasDispersionProductBound ∧ ¬PhysicsLogic.Papers.Robertson1929.RobertsonFramework.Countermodels.boundedProductWithWrongCommutator.RetainsPrintedCommutatorFormulas := PhysicsLogic.Papers.Robertson1929.RobertsonFramework.Countermodels.named_general_uncertainty_relations_do_not_supply_printed_formulas example {T : PhysicsLogic.Papers.Robertson1929.RobertsonTypes} (framework : PhysicsLogic.Papers.Robertson1929.RobertsonFramework T) (general : framework.EstablishedGeneralCommutatorUncertainty) : framework.toStateDispersionUncertaintyContext |>.EstablishedStateDispersionUncertaintyAt framework.sharedStateDispersionModel := framework.robertson_general_result_realizes_shared_state_dispersion_uncertainty general section RobertsonConcreteOperatorReuse open PhysicsLogic.Papers.Robertson1929 open PhysicsLogic.Papers.Robertson1929.FiniteRobertsonOperatorReadout open PhysicsLogic.Papers.Robertson1929.FiniteRobertsonAngularReadout open PhysicsLogic.Framework.StateDispersionUncertaintyContext.Countermodels example {T : RobertsonTypes} (framework : RobertsonFramework T) (general : framework.EstablishedGeneralCommutatorUncertainty) : framework.NormalizedPreparedStateFormulaAt := framework.established_general_uncertainty_retains_normalized_wavefunction general example {T : RobertsonTypes} (framework : RobertsonFramework T) (general : framework.EstablishedGeneralCommutatorUncertainty) : framework.CenteredSchwarzProofFormulaPackageAt := framework.established_general_uncertainty_retains_centered_schwarz_formulas general example {T : RobertsonTypes} (framework : RobertsonFramework T) (general : framework.EstablishedGeneralCommutatorUncertainty) : framework.CenteredCommutatorReductionFormulaAt := framework.established_general_uncertainty_retains_centered_commutator_reduction general example {T : RobertsonTypes} (framework : RobertsonFramework T) (general : framework.EstablishedGeneralCommutatorUncertainty) : Complex.normSq framework.centeredSchwarzOverlapComplexValueAt ≤ 4 * (framework.rmsDispersionValueAt framework.dispersionA * framework.rmsDispersionValueAt framework.dispersionB) ^ 2 := framework.established_general_uncertainty_bounds_actual_centered_overlap general example {T : RobertsonTypes} (framework : RobertsonFramework T) (setup : framework.OneParticleLinearOperatorProofSetup) : framework.commutatorExpectationMagnitudeFormulaValueAt / 2 ≤ framework.rmsDispersionValueAt framework.dispersionA * framework.rmsDispersionValueAt framework.dispersionB := framework.proof_setup_derives_half_absolute_commutator_bound setup example {T : RobertsonTypes} (framework : RobertsonFramework T) (setup : framework.OneParticleLinearOperatorProofSetup) (calibration : framework.SchwarzCommutatorCalculation) : framework.GeneralCommutatorUncertaintyResult := framework.schwarz_commutator_calculation_derives_general_result setup calibration example : finiteRobertsonCommutator pauliObservableX pauliObservableY = fun row column => 2 * Complex.I * pauliObservableZ row column := pauli_observables_have_actual_nonzero_commutator example : saturatedPositiveBoundReadout.SourceRobertsonUncertaintyAt := spin_up_realizes_source_state_dependent_uncertainty example : noncommutingZeroBoundReadout.SourceRobertsonUncertaintyAt := equatorial_state_realizes_zero_bound_with_positive_dispersions example : negativeCommutatorPositiveBoundReadout.SourceRobertsonUncertaintyAt := negative_commutator_still_has_positive_absolute_bound example : saturatedPositiveBoundReadout.leftObservable = noncommutingZeroBoundReadout.leftObservable ∧ saturatedPositiveBoundReadout.rightObservable = noncommutingZeroBoundReadout.rightObservable ∧ saturatedPositiveBoundReadout.SourceRobertsonUncertaintyAt ∧ noncommutingZeroBoundReadout.SourceRobertsonUncertaintyAt ∧ saturatedPositiveBoundReadout.lowerBound = 1 ∧ noncommutingZeroBoundReadout.lowerBound = 0 := the_same_noncommuting_observables_have_state_dependent_bounds example : finiteRobertsonNormSquared equatorialComplexState = 1 ∧ unconjugatedRobertsonNormSquared equatorialComplexState ≠ 1 := removing_conjugation_destroys_complex_state_normalization example : (finiteRobertsonExpectation centeredEigenstateReadout.preparedState (centeredEigenstateReadout.leftObservable * centeredEigenstateReadout.leftObservable)).re = 1 ∧ finiteRobertsonVariance centeredEigenstateReadout.preparedState centeredEigenstateReadout.leftObservable = 0 := raw_second_moment_is_not_centered_eigenstate_variance example : positiveDispersionWithoutApparatusTradeoff.SourceStateDispersionAt ∧ ¬positiveDispersionWithoutApparatusTradeoff.ApparatusProductTradeoffAt := state_dispersion_uncertainty_does_not_imply_apparatus_disturbance example : finiteRobertsonAngularCommutator spinOneAngularX spinOneAngularY = fun row column => Complex.I * spinOneAngularZ row column := spin_one_generators_realize_source_angular_commutator example : finiteRobertsonAngularExpectation spinOneZeroMagneticState spinOneAngularZ = 0 ∧ finiteRobertsonAngularExpectation spinOneExtremalMagneticState spinOneAngularZ = 1 := ⟨zero_magnetic_angular_state_has_source_component_means.2.2, extremal_magnetic_angular_state_has_source_component_means.2.2⟩ example : ¬SourceAngularVarianceSaturationAt spinOneZeroMagneticState ∧ SourceAngularVarianceSaturationAt spinOneExtremalMagneticState := ⟨zero_magnetic_angular_state_does_not_saturate_source_bound, extremal_magnetic_angular_state_saturates_source_bound⟩ example : Real.sqrt (1 / 2 : Real) * Real.sqrt (1 / 2 : Real) = 1 / 2 := extremal_magnetic_rms_product_saturates_source_angular_bound example : saturatedPositiveBoundReadout.leftVariance = physicalRobertson1929ExtremalAngularReadout.leftObservableVarianceValue ∧ saturatedPositiveBoundReadout.rightVariance = physicalRobertson1929ExtremalAngularReadout.rightObservableVarianceValue ∧ saturatedPositiveBoundReadout.lowerBound = physicalRobertson1929ExtremalAngularReadout.operatorLowerBoundValue := saturated_operator_state_agrees_with_shared_robertson_readout example : (finiteRobertsonAngularExpectation spinOneZeroMagneticState spinOneAngularZ).re = physicalRobertson1929ZeroCommutatorReadout.angularThirdComponentMeanValue ∧ finiteRobertsonAngularVariance spinOneZeroMagneticState spinOneAngularX = physicalRobertson1929ZeroCommutatorReadout.angularXVarianceValue ∧ (finiteRobertsonAngularExpectation spinOneZeroMagneticState spinOneTotalAngularSquare).re = physicalRobertson1929ZeroCommutatorReadout.angularTotalSquareValue ∧ physicalRobertson1929ZeroCommutatorReadout.operatorLowerBoundValue = 0 := zero_magnetic_spin_one_agrees_with_shared_robertson_readout end RobertsonConcreteOperatorReuse example : PhysicsLogic.Papers.Ozawa2003.Framework.Countermodels.preparedDispersionWithoutDeviceTradeoff.PreparedStateDispersionRelation ∧ ¬PhysicsLogic.Papers.Ozawa2003.Framework.Countermodels.preparedDispersionWithoutDeviceTradeoff.DeviceProductTradeoff := PhysicsLogic.Papers.Ozawa2003.Framework.Countermodels.prepared_state_dispersion_does_not_supply_device_error_tradeoff example : ¬PhysicsLogic.Papers.Ozawa2003.Framework.Countermodels.generalizedButNotProductTradeoff.EquationTwoProductOnlyTradeoffAt ∧ PhysicsLogic.Papers.Ozawa2003.Framework.Countermodels.generalizedButNotProductTradeoff.EquationFourteenUniversalCorrelationTradeoffAt ∧ PhysicsLogic.Papers.Ozawa2003.Framework.Countermodels.generalizedButNotProductTradeoff.EquationFifteenGeneralizedTradeoffAt := PhysicsLogic.Papers.Ozawa2003.Framework.Countermodels.product_only_violation_does_not_refute_generalized_tradeoff example {T : PhysicsLogic.Papers.Ozawa2003.OzawaTypes} (framework : PhysicsLogic.Papers.Ozawa2003.Framework T) (definitions : framework.EstablishedOperationalDefinitions) (universal : framework.EstablishedUniversalCorrelationTradeoff) (generalized : framework.EstablishedGeneralizedTradeoff) (formulations : framework.EstablishedUncertaintyFormulationDistinction) : framework.toOperationalErrorDisturbanceContext |>.EstablishedOperationalErrorDisturbanceAt framework.sharedOperationalErrorDisturbanceModel := framework.universal_measurement_results_realize_shared_operational_tradeoff definitions universal generalized formulations example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Robertson1929.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Robertson1929.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Ozawa2003.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Ozawa2003.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.WegnerHoughton1973.SharpShellFormulaReadout.Countermodels.NamedExactGeneratorOutputAt PhysicsLogic.Papers.WegnerHoughton1973.SharpShellFormulaReadout.Countermodels.wrongNamedGeneratorOutput ∧ ¬PhysicsLogic.Papers.WegnerHoughton1973.SharpShellFormulaReadout.Countermodels.wrongNamedGeneratorOutput.EquationTwentyAt := PhysicsLogic.Papers.WegnerHoughton1973.SharpShellFormulaReadout.Countermodels.named_exact_generator_output_does_not_supply_equation_twenty example {T : PhysicsLogic.Papers.WegnerHoughton1973.WegnerHoughtonTypes} (framework : PhysicsLogic.Papers.WegnerHoughton1973.Framework T) (route : framework.ExactGeneratorRouteApplicable) : framework.toSharedExactFlowContext |>.EstablishedExactRegulatedScaleFlowAt framework.sharedExactFlowModel := framework.wegner_houghton_equation_realizes_shared_exact_regulated_scale_flow route example {T : PhysicsLogic.Papers.WegnerHoughton1973.WegnerHoughtonTypes} (framework : PhysicsLogic.Papers.WegnerHoughton1973.Framework T) (established : framework.EstablishedExactRenormalizationGenerator) (scope : framework.FiniteStepScope) : (framework.toScaleFlowIntegrationContext |>.ExactInfinitesimalGeneratorAt framework.exactGeneratorEquation framework.effectiveHamiltonianAt) ∧ framework.finiteCutoffIntegrationExpectedToYieldRecursionsAt framework.exactGeneratorEquation framework.baseRegime ∧ ¬(framework.toScaleFlowIntegrationContext |>.IntegratedScaleFlowAt framework.exactGeneratorEquation framework.effectiveHamiltonianAt) := framework.finite_step_scope_realizes_generator_without_integrated_scale_flow established scope example : PhysicsLogic.Papers.Wetterich1993.ExactFlowFormulaReadout.Countermodels.NamedExactFlowOutputAt PhysicsLogic.Papers.Wetterich1993.ExactFlowFormulaReadout.Countermodels.wrongNamedOutput ∧ ¬PhysicsLogic.Papers.Wetterich1993.ExactFlowFormulaReadout.Countermodels.wrongNamedOutput.ExactEvolutionEquationAt := PhysicsLogic.Papers.Wetterich1993.ExactFlowFormulaReadout.Countermodels.named_exact_flow_output_does_not_supply_equation_two example {T : PhysicsLogic.Papers.Wetterich1993.WetterichTypes} (framework : PhysicsLogic.Papers.Wetterich1993.Framework T) (route : framework.ExactFlowRouteApplicable) : framework.toSharedExactFlowContext |>.EstablishedExactRegulatedScaleFlowAt framework.sharedExactFlowModel := framework.wetterich_exact_flow_realizes_shared_exact_regulated_scale_flow route example {T : PhysicsLogic.Papers.Wetterich1993.WetterichTypes} (framework : PhysicsLogic.Papers.Wetterich1993.Framework T) (route : framework.EndpointRouteApplicable) : framework.toScaleFlowIntegrationContext |>.EstablishedIntegratedScaleFlowAt framework.CompleteSolutionConditions framework.exactFlowEquation framework.averageActionAt := framework.endpoint_route_establishes_shared_integrated_scale_flow route example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.WegnerHoughton1973.Fidelity.countermodelMatrix := PhysicsLogic.Papers.WegnerHoughton1973.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Wetterich1993.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Wetterich1993.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.Feynman1948.Framework.Countermodels.feynmanPhaseWithoutCompositionReadout.HasLocalActionPhaseAt ∧ ¬PhysicsLogic.Papers.Feynman1948.Framework.Countermodels.feynmanPhaseWithoutCompositionReadout.HasIntermediateCompositionAt := PhysicsLogic.Papers.Feynman1948.Framework.Countermodels.action_phase_postulate_does_not_define_path_measure_or_sum example {T : PhysicsLogic.Papers.Feynman1948.FeynmanTypes} (framework : PhysicsLogic.Papers.Feynman1948.Framework T) (paths : framework.EstablishedPathAmplitudePostulates) (wave : framework.EstablishedWaveFunctionFactorization) (classical : framework.EstablishedClassicalLimit) : framework.selectedActionPhaseCompositionContext |>.EstablishedActionPhaseCompositionAt framework.selectedActionPhaseCompositionModel := framework.feynman_routes_establish_shared_action_phase_composition paths wave classical example : PhysicsLogic.Papers.Hawking1975.CollapseFramework.Countermodels.HeuristicPairPictureAt PhysicsLogic.Papers.Hawking1975.CollapseFramework.Countermodels.pairPictureWithoutThermalCalculation ∧ ¬PhysicsLogic.Papers.Hawking1975.CollapseFramework.Countermodels.ThermalOccupationCalculationAt PhysicsLogic.Papers.Hawking1975.CollapseFramework.Countermodels.pairPictureWithoutThermalCalculation := PhysicsLogic.Papers.Hawking1975.CollapseFramework.Countermodels.the_heuristic_pair_picture_does_not_supply_the_thermal_calculation example : PhysicsLogic.Papers.Hawking1975.CollapseFramework.Countermodels.QuasiStationaryEvaporationAt PhysicsLogic.Papers.Hawking1975.CollapseFramework.Countermodels.quasiStationaryWithoutControlledEndpoint ∧ ¬PhysicsLogic.Papers.Hawking1975.CollapseFramework.Countermodels.ControlledClassicalEndpointAt PhysicsLogic.Papers.Hawking1975.CollapseFramework.Countermodels.quasiStationaryWithoutControlledEndpoint := PhysicsLogic.Papers.Hawking1975.CollapseFramework.Countermodels.quasi_stationary_evaporation_does_not_determine_the_planck_endpoint example {T : PhysicsLogic.Papers.Hawking1975.HawkingTypes} (framework : PhysicsLogic.Papers.Hawking1975.CollapseFramework T) (semiclassical : framework.SemiclassicalResult) (setup : framework.CollapseSetup) (calculation : framework.BogoliubovCalculation) (interpretation : framework.ParticleInterpretation) : framework.toBogoliubovParticleProductionContext |>.EstablishedBogoliubovParticleProductionAt framework.selectedBogoliubovProductionModel := framework.hawking_collapse_route_realizes_shared_particle_production semiclassical setup calculation interpretation example : PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.positiveNormalized PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.normalizedButNotRepresentable 1 ∧ ¬PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.representable PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.normalizedButNotRepresentable := PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.positive_normalized_density_need_not_be_ground_state_representable example : PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.nearlyUniformRegime PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.nearlyUniformSystem ∧ PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.approximationExact PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.nearlyUniformSystem ∧ PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.slowlyVaryingRegime PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.slowlyVaryingSystem ∧ PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.approximationExact PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.slowlyVaryingSystem ∧ ¬PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.approximationExact PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.actualSystem := PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework.Countermodels.exactness_in_two_limits_does_not_imply_actual_system_accuracy example {Potential GroundState Density Approximation : Type} (framework : PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework Potential GroundState Density Approximation) (correct : GroundState) (potential : Potential) (particleNumber : Nat) (correctGround : framework.groundStateFor correct potential) (correctNondegenerate : framework.nondegenerateGroundStateFor correct potential) (correctParticleNumber : framework.particleNumber (framework.densityOf correct) = particleNumber) (groundEnergyEquation : framework.GroundStateEnergyEquation) (universal : framework.UniversalFunctionalConstruction) (wavefunctionVariational : framework.WavefunctionVariationalPrinciple) : (framework.toSharedDensityFunctionalContext correct potential particleNumber) |>.EstablishedGroundDensityVariationalAt (framework.sharedGroundDensityModel correct potential particleNumber) := framework.ground_density_variational_route_establishes_shared_context correct potential particleNumber correctGround correctNondegenerate correctParticleNumber groundEnergyEquation universal wavefunctionVariational example {Potential GroundState Density Approximation : Type} (framework : PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework Potential GroundState Density Approximation) (state : GroundState) (potential : Potential) (particleNumber : Nat) (setup : framework.SharedGroundDensityGroundingSetup state potential particleNumber) : (framework.sharedGroundDensityModel state potential particleNumber) |>.toAdmissibleDensityCompletenessContext.CompleteAt (framework.sharedGroundDensityModel state potential particleNumber) particleNumber := framework.ground_density_setup_realizes_scoped_descriptive_completeness state potential particleNumber setup example : (PhysicsLogic.Framework.DensityFunctionalContext.Countermodels.scopedDensityDomainModel |>.toAdmissibleDensityCompletenessContext.CompleteAt PhysicsLogic.Framework.DensityFunctionalContext.Countermodels.scopedDensityDomainModel 1) ∧ ¬(PhysicsLogic.Framework.DensityFunctionalContext.Countermodels.scopedDensityDomainModel |>.toAdmissibleDensityCompletenessContext.physicallyAdmissible .competitor 1) := PhysicsLogic.Framework.DensityFunctionalContext.Countermodels.scoped_descriptive_completeness_does_not_admit_an_excluded_density example : PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework.Countermodels.ReducedOldInflationExitReadout.HasSufficientExpansion PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework.Countermodels.outpacedBubbleGrowth ∧ ¬PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework.Countermodels.ReducedOldInflationExitReadout.TransitionCompletes PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework.Countermodels.outpacedBubbleGrowth := PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework.Countermodels.exponential_expansion_does_not_imply_transition_completion example : PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework.Countermodels.ReducedDeferredCosmologyReadout.SolvesSelectedTargets PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework.Countermodels.unresolvedPostInflationPhysics ∧ ¬PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework.Countermodels.ReducedDeferredCosmologyReadout.BaryogenesisAnalyzed PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework.Countermodels.unresolvedPostInflationPhysics := by obtain ⟨solved, _thermalization, notBaryogenesis⟩ := PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework.Countermodels.solving_selected_targets_does_not_complete_deferred_calculations exact ⟨solved, notBaryogenesis⟩ namespace OriginalNewInflationPhysicalReuseAudit open PhysicsLogic.Papers.AlbrechtSteinhardt1982 open PhysicsLogic.Papers.AlbrechtSteinhardt1982.SourcePhysics open PhysicsLogic.Framework.BubbleTransitionCompletionContext.Countermodels example : sourceRadiativePotential sourceRadiativePotentialReadout = -19 + 16 * Real.log 4 := radiative_source_model_retains_logarithm_and_thermal_integral example : sourceRadiativePotentialWithoutThermalContribution sourceRadiativePotentialReadout = -20 + 16 * Real.log 4 ∧ sourceRadiativePotentialWithoutThermalContribution sourceRadiativePotentialReadout ≠ sourceRadiativePotential sourceRadiativePotentialReadout := erasing_thermal_integral_changes_source_potential example : sourceSubcriticalBubbleReadout.SourceLargeConvertedFractionAt ∧ ¬sourceSubcriticalBubbleReadout.SourceConnectedPercolationAt := large_new_phase_fraction_does_not_imply_connected_bubbles example : sourceMissingPercolationCriterionReadout.SourcePositiveNucleationAt ∧ ¬sourceMissingPercolationCriterionReadout.SourceSubcriticalPercolationAt := positive_bubble_rate_does_not_supply_independent_percolation_threshold example : sourceSuccessfulFluctuationPatch.SourceHubbleDragSlowEvolutionAt ∧ sourceSuccessfulFluctuationPatch.SourceEFoldingRadiusFormulaAt := ⟨source_fluctuation_patch_retains_hubble_drag_equation, source_fluctuation_patch_retains_exact_efolding_radius_formula⟩ example : sourceAcceleratingFluctuationPatch.SourceHubbleDragSlowEvolutionAt ∧ ¬(3 * sourceAcceleratingFluctuationPatch.hubble * sourceAcceleratingFluctuationPatch.fieldVelocity + sourceAcceleratingFluctuationPatch.potentialSlope = 0) := full_klein_gordon_drag_does_not_imply_acceleration_free_slow_roll example : sourceSixtyEFoldingsWithoutContainment.SourceManyActualEFoldingsAt ∧ ¬sourceSixtyEFoldingsWithoutContainment.SourceObservedContainmentAt := sixty_actual_exponential_efoldings_need_not_contain_observed_region example : sourceSuccessfulFluctuationPatch.SourceNoNonthermalMonopolesAt ∧ ¬sourceSuccessfulFluctuationPatch.SourceNoMonopolesAt := eliminating_nonthermal_monopoles_allows_thermal_monopoles example : sourceSuccessfulFluctuationPatch.SourceNoObservedDomainWallsAt ∧ ¬sourceSuccessfulFluctuationPatch.SourceNoGlobalDomainWallsAt := no_observed_domain_wall_does_not_mean_no_global_domain_walls example : sourceSuccessfulFluctuationPatch.SourceSelectedCosmologicalTargetsAt ∧ ¬sourceSuccessfulFluctuationPatch.SourceCurvatureAnalyzedAt ∧ ¬sourceSuccessfulFluctuationPatch.SourcePerturbationSpectrumCalculatedAt := successful_source_targets_leave_curvature_and_perturbations_open example : sourceRadiativePotential sourceRadiativePotentialReadout = physicalAlbrechtSteinhardtRadiativePotentialReadout.radiativePotentialValue := source_radiative_potential_agrees_with_shared_bubble_framework example {T : CosmologyTypes} (framework : NewInflationFramework T) (setup : framework.StandardNucleationSetup) (calculation : framework.StandardNucleationCalculation) : framework.SourceSteepestDescentTrustedRegimeAt framework.criticalTemperature ∧ framework.SourceSteepestDescentRateFormulaAt framework.criticalTemperature := framework.established_nucleation_retains_temperature_fourth_saddle_rate setup calculation example {T : CosmologyTypes} (framework : NewInflationFramework T) (region : T.Region) (field : T.FieldValue) (setup : framework.ThermalPotentialSetup) (breakdown : framework.breakdownOccursBeforeTransitionCompletion framework.steepestDescentBreakdownTemperature) (calculation : framework.FluctuationRegimeCalculation region field) : framework.SourceSteepestDescentBreakdownFormulaAt ∧ framework.SourceThermalFluctuationScalingFormulaAt region field := framework.established_fluctuation_region_retains_breakdown_and_thermal_scaling region field setup breakdown calculation example {T : CosmologyTypes} (framework : NewInflationFramework T) (initial : T.InitialCondition) (region : T.Region) (field : T.FieldValue) (constant : framework.fieldApproximatelyConstantIn field region) (motion : framework.fieldMovesTowardBrokenMinimum field framework.brokenPhase) (flat : framework.effectivePotentialExtremelyFlatAlongMotion framework.thermalPotential field framework.brokenPhase) (calculation : framework.SlowFieldExpansionCalculation initial region field) : framework.SourceHubbleDragSlowFieldFormulaAt region ∧ framework.SourceInflationaryEFoldingFormulaAt region := framework.established_slow_expansion_retains_hubble_drag_and_efolding_formulas initial region field constant motion flat calculation example {T : CosmologyTypes} (framework : NewInflationFramework T) (region : T.Region) (dynamics : framework.SourceHubbleDragSlowFieldFormulaAt region) (negligibleAcceleration : framework.fieldAccelerationValueAt region = 0) : 3 * framework.fluctuationHubbleValueAt region * framework.fieldVelocityValueAt region + framework.potentialSlopeValueAt region = 0 := framework.acceleration_free_hubble_drag_requires_an_additional_assumption region dynamics negligibleAcceleration example {T : CosmologyTypes} (framework : NewInflationFramework T) (initial : T.InitialCondition) (region : T.Region) (initialSize : framework.regionSizeOfInverseTemperatureOrder region framework.steepestDescentBreakdownTemperature) (slow : framework.fieldEvolutionSlowIn region) (efoldings : framework.manyEFoldingsBeforeFieldChanges region initial) (calculation : framework.RegionSizeCalculation initial region) : framework.regionContainsObservedUniverse region initial ∧ framework.SourceObservedRegionRadiusFormulaAt region := framework.established_observed_region_retains_exponential_radius_and_tuning initial region initialSize slow efoldings calculation end OriginalNewInflationPhysicalReuseAudit example {T : PhysicsLogic.Papers.AlbrechtSteinhardt1982.CosmologyTypes} (framework : PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework T) (initial : T.InitialCondition) (firstOrder : framework.firstOrderTransition framework.symmetricPhase framework.brokenPhase) (enoughExpansion : framework.sufficientExpansionForCosmologicalPuzzles initial) (outpaced : framework.expansionDominatesBubbleProductionAndGrowth initial) (calculation : framework.OldInflationFailureCalculation initial) : framework.toBubbleTransitionCompletionContext initial |>.EstablishedBubbleCompletionObstructionAt initial framework.selectedRate framework.brokenPhase := framework.old_inflation_realizes_shared_bubble_completion_obstruction initial firstOrder enoughExpansion outpaced calculation example : PhysicsLogic.Papers.Penrose1965.CollapseFramework.Countermodels.futureNullIncomplete PhysicsLogic.Papers.Penrose1965.CollapseFramework.Countermodels.finiteAffineIncompleteWithBoundedCurvature ∧ ¬PhysicsLogic.Papers.Penrose1965.CollapseFramework.Countermodels.curvatureDiverges PhysicsLogic.Papers.Penrose1965.CollapseFramework.Countermodels.finiteAffineIncompleteWithBoundedCurvature := by obtain ⟨incomplete, _bounded, notDivergent, _blackHole⟩ := PhysicsLogic.Papers.Penrose1965.CollapseFramework.Countermodels.null_incompleteness_does_not_supply_curvature_or_black_hole_claims exact ⟨incomplete, notDivergent⟩ example (readout : PhysicsLogic.Papers.Penrose1965.TrappedSurfaceNullGeometryReadout) (geometry : readout.SourceGeometryAt) : ∀ point family, readout.expansionAt point family < 0 := readout.future_trapped_surface_requires_both_negative_null_expansions geometry example (readout : PhysicsLogic.Papers.Penrose1965.CauchyProjectionDegreeReadout) (transport : readout.SourceTransportAt) (compactImage : readout.CompactProjectionImageAt) : False := readout.compact_degree_one_projection_contradicts_noncompact_cauchy_surface transport compactImage example {T : PhysicsLogic.Papers.Penrose1965.PenroseTypes} (framework : PhysicsLogic.Papers.Penrose1965.CollapseFramework T) (degree : framework.NoncompactCauchyDegreeObstruction) : framework.toTrappedSurfaceIncompletenessContext |>.NoncompactCauchyTopologyObstructionAt framework.incompletenessModel := framework.original_degree_projection_realizes_shared_topology_obstruction degree example {T : PhysicsLogic.Papers.Penrose1965.PenroseTypes} (framework : PhysicsLogic.Papers.Penrose1965.CollapseFramework T) (global : framework.GlobalCausalSetup) (trapped : framework.FutureTrappedSurfaceSetup) (focusing : framework.NullFocusingCalculation) (compactBoundary : framework.CompactBoundaryCalculation) (topology : framework.NoncompactCauchyDegreeObstruction) : framework.toTrappedSurfaceIncompletenessContext |>.RouteStructuralDirectionalNullIncompletenessExplanationAt framework.incompletenessModel := framework.penrose_inputs_realize_structural_incompleteness_explanation global trapped focusing compactBoundary topology example : PhysicsLogic.Papers.Kubo1957.ResponseFramework.Countermodels.typedCorrelationPredictionAcrossSelectedSector PhysicsLogic.Papers.Kubo1957.ResponseFramework.Countermodels.typedRelationWithoutTheoremFiveFormula ∧ ¬PhysicsLogic.Papers.Kubo1957.ResponseFramework.Countermodels.theoremFiveTensorFormulaAcrossSelectedSector PhysicsLogic.Papers.Kubo1957.ResponseFramework.Countermodels.typedRelationWithoutTheoremFiveFormula := PhysicsLogic.Papers.Kubo1957.ResponseFramework.Countermodels.typed_correlation_prediction_does_not_supply_theorem_five_formula example : PhysicsLogic.Papers.Kubo1957.ResponseFramework.Countermodels.fieldReciprocityWithoutFrequencyParity.HasFieldReversedReciprocityAt () ∧ ¬PhysicsLogic.Papers.Kubo1957.ResponseFramework.Countermodels.fieldReciprocityWithoutFrequencyParity.HasFrequencyParityAt () := PhysicsLogic.Papers.Kubo1957.ResponseFramework.Countermodels.field_reciprocity_does_not_supply_frequency_parity example {T : PhysicsLogic.Papers.Kubo1957.KuboTypes} (framework : PhysicsLogic.Papers.Kubo1957.ResponseFramework T) (application : framework.ExactApplicationConclusion) (setup : framework.ConductivityFDTSetup) (calculation : framework.ConductivityFDTCalculation) : framework.toEquilibriumNoiseDissipationContext |>.EstablishedEquilibriumNoiseDissipationAt framework.equilibriumNoiseDissipationModel := framework.kubo_conductivity_realizes_shared_noise_dissipation application setup calculation example {T : PhysicsLogic.Papers.Kubo1957.KuboTypes} (framework : PhysicsLogic.Papers.Kubo1957.ResponseFramework T) (responseSource : framework.kuboLinearResponseTransportDerivationObtains .canonicalResponseRelaxationAndCorrelationAdmittance) (correlation : framework.CorrelationResponseConclusion) (applicationSetup : framework.ApplicationSetup) (applicationCalculation : framework.ApplicationCalculation) (fdtSetup : framework.ConductivityFDTSetup) (fdtCalculation : framework.ConductivityFDTCalculation) : framework.kuboResponseNoiseHandoff |>.SourceSensitiveResponseGroundedNoiseDissipationAt framework.toLinearResponseTransportContext framework.toEquilibriumNoiseDissipationContext .canonicalResponseRelaxationAndCorrelationAdmittance framework.kuboLinearResponseTransportDerivationObtains .responseCorrelationConductivityTheorem framework.KuboNoiseDissipationDerivationObtains .correlationConductivityAndSymmetrizedCurrentSpectrum framework.kuboResponseNoiseHandoffDerivationObtains := framework |>.kubo_response_and_conductivity_inputs_realize_source_sensitive_noise_composition responseSource correlation applicationSetup applicationCalculation fdtSetup fdtCalculation example {T : PhysicsLogic.Papers.Kubo1957.KuboTypes} (framework : PhysicsLogic.Papers.Kubo1957.ResponseFramework T) (sourceSensitive : framework.kuboResponseNoiseHandoff |>.SourceSensitiveResponseGroundedNoiseDissipationAt framework.toLinearResponseTransportContext framework.toEquilibriumNoiseDissipationContext .canonicalResponseRelaxationAndCorrelationAdmittance framework.kuboLinearResponseTransportDerivationObtains .responseCorrelationConductivityTheorem framework.KuboNoiseDissipationDerivationObtains .correlationConductivityAndSymmetrizedCurrentSpectrum framework.kuboResponseNoiseHandoffDerivationObtains) (frequency : T.Frequency) (selected : framework.selectedFrequencySector frequency) : framework.ConductivityTheoremFiveFormulaAt framework.conductionSystem framework.leftComponent framework.rightComponent framework.selectedTemperature frequency /\ framework.symmetrizedCurrentCorrelationSpectrumAt (framework.currentCorrelationAt framework.conductionSystem framework.leftComponent framework.rightComponent frequency) framework.conductionSystem framework.leftComponent framework.rightComponent frequency := framework |>.source_sensitive_kubo_response_noise_composition_retains_exact_formula_and_symmetrized_spectrum sourceSensitive |>.1 frequency selected example {T : PhysicsLogic.Papers.CallenWelton1951.CallenWeltonTypes} (framework : PhysicsLogic.Papers.CallenWelton1951.GeneralizedNoiseFramework T) (relation : framework.EstablishedGeneralizedNoiseDissipation) : framework.toEquilibriumNoiseDissipationContext |>.EstablishedEquilibriumNoiseDissipationAt framework.equilibriumNoiseDissipationModel := framework.callen_welton_relation_realizes_shared_noise_dissipation relation example : PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework.Countermodels.HasVariationalTrialSupportAt PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework.Countermodels.supportedTrialStateWithNonzeroResidual ∧ ¬PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework.Countermodels.IsExactGroundStateAt PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework.Countermodels.supportedTrialStateWithNonzeroResidual := PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework.Countermodels.variational_support_does_not_establish_exact_ground_state example : PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework.Countermodels.HasIncompressibleFluidAt PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework.Countermodels.incompressibleFluidWithoutLocalizedPlateau ∧ ¬PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework.Countermodels.HasHallPlateauAt PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework.Countermodels.incompressibleFluidWithoutLocalizedPlateau := by obtain ⟨fluid, _localized, notPlateau⟩ := PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework.Countermodels.incompressibility_alone_does_not_establish_fractional_hall_plateau exact ⟨fluid, notPlateau⟩ example {T : PhysicsLogic.Papers.BardeenCooperSchrieffer1957.BCSTypes} (framework : PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework T) (setup : framework.ZeroTemperatureGapSetup) (calculation : framework.ZeroTemperatureGapCalculation) : 0 < framework.gapMagnitude framework.zeroTemperatureGap ∧ framework.gapMagnitude framework.zeroTemperatureGap ≤ framework.quasiparticleEnergyValue framework.selectedQuasiparticle ∧ 0 < framework.pairBreakingThresholdEnergyValue framework.selectedPairBreakingExcitation ∧ framework.PairBreakingThresholdIsAttainedMinimumAt := framework.zero_temperature_calculation_derives_attained_pair_breaking_minimum setup calculation example {T : PhysicsLogic.Papers.BardeenCooperSchrieffer1957.BCSTypes} (framework : PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework T) (setup : framework.ZeroTemperatureGapSetup) (calculation : framework.ZeroTemperatureGapCalculation) (firstNormalEnergy secondNormalEnergy : Real) : framework.pairBreakingThresholdEnergyValue framework.selectedPairBreakingExcitation ≤ framework.pairExcitationEnergyValueAt firstNormalEnergy secondNormalEnergy := (framework.zero_temperature_calculation_derives_attained_pair_breaking_minimum setup calculation).2.2.2.1 firstNormalEnergy secondNormalEnergy example : (PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework.Countermodels.namedPositiveQuasiparticleBelowClaimedThreshold.NamedQuasiparticleDispersion ∧ PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework.Countermodels.namedPositiveQuasiparticleBelowClaimedThreshold.EquationTwoFortyNineThresholdValue) ∧ ¬PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework.Countermodels.namedPositiveQuasiparticleBelowClaimedThreshold.PairThresholdBoundsTwoQuasiparticles := PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework.Countermodels.named_threshold_does_not_exclude_subgap_pair_excitations example : PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework.Countermodels.netAttractionWithoutSelectedPairChannel.NetAttraction ∧ ¬PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework.Countermodels.netAttractionWithoutSelectedPairChannel.SelectedPairChannel := by obtain ⟨attraction, _amplitude, _momentum, _spin, notSelected⟩ := PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework.Countermodels.net_attraction_alone_does_not_supply_selected_pair_channel exact ⟨attraction, notSelected⟩ example : PhysicsLogic.Papers.AharonovBohm1959.ElectromagneticPotentialFramework.Countermodels.hasNamedMagneticPhaseCalculation PhysicsLogic.Papers.AharonovBohm1959.ElectromagneticPotentialFramework.Countermodels.wrongPrintedShieldedFluxPhaseModel ∧ ¬PhysicsLogic.Papers.AharonovBohm1959.ElectromagneticPotentialFramework.Countermodels.realizesPrintedShieldedFluxPhaseFormula PhysicsLogic.Papers.AharonovBohm1959.ElectromagneticPotentialFramework.Countermodels.wrongPrintedShieldedFluxPhaseModel := PhysicsLogic.Papers.AharonovBohm1959.ElectromagneticPotentialFramework.Countermodels.named_magnetic_phase_calculation_does_not_supply_printed_flux_formula example : PhysicsLogic.Papers.Berry1984.GeometricalPhaseFramework.Countermodels.hasNamedCurvatureFlux PhysicsLogic.Papers.Berry1984.GeometricalPhaseFramework.Countermodels.wrongPrintedPhaseFormulaModel ∧ ¬PhysicsLogic.Papers.Berry1984.GeometricalPhaseFramework.Countermodels.realizesPrintedEquationsSixAndNine PhysicsLogic.Papers.Berry1984.GeometricalPhaseFramework.Countermodels.wrongPrintedPhaseFormulaModel := PhysicsLogic.Papers.Berry1984.GeometricalPhaseFramework.Countermodels.named_curvature_flux_does_not_supply_printed_phase_formula example : PhysicsLogic.Papers.Hannay1985.AngleHolonomyFramework.Countermodels.namedAngleHolonomyOutputWithoutEquations PhysicsLogic.Papers.Hannay1985.AngleHolonomyFramework.Countermodels.wrongCurlHannayAngleFormulaReadout ∧ ¬PhysicsLogic.Papers.Hannay1985.AngleHolonomyFramework.Countermodels.wrongCurlHannayAngleFormulaReadout.OriginalEquationsOneThroughNineFormulaPackageAt := PhysicsLogic.Papers.Hannay1985.AngleHolonomyFramework.Countermodels.named_angle_holonomy_output_does_not_supply_equations_one_through_nine example : PhysicsLogic.Papers.AndersonLocalization1958.FirstOrderLocatorFormulaReadout.Countermodels.NamedFirstOrderLocatorDistributionAt PhysicsLogic.Papers.AndersonLocalization1958.FirstOrderLocatorFormulaReadout.Countermodels.wrongNamedLocatorDistribution ∧ ¬PhysicsLogic.Papers.AndersonLocalization1958.FirstOrderLocatorFormulaReadout.Countermodels.wrongNamedLocatorDistribution.SourceFormulaAt := PhysicsLogic.Papers.AndersonLocalization1958.FirstOrderLocatorFormulaReadout.Countermodels.named_first_order_locator_distribution_does_not_supply_equations_nine_and_eleven example : PhysicsLogic.Papers.AndersonLocalization1958.Framework.Countermodels.ReducedLocalizedTransportReadout.HasUnboundedSpatialDecayAt PhysicsLogic.Papers.AndersonLocalization1958.Framework.Countermodels.localizedAmplitudeWithoutZeroTransportBridge ∧ PhysicsLogic.Papers.AndersonLocalization1958.Framework.Countermodels.ReducedLocalizedTransportReadout.HasRetainedInitialAmplitudeAt PhysicsLogic.Papers.AndersonLocalization1958.Framework.Countermodels.localizedAmplitudeWithoutZeroTransportBridge ∧ ¬PhysicsLogic.Papers.AndersonLocalization1958.Framework.Countermodels.ReducedLocalizedTransportReadout.HasZeroTransportRateAt PhysicsLogic.Papers.AndersonLocalization1958.Framework.Countermodels.localizedAmplitudeWithoutZeroTransportBridge := PhysicsLogic.Papers.AndersonLocalization1958.Framework.Countermodels.localized_amplitude_and_retained_initial_site_do_not_set_transport_rate example {T : PhysicsLogic.Papers.AndersonLocalization1958.AndersonLocalizationTypes} (framework : PhysicsLogic.Papers.AndersonLocalization1958.Framework T) (result : framework.EstablishedShortRangeNoDiffusion) : framework.transportRateValueAt framework.shortRangeTransportRate = 0 := framework.established_no_diffusion_has_zero_isolated_transport_rate result example : PhysicsLogic.Framework.ResolventDecayLocalizationContext.Countermodels.completedContext.EstablishedResolventDecayLocalizationAt PhysicsLogic.Framework.ResolventDecayLocalizationContext.Countermodels.completedModel ∧ ¬PhysicsLogic.Framework.ResolventDecayLocalizationContext.Countermodels.completedModel.unqualifiedAllStateLocalizationEstablishedAt () () := PhysicsLogic.Framework.ResolventDecayLocalizationContext.Countermodels.selected_localization_does_not_establish_unqualified_all_state_localization example : PhysicsLogic.Framework.FieldFreeMagneticInterferenceContext.Countermodels.phaseWithoutInterferenceReadout.ClosedCirculationCalculationAt PhysicsLogic.Framework.FieldFreeMagneticInterferenceContext.Countermodels.phaseWithoutInterferenceReadoutModel ∧ ¬PhysicsLogic.Framework.FieldFreeMagneticInterferenceContext.Countermodels.phaseWithoutInterferenceReadout.EstablishedFieldFreeMagneticInterferenceAt PhysicsLogic.Framework.FieldFreeMagneticInterferenceContext.Countermodels.phaseWithoutInterferenceReadoutModel := by obtain ⟨_setup, phase, notReadout⟩ := PhysicsLogic.Framework.FieldFreeMagneticInterferenceContext.Countermodels.calculated_phase_does_not_supply_interference_readout exact ⟨phase, notReadout⟩ example : PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.basisOnlyContext.SelectedReplicationSetupAt PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.basisOnlyModel ∧ ¬PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.basisOnlyContext.SuperpositionLinearityCalculationAt PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.basisOnlyModel := PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.copying_two_basis_states_does_not_supply_superposition_calculation example : PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.nondeterministicContext.SuperpositionOutputMismatchCalculationAt PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.nondeterministicModel ∧ ¬PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.nondeterministicContext.SelectedEvolutionOutputUniqueAt PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.nondeterministicModel := by obtain ⟨_setup, _linearity, mismatch, _replicates, notUnique, _notApplicable, _notEstablished⟩ := PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.basis_linearity_and_mismatch_without_output_uniqueness_allow_replication exact ⟨mismatch, notUnique⟩ example : PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework.Countermodels.opposingCurrentWithoutFieldExpulsion.OpposingCurrent ∧ ¬PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework.Countermodels.opposingCurrentWithoutFieldExpulsion.FieldExpelled := PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework.Countermodels.opposing_current_relation_requires_field_expulsion_bridge example : PhysicsLogic.Papers.Bell1964.BellFramework.ViolatesBell1964Inequality PhysicsLogic.Papers.Bell1964.realCorrelationOrder PhysicsLogic.Papers.Bell1964.singletCorrelation := PhysicsLogic.Papers.Bell1964.BellFramework.concrete_singlet_correlation_violates_bell_1964 example : Nonempty (PhysicsLogic.Papers.Bell1964.LocalResponses Bool Empty) ∧ ¬PhysicsLogic.Papers.Bell1964.BellFramework.PredeterminedLocalResponses Bool Empty := PhysicsLogic.Papers.Bell1964.BellFramework.Countermodels.empty_hidden_state_response_functions_do_not_establish_predetermination example : Nonempty (PhysicsLogic.Papers.Bell1964.LocalResponses Empty Unit) ∧ ¬PhysicsLogic.Papers.Bell1964.BellFramework.PredeterminedLocalResponses Empty Unit := PhysicsLogic.Papers.Bell1964.BellFramework.Countermodels.empty_setting_response_functions_do_not_establish_predetermination example : Nonempty (PhysicsLogic.Papers.Bell1964.LocalResponses Unit Unit) ∧ ¬PhysicsLogic.Papers.Bell1964.BellFramework.PredeterminedLocalResponses Unit Unit := PhysicsLogic.Papers.Bell1964.BellFramework.Countermodels.singleton_setting_response_functions_do_not_establish_predetermination namespace BellPredeterminationPhysicalCompositionAudit open PhysicsLogic.Papers.Bell1964 open PhysicsLogic.Papers.Bell1964.BellFramework open PhysicsLogic.Philosophy.DescriptiveCompleteness universe u v w x y example {Setting : Type u} {HiddenState : Type v} {perfectPredictability operationalLocality : Prop} (principle : EPRPredeterminationPrinciple Setting HiddenState perfectPredictability operationalLocality) (predictable : perfectPredictability) (locality : operationalLocality) : Nonempty HiddenState ∧ ∃ firstSetting alternativeSetting : Setting, firstSetting ≠ alternativeSetting := epr_predetermination_exposes_inhabited_state_and_distinct_settings principle predictable locality example {Setting : Type u} {HiddenState : Type v} {Model : Type w} {Value : Type x} {Error : Type y} (framework : BellFramework Setting HiddenState Model Value Error) (model : Model) (reality : RealityCriterionContext Setting (MatchingSpinMeasurement Setting) BellCorrelationStage) (procedureAvailable : MatchingSpinMeasurement Setting -> Prop) (component : Setting) {experimentSetup operationalLocality : Prop} (experiment : experimentSetup) (locality : operationalLocality) (matchingMeasurements : MatchingSpinMeasurementsAvailable procedureAvailable) (quantumPerfect : framework.QuantumPerfectAnticorrelation) (predictionPrinciple : framework.PerfectAnticorrelationGivesPredictability reality procedureAvailable) (nonDisturbancePrinciple : OperationalLocalityGivesRemoteNonDisturbance reality operationalLocality) (eprPrinciple : EPRPredeterminationPrinciple Setting HiddenState (EveryChosenRemoteSpinResultPredictable reality) operationalLocality) (realization : framework.EPRPredeterminedResponsesRealizeLocalModel model operationalLocality) : (framework.remoteCorrelationModel reality procedureAvailable experimentSetup operationalLocality |>.NonvacuousRemoteCorrelationPredictionAt) ∧ ∃ localResponses : LocalResponses Setting HiddenState, (PhysicsLogic.Framework.DeterministicLocalResponse.deterministicLocalResponseCalculationModelAt (framework.bipartiteResponses model) localResponses) |>.NonvacuousDeterministicLocalResponseAt := framework |>.bell_remote_prediction_and_predetermination_realize_nonvacuous_local_response model reality procedureAvailable component experiment locality matchingMeasurements quantumPerfect predictionPrinciple nonDisturbancePrinciple eprPrinciple realization end BellPredeterminationPhysicalCompositionAudit example : PhysicsLogic.Papers.Bell1966.Countermodels.CompatibleAdditive ∧ ¬PhysicsLogic.Papers.Bell1966.Countermodels.UnrestrictedAdditive := PhysicsLogic.Papers.Bell1966.Countermodels.commuting_additivity_does_not_imply_incompatible_additivity example : PhysicsLogic.Papers.Bell1966.Countermodels.OutcomeDeterminateInEveryContext ∧ PhysicsLogic.Papers.Bell1966.Countermodels.deterministicContextualScenario.ContextualAt () .selected := PhysicsLogic.Papers.Bell1966.Countermodels.contextuality_does_not_imply_indeterminism example : PhysicsLogic.Papers.EPR1935.CompletenessFramework.Countermodels.certaintyWithoutNonDisturbance.predictableWithCertainty () () () ∧ ¬PhysicsLogic.Papers.EPR1935.CompletenessFramework.Countermodels.certaintyWithoutNonDisturbance.leavesUndisturbed () () () := by obtain ⟨certainty, notUndisturbed, _notAdmissible⟩ := PhysicsLogic.Papers.EPR1935.CompletenessFramework.Countermodels.certainty_alone_does_not_apply_the_reality_criterion exact ⟨certainty, notUndisturbed⟩ example : PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels.correlationWithoutPredictionBridge.CorrelationCalculation ∧ ¬PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels.correlationWithoutPredictionBridge.PredictionReadoutBridge := by obtain ⟨_setup, correlation, notPrediction⟩ := PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels.a_correlation_calculation_does_not_supply_prediction_readout exact ⟨correlation, notPrediction⟩ example : PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels.selectedSemanticsModel.NonvacuousRemoteCorrelationPredictionAt ∧ (PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels.selectedSemanticsModel.observation |>.MatchesRealityCriterion PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels.admittedSelectedSemanticsReality) ∧ (PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels.selectedSemanticsModel.observation |>.MatchesRealityCriterion PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels.rejectedSelectedSemanticsReality) ∧ PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels.admittedSelectedSemanticsReality.physicallyAdmissible false false ∧ ¬PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels.rejectedSelectedSemanticsReality.physicallyAdmissible false false := PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels.physical_admissibility_is_downstream_of_nonvacuous_remote_prediction example : let model := (selectedReadoutContext classicallyCorrelatedSeparable).selectedModel model.ClassicallyCorrelatedAt ∧ ¬model.EntangledAt := by obtain ⟨_setup, correlation, _separable, notEntangled⟩ := classical_correlation_does_not_imply_entanglement exact ⟨correlation, notEntangled⟩ example : let model := (selectedReadoutContext classicallyCorrelatedSeparable).selectedModel model.PositiveMarginalEntropyAt ∧ model.SeparableAt ∧ ¬model.PureJointStateAt := by obtain ⟨_setup, positive, separable, mixed, _notEntangled⟩ := mixed_joint_state_with_positive_marginal_can_be_separable exact ⟨positive, separable, mixed⟩ example : let operational := finiteOperationalContext nonseparableWithoutSignalOrBellViolation operational.composition.NonseparableCompositeAt nonseparableWithoutSignalOrBellViolation nonseparableWithoutSignalOrBellViolation nonseparableWithoutSignalOrBellViolation 0 1 .selected /\ ¬operational.controlledSignalAt nonseparableWithoutSignalOrBellViolation 0 1 .selected /\ ¬operational.bellViolationAt nonseparableWithoutSignalOrBellViolation 0 1 .selected := quantum_nonseparability_implies_neither_signalling_nor_bell_violation example : (finiteCompositionContext classicallySignallingControl).ClassicallyCorrelatedSeparableCompositeAt classicallySignallingControl classicallySignallingControl classicallySignallingControl 0 1 .control ∧ (finiteOperationalContext classicallySignallingControl).controlledSignalAt classicallySignallingControl 0 1 .control ∧ (finiteOperationalContext bellViolatingControl).bellViolationAt bellViolatingControl 0 1 .control := by obtain ⟨_selected, _wrongRegime, classical, signal, _notBell, _entangled, bell, _noSignal⟩ := finite_composition_roles_distinguish_preparations_regimes_signals_and_bell_controls exact ⟨classical, signal, bell⟩ example : classicallyDistinctGlobalContext.jointStateOfCompositeAt .left .selected .selected ∧ classicallyDistinctGlobalContext.jointStateOfCompositeAt .right .selected .selected ∧ identicalLocalDescriptionContext.localDescriptionAt .left .first .shared .selected ∧ identicalLocalDescriptionContext.localDescriptionAt .right .first .shared .selected ∧ ¬identicalLocalDescriptionContext.physicallyEquivalentAt .left .right .selected := by obtain ⟨_valid, _wrongRegime, left, right, _excludedState, _first, _second, _excludedSubsystem, sharedLeft, sharedRight, _wrongDescription, _controlValid, _controlJoint, _controlLocal⟩ := holistic_composition_roles_distinguish_descriptions_states_subsystems_and_controls exact ⟨left, right, sharedLeft, sharedRight, id⟩ example {T : PhysicsLogic.Papers.CoffmanKunduWootters2000.CKWTypes} (framework : PhysicsLogic.Papers.CoffmanKunduWootters2000.DistributedEntanglementFramework T) (opening : framework.OpeningScope) (pure : framework.PureStateSelection) (definition : framework.TangleDefinitionCalculation) (identification : framework.SelectedABTangleReadoutAt) (positive : 0 < framework.tangleAB framework.selectedPureState) (bridge : framework.SelectedABSeparabilityTangleBridgeAt) : framework.toBipartiteEntanglementContext |>.EstablishedBipartiteEntanglementAt framework.sharedBipartiteEntanglementModel := framework.selected_pair_tangle_establishes_shared_bipartite_entanglement opening pure definition identification positive bridge example {T : PhysicsLogic.Papers.CoffmanKunduWootters2000.CKWTypes} (framework : PhysicsLogic.Papers.CoffmanKunduWootters2000.DistributedEntanglementFramework T) (opening : framework.OpeningScope) (pure : framework.PureStateSelection) (definition : framework.TangleDefinitionCalculation) (identification : framework.SelectedABTangleReadoutAt) (positive : 0 < framework.tangleAB framework.selectedPureState) (bridge : framework.SelectedABSeparabilityTangleBridgeAt) : framework.toBipartiteEntanglementContext |>.SourceStructuralBipartiteEntanglementExplanationAt framework.sharedBipartiteEntanglementModel .selectedABSpinFlipTangleWitness framework.CoffmanKunduWoottersBipartiteEntanglementDerivationObtains := framework.selected_pair_tangle_realizes_source_structural_bipartite_entanglement_explanation opening pure definition identification positive bridge example {T : PhysicsLogic.Papers.CoffmanKunduWootters2000.CKWTypes} (framework : PhysicsLogic.Papers.CoffmanKunduWootters2000.DistributedEntanglementFramework T) (opening : framework.OpeningScope) (pure : framework.PureStateSelection) (definition : framework.TangleDefinitionCalculation) (identification : framework.SelectedABTangleReadoutAt) (positive : 0 < framework.tangleAB framework.selectedPureState) (bridge : framework.SelectedABSeparabilityTangleBridgeAt) : framework.selectedPairQuantumNonseparabilityContext |>.NonseparableCompositeAt framework.selectedPureState framework.selectedPureState framework.pairAB framework.qubitA framework.qubitB framework.selectedRegime := framework.positive_pair_tangle_realizes_paradigm_quantum_nonseparability opening pure definition identification positive bridge example {Mode Observer : Type} (framework : PhysicsLogic.Papers.AMPS2012.Framework Mode Observer) (old : framework.IsOldBlackHole) (postulate2 : framework.Postulate2) (postulate4 : framework.Postulate4) (accessible : framework.SelectedModeIsAccessibleLateRadiation) (thermality : framework.SemiclassicalExteriorThermality) (noDrama : framework.SemiclassicalNoDramaInteriorPairPrinciple) (bridge : framework.PureInteriorPairSeparabilityEntropyBridgeAt) : framework.toBipartiteEntanglementContext |>.EstablishedBipartiteEntanglementAt framework.sharedBipartiteEntanglementModel := framework.pure_thermal_interior_pair_establishes_shared_bipartite_entanglement old postulate2 postulate4 accessible thermality noDrama bridge example {Mode Observer : Type} (framework : PhysicsLogic.Papers.AMPS2012.Framework Mode Observer) (old : framework.IsOldBlackHole) (postulate2 : framework.Postulate2) (postulate4 : framework.Postulate4) (accessible : framework.SelectedModeIsAccessibleLateRadiation) (thermality : framework.SemiclassicalExteriorThermality) (noDrama : framework.SemiclassicalNoDramaInteriorPairPrinciple) (bridge : framework.PureInteriorPairSeparabilityEntropyBridgeAt) : framework.toBipartiteEntanglementContext |>.SourceStructuralBipartiteEntanglementExplanationAt framework.sharedBipartiteEntanglementModel .oldHoleThermalPureInteriorPair framework.AMPSBipartiteEntanglementDerivationObtains := framework.thermal_no_drama_route_realizes_source_structural_bipartite_entanglement_explanation old postulate2 postulate4 accessible thermality noDrama bridge example {Mode Observer : Type} (framework : PhysicsLogic.Papers.AMPS2012.Framework Mode Observer) (old : framework.IsOldBlackHole) (postulate2 : framework.Postulate2) (postulate4 : framework.Postulate4) (accessible : framework.SelectedModeIsAccessibleLateRadiation) (thermality : framework.SemiclassicalExteriorThermality) (noDrama : framework.SemiclassicalNoDramaInteriorPairPrinciple) (bridge : framework.PureInteriorPairSeparabilityEntropyBridgeAt) : framework.interiorPairQuantumNonseparabilityContext |>.NonseparableCompositeAt framework.entropy framework.entropy framework.entropy (.outgoingLateMode framework.selectedMode) (.interiorPartner framework.selectedMode) framework.age := framework.thermal_pure_pair_realizes_paradigm_quantum_nonseparability old postulate2 postulate4 accessible thermality noDrama bridge example (data : PhysicsLogic.Papers.AMPS2012.EntropyData) (decrease : data.DecreasesAfterEmission) (purity : data.InteriorPairIsPure) (separation : data.PurePairSeparatesFromEarly) (nonnegative : data.LateModeEntropyIsNonnegative) (subadditivity : data.StrongSubadditivity) : False := data.strong_subadditivity_conflict decrease purity separation nonnegative subadditivity example {Mode Observer : Type} (framework : PhysicsLogic.Papers.AMPS2012.Framework Mode Observer) (incompatible : ¬(framework.Postulate1 ∧ framework.Postulate2 ∧ framework.Postulate4)) : framework.InformationLossResolution ∨ framework.NovelExteriorDynamicsResolution ∨ framework.FirewallResolution := framework.resolution_trilemma incompatible example : PhysicsLogic.Papers.Maldacena1997.D3Framework.Countermodels.positiveCouplingEvidenceWithoutDuality.CoreDualityEvidence ∧ ¬PhysicsLogic.Papers.Maldacena1997.D3Framework.Countermodels.positiveCouplingEvidenceWithoutDuality.FullQuantumDualityConjecture := by have witness := PhysicsLogic.Papers.Maldacena1997.D3Framework.Countermodels.matched_symmetry_and_coupling_evidence_do_not_establish_duality exact ⟨witness.1, witness.2.2⟩ example : ¬PhysicsLogic.Papers.SeibergWitten1994.ElectromagneticCharge.MutuallyLocal PhysicsLogic.Papers.SeibergWitten1994.ElectromagneticCharge.monopole PhysicsLogic.Papers.SeibergWitten1994.ElectromagneticCharge.dyon := PhysicsLogic.Papers.SeibergWitten1994.ElectromagneticCharge.monopole_and_dyon_are_mutually_nonlocal example : ([1, -1] : List Int).sum = 0 ∧ (1 : Int) ∈ ([1, -1] : List Int) ∧ (-1 : Int) ∈ ([1, -1] : List Int) := PhysicsLogic.Papers.KosterlitzThouless1973.Countermodels.a_vortex_antivortex_pair_is_neutral_and_contains_both_signs example : let contour := PhysicsLogic.Papers.KosterlitzThouless1973.SolidDislocationContourReadout.sourceWitness contour.boundPairClosureFailure < contour.freeDislocationClosureFailure := PhysicsLogic.Papers.KosterlitzThouless1973.SolidDislocationContourReadout.source_witness_realizes_distinct_solid_dislocation_scalings.2 example : PhysicsLogic.Papers.KosterlitzThouless1973.ChargedSuperfluidTopologicalExclusion := PhysicsLogic.Papers.KosterlitzThouless1973.charged_superfluid_screening_excludes_logarithmic_vortex_protection example : PhysicsLogic.Papers.KosterlitzThouless1973.HeisenbergTopologicalExclusion := PhysicsLogic.Papers.KosterlitzThouless1973.heisenberg_third_component_excludes_protected_planar_winding example : PhysicsLogic.Papers.KosterlitzThouless1973.Fidelity.countermodelMatrix .rescaledScreeningEquationNotOriginalUnfrozenEquation := PhysicsLogic.Papers.KosterlitzThouless1973.Fidelity.countermodel_matrix_is_realized .rescaledScreeningEquationNotOriginalUnfrozenEquation example : PhysicsLogic.Papers.TKNN1982.PaperFramework.Countermodels.curvatureWithoutPhasePatching.HasKuboCurvatureIdentityAt ∧ ¬PhysicsLogic.Papers.TKNN1982.PaperFramework.Countermodels.curvatureWithoutPhasePatching.HasPhasePatchingIntegralityAt := by obtain ⟨_gapped, curvature, notIntegral⟩ := PhysicsLogic.Papers.TKNN1982.PaperFramework.Countermodels.a_curvature_formula_without_phase_patching_does_not_supply_integrality exact ⟨curvature, notIntegral⟩ example : PhysicsLogic.Papers.TKNN1982.PaperFramework.Countermodels.conditionalInvariantWithClosedGap.HasConditionalInvariantStabilityAt ∧ ¬0 < PhysicsLogic.Papers.TKNN1982.PaperFramework.Countermodels.conditionalInvariantWithClosedGap.minimumDeformationGapValue := PhysicsLogic.Papers.TKNN1982.PaperFramework.Countermodels.invariant_stability_if_a_gap_stays_open_does_not_prove_gap_nonclosure example : PhysicsLogic.Papers.Steane1996.QuantumComputerFramework.Countermodels.equalDistanceWithoutGeneralPairs.HasEqualDistanceExistence ∧ ¬PhysicsLogic.Papers.Steane1996.QuantumComputerFramework.Countermodels.equalDistanceWithoutGeneralPairs.HasGeneralRelatedCodeExistence := PhysicsLogic.Papers.Steane1996.QuantumComputerFramework.Countermodels.equal_distance_existence_does_not_logically_discharge_general_conjecture example : PhysicsLogic.Papers.CalderbankShor1996.AsymptoticEntropyRateReadout.Countermodels.NamedNonnegativeCodeRateAt PhysicsLogic.Papers.CalderbankShor1996.AsymptoticEntropyRateReadout.Countermodels.namedRateWithoutPrintedFormula ∧ ¬PhysicsLogic.Papers.CalderbankShor1996.AsymptoticEntropyRateReadout.SourceFormulaAt PhysicsLogic.Papers.CalderbankShor1996.AsymptoticEntropyRateReadout.Countermodels.namedRateWithoutPrintedFormula := PhysicsLogic.Papers.CalderbankShor1996.AsymptoticEntropyRateReadout.Countermodels.named_quantum_code_rate_does_not_supply_printed_entropy_formula example : PhysicsLogic.Papers.CalderbankShor1996.AsymptoticEntropyRateReadout.GilbertVarshamovQuantumRateBoundAt PhysicsLogic.Papers.CalderbankShor1996.AsymptoticEntropyRateReadout.Countermodels.countingBoundWithoutAbstractErrorFormula ∧ ¬PhysicsLogic.Papers.CalderbankShor1996.AsymptoticEntropyRateReadout.AbstractCorrectableErrorRateFormulaAt PhysicsLogic.Papers.CalderbankShor1996.AsymptoticEntropyRateReadout.Countermodels.countingBoundWithoutAbstractErrorFormula := PhysicsLogic.Papers.CalderbankShor1996.AsymptoticEntropyRateReadout.Countermodels.counting_entropy_bound_does_not_supply_abstract_error_rate_formula example : PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.NamedCodeRateAt PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.namedRateWithoutPrintedBound ∧ ¬PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.EstablishedUpperSourceFormulaAt PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.namedRateWithoutPrintedBound := PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.named_quantum_rate_does_not_supply_hamming_entropy_formula example : PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.ReducedRateExistenceReadout.HasNumericalQuantumRateWindow PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.conditionalRateWindowWithoutExistenceOrCapacity ∧ PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.ReducedRateExistenceReadout.HasGeneralRelatedCodeConjecture PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.conditionalRateWindowWithoutExistenceOrCapacity ∧ ¬PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.ReducedRateExistenceReadout.HasProvedGeneralRelatedCodeExistence PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.conditionalRateWindowWithoutExistenceOrCapacity := PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.conditional_entropy_window_does_not_prove_general_code_existence example : PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.ReducedRateExistenceReadout.HasNumericalQuantumRateWindow PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.conditionalRateWindowWithoutExistenceOrCapacity ∧ ¬PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.ReducedRateExistenceReadout.HasEstablishedNoisyChannelCapacity PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.conditionalRateWindowWithoutExistenceOrCapacity := PhysicsLogic.Papers.Steane1996.BinaryEntropyRateBoundsReadout.Countermodels.entropy_rate_window_does_not_establish_noisy_channel_capacity example : PhysicsLogic.Framework.ExactQuantumErrorRecoveryContext.Countermodels.amplitudeOnly.EstablishedAmplitudeRecoveryAt (PhysicsLogic.Framework.ExactQuantumErrorRecoveryContext.Countermodels.testModel PhysicsLogic.Framework.ExactQuantumErrorRecoveryContext.Countermodels.amplitudeOnly) ∧ ¬PhysicsLogic.Framework.ExactQuantumErrorRecoveryContext.Countermodels.amplitudeOnly.EstablishedTwoBasisRecoveryAt (PhysicsLogic.Framework.ExactQuantumErrorRecoveryContext.Countermodels.testModel PhysicsLogic.Framework.ExactQuantumErrorRecoveryContext.Countermodels.amplitudeOnly) := PhysicsLogic.Framework.ExactQuantumErrorRecoveryContext.Countermodels.amplitude_recovery_does_not_supply_phase_recovery example : PhysicsLogic.Papers.Wilson1974.LatticeGaugeFramework.Countermodels.boundaryReducedWilsonGaugeReadout.HasIsolatedLoopAreaLawAt ∧ ¬PhysicsLogic.Papers.Wilson1974.LatticeGaugeFramework.Countermodels.boundaryReducedWilsonGaugeReadout.HasNoFreeQuarkInferenceAt := PhysicsLogic.Papers.Wilson1974.LatticeGaugeFramework.Countermodels.area_law_statement_does_not_by_itself_supply_confinement example : PhysicsLogic.Papers.Wilson1974.LatticeGaugeFramework.Countermodels.boundaryReducedWilsonGaugeReadout.HasRegulatedConfinementResultAt ∧ ¬PhysicsLogic.Papers.Wilson1974.LatticeGaugeFramework.Countermodels.boundaryReducedWilsonGaugeReadout.HasContinuumConfinementTransferAt := PhysicsLogic.Papers.Wilson1974.LatticeGaugeFramework.Countermodels.regulated_result_does_not_logically_supply_continuum_transfer example : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.pairwiseBalanced PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.balancedDisconnectedUpdate ∧ ¬PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.configurationAccessible PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.balancedDisconnectedUpdate := PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.pairwise_balance_does_not_supply_configuration_accessibility example : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.disconnectedTwoStateGlobalDetailedBalanceAt ∧ ¬PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.disconnectedTwoStateFinitePositiveReachabilityAt false true := PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.global_detailed_balance_does_not_connect_disconnected_markov_components example {T : PhysicsLogic.Papers.Metropolis1953.MetropolisTypes} (framework : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework T) (balance : framework.PairwiseBalanceResult) (source target : T.Configuration) (sourceAdmissible : framework.canonicalConfigurationAdmissibleAt framework.finitePeriodicDescription source) (targetAdmissible : framework.canonicalConfigurationAdmissibleAt framework.finitePeriodicDescription target) : framework.FinitePositiveUpdateReachabilityAt framework.selectedUpdate source target := framework.established_metropolis_balance_connects_every_admissible_configuration balance source target sourceAdmissible targetAdmissible example : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.finiteEquationOfStateEvidence PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.finitePressureEvidenceWithSizeShift ∧ ¬PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.exactThermodynamicLimit PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.finitePressureEvidenceWithSizeShift := PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.finite_equation_of_state_evidence_does_not_establish_thermodynamic_limit example : PhysicsLogic.Papers.Higgs1964.Framework.Countermodels.RealizesLinearizedClassicalMassAt PhysicsLogic.Papers.Higgs1964.Framework.Countermodels.classicalMassWithoutQuantizedProof ∧ ¬PhysicsLogic.Papers.Higgs1964.Framework.Countermodels.RealizesQuantizedVectorMassPoleAt PhysicsLogic.Papers.Higgs1964.Framework.Countermodels.classicalMassWithoutQuantizedProof := PhysicsLogic.Papers.Higgs1964.Framework.Countermodels.linearized_classical_mass_does_not_supply_a_quantized_theory_proof example : PhysicsLogic.Papers.Higgs1964.Framework.Countermodels.RealizesU1VectorMassAt PhysicsLogic.Papers.Higgs1964.Framework.Countermodels.u1MassWithoutSemisimplePattern ∧ ¬PhysicsLogic.Papers.Higgs1964.Framework.Countermodels.RealizesSemisimpleMassPatternAt PhysicsLogic.Papers.Higgs1964.Framework.Countermodels.u1MassWithoutSemisimplePattern := PhysicsLogic.Papers.Higgs1964.Framework.Countermodels.u1_mass_does_not_supply_a_semisimple_component_mass example {T : PhysicsLogic.Papers.Mayer1949.MayerTypes} (framework : PhysicsLogic.Papers.Mayer1949.ShellFramework T) (levelOrder : framework.SpinOrbitLevelOrderOutputs) : framework.SelectedHigherJBindingEnergyOrdersAt := framework.level_order_derives_higher_j_partner_binding_energy_preferences levelOrder example {T : PhysicsLogic.Papers.Mayer1949.MayerTypes} (framework : PhysicsLogic.Papers.Mayer1949.ShellFramework T) (closure : framework.EstablishedShellClosure) : framework.SelectedHigherJBindingEnergyOrdersAt := framework.established_shell_closure_retains_higher_j_binding_energy_preferences closure example {T : PhysicsLogic.Papers.Mayer1949.MayerTypes} (framework : PhysicsLogic.Papers.Mayer1949.ShellFramework T) (closure : framework.EstablishedShellClosure) : 0 < framework.spinOrbitPartnerBindingGapValueAt framework.strongSpinOrbitScheme framework.fSevenHalf framework.fFiveHalf := (framework.established_shell_closure_retains_strictly_increasing_spin_orbit_binding_gaps closure).1 example : PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.hasCorrectHigherJPartnerAssignment PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.correctlyAssignedReversedBindingPartners ∧ ¬PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.hasHigherJPartnerBindingPreference PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.correctlyAssignedReversedBindingPartners := PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.higher_j_quantum_numbers_do_not_determine_binding_energy_order example : PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.ReducedAngularMomentumSplittingReadout.HasPositiveHigherJPreferencesAt PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.positivePartnerPreferencesWithNonmonotonicSplitting ∧ ¬PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.ReducedAngularMomentumSplittingReadout.HasIncreasingSpinOrbitSplittingAt PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.positivePartnerPreferencesWithNonmonotonicSplitting := PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.positive_higher_j_preferences_do_not_supply_increasing_spin_orbit_splitting example : PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.hasObservedHigherClosures PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.unassembledMayerFilling ∧ ¬PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.hasFavoredWithinShellOrder PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.unassembledMayerFilling := PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.observed_magic_numbers_do_not_supply_a_level_ordering example : PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.fitsManyKnownAssignments PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.limitedMayerEvidenceScope ∧ ¬PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.hasExactParticleModel PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.limitedMayerEvidenceScope := PhysicsLogic.Papers.Mayer1949.ShellFramework.Countermodels.a_shell_assignment_can_fit_many_data_without_being_exact example : PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.HasOverlappingQuotedProbableErrorBands PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.reportedButUnadoptedEvidenceReadout ∧ ¬PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.HasUniquelySelectedCompatibleCoefficient PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.reportedButUnadoptedEvidenceReadout := PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.overlapping_reported_probable_error_bands_do_not_select_a_unique_coefficient example : PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.ReducedVirgoClusterDistanceReadout.HasInhabitedVirgoSubsetAt PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.correctlyAssignedVirgoSubsetWithMoreDistantParent ∧ ¬PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.ReducedVirgoClusterDistanceReadout.HasFarthestVirgoAssignmentAt PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.correctlyAssignedVirgoSubsetWithMoreDistantParent := PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.two_megaparsec_virgo_subset_does_not_imply_farthest_parent_members example {T : PhysicsLogic.Papers.Hubble1929.Hubble1929Types} (framework : PhysicsLogic.Papers.Hubble1929.NebularRelationFramework T) (validation : framework.EstablishedIndependentValidation) : ∃ object, framework.objectBelongsToSampleAt object framework.virgoSample framework.observationalRegime := (framework.validation_retains_inhabited_two_megaparsec_farthest_virgo_sample validation).1 example {T : PhysicsLogic.Papers.Hubble1929.Hubble1929Types} (framework : PhysicsLogic.Papers.Hubble1929.NebularRelationFramework T) (validation : framework.EstablishedIndependentValidation) : framework.radialVelocityKilometersPerSecondValueAt framework.velocityData framework.lowVelocityObject = -25 := (framework.validation_retains_corrected_ngc_404_velocity_and_peculiar_bound validation).1 namespace HubbleObserverCorrectionClient open PhysicsLogic.Papers.Hubble1929 /-- Independently constructed ray and calibration, not Hubble's tabulated objects. This client uses the actual paper-local observer and residual API. -/ def solar : SolarReflexReadout := ⟨14, -16, 9⟩ noncomputable def readout (coefficient distance : Real) : RadialVelocityReadout := ⟨44, solar.atDirection (Real.pi / 2) 0, coefficient * distance⟩ theorem the_selected_ray_has_the_computed_solar_correction : solar.atDirection (Real.pi / 2) 0 = -16 := by norm_num [solar, SolarReflexReadout.atDirection] theorem a_different_sky_direction_uses_a_different_cartesian_component : solar.atDirection 0 (Real.pi / 2) = 9 := by norm_num [solar, SolarReflexReadout.atDirection] theorem the_independent_readout_has_nonzero_observed_corrected_and_peculiar_velocities : (readout 80 3).observed = 44 ∧ (readout 80 3).corrected = 60 ∧ (readout 80 3).distanceEffect = 240 ∧ (readout 80 3).peculiar = -180 := by dsimp [readout, RadialVelocityReadout.corrected, RadialVelocityReadout.peculiar] rw [the_selected_ray_has_the_computed_solar_correction] norm_num theorem the_source_equation_has_an_actual_nonzero_residual : (readout 80 3).EquationOfConditionAt (-180) := by dsimp [RadialVelocityReadout.EquationOfConditionAt, readout] rw [the_selected_ray_has_the_computed_solar_correction] norm_num theorem the_library_identifies_the_residual_after_solar_subtraction : (-180 : Real) = (readout 80 3).peculiar := (readout 80 3).the_fitted_residual_is_the_solar_corrected_peculiar_velocity (-180) the_source_equation_has_an_actual_nonzero_residual theorem the_library_retains_the_plotted_distance_plus_residual : (readout 80 3).corrected = (readout 80 3).distanceEffect - 180 := by simpa only [sub_eq_add_neg] using (readout 80 3).corrected_velocity_is_distance_effect_plus_residual (-180) the_source_equation_has_an_actual_nonzero_residual theorem the_independent_relative_smallness_input_is_realized : (readout 80 3).RelativeSmallnessAt (1 / 4) := by dsimp [RadialVelocityReadout.RelativeSmallnessAt, RadialVelocityReadout.corrected, readout] rw [the_selected_ray_has_the_computed_solar_correction] norm_num theorem the_library_derives_the_relative_peculiar_bound : (readout 80 3).ComparablePeculiarMotionAt (1 / 4) := (readout 80 3).relative_smallness_of_corrected_velocity_bounds_the_peculiar_motion (1 / 4) the_independent_relative_smallness_input_is_realized theorem the_derived_negative_residual_is_smaller_than_the_distance_effect : 0 < |(readout 80 3).peculiar| ∧ |(readout 80 3).peculiar| < (readout 80 3).distanceEffect := by apply (readout 80 3).positive_small_corrected_velocity_gives_less_than_the_full_distance_effect (1 / 4) the_independent_relative_smallness_input_is_realized rw [the_independent_readout_has_nonzero_observed_corrected_and_peculiar_velocities.2.1] norm_num theorem a_consistent_observer_change_preserves_both_physical_residual_readouts : ((readout 80 3).changeObserver 75).corrected = (readout 80 3).corrected ∧ ((readout 80 3).changeObserver 75).peculiar = (readout 80 3).peculiar := (readout 80 3).consistent_observer_change_preserves_corrected_and_peculiar_velocities 75 theorem the_same_relative_smallness_input_survives_the_observer_change : ((readout 80 3).changeObserver 75).RelativeSmallnessAt (1 / 4) := ((readout 80 3).consistent_observer_change_preserves_relative_smallness 75 (1 / 4)).mpr the_independent_relative_smallness_input_is_realized theorem the_changed_observer_can_measure_a_negative_velocity_with_positive_corrected_velocity : ((readout 80 3).changeObserver 75).observed = -31 ∧ ((readout 80 3).changeObserver 75).corrected = 60 := by constructor · norm_num [RadialVelocityReadout.changeObserver, readout] · rw [a_consistent_observer_change_preserves_both_physical_residual_readouts.1] exact the_independent_readout_has_nonzero_observed_corrected_and_peculiar_velocities.2.1 theorem a_distance_scale_change_can_preserve_the_velocity_prediction : (readout 80 3).distanceEffect = (readout 40 6).distanceEffect ∧ (readout 80 3).peculiar = (readout 40 6).peculiar ∧ (3 : Real) ≠ 6 := by norm_num [readout, RadialVelocityReadout.peculiar] end HubbleObserverCorrectionClient example : PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.ReducedLowVelocityDistanceEffectReadout.HasSmallObservedVelocityAt PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.positiveLowVelocityWithoutDominantPeculiarMotion ∧ ¬PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.ReducedLowVelocityDistanceEffectReadout.HasDominantPeculiarMotionAt PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.positiveLowVelocityWithoutDominantPeculiarMotion := PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.small_unsigned_velocity_does_not_supply_dominant_peculiar_motion example : PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.HasRoughRestrictedRelation PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.reportedButUnadoptedEvidenceReadout ∧ ¬PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.HasExactUniversalLaw PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.reportedButUnadoptedEvidenceReadout := PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.a_rough_restricted_relation_is_not_an_exact_universal_law example : PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.HasPossibleDeSitterEffect PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.reportedButUnadoptedEvidenceReadout ∧ ¬PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.HasEstablishedDeSitterExplanation PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.reportedButUnadoptedEvidenceReadout := PhysicsLogic.Papers.Hubble1929.NebularRelationFramework.Countermodels.a_possible_de_sitter_effect_is_not_an_established_explanation example {T : PhysicsLogic.Papers.LandauSuperfluidity1941.LandauTypes} (framework : PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework T) (dispersion : framework.RotonDispersionFormulaAt framework.rotonClass framework.elementarySpectrum framework.helium framework.zeroTemperatureRegime) : framework.RotonEnergyPerMomentumMinimumAt framework.rotonClass framework.elementarySpectrum framework.helium framework.zeroTemperatureRegime := framework.roton_dispersion_supplies_attained_energy_per_momentum_minimum framework.rotonClass framework.elementarySpectrum framework.helium framework.zeroTemperatureRegime dispersion example : PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.retainsPhononRotonDispersionFormulas PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.elementarySpectrumWithoutHydrodynamics.spectrum ∧ ¬PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.TwoVelocityHydrodynamicsAvailable PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.elementarySpectrumWithoutHydrodynamics := PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.an_elementary_spectrum_does_not_supply_two_velocity_hydrodynamics example : PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.SelectedWallChannelStable PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.stableWallWithVortexDissipation ∧ ¬PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.AllBoundaryAndVortexDissipationAbsent PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.stableWallWithVortexDissipation := PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.selected_wall_channel_stability_does_not_exclude_every_dissipation_mechanism example : PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.PositiveQuadraticRotonMomentumAvailable PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.quadraticRotonAtNonminimalMomentum ∧ ¬PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.SelectedRotonMomentumAttainsMinimum PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.quadraticRotonAtNonminimalMomentum := PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework.Countermodels.one_quadratic_roton_momentum_does_not_attain_critical_threshold example : PhysicsLogic.Papers.Lorenz1963.LorenzFramework.Countermodels.FiniteNumericalEvidence PhysicsLogic.Papers.Lorenz1963.LorenzFramework.Countermodels.finiteIrregularReportWithPeriodicContinuation ∧ ¬PhysicsLogic.Papers.Lorenz1963.LorenzFramework.Countermodels.ContinuumNonperiodicityProven PhysicsLogic.Papers.Lorenz1963.LorenzFramework.Countermodels.finiteIrregularReportWithPeriodicContinuation := PhysicsLogic.Papers.Lorenz1963.LorenzFramework.Countermodels.finite_numerical_evidence_does_not_prove_continuum_nonperiodicity example : PhysicsLogic.Papers.Lorenz1963.LorenzFramework.Countermodels.FiniteModelInstabilityObserved PhysicsLogic.Papers.Lorenz1963.LorenzFramework.Countermodels.finiteInstabilityWithOpenAtmosphericScope ∧ ¬PhysicsLogic.Papers.Lorenz1963.LorenzFramework.Countermodels.AtmosphericApplicabilityEstablished PhysicsLogic.Papers.Lorenz1963.LorenzFramework.Countermodels.finiteInstabilityWithOpenAtmosphericScope := PhysicsLogic.Papers.Lorenz1963.LorenzFramework.Countermodels.finite_model_instability_does_not_establish_atmospheric_applicability example {T : PhysicsLogic.Papers.Kadanoff1966.KadanoffTypes} (framework : PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework T) (relations : framework.EquationTwentyTwoMagnetizationDerivativeRelationsAt) (epsilon field : Real) : framework.magnetizationValueAt epsilon field = 0 ↔ framework.averageMagnetizationValueAt epsilon field = 0 := framework.equation_twenty_two_magnetization_vanishes_iff_average_spin_vanishes relations epsilon field example {T : PhysicsLogic.Papers.Kadanoff1966.KadanoffTypes} (framework : PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework T) (relation : framework.EquationTwentyThreeSusceptibilityDerivativeRelationAt) (epsilon field : Real) : framework.susceptibilityValueAt epsilon field = 0 ↔ framework.magnetizationFieldDerivativeValueAt epsilon field = 0 := framework.equation_twenty_three_susceptibility_vanishes_iff_derivative_vanishes relation epsilon field example : PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.ReducedObservableNormalizationReadout.HasPointwiseNonzeroProportionalityAt PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.pointwiseProportionalityWithoutFixedNormalization ∧ ¬PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.ReducedObservableNormalizationReadout.HasFixedNonzeroNormalizationAt PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.pointwiseProportionalityWithoutFixedNormalization := PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.pointwise_proportionality_does_not_supply_fixed_observable_normalization example : PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.ReducedObservableNormalizationReadout.HasFixedNonzeroNormalizationAt PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.fixedNonunitObservableNormalization ∧ ¬PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.ReducedObservableNormalizationReadout.HasLiteralObservableEqualityAt PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.fixedNonunitObservableNormalization := PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.fixed_nonzero_observable_normalization_does_not_imply_literal_equality example : PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.retainsEquationsFourteen PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.unequalAboveBelowExponentModel ∧ ¬PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.retainsEquationFifteen PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.unequalAboveBelowExponentModel := PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.equations_fourteen_do_not_supply_tentative_equation_fifteen example : PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.oneSidedLogarithmicCompensation.RegularCompensationAvailable ∧ ¬PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.oneSidedLogarithmicCompensation.LogarithmicAmplitudeEqualityAvailable := PhysicsLogic.Papers.Kadanoff1966.CellScalingFramework.Countermodels.one_sided_logarithmic_branch_does_not_supply_amplitude_equality example : PhysicsLogic.Papers.KochenSpecker1967.Countermodels.StatisticalAgreement ∧ ¬PhysicsLogic.Papers.KochenSpecker1967.Countermodels.FunctionalPreservation := PhysicsLogic.Papers.KochenSpecker1967.Countermodels.statistical_agreement_does_not_preserve_observable_functions example : (∃ context, PhysicsLogic.Papers.KochenSpecker1967.Countermodels.completeButUntestable context) ∧ ¬∀ context, PhysicsLogic.Papers.KochenSpecker1967.Countermodels.completeButUntestable context → PhysicsLogic.Papers.KochenSpecker1967.Countermodels.testable context := PhysicsLogic.Papers.KochenSpecker1967.Countermodels.coloring_failure_without_measurement_bridge_is_not_a_test example : PhysicsLogic.Papers.Chandrasekhar1931.Framework.Countermodels.uniqueMassCalculated PhysicsLogic.Papers.Chandrasekhar1931.Framework.Countermodels.uniqueMassWithoutUpperLimit ∧ ¬PhysicsLogic.Papers.Chandrasekhar1931.Framework.Countermodels.upperLimitInterpreted PhysicsLogic.Papers.Chandrasekhar1931.Framework.Countermodels.uniqueMassWithoutUpperLimit := PhysicsLogic.Papers.Chandrasekhar1931.Framework.Countermodels.unique_extreme_mass_does_not_definitionally_supply_upper_limit example : PhysicsLogic.Papers.Chandrasekhar1931.Framework.Countermodels.technicalRoutesComplete PhysicsLogic.Papers.Chandrasekhar1931.Framework.Countermodels.technicalWhiteDwarfWithoutWorldGrounding ∧ ¬PhysicsLogic.Papers.Chandrasekhar1931.Framework.Countermodels.theoryWorldGrounded PhysicsLogic.Papers.Chandrasekhar1931.Framework.Countermodels.technicalWhiteDwarfWithoutWorldGrounding := PhysicsLogic.Papers.Chandrasekhar1931.Framework.Countermodels.completed_technical_routes_do_not_supply_theory_model_grounding example : PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.microscopicPathRelation ∧ ¬PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.stateDetailedBalance := PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.path_microscopic_reversibility_does_not_supply_state_detailed_balance example : PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.DrivenThreeStateMarkovReadout.PhysicalForwardReverseMarkovProcessAt PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.circulatingDrivenMarkovReadout ∧ PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.DrivenThreeStateMarkovReadout.selectedClockwisePopulationCurrent PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.circulatingDrivenMarkovReadout = (1 / 9 : Real) := by obtain ⟨process, current, _⟩ := PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.circulating_driven_markov_process_retains_nonzero_probability_current exact ⟨process, current⟩ example : PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.DrivenThreeStateMarkovReadout.PhysicalForwardReverseMarkovProcessAt PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.balancedDrivenMarkovReadout ∧ PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.DrivenThreeStateMarkovReadout.SameProtocolStateDetailedBalanceAt PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.balancedDrivenMarkovReadout := by obtain ⟨process, balance, _⟩ := PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.microscopic_reversibility_also_allows_equilibrium_detailed_balance exact ⟨process, balance⟩ example : PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.BareEntropyIdentity PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.identityWithUnrelatedNegativeMean ∧ ¬PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.BareMeanEntropySecondLaw PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.identityWithUnrelatedNegativeMean := PhysicsLogic.Papers.Crooks1999.FluctuationFramework.Countermodels.bare_entropy_identity_does_not_supply_mean_second_law example : PhysicsLogic.Papers.Nyquist1928.ThermalNoiseFramework.Countermodels.hasNamedClassicalNoiseRelation PhysicsLogic.Papers.Nyquist1928.ThermalNoiseFramework.Countermodels.wrongEquationOneModel ∧ ¬PhysicsLogic.Papers.Nyquist1928.ThermalNoiseFramework.Countermodels.realizesEquationOne PhysicsLogic.Papers.Nyquist1928.ThermalNoiseFramework.Countermodels.wrongEquationOneModel := PhysicsLogic.Papers.Nyquist1928.ThermalNoiseFramework.Countermodels.named_classical_noise_relation_does_not_supply_equation_one example : PhysicsLogic.Papers.Nyquist1928.PlanckReplacementFormulaReadout.Countermodels.NamedPlanckReplacementAt PhysicsLogic.Papers.Nyquist1928.PlanckReplacementFormulaReadout.Countermodels.wrongNamedPlanckReplacementReadout ∧ ¬PhysicsLogic.Papers.Nyquist1928.PlanckReplacementFormulaReadout.Countermodels.wrongNamedPlanckReplacementReadout.SourceFormulaAt := PhysicsLogic.Papers.Nyquist1928.PlanckReplacementFormulaReadout.Countermodels.named_planck_replacement_does_not_supply_equations_seven_and_eight example : PhysicsLogic.Papers.Nyquist1928.PlanckReplacementFormulaReadout.Countermodels.zeroPointFreePlanckReplacementReadout.SourceFormulaAt ∧ ¬PhysicsLogic.Papers.Nyquist1928.PlanckReplacementFormulaReadout.Countermodels.zeroPointFreePlanckReplacementReadout.IncludesHalfQuantumZeroPointAt := PhysicsLogic.Papers.Nyquist1928.PlanckReplacementFormulaReadout.Countermodels.nyquist_planck_formula_does_not_include_callen_welton_zero_point example (nyquist : PhysicsLogic.Papers.Nyquist1928.PlanckReplacementFormulaReadout Unit) (callen : PhysicsLogic.Papers.CallenWelton1951.GeneralizedNyquistFormulaReadout Unit) (nyquistDomain : nyquist.FormulaDomainAt) (nyquistEnergy : nyquist.EquationSevenPlanckOscillatorEnergyAt) (callenDomain : callen.FormulaDomainAt) (callenEnergy : callen.EquationFourNineOscillatorMeanEnergyAt) (sameQuantum : nyquist.planckConstantValue * nyquist.frequencyValueAt () = callen.reducedPlanckConstantValue * callen.frequencyValueAt ()) (sameThermalScale : nyquist.boltzmannConstantValue * nyquist.temperatureValue = callen.boltzmannConstantValue * callen.selectedTemperatureValue) (sameReportedEnergy : nyquist.thermalOscillatorEnergyValueAt () = callen.oscillatorMeanEnergyValueAt ()) : False := by have nyquistSector : nyquist.selectedFrequencySector () := by simpa using nyquistDomain.2.2.2.1 have callenFrequencyPositive : 0 < callen.frequencyValueAt () := by simpa using callenDomain.2.2.2 have nyquistValue := nyquistEnergy () nyquistSector have callenValue := callenEnergy () callenFrequencyPositive rw [sameQuantum, sameThermalScale] at nyquistValue rw [sameReportedEnergy] at nyquistValue have planckPositive := callenDomain.1 have zeroPointPositive : 0 < callen.reducedPlanckConstantValue * callen.frequencyValueAt () / 2 := by positivity linarith example : PhysicsLogic.Papers.CallenWelton1951.GeneralizedNyquistFormulaReadout.Countermodels.NamedGeneralizedNyquistOutputAt PhysicsLogic.Papers.CallenWelton1951.GeneralizedNyquistFormulaReadout.Countermodels.wrongNamedGeneralizedNyquistReadout ∧ ¬PhysicsLogic.Papers.CallenWelton1951.GeneralizedNyquistFormulaReadout.Countermodels.wrongNamedGeneralizedNyquistReadout.SourceFormulaAt := PhysicsLogic.Papers.CallenWelton1951.GeneralizedNyquistFormulaReadout.Countermodels.named_generalized_relation_does_not_supply_printed_nyquist_formula example {T : PhysicsLogic.Papers.OppenheimerVolkoff1939.OppenheimerVolkoffTypes} (readout : PhysicsLogic.Papers.OppenheimerVolkoff1939.ColdNeutronEquationOfStateFormulaReadout T) (massPositive : 0 < readout.neutronRestMassValue) (lightPositive : 0 < readout.lightSpeedValue) (momentumNonnegative : 0 ≤ readout.maximumMomentumValue) (formula : readout.EquationsFourteenAndFifteenFermiParameterAt) : readout.fermiParameterValue = 0 ↔ readout.maximumMomentumValue = 0 := readout.equation_fourteen_zero_parameter_iff_zero_maximum_momentum massPositive lightPositive momentumNonnegative formula example {T : PhysicsLogic.Papers.OppenheimerVolkoff1939.OppenheimerVolkoffTypes} (readout : PhysicsLogic.Papers.OppenheimerVolkoff1939.ColdNeutronEquationOfStateFormulaReadout T) (inputs : readout.SurfaceSourceInputsAt) : readout.maximumMomentumValue = 0 ∧ readout.densityValue = 0 ∧ readout.pressureValue = 0 ∧ readout.properParticleNumberDensityValue = 0 ∧ readout.dimensionlessEnclosedMassRadialDerivativeValue = 0 := by rcases readout.surface_source_inputs_derive_vacuum_boundary inputs with ⟨_domain, _zeroParameter, zeroMomentum, zeroDensity, zeroPressure, zeroParticleDensity, zeroMassDerivative, _zeroDenominator⟩ exact ⟨zeroMomentum, zeroDensity, zeroPressure, zeroParticleDensity, zeroMassDerivative⟩ example {T : PhysicsLogic.Papers.OppenheimerVolkoff1939.OppenheimerVolkoffTypes} (framework : PhysicsLogic.Papers.OppenheimerVolkoff1939.Framework T) (family : framework.EstablishedColdNeutronEquilibriumFamily) : framework.SelectedColdNeutronVacuumSurfaceAt := framework.cold_neutron_family_derives_vacuum_surface family example : PhysicsLogic.Papers.OppenheimerVolkoff1939.ColdNeutronEquationOfStateFormulaReadout.Countermodels.zeroFermiVacuumSurfaceReadout.SurfaceSourceInputsAt ∧ ¬PhysicsLogic.Papers.OppenheimerVolkoff1939.ColdNeutronEquationOfStateFormulaReadout.Countermodels.zeroFermiVacuumSurfaceReadout.FormulaDomainAt := PhysicsLogic.Papers.OppenheimerVolkoff1939.ColdNeutronEquationOfStateFormulaReadout.Countermodels.physical_vacuum_surface_is_not_a_regular_fermi_interior example : PhysicsLogic.Papers.OppenheimerVolkoff1939.Framework.Countermodels.coldFermiEquationOfState PhysicsLogic.Papers.OppenheimerVolkoff1939.Framework.Countermodels.coldFermiEquationWithoutMassCurve ∧ ¬PhysicsLogic.Papers.OppenheimerVolkoff1939.Framework.Countermodels.maximumMassCurveCalculated PhysicsLogic.Papers.OppenheimerVolkoff1939.Framework.Countermodels.coldFermiEquationWithoutMassCurve := PhysicsLogic.Papers.OppenheimerVolkoff1939.Framework.Countermodels.cold_neutron_equation_of_state_does_not_supply_mass_curve example : PhysicsLogic.Papers.OppenheimerVolkoff1939.Framework.Countermodels.lacksStaticEquilibrium PhysicsLogic.Papers.OppenheimerVolkoff1939.Framework.Countermodels.noStaticSolutionWithoutHorizonOrFiniteTimeCollapse ∧ ¬PhysicsLogic.Papers.OppenheimerVolkoff1939.Framework.Countermodels.establishesHorizonOrFiniteTimeCollapse PhysicsLogic.Papers.OppenheimerVolkoff1939.Framework.Countermodels.noStaticSolutionWithoutHorizonOrFiniteTimeCollapse := PhysicsLogic.Papers.OppenheimerVolkoff1939.Framework.Countermodels.no_static_equilibrium_does_not_prove_horizon_or_finite_time_collapse example : PhysicsLogic.Papers.MerminWagner1966.NoMagneticOrderAt PhysicsLogic.Papers.MerminWagner1966.magneticZeroWithOtherOrderModel .ferromagnetic ∧ ¬PhysicsLogic.Papers.MerminWagner1966.NoOtherOrder PhysicsLogic.Papers.MerminWagner1966.magneticZeroWithOtherOrderModel := by obtain ⟨magnetic, _notCrystalline, notOther, _susceptibility⟩ := PhysicsLogic.Papers.MerminWagner1966.no_magnetic_order_does_not_exclude_other_order_or_susceptibility exact ⟨magnetic, notOther⟩ namespace ThermodynamicLimitPhysicalReuseAudit open PhysicsLogic.Paradigm.ThermodynamicLimit.Countermodels example : singularLimit.finiteDescription .small = .finiteSmall /\ singularLimit.finiteDescription .large = .finiteLarge /\ singularLimit.regularAt .finiteSmall .critical /\ singularLimit.regularAt .finiteLarge .critical /\ singularLimit.singularAt .limit .critical /\ Not (singularLimit.singularAt .limit .offCritical) /\ singularLimit.contains .criticalNeighborhood .critical /\ Not (singularLimit.contains .offCriticalNeighborhood .critical) /\ singularLimit.exceptionalAt .large .critical /\ Not (singularLimit.exceptionalAt .small .critical) := finite_volume_phase_model_distinguishes_sizes_controls_and_regions example : unbridgedUniformFamily.volumeAdmissible .small /\ unbridgedUniformFamily.volumeAdmissible .large /\ unbridgedUniformFamily.finiteHasPropertyAt .finiteSmall .boundedCorrelation /\ unbridgedUniformFamily.finiteHasPropertyAt .finiteLarge .boundedCorrelation /\ Not (unbridgedUniformFamily.limitHasPropertyAt .limit .boundedCorrelation) /\ unbridgedUniformFamily.limitHasPropertyAt .limit .spontaneousMagnetization := finite_volume_family_distinguishes_sizes_and_limit_properties example : directInfiniteWithoutFiniteRoute.volumeAdmissible .large /\ directInfiniteWithoutFiniteRoute.DirectInfinitePropertyAt .spontaneousMagnetization /\ Not (directInfiniteWithoutFiniteRoute.finiteFamilyAdmissibleFor directInfiniteWithoutFiniteRoute.finiteDescription .spontaneousMagnetization) := direct_infinite_property_retains_a_distinct_finite_volume end ThermodynamicLimitPhysicalReuseAudit example : PhysicsLogic.Papers.Friedmann1922.Countermodels.ReducedFriedmannCurvatureReadout.HasIllustrativePeriodAt PhysicsLogic.Papers.Friedmann1922.Countermodels.completeReducedFriedmannCurvatureReadout ∧ ¬PhysicsLogic.Papers.Friedmann1922.Countermodels.ReducedFriedmannCurvatureReadout.HasActualWorldDeterminationAt PhysicsLogic.Papers.Friedmann1922.Countermodels.completeReducedFriedmannCurvatureReadout := PhysicsLogic.Papers.Friedmann1922.Countermodels.an_illustrative_period_does_not_determine_the_actual_world example : PhysicsLogic.Papers.Guth1981.InflationFramework.Countermodels.successfulInflationSolution.Solves .supercoolingExpansion .falseVacuumUniverse (fun target => target = .horizon ∨ target = .flatness) ∧ ¬PhysicsLogic.Papers.Guth1981.InflationFramework.Countermodels.nonviableInflation.Viable .supercoolingExpansion .falseVacuumUniverse := PhysicsLogic.Papers.Guth1981.InflationFramework.Countermodels.solving_horizon_and_flatness_does_not_imply_viability example : PhysicsLogic.Papers.Guth1981.InflationFramework.Countermodels.uniformBubblesWithUnresolvedObligations.ConditionallyRepairsBubbleExit ∧ ¬PhysicsLogic.Papers.Guth1981.InflationFramework.Countermodels.uniformBubblesWithUnresolvedObligations.ViableModificationEstablished := by obtain ⟨repair, _notTrigger, _notParticle, notViable⟩ := PhysicsLogic.Papers.Guth1981.InflationFramework.Countermodels.conditional_bubble_repair_does_not_establish_modified_viability exact ⟨repair, notViable⟩ example : PhysicsLogic.Papers.Guth1981.MonopoleHorizonFormulaReadout.Countermodels.NamedMonopoleHorizonBoundAt PhysicsLogic.Papers.Guth1981.MonopoleHorizonFormulaReadout.Countermodels.namedBoundWithoutPrintedEquations ∧ ¬PhysicsLogic.Papers.Guth1981.MonopoleHorizonFormulaReadout.Countermodels.namedBoundWithoutPrintedEquations.SourceFormulaAt := PhysicsLogic.Papers.Guth1981.MonopoleHorizonFormulaReadout.Countermodels.named_monopole_horizon_bound_does_not_supply_printed_formulas example : PhysicsLogic.Papers.Guth1981.MonopoleHorizonFormulaReadout.Countermodels.horizonRadiusWithoutMonopoleDensity.EquationFourOneBubbleHorizonBoundAt ∧ ¬PhysicsLogic.Papers.Guth1981.MonopoleHorizonFormulaReadout.Countermodels.horizonRadiusWithoutMonopoleDensity.EquationFourTwoMonopoleDensityEstimateAt := PhysicsLogic.Papers.Guth1981.MonopoleHorizonFormulaReadout.Countermodels.bubble_horizon_bound_does_not_supply_monopole_density_estimate example : PhysicsLogic.Papers.Guth1981.MonopoleHorizonFormulaReadout.Countermodels.densityEstimateWithoutPreskillObservation.EquationFourOneBubbleHorizonBoundAt ∧ PhysicsLogic.Papers.Guth1981.MonopoleHorizonFormulaReadout.Countermodels.densityEstimateWithoutPreskillObservation.EquationFourTwoMonopoleDensityEstimateAt ∧ ¬PhysicsLogic.Papers.Guth1981.MonopoleHorizonFormulaReadout.Countermodels.densityEstimateWithoutPreskillObservation.EquationFourThreePreskillAbundanceBoundAt := PhysicsLogic.Papers.Guth1981.MonopoleHorizonFormulaReadout.Countermodels.horizon_density_estimate_does_not_supply_preskill_observational_bound example : PhysicsLogic.Papers.Guth1981.StrongTransitionMonopoleRevisionReadout.Countermodels.revisedBoundWithoutCoarseEstimate.SourceRevisionAt ∧ ¬PhysicsLogic.Papers.Guth1981.StrongTransitionMonopoleRevisionReadout.Countermodels.revisedBoundWithoutCoarseEstimate.CoarseHorizonEstimateSatisfiedAt := PhysicsLogic.Papers.Guth1981.StrongTransitionMonopoleRevisionReadout.Countermodels.revised_strong_transition_bound_does_not_imply_coarse_horizon_estimate example : PhysicsLogic.Papers.LandauDamping1946.PlasmaFramework.Countermodels.HasCollectiveFieldDampingAt PhysicsLogic.Papers.LandauDamping1946.PlasmaFramework.Countermodels.dampedFieldWithUndampedDistribution ∧ ¬PhysicsLogic.Papers.LandauDamping1946.PlasmaFramework.Countermodels.HasVelocityDistributionDampingAt PhysicsLogic.Papers.LandauDamping1946.PlasmaFramework.Countermodels.dampedFieldWithUndampedDistribution := by obtain ⟨field, _velocityPole, notDistribution⟩ := PhysicsLogic.Papers.LandauDamping1946.PlasmaFramework.Countermodels.collective_field_damping_does_not_entail_distribution_damping exact ⟨field, notDistribution⟩ example : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ChiralTwoNucleonFramework.Countermodels.HasOneSuccessfulPredictionAt PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ChiralTwoNucleonFramework.Countermodels.reducedTwoNucleonPredictionScope ∧ ¬PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ChiralTwoNucleonFramework.Countermodels.HasCutoffIndependentPredictionAt PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ChiralTwoNucleonFramework.Countermodels.reducedTwoNucleonPredictionScope := PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ChiralTwoNucleonFramework.Countermodels.one_successful_prediction_does_not_establish_cutoff_independence example : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ProjectedContactFormulaReadout.originalNNLOSharpContactReadout.SourceFormulaAt := PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ProjectedContactFormulaReadout.original_nnlo_sharp_contact_formulas_are_nonvacuously_realized example : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ProjectedContactFormulaReadout.originalNNLOSharpContactReadout.ReciprocalSDMixingFormulasAt ∧ PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ProjectedContactFormulaReadout.originalNNLOSharpContactReadout.dFromSPotentialValue ≠ PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ProjectedContactFormulaReadout.originalNNLOSharpContactReadout.sFromDPotentialValue := PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ProjectedContactFormulaReadout.common_sd_contact_coefficient_need_not_give_equal_transition_values example {Value : Type} (framework : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ChiralTwoNucleonFramework Value) (calibration : framework.LowPartialWaveCalibrationResult) : framework.SelectedProjectedContactFormulaAt ∧ framework.projectedContactFormulaReadout.totalContactParameterCountValue = 9 := framework.calibrated_contacts_retain_printed_projected_channel_formulas calibration example : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.DeuteronObservableTableReadout.originalExponentialDeuteronReadout.SourceTableAt := PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.DeuteronObservableTableReadout.original_deuteron_table_is_nonvacuously_realized example : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.DeuteronObservableTableReadout.Countermodels.HasImprovedNNLOBindingEnergyAt PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.DeuteronObservableTableReadout.originalExponentialDeuteronReadout ∧ ¬PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.DeuteronObservableTableReadout.Countermodels.HasImprovedNNLOQuadrupoleMomentAt PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.DeuteronObservableTableReadout.originalExponentialDeuteronReadout := PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.DeuteronObservableTableReadout.Countermodels.improved_deuteron_binding_does_not_imply_improved_quadrupole_moment example {Value : Type} (framework : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.ChiralTwoNucleonFramework Value) (result : framework.DeuteronResult) : framework.deuteronObservableTableReadout.experimentalDStateProbabilityMeasurementCount = 0 := (framework.scoped_deuteron_result_retains_printed_values_and_physical_scope result).2.2.2.1 example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.Fidelity.countermodelMatrix := PhysicsLogic.Papers.EpelbaumGloeckleMeissner1999.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.Jaynes1957.JaynesFramework.Countermodels.ProbabilityRoleReadout.HasEpistemicAssignment PhysicsLogic.Papers.Jaynes1957.JaynesFramework.Countermodels.epistemicAssignmentWithoutPhysicalTrials ∧ ¬PhysicsLogic.Papers.Jaynes1957.JaynesFramework.Countermodels.ProbabilityRoleReadout.HasObjectiveChanceEvidence PhysicsLogic.Papers.Jaynes1957.JaynesFramework.Countermodels.epistemicAssignmentWithoutPhysicalTrials := by obtain ⟨assignment, _value, _noTrials, notChance⟩ := PhysicsLogic.Papers.Jaynes1957.JaynesFramework.Countermodels.maximum_entropy_epistemic_probability_does_not_establish_objective_chance exact ⟨assignment, notChance⟩ example : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels.normalizedIndependentSteps PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels.independentButBiasedSteps ∧ ¬PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels.zeroDriftStepSymmetry PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels.independentButBiasedSteps := PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels.independent_steps_do_not_supply_zero_drift_symmetry example : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels.positiveFrequencyDensityAvailable PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels.positiveFrequencyDensityWithoutPredictiveUse ∧ ¬PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels.predictiveFrequencyUse PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels.positiveFrequencyDensityWithoutPredictiveUse := PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels.positive_frequency_density_does_not_supply_predictive_use example : PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.prandtlFrictionContrastReadout.HasSmallFrictionLayerContrastAt ∧ ¬PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.prandtlFrictionContrastReadout.HasGloballyNegligibleFrictionAt := PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.small_viscosity_does_not_make_viscous_effects_globally_negligible example : PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.prandtlTangentialWallMatchingWithPenetration.HasWallTangentialMatchingAt ∧ PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.prandtlTangentialWallMatchingWithPenetration.HasOuterTangentialMatchingAt ∧ ¬PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.prandtlTangentialWallMatchingWithPenetration.HasWallNormalMatchingAt := PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.tangential_wall_adherence_does_not_prevent_normal_penetration example : PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.prandtlWallMatchingWithoutOuterRecovery.HasWallTangentialMatchingAt ∧ PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.prandtlWallMatchingWithoutOuterRecovery.HasWallNormalMatchingAt ∧ ¬PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.prandtlWallMatchingWithoutOuterRecovery.HasOuterTangentialMatchingAt := PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.wall_adherence_does_not_supply_outer_velocity_recovery example : PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.prandtlMovingWallWithOuterRecovery.HasVelocityBoundaryConditionsAt ∧ ¬PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.prandtlMovingWallWithOuterRecovery.HasStationaryWallAt := PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.prandtl_wall_matching_does_not_require_a_stationary_surface example : PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.HasPositiveCoordinatePressureGradientAt PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.reversedFlowWithPositiveCoordinatePressureGradient ∧ PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.HasConsistentFlowDirectedPressureProbeAt PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.reversedFlowWithPositiveCoordinatePressureGradient ∧ ¬PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.HasFlowDirectedAdversePressureRiseAt PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.reversedFlowWithPositiveCoordinatePressureGradient := PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework.Countermodels.positive_coordinate_pressure_gradient_need_not_rise_along_reversed_flow example {T : PhysicsLogic.Papers.Prandtl1904.Prandtl1904Types} (framework : PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework T) (separation : framework.EstablishedSeparationResult) : framework.FlowDirectedAdversePressureRiseAt framework.selectedSeparationLocation := (framework.established_separation_applies_pressure_necessity_to_actual_flow separation).1 example {T : PhysicsLogic.Papers.Prandtl1904.Prandtl1904Types} (framework : PhysicsLogic.Papers.Prandtl1904.BoundaryLayerFramework T) (reduction : framework.EstablishedBoundaryLayerReduction) : framework.thinViscousBoundaryLayerReductionModel |>.VelocityMatchedBoundaryLayerReductionAt framework.selectedBoundaryVelocityMatchingReadout := framework.established_prandtl_reduction_realizes_velocity_matched_boundary_layer reduction example : PhysicsLogic.Framework.ResonantTransitionSpectroscopyContext.Countermodels.responseWithoutRepresentationModel.drivenResponseCalculatedAt () () () () () () () ∧ ¬PhysicsLogic.Framework.ResonantTransitionSpectroscopyContext.Countermodels.responseWithoutRepresentationModel.responseRepresentsTransitionAt () () () () () := PhysicsLogic.Framework.ResonantTransitionSpectroscopyContext.Countermodels.calculated_response_does_not_supply_transition_representation example : PhysicsLogic.Framework.TwoStateTransitionOutcomeReadout.Countermodels.complementaryButNegativeTransition.transitionProbabilityValue + PhysicsLogic.Framework.TwoStateTransitionOutcomeReadout.Countermodels.complementaryButNegativeTransition.remainingProbabilityValue = 1 ∧ ¬PhysicsLogic.Framework.TwoStateTransitionOutcomeReadout.Countermodels.complementaryButNegativeTransition.NormalizedOutcomeProbabilitiesAt := PhysicsLogic.Framework.TwoStateTransitionOutcomeReadout.Countermodels.complementary_outcomes_need_not_be_physical_probabilities example (readout : PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout) (formula : readout.EquationTwoWeakOscillatingTransitionAt) : readout.PhysicalFormulaDomainAt := readout.equation_two_derives_admissible_transition_probability formula example (readout : PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout) (domain : readout.PhysicalFieldFrequencyDomainAt) : readout.equationTwoTransitionEnvelopeValueAt = 1 ↔ readout.driveFrequency = readout.larmorFrequency := readout.transition_envelope_is_maximal_iff_drive_matches_larmor domain example : PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout.Countermodels.positiveFieldsWithUnphysicalProbability.PhysicalFieldFrequencyDomainAt ∧ ¬PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout.Countermodels.positiveFieldsWithUnphysicalProbability.PhysicalFormulaDomainAt := by obtain ⟨positiveFields, _positiveDetuning, notAdmissible⟩ := PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout.Countermodels.positive_fields_do_not_imply_admissible_transition_probability exact ⟨positiveFields, notAdmissible⟩ example : PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout.Countermodels.NamedResonanceOutputsAt PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout.Countermodels.wrongNamedOutputs ∧ ¬PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout.Countermodels.wrongNamedOutputs.EquationOneLarmorFrequencyAt := by obtain ⟨named, notLarmor, _notTransition⟩ := PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout.Countermodels.named_resonance_outputs_do_not_supply_equations_one_and_two exact ⟨named, notLarmor⟩ example : (0 ≤ PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout.Countermodels.boundedTransitionWithoutOutcomeComplement.transitionProbabilityValue ∧ PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout.Countermodels.boundedTransitionWithoutOutcomeComplement.transitionProbabilityValue ≤ 1) ∧ ¬PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout.Countermodels.boundedTransitionWithoutOutcomeComplement.NormalizedOutcomeProbabilitiesAt := PhysicsLogic.Papers.Rabi1939.ResonanceFormulaReadout.Countermodels.bounded_transition_probability_does_not_supply_outcome_complement example {T : PhysicsLogic.Papers.Rabi1939.RabiTypes} (framework : PhysicsLogic.Papers.Rabi1939.MolecularBeamResonanceFramework T) (complete : framework.CompleteRabiConclusion) : framework.resonantTransitionSpectroscopyModel |>.NonvacuousNormalizedResonantTransitionSpectroscopyAt framework.selectedResonanceFormulaReadout.selectedTransitionOutcomeReadout := framework.complete_rabi_result_exhibits_normalized_shared_resonant_spectroscopy complete namespace MathematicsAndNaturePhysicalReuseAudit open PhysicsLogic.Philosophy.MathematicsAndNature.Countermodels example : modelRepresentsPhysicalState .selectedModel .preparedSample /\ modelRepresentsPhysicalState .controlModel .controlSample /\ Not (modelRepresentsPhysicalState .selectedModel .controlSample) /\ coordinateRepresentsPhysicalQuantity .chargeCoordinate .electricCharge /\ coordinateRepresentsPhysicalQuantity .temperatureCoordinate .backgroundTemperature /\ Not (coordinateRepresentsPhysicalQuantity .chargeCoordinate .backgroundTemperature) := model_state_and_quantity_representation_distinguish_physical_roles example : assignedButPhysicallyWrong.physicallyHasValueAt .electricCharge .preparedSample () .actual () /\ Not (assignedButPhysicallyWrong.physicallyHasValueAt .electricCharge .preparedSample () .spurious ()) := selected_physical_charge_distinguishes_actual_and_spurious_values end MathematicsAndNaturePhysicalReuseAudit namespace ObservabilityPhysicalReuseAudit open PhysicsLogic.Philosophy.Observability.Countermodels example : observerRelativeAccess.OperationallyAccessible .inside .resonanceFrequency .selected .selected /\ Not (observerRelativeAccess.OperationallyAccessible .inside .backgroundTemperature .selected .selected) /\ Not (observerRelativeAccess.OperationallyAccessible .outside .resonanceFrequency .selected .selected) := selected_operational_access_distinguishes_quantities_and_observers example : existingPhysicalQuantity.ObserverIndependentExistence .resonanceFrequency .selected .selected /\ Not (existingPhysicalQuantity.ObserverIndependentExistence .backgroundTemperature .selected .selected) := selected_physical_existence_distinguishes_observable_quantities example : observerRelativeAccess.OperationallyAccessible .inside .backgroundTemperature .control .control ∧ noMeasurementPerformance.MeasurementPerformedAt .inside .backgroundTemperature .control .control ∧ meaningWithoutProcedure.ProcedureGroundedMeaningAt .backgroundTemperature .control .control := by obtain ⟨_selected, controlAccess, _notCrossed, _performed, controlPerformance, _exists, grounded⟩ := operational_roles_distinguish_systems_regimes_quantities_and_control_procedures exact ⟨controlAccess, controlPerformance, grounded⟩ end ObservabilityPhysicalReuseAudit example : PhysicsLogic.Paradigm.SpacetimeAndBackground.Countermodels.covariantLaw.LawCovariant .selected ∧ ¬∃ candidate model, PhysicsLogic.Paradigm.SpacetimeAndBackground.Countermodels.fixedBackground.governedByFieldEquation candidate model := PhysicsLogic.Papers.Einstein1916.RelativityFramework.Countermodels.general_covariance_does_not_select_the_sourced_field_equation example : PhysicsLogic.Papers.EinsteinSpecialRelativity1905.Countermodels.satisfiesStationarySphericalLightEquation PhysicsLogic.Papers.EinsteinSpecialRelativity1905.Countermodels.doubledLightConeReport ∧ PhysicsLogic.Papers.EinsteinSpecialRelativity1905.Countermodels.satisfiesMovingSphericalLightEquation PhysicsLogic.Papers.EinsteinSpecialRelativity1905.Countermodels.doubledLightConeReport ∧ ¬PhysicsLogic.Papers.EinsteinSpecialRelativity1905.Countermodels.hasUnitCoordinateScale PhysicsLogic.Papers.EinsteinSpecialRelativity1905.Countermodels.doubledLightConeReport := by obtain ⟨_positive, stationary, moving, notNormalized⟩ := PhysicsLogic.Papers.EinsteinSpecialRelativity1905.Countermodels.light_constancy_alone_does_not_supply_the_lorentz_transformation exact ⟨stationary, moving, notNormalized⟩ namespace ObserverPhysicalReuseAudit open PhysicsLogic.Philosophy.Observer.Countermodels example : noObserverRoles.physicalSystemAt .target .selected /\ noObserverRoles.physicalSystemAt .instrument .selected /\ Not (noObserverRoles.ObserverScopeInhabitedAt .selected) := physical_systems_can_exist_without_an_occupied_observer_role example : physicalObserverRoles.physicalSystemAt .target .selected /\ physicalObserverRoles.observerSystemAt .instrument .selected /\ physicalObserverRoles.observerSystemAt .person .selected /\ Not (physicalObserverRoles.observerSystemAt .target .selected) /\ Not (physicalObserverRoles.consciousSystemAt .instrument .selected) /\ physicalObserverRoles.consciousSystemAt .person .selected := selected_roles_distinguish_targets_instruments_and_conscious_observers example : frameDependentCoordinates.physicalValue .targetParticle .position .invariantSeparation /\ Not (frameDependentCoordinates.physicalValue .referenceParticle .position .invariantSeparation) /\ Not (frameDependentCoordinates.physicalValue .targetParticle .position .alternativeSeparation) /\ frameDependentCoordinates.coordinateValue .left .targetParticle .position .zero /\ frameDependentCoordinates.coordinateValue .right .targetParticle .position .one /\ Not (frameDependentCoordinates.represents .left .invariantSeparation .one) /\ Not (frameDependentCoordinates.represents .right .invariantSeparation .zero) := frame_representation_distinguishes_entities_values_and_coordinates example : frameOnlyOperationalContext.physicalSystemAt .left .selected /\ frameOnlyOperationalContext.interactsWithAt .idle .left .referenceParticle .selected /\ Not (frameOnlyOperationalContext.HasOperationalObserverOfAt .targetParticle .selected) := frame_has_physical_presence_without_target_observer_agency example : noObserverRoles.ObserverScopeInhabitedAt .control ∧ operationalObserverRoles.OperationalObserverOfAt .control .control .control .control .control := by obtain ⟨controlScope, _physical, controlObservation, _frame, _excluded⟩ := observer_roles_distinguish_regimes_operational_records_and_control_observers exact ⟨controlScope, controlObservation⟩ end ObserverPhysicalReuseAudit namespace LocalityPhysicalReuseAudit open PhysicsLogic.Paradigm.Locality.Countermodels example : prBox.possible true true .selected false true ∧ ¬prBox.possible true true .selected false false ∧ prBox.FactorizesAt .control ∧ totalButSignalling.NoSignallingAt .control := by obtain ⟨selected, excluded, factorized, _holeTotal, _holeNoSignalling, _signallingFactorized, signallingNoSignalling⟩ := locality_roles_distinguish_settings_outcomes_hidden_regimes_and_control_factorization exact ⟨selected, excluded, factorized, signallingNoSignalling⟩ end LocalityPhysicalReuseAudit namespace QuantumInterpretivePhysicalReuseAudit open PhysicsLogic.Paradigm.QuantumInterpretiveCommitments.Countermodels example : testPredictions.predicts .universal .spin .up ∧ ¬testPredictions.predicts .universal .spin .down ∧ linearContext.branchComponent .left .universal .selected ∧ linearContext.branchComponent .right .universal .selected := by obtain ⟨_low, _high, _excludedState, _controlState, _baselineLow, _baselineHigh, _spin, _excludedObservable, up, notDown, _controlPrediction, _excludedSystem, _hidden, _wrongRegime, left, right, _controlRecord, _unrecorded, _definite, _observer, _nonconscious, _consciousControl⟩ := quantum_interpretive_roles_distinguish_states_observables_records_and_control_regimes exact ⟨up, notDown, left, right⟩ end QuantumInterpretivePhysicalReuseAudit example : PhysicsLogic.Papers.EinsteinSpecialRelativity1905.Countermodels.kinematicsWithoutElectronDynamics.HasParallelVelocityComposition ∧ ¬PhysicsLogic.Papers.EinsteinSpecialRelativity1905.Countermodels.kinematicsWithoutElectronDynamics.HasSectionTenElectronDynamics := by obtain ⟨composition, _longitudinal, _transverse, notDynamics⟩ := PhysicsLogic.Papers.EinsteinSpecialRelativity1905.Countermodels.velocity_composition_does_not_supply_electron_dynamics exact ⟨composition, notDynamics⟩ example : PhysicsLogic.Papers.HamiltonGeneralMethod1834.Countermodels.ReducedHamiltonMethodReadout.HasStationaryActionAt PhysicsLogic.Papers.HamiltonGeneralMethod1834.Countermodels.stationaryActionWithoutVaryingAction ∧ ¬PhysicsLogic.Papers.HamiltonGeneralMethod1834.Countermodels.ReducedHamiltonMethodReadout.HasVaryingActionAt PhysicsLogic.Papers.HamiltonGeneralMethod1834.Countermodels.stationaryActionWithoutVaryingAction := PhysicsLogic.Papers.HamiltonGeneralMethod1834.Countermodels.stationary_action_does_not_supply_varying_action example : PhysicsLogic.Papers.Noether1918.VariationalFramework.Countermodels.FiniteDivergenceReadout.HasFiniteInvariantGroup PhysicsLogic.Papers.Noether1918.VariationalFramework.Countermodels.finiteOnlyReadout.finite ∧ ¬PhysicsLogic.Papers.Noether1918.VariationalFramework.Countermodels.VariationalGroupScopeReadout.HasArbitraryFunctionIdentities PhysicsLogic.Papers.Noether1918.VariationalFramework.Countermodels.finiteOnlyReadout := by obtain ⟨finite, _relations, _formula, _noFunctions, notIdentities⟩ := PhysicsLogic.Papers.Noether1918.VariationalFramework.Countermodels.finite_invariance_does_not_imply_arbitrary_function_identities exact ⟨finite, notIdentities⟩ example : PhysicsLogic.Papers.Noether1918.VariationalFramework.Countermodels.EquationHistoryReadout.HasSourcefulFieldEquationBalance PhysicsLogic.Papers.Noether1918.VariationalFramework.Countermodels.offShellBalanceReadout ∧ ¬PhysicsLogic.Papers.Noether1918.VariationalFramework.Countermodels.EquationHistoryReadout.HasOnShellDivergenceEquations PhysicsLogic.Papers.Noether1918.VariationalFramework.Countermodels.offShellBalanceReadout := by obtain ⟨_identification, balance, notConserved⟩ := PhysicsLogic.Papers.Noether1918.VariationalFramework.Countermodels.field_equation_balance_is_not_on_shell_divergence_equation exact ⟨balance, notConserved⟩ example : PhysicsLogic.Papers.Onsager1931.ReciprocalRelationsFramework.Countermodels.EntropyProductionNonnegative PhysicsLogic.Papers.Onsager1931.ReciprocalRelationsFramework.Countermodels.positiveEntropyWithoutReciprocity ∧ ¬PhysicsLogic.Papers.Onsager1931.ReciprocalRelationsFramework.Countermodels.ReciprocalCrossCoefficients PhysicsLogic.Papers.Onsager1931.ReciprocalRelationsFramework.Countermodels.positiveEntropyWithoutReciprocity := PhysicsLogic.Papers.Onsager1931.ReciprocalRelationsFramework.Countermodels.positive_entropy_production_does_not_imply_reciprocity example : PhysicsLogic.Papers.Onsager1931.ReciprocalRelationsFramework.Countermodels.onsagerMagneticReversalReadout.HasFieldReversedParityReciprocityAt ∧ ¬PhysicsLogic.Papers.Onsager1931.ReciprocalRelationsFramework.Countermodels.onsagerMagneticReversalReadout.HasSameFieldParityReciprocityAt := by obtain ⟨_matches, reversed, _notSameBalance, notSameReciprocity⟩ := PhysicsLogic.Papers.Onsager1931.ReciprocalRelationsFramework.Countermodels.onsager_realizes_shared_field_reversal_boundary exact ⟨reversed, notSameReciprocity⟩ example : PhysicsLogic.Papers.Casimir1945.Countermodels.FiveSelectedReciprocityRelations PhysicsLogic.Papers.Casimir1945.Countermodels.wrongNamedReciprocityReadout ∧ ¬PhysicsLogic.Papers.Casimir1945.Countermodels.RetainsSelectedEquationEighteenFormulas PhysicsLogic.Papers.Casimir1945.Countermodels.wrongNamedReciprocityReadout := PhysicsLogic.Papers.Casimir1945.Countermodels.named_reciprocity_output_does_not_supply_selected_equations_eighteen example : PhysicsLogic.Framework.ReciprocalTransport.Countermodels.oppositeParityContext.FieldReversedParityReciprocityAt (PhysicsLogic.Framework.ReciprocalTransport.Countermodels.oppositeParityModel .zero) ∧ (PhysicsLogic.Framework.ReciprocalTransport.Countermodels.oppositeParityModel .zero).kineticCoefficient .first .second .zero ≠ (PhysicsLogic.Framework.ReciprocalTransport.Countermodels.oppositeParityModel .zero).kineticCoefficient .second .first .zero := PhysicsLogic.Framework.ReciprocalTransport.Countermodels.opposite_parity_reciprocity_is_not_ordinary_reciprocity example : PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.densityMap PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.TestPotential.reference = PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.densityMap PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.TestPotential.shifted ∧ PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.TestPotential.reference ≠ PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.TestPotential.shifted := by obtain ⟨sameDensity, _gaugeEquivalent, _timeOnlyShift, distinct⟩ := PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.gauge_qualified_uniqueness_does_not_identify_potentials exact ⟨sameDensity, distinct⟩ example : PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.stationaryAction PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.stationarySaddleAction ∧ ¬PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.minimumPrincipleAvailable PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.stationarySaddleAction := by obtain ⟨stationary, _negativeHessian, _lowerNearbyAction, notMinimum⟩ := PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework.Countermodels.stationary_action_does_not_supply_a_minimum_principle exact ⟨stationary, notMinimum⟩ example : PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.HasReportedPositiveCoefficientMagnitudes PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.positiveMagnitudesWithWrongDirectionalSigns ∧ PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.realizesPrintedAsymmetryParameterFormula PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.positiveMagnitudesWithWrongDirectionalSigns ∧ ¬PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.HasNegativeDirectionalCoefficientSigns PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.positiveMagnitudesWithWrongDirectionalSigns := PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.positive_rough_magnitudes_and_printed_formula_do_not_determine_directional_sign example : PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.HasObservedNegativeAsymmetryCoefficients PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.negativeCoefficientsWithoutTensorSignInference ∧ ¬PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.HasOppositeTensorCouplingSigns PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.negativeCoefficientsWithoutTensorSignInference := PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.negative_asymmetry_coefficients_do_not_supply_tensor_coupling_sign_bridge example {T : PhysicsLogic.Papers.WuEtAl1957.Wu1957Types} (framework : PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework T) (asymmetry : framework.EstablishedBetaAsymmetry) (formula : framework.PrintedAsymmetryParameterFormula) : framework.coefficientRealValueAt framework.alphaCoefficient < 0 ∧ framework.coefficientRealValueAt framework.betaCoefficient < 0 := by have signed := framework |>.negative_alpha_and_positive_polarization_force_negative_signed_beta asymmetry formula exact ⟨signed.1, signed.2.1⟩ example : PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.GammaAnisotropyCalibratesPolarization PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.gammaCalibratedWithoutBetaRun ∧ ¬PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.BetaMeasurementPerformed PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.gammaCalibratedWithoutBetaRun := PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.Countermodels.gamma_calibration_does_not_supply_a_beta_measurement example : PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.selectedReadout.forwardCountRate = 3 ∧ PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.selectedReadout.reflectedCountRate = 5 ∧ PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.selectedReadout.calibrationSignal = 1 ∧ PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.selectedReadout.fieldReversalResidual = 0 := PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.selected_readout_retains_count_calibration_and_control_values example : PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.detectedExperiment.SelectedAlternativesExcludedAt PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.selectedAlternative () PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.selectedReadout () ∧ PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.detectedExperiment.AlternativeRemainsOpenAt PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.TestAlternative.detectorDrift () PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.selectedReadout () := PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels.selected_field_reversal_control_leaves_detector_drift_open example {T : PhysicsLogic.Papers.WuEtAl1957.Wu1957Types} (framework : PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework T) (calibration : framework.PolarizationCalibrationResult) : framework.figureTwoCountReadout.coldGammaAnisotropyValue = (framework.figureTwoCountReadout.coldEquatorialGammaRateValue - framework.figureTwoCountReadout.coldPolarGammaRateValue) / framework.figureTwoCountReadout.coldEquatorialGammaRateValue := (framework.polarization_calibration_retains_figure_two_gamma_quotient calibration).2.1 example {T : PhysicsLogic.Papers.WuEtAl1957.Wu1957Types} (framework : PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework T) (asymmetry : framework.EstablishedBetaAsymmetry) : framework.figureTwoCountReadout.upwardFieldColdBetaRateValue / framework.figureTwoCountReadout.warmBetaNormalizationValue < 1 ∧ 1 < framework.figureTwoCountReadout.downwardFieldColdBetaRateValue / framework.figureTwoCountReadout.warmBetaNormalizationValue ∧ framework.figureTwoCountReadout.upwardFieldWarmBetaRateValue = framework.figureTwoCountReadout.warmBetaNormalizationValue ∧ framework.figureTwoCountReadout.downwardFieldWarmBetaRateValue = framework.figureTwoCountReadout.warmBetaNormalizationValue := (framework.beta_asymmetry_retains_figure_two_field_reversal asymmetry).2 example {T : PhysicsLogic.Papers.WuEtAl1957.Wu1957Types} (framework : PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework T) (controls : framework.ArtifactControlObservations) : framework.figureTwoCountReadout.unpolarizedDiskAsymmetryValue = 0 ∧ framework.figureTwoCountReadout.rubbedSaltAsymmetryValue = 0 := framework.artifact_observations_retain_both_null_control_readouts controls example : PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.NamedPositiveGammaAnisotropyAt PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.namedGammaSignalWithWrongQuotient ∧ ¬PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.PrintedGammaAnisotropyFigureAt PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.namedGammaSignalWithWrongQuotient := PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.positive_gamma_signal_does_not_supply_figure_two_quotient example : PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.NamedColdFieldAsymmetryAt PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.coldBranchesWithUnequalWarmRates ∧ ¬PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.PrintedBetaFieldReversalFigureAt PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.coldBranchesWithUnequalWarmRates := PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.cold_field_asymmetry_does_not_supply_warm_rate_convergence example : PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.PrintedGammaAnisotropyFigureAt PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.polarizedFigureWithNonNullControls ∧ PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.PrintedBetaFieldReversalFigureAt PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.polarizedFigureWithNonNullControls ∧ ¬PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.NullUnpolarizedControlReadoutsAt PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.polarizedFigureWithNonNullControls := PhysicsLogic.Papers.WuEtAl1957.FigureTwoCountReadout.Countermodels.polarized_field_figures_do_not_supply_null_control_specimens example {T : PhysicsLogic.Papers.WuEtAl1957.Wu1957Types} (framework : PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework T) (calibration : framework.PolarizationCalibrationResult) : framework.toReferenceCalibrationContext.ReferenceCalibrationRunAt framework.gammaPolarizationProcedure framework.nuclearPolarizationQuantity framework.roseGorterPreparation framework.calibratedPolarizationValue framework.cobalt60Sample framework.experimentalRegime := (framework.selected_gamma_run_realizes_nonvacuous_reference_calibration calibration).1 example {T : PhysicsLogic.Papers.Bohm1952.BohmTypes} (framework : PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework T) (run : framework.StandardMomentumMeasurementRun) (calculation : framework.ContextualObservableCalculation) : ¬framework.valueIndicationContext.FaithfullyMeasuresAt framework.standardProcedure framework.actualMomentumQuantity framework.electronSystem framework.measurementRegime := by have unfaithful := (framework |>.standard_momentum_observable_need_not_reveal_premeasurement_particle_momentum run calculation).2.2.1 exact PhysicsLogic.Philosophy.Measurement.ValueIndicationContext.unfaithful_measurement_refutes_faithfulness framework.valueIndicationContext framework.standardProcedure framework.actualMomentumQuantity framework.electronSystem framework.measurementRegime unfaithful example {T : PhysicsLogic.Papers.PenziasWilson1965.PenziasWilsonTypes} (framework : PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework T) (accounting : framework.EstablishedCalibrationAndAccounting) : framework.PositiveResidualThroughoutReportedUncertaintyAt := framework.established_accounting_retains_uncertainty_robust_positive_excess accounting example : PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework.Countermodels.ReducedAntennaTemperatureAccountingReadout.HasPositiveCentralResidualAt PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework.Countermodels.positiveCentralResidualWithOverlappingZeroUncertainty ∧ ¬PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework.Countermodels.ReducedAntennaTemperatureAccountingReadout.HasPositiveResidualIntervalAt PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework.Countermodels.positiveCentralResidualWithOverlappingZeroUncertainty := PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework.Countermodels.positive_central_residual_without_uncertainty_control_can_include_zero example : PhysicsLogic.Papers.PenziasWilson1965.ObservedSkyUniformityReadout.IsotropicWithinObservedDirectionsAt PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework.Countermodels.observedUniformityWithUnseenDirectionalAndFutureVariation ∧ ¬PhysicsLogic.Papers.PenziasWilson1965.ObservedSkyUniformityReadout.observedDirectionAt PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework.Countermodels.observedUniformityWithUnseenDirectionalAndFutureVariation 2 := by obtain ⟨observed, notObserved, _outsideTolerance⟩ := PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework.Countermodels.observed_directional_agreement_does_not_establish_global_isotropy exact ⟨observed, notObserved⟩ example {T : PhysicsLogic.Papers.PenziasWilson1965.PenziasWilsonTypes} (framework : PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework T) (characterization : framework.EstablishedSkyCharacterization) : framework.observedSkyUniformityReadout.lastObservedMonth = framework.observedSkyUniformityReadout.firstObservedMonth + 9 := framework.established_sky_characterization_retains_ten_month_temporal_scope characterization example {T : PhysicsLogic.Papers.PenziasWilson1965.PenziasWilsonTypes} (framework : PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework T) (setup : framework.HornRadiometerSetup) : framework.toCompanionEvidentialStatusContext.supportsAt framework.zenithObservation .companionCosmologicalBackground framework.measurementRegime ↔ framework.companionSupportAssessmentAt framework.zenithObservation framework.excessAntennaTemperature framework.measurementRegime := framework.companion_support_iff_separate_assessment setup example : PhysicsLogic.Papers.PoundRebka1960.PoundRebkaFramework.Countermodels.reversalRecordsObtained PhysicsLogic.Papers.PoundRebka1960.PoundRebkaFramework.Countermodels.reversalWithDriftingOffset ∧ ¬PhysicsLogic.Papers.PoundRebka1960.PoundRebkaFramework.Countermodels.offsetUnchangedByInversion PhysicsLogic.Papers.PoundRebka1960.PoundRebkaFramework.Countermodels.reversalWithDriftingOffset := PhysicsLogic.Papers.PoundRebka1960.PoundRebkaFramework.Countermodels.reversal_observation_does_not_supply_inversion_stability example : PhysicsLogic.Papers.PoundRebka1960.PoundRebkaFramework.Countermodels.modelSupportWithoutTheoryTransfer.SupportsModel 0 0 0 ∧ ¬PhysicsLogic.Papers.PoundRebka1960.PoundRebkaFramework.Countermodels.modelSupportWithoutTheoryTransfer.SupportsTheory 0 0 0 := by obtain ⟨modelSupport, notTheory, _notTransfer⟩ := PhysicsLogic.Papers.PoundRebka1960.PoundRebkaFramework.Countermodels.redshift_model_support_does_not_supply_gravity_theory_support_without_transfer exact ⟨modelSupport, notTheory⟩ namespace ProbabilityRolesPhysicalReuseAudit open PhysicsLogic.Philosophy.ProbabilityRoles open PhysicsLogic.Philosophy.ProbabilityRoles.Countermodels example : predictiveAssignment.predictiveProbability .scatteringModel .selected .high .declared ∧ absentObjectiveChance.objectiveChance .decayingNucleus .omitted .low .declared ∧ ¬absentObjectiveChance.objectiveChance .decayingNucleus .selected .high .declared ∧ ¬predictiveAssignment.predictiveProbability .controlModel .selected .high .declared := predictive_and_objective_roles_distinguish_events_and_states example : assignedEnsembleProbability.ensembleProbability .preparedPopulation .selected .high .declared ∧ absentSampleFrequency.ensembleFrequency .detectedSample .omitted .low .declared ∧ ¬absentSampleFrequency.ensembleFrequency .detectedSample .selected .high .declared ∧ ¬assignedEnsembleProbability.ensembleProbability .controlPopulation .selected .high .declared := population_and_sample_roles_distinguish_assignment_from_frequency example : typicalSelectionButNotUniversal.NonvacuousTypicalPropertyFor .equilibriumMeasure .equilibriumBehavior ∧ typicalSelectionButNotUniversal.hasProperty .exceptional .exceptionalBehavior ∧ ¬typicalSelectionButNotUniversal.UniversalProperty .equilibriumBehavior := selected_typical_property_has_an_actual_typical_state example : typicalPropertyWithoutSelectedState.propertyTypicalFor .equilibriumMeasure .equilibriumBehavior ∧ typicalPropertyWithoutSelectedState.TypicalPropertyBridge ∧ ¬typicalPropertyWithoutSelectedState.TypicalStateScopeInhabitedFor .equilibriumMeasure ∧ ¬typicalPropertyWithoutSelectedState.NonvacuousTypicalPropertyFor .equilibriumMeasure .equilibriumBehavior := declared_typical_property_without_selected_state_is_vacuous end ProbabilityRolesPhysicalReuseAudit example : PhysicsLogic.Papers.BornCollisions1926.Countermodels.weightCalculatedFromAmplitude PhysicsLogic.Papers.BornCollisions1926.Countermodels.calculatedButUninterpreted ∧ ¬PhysicsLogic.Papers.BornCollisions1926.Countermodels.weightRepresentsOutcomeProbability PhysicsLogic.Papers.BornCollisions1926.Countermodels.calculatedButUninterpreted := by obtain ⟨_representation, calculated, notInterpreted⟩ := PhysicsLogic.Papers.BornCollisions1926.Countermodels.calculated_weight_does_not_supply_probability_interpretation exact ⟨calculated, notInterpreted⟩ example : PhysicsLogic.Philosophy.Probability.ProbabilisticDescription.PredictivelyBranchesAt PhysicsLogic.Philosophy.Probability.Countermodels.unsoundBranchingDescription .overinclusive .selected ∧ ¬PhysicsLogic.Philosophy.Probability.PossibilityContext.OnticallyBranchesAt PhysicsLogic.Philosophy.Probability.Countermodels.deterministicPossibility .selected := PhysicsLogic.Papers.BornCollisions1926.Countermodels.predictive_branching_does_not_by_itself_imply_ontic_branching example {Horn : PhysicsLogic.Papers.PenziasWilson1965.PenziasWilsonTypes} (horn : PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework Horn) (hornSetup : horn.HornRadiometerSetup) (hornCalibration : horn.ReferenceCalibrationCalculation) (foregroundAccounting : horn.ForegroundAccountingCalculation) (hornUncertainty : horn.ResidualUncertaintyCalculation) (hornRobustness : horn.InstrumentAndSkyRobustnessCalculation) (hornInterpretation : horn.ResidualDetectionInterpretationCalculation) (hornBridge : horn.ResidualDetectionBridgeCalculation) {Firas : PhysicsLogic.Papers.FixsenEtAl1996.FixsenTypes} (firas : PhysicsLogic.Papers.FixsenEtAl1996.FirasFramework Firas) (firasSetup : firas.FirasObservationSetup) (firasDetection : firas.toCalibratedResidualBackgroundContext |>.EstablishedCalibratedResidualBackgroundDetectionAt firas.sharedBackgroundModel) (blackbody : firas.EstablishedBlackbodySpectrum) (temperatureScale : firas.TemperatureScaleCalculation) : horn.RecordedResidualTemperatureEstimate ∧ firas.RecordedCMBRTemperatureEstimate := ⟨horn.calibrated_record_and_representation_record_residual_estimate hornSetup hornCalibration foregroundAccounting hornUncertainty hornRobustness hornInterpretation hornBridge, firas.calibrated_full_mission_records_cmbr_temperature_estimate firasSetup firasDetection blackbody temperatureScale⟩ example {Everett : PhysicsLogic.Papers.Everett1957.EverettTypes} (everett : PhysicsLogic.Papers.Everett1957.RelativeStateFramework Everett) (externalSetup : everett.ExternalObservationProblemSetup) (everettSetup : everett.GoodMeasurementSetup) (everettCalculation : everett.LinearMeasurementCalculation) (everettInterpretation : everett.RelativeRecordInterpretation) (everettRealization : everett.FirstCoherentRecordRealizationInputs) {Rovelli : PhysicsLogic.Papers.Rovelli1996.RovelliTypes} (rovelli : PhysicsLogic.Papers.Rovelli1996.RelationalFramework Rovelli) (rovelliSetup : rovelli.ThirdPersonMeasurementSetup) (rovelliCalculation : rovelli.ThirdPersonLinearCalculation) (rovelliInterpretation : rovelli.RelativeRecordInterpretation) (relativeObserver : rovelli.ObserverRelativeDescriptionSetup) (rovelliRecordCalibration : rovelli.recordProcess.MeasurementCalibration) (rovelliStateCoordinates : rovelli.measuredRecordCoefficientsAt rovelli.secondObserver rovelli.correlatedState = rovelli.recordProcess.measuredState) (rovelliFirstSupported : rovelli.recordProcess.inputAmplitude false ≠ 0) : everett.toOperationalObserverContext.OperationalObserverOfAt everett.observerSystem everett.objectSystem everett.measurementInteraction everett.firstRecord everett.measurementRegime ∧ rovelli.toOperationalObserverContext.OperationalObserverOfAt rovelli.firstObserver rovelli.targetSystem rovelli.measurementInteraction rovelli.firstRecord rovelli.measurementRegime := ⟨everett.shared_relative_measurement_realizes_operational_observer_role externalSetup everettSetup everettCalculation everettInterpretation everettRealization, rovelli.shared_relative_measurement_realizes_operational_observer_role rovelliSetup rovelliCalculation rovelliInterpretation relativeObserver rovelliRecordCalibration rovelliStateCoordinates rovelliFirstSupported⟩ example {T : PhysicsLogic.Papers.EvansCohenMorriss1993.EvansCohenMorrissTypes} (framework : PhysicsLogic.Papers.EvansCohenMorriss1993.Framework T) (measure : framework.EstablishedSegmentMeasureProposal) (pairing : framework.EstablishedFixedShearPair) (ratio : framework.EstablishedProbabilityRatio) (secondLaw : framework.EstablishedSecondLawAssessment) (scope : framework.EstablishedScopeAssessment) : framework.selectedEntropyProductionFluctuationModel |>.reversingSymmetryAt framework.selectedEntropyProductionFluctuationModel.dynamics framework.selectedEntropyProductionFluctuationModel.system framework.selectedEntropyProductionFluctuationModel.regime := by have arrow := framework.ecm_routes_establish_selected_reversible_temporal_asymmetry measure pairing ratio secondLaw scope exact TemporalAsymmetryContext.temporal_asymmetry_preserves_reversible_dynamics framework.selectedEntropyProductionFluctuationModel.selectedTemporalAsymmetryContext framework.selectedEntropyProductionFluctuationModel.system framework.selectedEntropyProductionFluctuationModel.dynamics framework.selectedEntropyProductionFluctuationModel.probabilityConstruction framework.selectedEntropyProductionFluctuationModel.positiveWindow framework.selectedEntropyProductionFluctuationModel.negativeWindow framework.selectedEntropyProductionFluctuationModel.positiveProbability framework.selectedEntropyProductionFluctuationModel.negativeProbability framework.selectedEntropyProductionFluctuationModel.regime arrow example {T : PhysicsLogic.Papers.GallavottiCohen1995.GallavottiCohenTypes} (framework : PhysicsLogic.Papers.GallavottiCohen1995.Framework T) (dynamics : framework.EstablishedDissipativeReversibleSetup) (chaotic : framework.EstablishedChaoticHypothesis) (srb : framework.EstablishedSRBConstruction) (symmetry : framework.EstablishedFluctuationSymmetry) (scope : framework.EstablishedScopeAssessment) : framework.selectedEntropyProductionFluctuationModel |>.reversingSymmetryAt framework.selectedEntropyProductionFluctuationModel.dynamics framework.selectedEntropyProductionFluctuationModel.system framework.selectedEntropyProductionFluctuationModel.regime := by have arrow := framework.gc_routes_establish_selected_reversible_temporal_asymmetry dynamics chaotic srb symmetry scope exact TemporalAsymmetryContext.temporal_asymmetry_preserves_reversible_dynamics framework.selectedEntropyProductionFluctuationModel.selectedTemporalAsymmetryContext framework.selectedEntropyProductionFluctuationModel.system framework.selectedEntropyProductionFluctuationModel.dynamics framework.selectedEntropyProductionFluctuationModel.probabilityConstruction framework.selectedEntropyProductionFluctuationModel.positiveWindow framework.selectedEntropyProductionFluctuationModel.negativeWindow framework.selectedEntropyProductionFluctuationModel.positiveProbability framework.selectedEntropyProductionFluctuationModel.negativeProbability framework.selectedEntropyProductionFluctuationModel.regime arrow namespace TemporalArrowPhysicalReuseAudit open PhysicsLogic.Paradigm.FluctuationsAndArrow.Countermodels example : temporalPhysicalContext.reversingDynamicsAt .reversible .selected .selected ∧ ¬temporalPhysicalContext.reversingDynamicsAt .irreversible .selected .selected ∧ reversibleUnbiasedTemporalContext.DirectedTemporalAsymmetryAt .control .control .control .control .control .control .control .control := by obtain ⟨reversible, notIrreversible, _prepared, _unselected, _paired, _notReversed, _represented, _wrongWeight, _selectedArrow, controlArrow, _unselectedControl, _irreversibleControl, _unpairedControl, _unrepresentedControl, _equalControl⟩ := temporal_roles_distinguish_dynamics_preparations_observations_weights_and_controls exact ⟨reversible, notIrreversible, controlArrow⟩ end TemporalArrowPhysicalReuseAudit example {System Run State Time : Type} (context : PhysicsLogic.Framework.ValidatedNumericalNearReturnContext System Run State Time) (calculation : PhysicsLogic.Framework.ValidatedNumericalNearReturnCalculationModel System Run State Time) (established : context.EstablishedValidatedNumericalNearReturnAt calculation) : (context.validatedNumericalPredictionModelAt calculation) |>.NonvacuousValidatedNumericalPredictionAt := context |>.established_near_return_exhibits_nonvacuous_validated_numerical_prediction established example : ((PhysicsLogic.Framework.ValidatedNumericalNearReturnContext.Countermodels.nonpositiveTimePrediction |>.validatedNumericalPredictionModelAt PhysicsLogic.Framework.ValidatedNumericalNearReturnContext.Countermodels.nonpositiveTimePredictionModel) |>.NonvacuousValidatedNumericalPredictionAt) ∧ ¬PhysicsLogic.Framework.ValidatedNumericalNearReturnContext.Countermodels.nonpositiveTimePredictionModel.NonvacuousValidatedNumericalNearReturnAt := PhysicsLogic.Framework.ValidatedNumericalNearReturnContext.Countermodels.validated_prediction_without_positive_time_does_not_establish_near_return open PhysicsLogic.Papers.FermiPastaUlam1955.NonlinearChainFramework.Countermodels in example : oscillatorModalHistory.ModeEnergyReturnAt 1 1 (1 / 100) ∧ oscillatorReturnContext.NearRecursAt oscillatorEnergyNear Real.pi oscillatorInitial ∧ ¬oscillatorReturnContext.RecursAt Real.pi oscillatorInitial := modal_near_return_does_not_establish_exact_full_state_recurrence open PhysicsLogic.Papers.FermiPastaUlam1955.NonlinearChainFramework.Countermodels in example : (∀ run : Fin 4, run.val < 3 → ¬(comparisonHistory run).ModalEquipartitionAt 0 30000) ∧ ¬(∀ run : Fin 4, ¬(comparisonHistory run).ModalEquipartitionAt 0 30000) := selected_run_evidence_does_not_establish_universal_nonthermalization 30000 example {T : PhysicsLogic.Papers.FermiPastaUlam1955.FPUTypes} (framework : PhysicsLogic.Papers.FermiPastaUlam1955.NonlinearChainFramework T) (setup : framework.FiniteNonlinearChainSetup framework.quadraticPureModeRun) (protocol : framework.NumericalProtocol framework.quadraticPureModeRun) (validation : framework.NumericalValidationCalculation framework.quadraticPureModeRun) (reliability : framework.NumericalReliabilityBridge framework.quadraticPureModeRun) (quadraticSetup : framework.QuadraticPureModeSetup) (calculation : framework.QuadraticRunCalculation) : framework.validatedNumericalModel |>.NonvacuousValidatedNumericalPredictionAt := framework.quadratic_run_exhibits_nonvacuous_shared_validated_numerical_prediction setup protocol validation reliability quadraticSetup calculation example : PhysicsLogic.Papers.KortewegDeVries1895.LongWaveFramework.Countermodels.ReducedLongWaveScopeReadout.HasFiniteOrderStationarySupport PhysicsLogic.Papers.KortewegDeVries1895.LongWaveFramework.Countermodels.longWaveScopeBoundaryReadout ∧ ¬PhysicsLogic.Papers.KortewegDeVries1895.LongWaveFramework.Countermodels.ReducedLongWaveScopeReadout.HasModernAnalyticConvergence PhysicsLogic.Papers.KortewegDeVries1895.LongWaveFramework.Countermodels.longWaveScopeBoundaryReadout := PhysicsLogic.Papers.KortewegDeVries1895.LongWaveFramework.Countermodels.higher_order_stationary_support_is_not_a_modern_convergence_theorem example {T : PhysicsLogic.Papers.KortewegDeVries1895.KdV1895Types} (framework : PhysicsLogic.Papers.KortewegDeVries1895.LongWaveFramework T) (equation : framework.EstablishedDeformationEquation) (stationary : framework.EstablishedStationarySolitaryWaves) : framework.toSolitaryTravelingWaveContext |>.DirectStructuralSolitaryTravelingWaveExplanationAt framework.selectedPositiveSolitaryWaveModel := framework.positive_solitary_wave_realizes_shared_structural_explanation equation stationary example {T : PhysicsLogic.Papers.ZabuskyKruskal1965.ZKTypes} (framework : PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework T) (setup : framework.IsolatedSolitonSetup) (calculation : framework.IsolatedSolitonCalculation) (interpretation : framework.IsolatedSolitonInterpretationBridge) : framework.toSolitaryTravelingWaveContext |>.DirectStructuralSolitaryTravelingWaveExplanationAt framework.selectedSolitaryWaveModel := framework.isolated_pulse_realizes_shared_structural_explanation setup calculation interpretation example : PhysicsLogic.Papers.BogoliubovSuperfluidity1947.WeakBoseGasFramework.Countermodels.HasQuasiparticleSpectrum PhysicsLogic.Papers.BogoliubovSuperfluidity1947.WeakBoseGasFramework.Countermodels.quasiparticleSpectrumWithoutOccupations ∧ ¬PhysicsLogic.Papers.BogoliubovSuperfluidity1947.WeakBoseGasFramework.Countermodels.HasEquilibriumOccupations PhysicsLogic.Papers.BogoliubovSuperfluidity1947.WeakBoseGasFramework.Countermodels.quasiparticleSpectrumWithoutOccupations := PhysicsLogic.Papers.BogoliubovSuperfluidity1947.WeakBoseGasFramework.Countermodels.quasiparticle_spectrum_does_not_supply_equilibrium_occupations example : PhysicsLogic.Papers.BogoliubovSuperfluidity1947.WeakBoseGasFramework.Countermodels.HasRealLiquidExtension PhysicsLogic.Papers.BogoliubovSuperfluidity1947.WeakBoseGasFramework.Countermodels.semiPhenomenologicalLiquidWithoutExactDerivation ∧ ¬PhysicsLogic.Papers.BogoliubovSuperfluidity1947.WeakBoseGasFramework.Countermodels.HasExactRealLiquidDerivation PhysicsLogic.Papers.BogoliubovSuperfluidity1947.WeakBoseGasFramework.Countermodels.semiPhenomenologicalLiquidWithoutExactDerivation := by obtain ⟨extension, _phenomenological, notExact⟩ := PhysicsLogic.Papers.BogoliubovSuperfluidity1947.WeakBoseGasFramework.Countermodels.a_proposed_low_density_extension_is_not_an_exact_real_liquid_derivation exact ⟨extension, notExact⟩ example {T : PhysicsLogic.Papers.BogoliubovSuperfluidity1947.BogoliubovTypes} (framework : PhysicsLogic.Papers.BogoliubovSuperfluidity1947.WeakBoseGasFramework T) (motion : framework.EstablishedFrictionlessMotion) : framework.toSubthresholdExcitationStabilityContext |>.EstablishedSubthresholdExcitationStabilityAt framework.sharedSubthresholdModel := framework.frictionless_motion_realizes_shared_excitation_stability motion example : PhysicsLogic.Papers.Kibble1976.CosmicDomainFramework.Countermodels.trivialSelectedLoop.vacuumPiOneGenerators > 0 ∧ ¬PhysicsLogic.Papers.Kibble1976.CosmicDomainFramework.Countermodels.LoopProfile.FormsString PhysicsLogic.Papers.Kibble1976.CosmicDomainFramework.Countermodels.trivialSelectedLoop := PhysicsLogic.Papers.Kibble1976.CosmicDomainFramework.Countermodels.nontrivial_pi_one_without_selected_noncontractible_loop_does_not_form_string example : PhysicsLogic.Papers.Kibble1976.FiniteTemperatureSymmetryBreakingReadout.Countermodels.NamedBrokenPhaseAt PhysicsLogic.Papers.Kibble1976.FiniteTemperatureSymmetryBreakingReadout.Countermodels.namedBrokenPhaseWithoutThermalPotential ∧ ¬PhysicsLogic.Papers.Kibble1976.FiniteTemperatureSymmetryBreakingReadout.Countermodels.namedBrokenPhaseWithoutThermalPotential.SourceFormulaAt := PhysicsLogic.Papers.Kibble1976.FiniteTemperatureSymmetryBreakingReadout.Countermodels.named_broken_phase_does_not_supply_printed_thermal_potential example : PhysicsLogic.Papers.Kibble1976.FiniteTemperatureSymmetryBreakingReadout.Countermodels.gaugeBlindThermalPotentialReadout.FormulaDomainAt ∧ PhysicsLogic.Papers.Kibble1976.FiniteTemperatureSymmetryBreakingReadout.Countermodels.GaugeBlindThermalPotentialAt PhysicsLogic.Papers.Kibble1976.FiniteTemperatureSymmetryBreakingReadout.Countermodels.gaugeBlindThermalPotentialReadout ∧ ¬PhysicsLogic.Papers.Kibble1976.FiniteTemperatureSymmetryBreakingReadout.Countermodels.gaugeBlindThermalPotentialReadout.EquationThreeThermalEffectivePotentialAt := PhysicsLogic.Papers.Kibble1976.FiniteTemperatureSymmetryBreakingReadout.Countermodels.gauge_blind_potential_does_not_supply_original_gauge_thermal_term example {T : PhysicsLogic.Papers.Kibble1976.Kibble1976Types} (framework : PhysicsLogic.Papers.Kibble1976.CosmicDomainFramework T) (setup : framework.SymmetryBreakingSetup) : 0 < framework.orderParameterMagnitudeValueAt framework.transition := framework.finite_temperature_formula_forces_nonzero_order_parameter (framework.symmetry_setup_retains_printed_thermal_potential_formulas setup) example {T : PhysicsLogic.Papers.Kibble1976.Kibble1976Types} (framework : PhysicsLogic.Papers.Kibble1976.CosmicDomainFramework T) (monopole : framework.EstablishedMonopoleFormation) : framework.toCausalDomainDefectFormationContext |>.DirectStructuralDefectFormationExplanationAt framework.sharedDefectFormationModel := framework.monopole_route_realizes_structural_defect_formation_explanation monopole example : PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.selectedCausalDomainFamily PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.TestDomain.firstDomain ∧ PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.selectedCausalDomainFamily PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.TestDomain.secondDomain ∧ PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.selectedCausalOrderParameter PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.TestDomain.firstDomain ≠ PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.selectedCausalOrderParameter PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.TestDomain.secondDomain ∧ ¬PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.selectedCausalDomainFamily PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.TestDomain.exterior := PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.selected_domain_family_retains_distinct_choices_and_excludes_exterior example : PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.independentWithoutAssembly.IndependentDomainSetupAt PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.independentWithoutAssemblyModel ∧ ¬PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.independentWithoutAssembly.CollectiveAssemblyAt PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.independentWithoutAssemblyModel := PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels.independent_domains_do_not_supply_collective_assembly example : PhysicsLogic.Papers.FeynmanTwoFluid1954.AtomicTwoFluidFramework.Countermodels.isVariationalUpperBound PhysicsLogic.Papers.FeynmanTwoFluid1954.AtomicTwoFluidFramework.Countermodels.strictVariationalUpperBoundReport ∧ ¬PhysicsLogic.Papers.FeynmanTwoFluid1954.AtomicTwoFluidFramework.Countermodels.isExactSectorEnergy PhysicsLogic.Papers.FeynmanTwoFluid1954.AtomicTwoFluidFramework.Countermodels.strictVariationalUpperBoundReport := PhysicsLogic.Papers.FeynmanTwoFluid1954.AtomicTwoFluidFramework.Countermodels.variational_upper_bound_does_not_supply_exact_sector_energy_value example {T : PhysicsLogic.Papers.FeynmanTwoFluid1954.FeynmanTypes} (framework : PhysicsLogic.Papers.FeynmanTwoFluid1954.AtomicTwoFluidFramework T) (spectrum : framework.EstablishedVariationalSpectrum) : framework.toMomentumResolvedVariationalBoundContext |>.EstablishedMomentumResolvedVariationalBoundAt framework.sharedVariationalModel := framework.variational_spectrum_realizes_shared_momentum_bound spectrum example : PhysicsLogic.Papers.Kerr1963.Framework.Countermodels.analyticExteriorEstablished PhysicsLogic.Papers.Kerr1963.Framework.Countermodels.exteriorWithExceptionalRing ∧ ¬PhysicsLogic.Papers.Kerr1963.Framework.Countermodels.globallyAnalytic PhysicsLogic.Papers.Kerr1963.Framework.Countermodels.exteriorWithExceptionalRing := PhysicsLogic.Papers.Kerr1963.Framework.Countermodels.analytic_exterior_does_not_establish_global_analyticity example {T : PhysicsLogic.Papers.Kerr1963.KerrTypes} (framework : PhysicsLogic.Papers.Kerr1963.Framework T) (stationary : framework.EstablishedStationaryAxisymmetricVacuumMetric) (asymptotic : framework.EstablishedAsymptoticAndAnalyticScope) : framework.selectedExactExteriorVacuumMetricContext |>.EstablishedExactExteriorVacuumMetricSolutionAt framework.selectedExactExteriorVacuumMetricModel := framework.stationary_and_asymptotic_routes_establish_shared_exterior_solution stationary asymptotic example : PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework.Countermodels.ClassicallyForbiddenBarrierAt PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework.Countermodels.forbiddenBarrierWithoutComplexEnergy ∧ ¬PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework.Countermodels.ComplexEnergyDampingAt PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework.Countermodels.forbiddenBarrierWithoutComplexEnergy := PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework.Countermodels.a_classically_forbidden_barrier_does_not_supply_the_complex_energy_step example (readout : PhysicsLogic.Papers.Gamow1928.SimpleBarrierFormulaReadout) (domain : readout.WaveDomainAt) (factor : readout.PostEquationFourTransmissionFactorAt) : 0 < readout.transmittedSuppressionValue ∧ readout.transmittedSuppressionValue < 1 := readout.transmission_factor_is_strictly_between_zero_and_one domain factor example (readout : PhysicsLogic.Papers.Gamow1928.SimpleBarrierFormulaReadout) (formulas : readout.SourceFormulaAt) : readout.decayRateValue = readout.EquationFivePrefactorAt * readout.transmittedSuppressionValue ^ 2 := readout.equation_five_rate_is_prefactor_times_squared_transmission formulas example {T : PhysicsLogic.Papers.Gamow1928.GamowTypes} (framework : PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework T) (penetration : framework.EstablishedRectangularBarrierPenetration) : framework.simpleBarrierFormulaReadout.transmittedSuppressionValue < framework.higherSimpleBarrierFormulaReadout.transmittedSuppressionValue := (framework.rectangular_penetration_derives_ordered_finite_barrier_transmission penetration).2.2.2.2.1 example (readout : PhysicsLogic.Papers.Gamow1928.SimpleBarrierFormulaReadout) (wave : readout.WaveFormulaAt) (calculation : readout.QuasistationaryCalculationAt) : readout.EquationFiveSimpleDecayFormulaAt := readout.quasistationary_inputs_derive_equation_five wave.1 calculation example (readout : PhysicsLogic.Papers.Gamow1928.SimpleBarrierFormulaReadout) (wave : readout.WaveFormulaAt) (calculation : readout.QuasistationaryCalculationAt) (time : Real) (nonnegative : 0 ≤ time) : readout.selectedExponentialSurvivalReadout.survivalProbabilityAt time = PhysicsLogic.Papers.Gamow1928.WaveMechanics.interiorProbability readout.planckConstantValue readout.complexEnergyValue readout.outgoingSpatialAmplitudeValueAt readout.outgoingBoundaryRadius time / readout.initialInteriorProbability := readout.selected_survival_is_actual_interior_probability_ratio (readout.wave_and_quasistationary_inputs_derive_source_formula wave calculation) time nonnegative example (readout : PhysicsLogic.Papers.Gamow1928.SimpleBarrierFormulaReadout) (wave : readout.WaveFormulaAt) (calculation : readout.QuasistationaryCalculationAt) (time : Real) : PhysicsLogic.Papers.Gamow1928.WaveMechanics.ConservesInteriorProbabilityAt readout.planckConstantValue readout.massValue readout.complexEnergyValue readout.outgoingSpatialAmplitudeValueAt readout.outgoingBoundaryRadius time := readout.source_formula_conserves_actual_interior_probability (readout.wave_and_quasistationary_inputs_derive_source_formula wave calculation) time example (readout : PhysicsLogic.Papers.Gamow1928.SimpleBarrierFormulaReadout) (wave : readout.WaveFormulaAt) (calculation : readout.QuasistationaryCalculationAt) : ¬PhysicsLogic.Papers.Gamow1928.WaveMechanics.ConservesInteriorProbabilityAt readout.planckConstantValue readout.massValue readout.realEnergyValue readout.outgoingSpatialAmplitudeValueAt readout.outgoingBoundaryRadius 0 := readout.source_formula_rejects_real_energy_with_the_same_outgoing_profile (readout.wave_and_quasistationary_inputs_derive_source_formula wave calculation) example {T : PhysicsLogic.Papers.Gamow1928.GamowTypes} (framework : PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework T) (calculation : framework.EquationSevenCalculationAt) : |framework.logarithmValueAt (framework.decayRateValueAt framework.higherDecayRate) - (framework.logRateInterceptValue + framework.theoreticalLogSlopeValue * framework.energyIncrementValue)| ≤ framework.modelLogErrorAllowanceValue + framework.linearizationLogErrorAllowanceValue ∧ deriv framework.referenceCoulombLogRateAt (framework.referenceEnergyValue + framework.energyIncrementValue) < framework.theoreticalLogSlopeValue := (framework.coulomb_derivative_and_remainders_derive_equation_seven calculation).2.2.2.2 example {T : PhysicsLogic.Papers.Gamow1928.GamowTypes} (framework : PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework T) (angular : T.AngularMomentum) (system : T.System) (regime : T.Regime) (mass : 0 < framework.alphaMassValue) (h : 0 < framework.planckConstantValue) (nonzero : framework.nonzeroAngularMomentumAt angular regime) : framework.nonzeroAngularMomentumActsAsRaisedPotentialAt angular system regime := framework.positive_angular_momentum_raises_the_actual_radial_potential angular system regime mass h nonzero example : PhysicsLogic.Papers.Gamow1928.SimpleBarrierFormulaReadout.Countermodels.OrderedSubBarrierTransmissionAt PhysicsLogic.Papers.Gamow1928.SimpleBarrierFormulaReadout.Countermodels.unequalWidthsReverseEnergyTransmissionOrdering ∧ ¬(PhysicsLogic.Papers.Gamow1928.SimpleBarrierFormulaReadout.Countermodels.unequalWidthsReverseEnergyTransmissionOrdering.lowerTransmissionValue < PhysicsLogic.Papers.Gamow1928.SimpleBarrierFormulaReadout.Countermodels.unequalWidthsReverseEnergyTransmissionOrdering.higherTransmissionValue) := PhysicsLogic.Papers.Gamow1928.SimpleBarrierFormulaReadout.Countermodels.energy_order_without_a_common_width_does_not_order_transmission example : 0 < PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework.Countermodels.gamowPositiveRateWithoutExponentialSurvival.decayRateValue ∧ ¬PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework.Countermodels.gamowPositiveRateWithoutExponentialSurvival.PositiveExponentialSurvivalLawAt := PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework.Countermodels.a_positive_decay_constant_does_not_supply_the_survival_law example {T : PhysicsLogic.Papers.Gamow1928.GamowTypes} (framework : PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework T) (damped : framework.EstablishedDampedOutgoingDecay) (radial : framework.EstablishedRadialDecayAt framework.lowerSelection) : framework.finiteBarrierDecayModel.NonvacuousMemorylessFiniteBarrierDecayAt framework.lowerSelection framework.simpleBarrierFormulaReadout.selectedExponentialSurvivalReadout := framework.damped_radial_decay_realizes_memoryless_finite_barrier_decay damped radial example {T : PhysicsLogic.Papers.Gamow1928.GamowTypes} (framework : PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework T) (comparison : framework.EstablishedGeigerNuttallComparison) : framework.toFiniteBarrierMetastableDecayContext |>.EstablishedFiniteBarrierMetastableDecayComparisonAt framework.finiteBarrierDecayModel := framework.local_geiger_nuttall_result_establishes_shared_finite_barrier_decay comparison example : PhysicsLogic.Papers.KolmogorovLocalStructure1941.LocalTurbulenceFramework.Countermodels.SimilarityReadout.HasFirstSimilarity PhysicsLogic.Papers.KolmogorovLocalStructure1941.LocalTurbulenceFramework.Countermodels.firstSimilarityOnly ∧ ¬PhysicsLogic.Papers.KolmogorovLocalStructure1941.LocalTurbulenceFramework.Countermodels.SimilarityReadout.HasSecondSimilarity PhysicsLogic.Papers.KolmogorovLocalStructure1941.LocalTurbulenceFramework.Countermodels.firstSimilarityOnly := by obtain ⟨first, _differentLaws, notSecond⟩ := PhysicsLogic.Papers.KolmogorovLocalStructure1941.LocalTurbulenceFramework.Countermodels.first_similarity_does_not_supply_second_similarity exact ⟨first, notSecond⟩ example {T : PhysicsLogic.Papers.KolmogorovLocalStructure1941.Kolmogorov1941Types} (framework : PhysicsLogic.Papers.KolmogorovLocalStructure1941.LocalTurbulenceFramework T) (inertial : framework.InertialRangeResult) : framework.toInertialDimensionalFormContext.OpenCoefficientDimensionalFormAt .twoThirdsSecondMoment framework.universalConstant PhysicsLogic.Papers.KolmogorovLocalStructure1941.LocalTurbulenceFramework.selectedInertialDimensionalInputs PhysicsLogic.Papers.KolmogorovLocalStructure1941.LocalTurbulenceFramework.selectedInertialDimensionalConstraints framework.longitudinalSecondMoment framework.inertialRangeRegime := framework.inertial_range_result_realizes_open_coefficient_dimensional_form inertial namespace DimensionalInputReuseAudit open PhysicsLogic.Paradigm.DimensionalAnalysis.DimensionalFormContext.Countermodels example : selectedInputs .length /\ selectedInputs .time /\ selectiveDimensionalContext.inputRelevantAt .omittedMassScale () .declared /\ Not (selectedInputs .omittedMassScale) := selected_input_family_retains_length_time_and_excludes_relevant_mass example : selectiveDimensionalContext.ScopedDimensionalFormAt .selected selectedInputs selectedConstraints () .declared /\ Not (selectiveDimensionalContext.SelectionCoversEveryRelevantInputAt selectedInputs () .declared) := scoped_dimensional_form_does_not_cover_every_relevant_input end DimensionalInputReuseAudit example : PhysicsLogic.Papers.Ramsey1950.Fidelity.VelocityAveragingReadout.HasResonantReinforcement PhysicsLogic.Papers.Ramsey1950.Fidelity.selectedVelocityWithoutBeamAverage ∧ ¬PhysicsLogic.Papers.Ramsey1950.Fidelity.VelocityAveragingReadout.HasBeamAverage PhysicsLogic.Papers.Ramsey1950.Fidelity.selectedVelocityWithoutBeamAverage := by norm_num [PhysicsLogic.Papers.Ramsey1950.Fidelity.VelocityAveragingReadout.HasSelectedTransition, PhysicsLogic.Papers.Ramsey1950.Fidelity.VelocityAveragingReadout.HasResonantReinforcement, PhysicsLogic.Papers.Ramsey1950.Fidelity.VelocityAveragingReadout.HasBeamAverage, PhysicsLogic.Papers.Ramsey1950.Fidelity.selectedVelocityWithoutBeamAverage] example : PhysicsLogic.Papers.Ramsey1950.MolecularBeamVelocityAverageReadout.Countermodels.unnormalizedMolecularBeamAverage.PhysicalVelocityDomainAt ∧ ¬PhysicsLogic.Papers.Ramsey1950.MolecularBeamVelocityAverageReadout.Countermodels.unnormalizedMolecularBeamAverage.NormalizedMolecularBeamWeightAt := by obtain ⟨domain, _resolved, _average, notNormalized⟩ := PhysicsLogic.Papers.Ramsey1950.MolecularBeamVelocityAverageReadout.Countermodels.bounded_velocity_resolved_probabilities_do_not_normalize_beam_weight exact ⟨domain, notNormalized⟩ example : (0 ≤ PhysicsLogic.Papers.Ramsey1950.MolecularBeamVelocityAverageReadout.Countermodels.namedAverageWithoutPositiveSelectedVelocity.selectedTransitionProbabilityValue ∧ PhysicsLogic.Papers.Ramsey1950.MolecularBeamVelocityAverageReadout.Countermodels.namedAverageWithoutPositiveSelectedVelocity.selectedTransitionProbabilityValue ≤ 1) ∧ ¬PhysicsLogic.Papers.Ramsey1950.MolecularBeamVelocityAverageReadout.Countermodels.namedAverageWithoutPositiveSelectedVelocity.PhysicalVelocityDomainAt := by obtain ⟨selected, _average, notPhysical⟩ := PhysicsLogic.Papers.Ramsey1950.MolecularBeamVelocityAverageReadout.Countermodels.named_beam_average_does_not_supply_positive_velocity_support exact ⟨selected, notPhysical⟩ example : 1 < PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.unnormalizedMixingWithSuperunitTransition.transitionProbabilityValue ∧ ¬(PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.unnormalizedMixingWithSuperunitTransition.sineMixingValue ^ 2 + PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.unnormalizedMixingWithSuperunitTransition.cosineMixingValue ^ 2 = 1) := by obtain ⟨_formula, superunit, notNormalized⟩ := PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.equation_twelve_without_normalized_mixing_can_exceed_probability_one exact ⟨superunit, notNormalized⟩ example : PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.offResonantSingleVelocityFullCycle.HasSelectedFullCycleAliasAt ∧ ¬PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.offResonantSingleVelocityFullCycle.HasExactFrequencyResonanceAt ∧ ¬PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.FreeEvolutionPhaseAliasReadout.HasVelocityIndependentFullCycleAliasAt PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.offResonantSingleVelocityFullCycle := by obtain ⟨aliasAtSelectedSpeed, _positiveComparison, _comparisonPhase, notExact, notUniversal⟩ := PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.single_velocity_full_cycle_is_neither_exact_nor_velocity_independent exact ⟨aliasAtSelectedSpeed, notExact, notUniversal⟩ example {T : PhysicsLogic.Papers.Ramsey1950.RamseyTypes} (framework : PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework T) (mechanism : framework.EstablishedResonanceMechanism) : framework.ExactAndOffResonantVelocityPhaseDiscriminationAt := framework.established_resonance_discriminates_exact_and_velocity_alias_phases mechanism example {T : PhysicsLogic.Papers.Ramsey1950.RamseyTypes} (framework : PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework T) (average : framework.EstablishedVelocityAveragedResonance) : 0 ≤ framework.probabilityValue framework.averagedTransitionProbability ∧ framework.probabilityValue framework.averagedTransitionProbability ≤ 1 := framework.established_velocity_average_is_an_admissible_probability average example {T : PhysicsLogic.Papers.Ramsey1950.RamseyTypes} (framework : PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework T) (complete : framework.CompleteRamseyConclusion) (transition : framework.SeparatedEvolutionCalculation) (mechanism : framework.ResonanceMechanismCalculation) (averaging : framework.VelocityDistributionAverageCalculation) (interval : framework.IntermediateIntervalAssessment) (lineShape : framework.PulseStrengthAndWidthCalculation) : framework.toResonantTransitionSpectroscopyContext |>.EstablishedResonantTransitionSpectroscopyAt framework.resonantTransitionSpectroscopyModel := framework.complete_ramsey_result_realizes_shared_resonant_spectroscopy complete transition mechanism averaging interval lineShape example {T : PhysicsLogic.Papers.Ramsey1950.RamseyTypes} (framework : PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework T) (complete : framework.CompleteRamseyConclusion) (transition : framework.SeparatedEvolutionCalculation) (mechanism : framework.ResonanceMechanismCalculation) (averaging : framework.VelocityDistributionAverageCalculation) (interval : framework.IntermediateIntervalAssessment) (lineShape : framework.PulseStrengthAndWidthCalculation) : framework.resonantTransitionSpectroscopyModel |>.NonvacuousNormalizedResonantTransitionSpectroscopyAt framework.selectedBeamAveragedOutcomeReadout := framework.complete_ramsey_result_exhibits_normalized_shared_resonant_spectroscopy complete transition mechanism averaging interval lineShape example {T : PhysicsLogic.Papers.Ramsey1950.RamseyTypes} (framework : PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework T) (setup : framework.IntermediateSplittingAccessSetup) : framework.intermediateSplittingAvailabilityContext |>.NonvacuousAvailableSectorAt framework.molecularBeam framework.selectedRegime (fun quantity => quantity = framework.intermediateMeanSplitting) := framework.separated_field_access_realizes_nonvacuous_available_splitting_sector setup example {T : PhysicsLogic.Papers.HaxelJensenSuess1949.HJSTypes} (framework : PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework T) (classification : framework.EstablishedLevelClassification) (calculation : framework.MagicClosureCalculation) : framework.SelectedMarkedHighJSuccessorGroupEnergyPlacementsAt := framework.marked_high_j_energy_inequalities_derive_successive_shell_regrouping classification calculation example {T : PhysicsLogic.Papers.HaxelJensenSuess1949.HJSTypes} (framework : PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework T) (closures : framework.EstablishedMagicClosures) : framework.SelectedMarkedHighJSuccessorGroupEnergyPlacementsAt := framework.established_magic_closures_retain_physical_successor_energy_placement closures example : PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework.Countermodels.hasSuccessiveOscillatorGroupLabels PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework.Countermodels.correctlyLabeledUnshiftedHighJEnergyReadout ∧ ¬PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework.Countermodels.hasPhysicalMarkedHighJEnergyPlacement PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework.Countermodels.correctlyLabeledUnshiftedHighJEnergyReadout := PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework.Countermodels.successive_group_labels_do_not_supply_marked_high_j_energy_placement example : PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework.Countermodels.hasPrintedOscillatorMultiplicity PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework.Countermodels.hjsMultiplicityWithoutReordering ∧ ¬PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework.Countermodels.hasMarkedHighJReordering PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework.Countermodels.hjsMultiplicityWithoutReordering := PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework.Countermodels.oscillator_multiplicities_do_not_supply_spin_orbit_reordering example {T : PhysicsLogic.Papers.HaxelJensenSuess1949.HJSTypes} (framework : PhysicsLogic.Papers.HaxelJensenSuess1949.ShellFramework T) (closures : framework.EstablishedMagicClosures) (complete : framework.CompleteScopedResult) : framework.toSpinOrbitShellClosureContext |>.EstablishedSpinOrbitShellClosureAt framework.spinOrbitShellClosureModel := framework.established_magic_closures_establish_shared_spin_orbit_closure closures complete example : PhysicsLogic.Papers.Nakano1956.ConductivityFramework.Countermodels.causalPulseResponseAvailable PhysicsLogic.Papers.Nakano1956.ConductivityFramework.Countermodels.causalWithoutThermalKernel ∧ ¬PhysicsLogic.Papers.Nakano1956.ConductivityFramework.Countermodels.thermalKernelFromCurrentCorrelation PhysicsLogic.Papers.Nakano1956.ConductivityFramework.Countermodels.causalWithoutThermalKernel := PhysicsLogic.Papers.Nakano1956.ConductivityFramework.Countermodels.causal_response_does_not_supply_thermal_kernel example {T : PhysicsLogic.Papers.Nakano1956.NakanoTypes} (framework : PhysicsLogic.Papers.Nakano1956.ConductivityFramework T) (response : framework.EstablishedCurrentResponse) (conductivity : framework.EstablishedStaticConductivity) : framework.toLinearResponseTransportContext |>.EstablishedLinearResponseTransportAt framework.linearResponseTransportModel := framework.established_response_and_conductivity_realize_shared_transport response conductivity example : PhysicsLogic.Papers.PerlmutterEtAl1999.SupernovaCosmologyFramework.Countermodels.positiveLambdaPreference PhysicsLogic.Papers.PerlmutterEtAl1999.SupernovaCosmologyFramework.Countermodels.preferenceWithoutOntology ∧ ¬PhysicsLogic.Papers.PerlmutterEtAl1999.SupernovaCosmologyFramework.Countermodels.uniqueCosmologicalOntology PhysicsLogic.Papers.PerlmutterEtAl1999.SupernovaCosmologyFramework.Countermodels.preferenceWithoutOntology := by obtain ⟨_sample, preference, noOntology⟩ := PhysicsLogic.Papers.PerlmutterEtAl1999.SupernovaCosmologyFramework.Countermodels.positive_lambda_preference_does_not_identify_unique_ontology exact ⟨preference, noOntology⟩ example {T : PhysicsLogic.Papers.PerlmutterEtAl1999.PerlmutterTypes} (framework : PhysicsLogic.Papers.PerlmutterEtAl1999.SupernovaCosmologyFramework T) (standardized : framework.EstablishedStandardizedMagnitudeData) (fit : framework.EstablishedPrimaryCosmologicalFit) (systematics : framework.EstablishedSystematicAssessment) : framework.toSharedStandardCandleAccelerationContext |>.EstablishedStandardCandleAccelerationInferenceAt framework.selectedStandardCandleAccelerationModel := framework.paper_results_realize_shared_standard_candle_acceleration_inference standardized fit systematics example : PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework.Countermodels.ClassicalConfinementAt PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework.Countermodels.classicallyConfinedWithoutOverlap ∧ ¬PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework.Countermodels.WavefunctionOverlapAt PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework.Countermodels.classicallyConfinedWithoutOverlap := PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework.Countermodels.classical_confinement_does_not_supply_wavefunction_overlap example : PhysicsLogic.Papers.GurneyCondon1928.WaveOverlapChanceReadout.Countermodels.finiteEscapeChanceWithoutPositiveObservationDuration.SmallFiniteEscapeChanceAt ∧ ¬PhysicsLogic.Papers.GurneyCondon1928.WaveOverlapChanceReadout.Countermodels.finiteEscapeChanceWithoutPositiveObservationDuration.ValueDomainAt := PhysicsLogic.Papers.GurneyCondon1928.WaveOverlapChanceReadout.Countermodels.finite_escape_chance_does_not_supply_a_positive_observation_duration example : 0 < PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework.Countermodels.positiveNuclearRateWithoutSurvivalLaw.decayRateValue ∧ ¬PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework.Countermodels.positiveNuclearRateWithoutSurvivalLaw.PositiveExponentialSurvivalLawAt := PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework.Countermodels.a_positive_escape_rate_does_not_supply_exponential_survival example : PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework.Countermodels.exponentialNuclearSurvivalWithAgeDependentChance.PositiveExponentialSurvivalLawAt ∧ ¬PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework.Countermodels.exponentialNuclearSurvivalWithAgeDependentChance.ConditionalFutureProbabilitiesConsistentAt := PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework.Countermodels.exponential_survival_without_conditioning_does_not_ensure_history_independence example {T : PhysicsLogic.Papers.GurneyCondon1928.GurneyCondonTypes} (framework : PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework T) (selection : PhysicsLogic.Framework.BarrierDecaySelection T.System T.State T.Channel T.Barrier T.Energy (PhysicsLogic.Papers.GurneyCondon1928.BarrierWavePair T) T.Rate T.Regime) (potential : T.Potential) (result : framework.EstablishedWaveOverlapDecayAt selection potential) : ∃ readout : PhysicsLogic.Framework.ExponentialMetastableSurvivalReadout, framework.exponentialSurvivalReadoutRepresentsAt readout selection.system selection.decayRate selection.regime ∧ framework.finiteBarrierDecayModel.NonvacuousMemorylessFiniteBarrierDecayAt selection readout := framework.overlap_decay_realizes_memoryless_finite_barrier_decay selection potential result example {T : PhysicsLogic.Papers.GurneyCondon1928.GurneyCondonTypes} (framework : PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework T) (comparison : framework.EstablishedOverlapRateComparison) : framework.toFiniteBarrierMetastableDecayContext |>.EstablishedFiniteBarrierMetastableDecayComparisonAt framework.finiteBarrierDecayModel := framework.local_overlap_comparison_establishes_shared_finite_barrier_decay comparison example : PhysicsLogic.Papers.GoldstoneSalamWeinberg1962.BrokenSymmetryFramework.Countermodels.brokenVacuumWithoutChargeBridge.HasBrokenVacuumAndLocalChargeData ∧ ¬PhysicsLogic.Papers.GoldstoneSalamWeinberg1962.BrokenSymmetryFramework.Countermodels.brokenVacuumWithoutChargeBridge.HasNonannihilatingIntegratedCharge := by obtain ⟨broken, notIntegrated, _products, _convergence⟩ := PhysicsLogic.Papers.GoldstoneSalamWeinberg1962.BrokenSymmetryFramework.Countermodels.broken_vacuum_and_nonzero_commutator_do_not_supply_nonannihilation_without_bridge exact ⟨broken, notIntegrated⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.GoldstoneSalamWeinberg1962.Fidelity.countermodelMatrix := PhysicsLogic.Papers.GoldstoneSalamWeinberg1962.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.GoldstoneSalamWeinberg1962.BrokenSymmetryTypes} (framework : PhysicsLogic.Papers.GoldstoneSalamWeinberg1962.BrokenSymmetryFramework T) (broken : framework.SelectedBrokenVacuumSetup) (assumptions : framework.GeneralCurrentAssumptions) (calculation : framework.GeneralCurrentSpectralCalculation) : framework.toBrokenSymmetrySpectrumContext |>.GoldstoneSpectrumRouteApplicableAt framework.brokenSymmetrySpectrumModel := framework.general_current_inputs_make_shared_goldstone_route_applicable broken assumptions calculation example : (∀ zero ∈ PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.finiteCircleZeros, PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.OnUnitCircle zero) ∧ (∀ zero ∈ PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.finiteCircleZeros, PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.SeparatedFromPositivePhysicalPoint zero) := PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.circle_location_does_not_imply_accumulation_or_transition example {T : PhysicsLogic.Papers.LeeYang1952.LeeYangTypes} (framework : PhysicsLogic.Papers.LeeYang1952.LeeYangFramework T) (description : T.Description) (size : T.Size) (activityValue : Real) (coefficients : framework.PositiveEquationFourCoefficientFormulaAt description size) (positive : 0 < activityValue) : 0 < framework.equationFourRealGrandPartitionPolynomialValueAt description size activityValue := framework.positive_activity_makes_equation_four_polynomial_strictly_positive description size activityValue coefficients positive example {T : PhysicsLogic.Papers.LeeYang1952.LeeYangTypes} (framework : PhysicsLogic.Papers.LeeYang1952.LeeYangFramework T) (description : T.Description) (setup : framework.FiniteVolumeSetup description) (formulaData : framework.FiniteGrandPartitionFormulaData description) (size : T.Size) (activity : T.Activity) (positive : framework.positiveRealActivity activity) : ¬framework.finitePartitionZeroAt (framework.finiteDescription description size) size activity := (framework.finite_partition_polynomial_excludes_positive_real_zeros description setup formulaData size activity positive).2.2 example : PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.PositiveFinitePolynomialCoefficients PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.positivelyLabeledNegativeActivityReadout ∧ PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.SelectedActivityIsLabeledPositive PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.positivelyLabeledNegativeActivityReadout ∧ PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.FinitePolynomialVanishesAtSelectedActivity PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.positivelyLabeledNegativeActivityReadout ∧ ¬PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.PositiveActivityMatchesValue PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.positivelyLabeledNegativeActivityReadout := PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels.positive_activity_label_without_positive_value_allows_a_partition_zero example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.LeeYang1952.Fidelity.countermodelMatrix := PhysicsLogic.Papers.LeeYang1952.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.LeeYang1952.LeeYangTypes} (framework : PhysicsLogic.Papers.LeeYang1952.LeeYangFramework T) (description : T.Description) (region : T.Region) (activity : T.Activity) (setup : framework.FiniteVolumeSetup description) (positive : framework.positiveRealActivity activity) (contained : framework.contains region activity) (regularity : framework.FiniteRegularityCalculationAt description activity) (localization : framework.TransitionRequiresZeroAccumulation description) (accumulation : framework.ZeroAccumulationSingularityBridgeOnPositiveAxis description) (phase : framework.SingularityPhaseBoundaryBridge description) (transition : framework.phaseTransitionAt description activity) : framework.toPartitionZeroPhaseBoundaryContext description region activity |>.EstablishedPartitionZeroPhaseBoundaryAt (framework.partitionZeroPhaseBoundaryModelAt description region activity) := framework.lee_yang_route_realizes_shared_partition_zero_correspondence description region activity setup positive contained regularity localization accumulation phase transition example : PhysicsLogic.Papers.RiessEtAl1998.SupernovaAccelerationFramework.Countermodels.calibratedDistancesAvailable PhysicsLogic.Papers.RiessEtAl1998.SupernovaAccelerationFramework.Countermodels.calibrationWithoutFit ∧ ¬PhysicsLogic.Papers.RiessEtAl1998.SupernovaAccelerationFramework.Countermodels.scopedCosmologicalFitAvailable PhysicsLogic.Papers.RiessEtAl1998.SupernovaAccelerationFramework.Countermodels.calibrationWithoutFit := PhysicsLogic.Papers.RiessEtAl1998.SupernovaAccelerationFramework.Countermodels.calibrated_distances_do_not_supply_a_cosmological_fit example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.RiessEtAl1998.Fidelity.countermodelMatrix := PhysicsLogic.Papers.RiessEtAl1998.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.RiessEtAl1998.RiessTypes} (framework : PhysicsLogic.Papers.RiessEtAl1998.SupernovaAccelerationFramework T) (distance : framework.EstablishedDistanceData) (inference : framework.EstablishedCosmologicalInference) (robustness : framework.EstablishedRobustnessAssessment) : framework.toSharedStandardCandleAccelerationContext |>.EstablishedStandardCandleAccelerationInferenceAt framework.selectedStandardCandleAccelerationModel := framework.paper_results_realize_shared_standard_candle_acceleration_inference distance inference robustness example : PhysicsLogic.Papers.Schwarzschild1916.PointMassFramework.Countermodels.sameFiniteApproximation PhysicsLogic.Papers.Schwarzschild1916.PointMassFramework.Countermodels.firstFiniteMetricCandidate ∧ PhysicsLogic.Papers.Schwarzschild1916.PointMassFramework.Countermodels.sameFiniteApproximation PhysicsLogic.Papers.Schwarzschild1916.PointMassFramework.Countermodels.secondFiniteMetricCandidate ∧ PhysicsLogic.Papers.Schwarzschild1916.PointMassFramework.Countermodels.firstFiniteMetricCandidate ≠ PhysicsLogic.Papers.Schwarzschild1916.PointMassFramework.Countermodels.secondFiniteMetricCandidate := PhysicsLogic.Papers.Schwarzschild1916.PointMassFramework.Countermodels.finite_order_agreement_does_not_identify_an_exact_metric example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Schwarzschild1916.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Schwarzschild1916.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Schwarzschild1916.SchwarzschildTypes} (framework : PhysicsLogic.Papers.Schwarzschild1916.PointMassFramework T) (metric : T.Metric) (source : T.Source) (worldline : T.Worldline) (situation : T.Situation) (solution : framework.ExactPointMassSolutionAt metric source situation) (free : framework.freeTestWorldlineAt worldline situation) (calculation : framework.OrbitalGeodesicCalculation metric worldline situation) : framework.selectedVacuumMetricDynamicsContext metric worldline situation |>.EstablishedVacuumMetricDynamicsAt (framework.selectedVacuumMetricDynamicsModel metric worldline situation) := framework.exact_vacuum_solution_and_orbit_realize_shared_metric_dynamics metric source worldline situation solution free calculation example : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework.Countermodels.balancedCollisionDirections PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework.Countermodels.balancedDirectionsWithFiniteExcess ∧ ¬PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework.Countermodels.exactCollisionCancellation PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework.Countermodels.balancedDirectionsWithFiniteExcess := PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework.Countermodels.balanced_collision_directions_do_not_imply_exact_cancellation example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.SmoluchowskiBrownian1906.Fidelity.countermodelMatrix := PhysicsLogic.Papers.SmoluchowskiBrownian1906.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianTypes} (framework : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework T) (smallSetup : framework.SmallParticleSetup) (validDiffusion : framework.validAt framework.kineticDescription framework.diffusionRegime) (persistentCalculation : framework.PersistentPathCalculation) (regimeBridge : framework.SpreadRegimeBridge) (analogy : framework.gasMoleculeAnalogyApplicable framework.selectedParticle framework.selectedMedium framework.diffusionRegime) (diffusionCalculation : framework.DiffusionCalculation) : framework.toRandomDisplacementDiffusionContext |>.EstablishedRandomDisplacementDiffusionAt framework.randomDisplacementModel := framework.smoluchowski_route_realizes_shared_random_displacement_diffusion smallSetup validDiffusion persistentCalculation regimeBridge analogy diffusionCalculation example : PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.blasiusFrictionContrastReadout.HasSmallFrictionLayerContrastAt ∧ ¬PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.blasiusFrictionContrastReadout.HasGloballyNegligibleFrictionAt := PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.small_friction_does_not_make_friction_globally_irrelevant example : PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.ReducedDirectionalPressureReadout.HasPositiveCoordinatePressureGradientAt PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.positiveCoordinateGradientOpposingPhysicalFlow ∧ ¬PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.ReducedDirectionalPressureReadout.HasFlowDirectedAdversePressureRiseAt PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.positiveCoordinateGradientOpposingPhysicalFlow := PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.positive_coordinate_gradient_does_not_supply_flow_directed_pressure_rise example : PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.ReducedWallShearReversalReadout.HasZeroWallShearAt PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.zeroWallShearWithoutNearWallReversal ∧ ¬PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.ReducedWallShearReversalReadout.HasNearWallFlowReversalAt PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.zeroWallShearWithoutNearWallReversal := PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.zero_wall_shear_does_not_supply_near_wall_flow_reversal example : PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.blasiusTangentialWallMatchingWithPenetration.HasWallTangentialMatchingAt ∧ PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.blasiusTangentialWallMatchingWithPenetration.HasOuterTangentialMatchingAt ∧ ¬PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.blasiusTangentialWallMatchingWithPenetration.HasWallNormalMatchingAt := PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.tangential_no_slip_does_not_supply_wall_impermeability example : PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.blasiusStationaryWallWithoutOuterRecovery.HasWallTangentialMatchingAt ∧ PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.blasiusStationaryWallWithoutOuterRecovery.HasWallNormalMatchingAt ∧ ¬PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.blasiusStationaryWallWithoutOuterRecovery.HasOuterTangentialMatchingAt := PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.stationary_no_slip_does_not_supply_outer_velocity_recovery example : PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.blasiusImpermeableOuterMatchWithTangentialSlip.HasWallNormalMatchingAt ∧ PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.blasiusImpermeableOuterMatchWithTangentialSlip.HasOuterTangentialMatchingAt ∧ ¬PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.blasiusImpermeableOuterMatchWithTangentialSlip.HasWallTangentialMatchingAt := PhysicsLogic.Papers.Blasius1908.BlasiusFramework.Countermodels.impermeability_and_outer_recovery_do_not_supply_tangential_no_slip example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Blasius1908.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Blasius1908.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Blasius1908.Blasius1908Types} (framework : PhysicsLogic.Papers.Blasius1908.BlasiusFramework T) (reduction : framework.EstablishedThinLayerReduction) : framework.toThinViscousBoundaryLayerReductionContext |>.EstablishedReductionAt framework.thinViscousBoundaryLayerReductionModel := framework.established_reduction_realizes_shared_boundary_layer_model reduction example {T : PhysicsLogic.Papers.Blasius1908.Blasius1908Types} (framework : PhysicsLogic.Papers.Blasius1908.BlasiusFramework T) (reduction : framework.EstablishedThinLayerReduction) : framework.thinViscousBoundaryLayerReductionModel |>.StationaryWallVelocityMatchedBoundaryLayerReductionAt framework.selectedBoundaryVelocityMatchingReadout := framework.established_blasius_reduction_realizes_stationary_velocity_matched_boundary_layer reduction example {T : PhysicsLogic.Papers.Blasius1908.Blasius1908Types} (framework : PhysicsLogic.Papers.Blasius1908.BlasiusFramework T) (separation : framework.EstablishedSeparationMechanism) : framework.FlowDirectedAdversePressureRiseAt framework.outerPressureProfile framework.selectedSteadySeparationLocation := framework.established_separation_retains_flow_directed_pressure_rise separation example {T : PhysicsLogic.Papers.Blasius1908.Blasius1908Types} (framework : PhysicsLogic.Papers.Blasius1908.BlasiusFramework T) (separation : framework.EstablishedSeparationMechanism) : framework.wallNormalTangentialVelocityDerivativeValueAt (framework.separationLocationCoordinateValueAt framework.selectedSteadySeparationLocation) 0 = 0 := framework.established_separation_retains_zero_wall_normal_velocity_derivative separation example {T : PhysicsLogic.Papers.Bethe1949.BetheTypes} (framework : PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework T) (assessment : framework.EstablishedNeutralCrossSectionAssessment) : framework.EquationSeventeenCrossSectionFormulaAt framework.neutronProtonTriplet framework.selectedCrossSectionEnergy ∧ framework.EquationSeventeenSineSquaredCrossSectionFormulaAt framework.neutronProtonTriplet framework.selectedCrossSectionEnergy := framework.established_neutral_cross_section_retains_both_equation_seventeen_forms assessment example {T : PhysicsLogic.Papers.Bethe1949.BetheTypes} (framework : PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework T) (assessment : framework.EstablishedNeutralCrossSectionAssessment) : framework.NeutralUnitarityBoundValueAt framework.neutronProtonTriplet framework.selectedCrossSectionEnergy ∧ framework.NeutralUnitarityMaximumValueAt framework.neutronProtonTriplet framework.selectedCrossSectionEnergy := framework.established_neutral_cross_section_retains_numerical_bound_and_maximum assessment example {T : PhysicsLogic.Papers.Bethe1949.BetheTypes} (framework : PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework T) (assessment : framework.EstablishedNeutralCrossSectionAssessment) : framework.NeutralUnitarityMaximumValueAt framework.neutronProtonTriplet framework.selectedCrossSectionEnergy ↔ framework.neutralPhaseFunctionValue framework.neutronProtonTriplet framework.selectedCrossSectionEnergy = 0 := framework.established_neutral_cross_section_characterizes_saturation_by_zero_phase assessment example {T : PhysicsLogic.Papers.Bethe1949.BetheTypes} (framework : PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework T) (assessment : framework.EstablishedNeutralCrossSectionAssessment) : framework.NeutralUnitarityMaximumValueAt framework.neutronProtonTriplet framework.selectedCrossSectionEnergy ↔ Real.cos (framework.neutralPhaseShiftValue framework.neutronProtonTriplet framework.selectedCrossSectionEnergy) = 0 := framework.established_neutral_cross_section_characterizes_physical_phase_shift_maximum assessment example {T : PhysicsLogic.Papers.Bethe1949.BetheTypes} (framework : PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework T) (channel : T.Channel) (energy : T.Energy) (waveNumberPositive : 0 < framework.waveNumberSquared energy) (formula : framework.EquationSeventeenCrossSectionFormulaAt channel energy) (phaseNonzero : framework.neutralPhaseFunctionValue channel energy ≠ 0) : framework.neutralCrossSectionValue channel energy < (4 * Real.pi) / framework.waveNumberSquared energy := framework.equation_seventeen_nonzero_phase_is_strictly_below_unitarity_maximum channel energy waveNumberPositive formula phaseNonzero example : PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.retainsEquationSeventeenCrossSectionValue PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.zeroWaveNumberEquationSeventeenModel ∧ ¬PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.hasPositiveWaveNumberAndCrossSection PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.zeroWaveNumberEquationSeventeenModel := PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.equation_seventeen_without_positive_wave_number_is_not_physical example : (PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.hasPositiveWaveNumberAndCrossSection PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.nonzeroPhaseBelowUnitarityMaximumModel ∧ PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.retainsEquationSeventeenCrossSectionValue PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.nonzeroPhaseBelowUnitarityMaximumModel) ∧ ¬PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.saturatesNeutralUnitarityMaximum PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.nonzeroPhaseBelowUnitarityMaximumModel := PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.equation_seventeen_without_zero_phase_does_not_reach_maximum example : PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.inconsistentWaveSquareWithExactDenominatorFormula.RetainsDenominatorCrossSectionFormula ∧ ¬PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.inconsistentWaveSquareWithExactDenominatorFormula.RetainsSineSquaredCrossSectionFormula := by obtain ⟨⟨_domain, denominator, _phase⟩, _noSquare, noSine⟩ := PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.exact_phase_function_without_wave_square_does_not_supply_sine_squared_formula exact ⟨denominator, noSine⟩ example : PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.inconsistentPhaseFunctionWithExactDenominatorFormula.RetainsDenominatorCrossSectionFormula ∧ ¬PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.inconsistentPhaseFunctionWithExactDenominatorFormula.RetainsSineSquaredCrossSectionFormula := by obtain ⟨⟨_domain, denominator, _wave⟩, _noPhase, noSine⟩ := PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.exact_wave_square_without_phase_readout_does_not_supply_sine_squared_formula exact ⟨denominator, noSine⟩ example : PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.exactComparisonWithoutReplacement.ExactComparisonIdentity ∧ ¬PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.exactComparisonWithoutReplacement.ControlledLowEnergyReplacement := PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework.Countermodels.exact_comparison_identity_does_not_supply_low_energy_replacement example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Bethe1949.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Bethe1949.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Bethe1949.BetheTypes} (framework : PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework T) (law : framework.EstablishedNeutralEffectiveRangeLaw) (accuracy : framework.EstablishedAccuracyAndShapeScope) : framework.toLowEnergyEffectiveRangeContext |>.EstablishedEffectiveRangeReductionAt framework.effectiveRangeReductionModel := framework.neutral_effective_range_routes_establish_shared_reduction law accuracy example : PhysicsLogic.Papers.Kitaev2001.MajoranaWireFramework.Countermodels.gappedFiniteChainWithTunneling PhysicsLogic.Papers.Kitaev2001.MajoranaWireFramework.Countermodels.gappedChainWithFiniteSplitting ∧ ¬PhysicsLogic.Papers.Kitaev2001.MajoranaWireFramework.Countermodels.exactFiniteDegeneracy PhysicsLogic.Papers.Kitaev2001.MajoranaWireFramework.Countermodels.gappedChainWithFiniteSplitting := PhysicsLogic.Papers.Kitaev2001.MajoranaWireFramework.Countermodels.a_bulk_gap_does_not_force_exact_finite_chain_degeneracy example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Kitaev2001.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Kitaev2001.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Kitaev2001.KitaevTypes} (framework : PhysicsLogic.Papers.Kitaev2001.MajoranaWireFramework T) (boundary : framework.BoundaryPhaseResult) : framework.toBoundaryMajoranaModeContext |>.EstablishedBoundaryMajoranaModeAt framework.boundaryMajoranaModel := framework.boundary_phase_result_establishes_shared_boundary_majorana boundary example : PhysicsLogic.Papers.Parker1969.ExpandingUniverseFramework.Countermodels.BogoliubovExpansionAt PhysicsLogic.Papers.Parker1969.ExpandingUniverseFramework.Countermodels.modeExpansionWithoutCreation ∧ ¬PhysicsLogic.Papers.Parker1969.ExpandingUniverseFramework.Countermodels.PhysicalParticleProductionAt PhysicsLogic.Papers.Parker1969.ExpandingUniverseFramework.Countermodels.modeExpansionWithoutCreation := PhysicsLogic.Papers.Parker1969.ExpandingUniverseFramework.Countermodels.a_bogoliubov_expansion_does_not_by_itself_establish_particle_creation example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Parker1969.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Parker1969.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Parker1969.ParkerTypes} (framework : PhysicsLogic.Papers.Parker1969.ExpandingUniverseFramework T) (setup : framework.BogoliubovSetup) (calculation : framework.BogoliubovCalculation) (interpretation : framework.BogoliubovParticleInterpretation) : framework.toBogoliubovParticleProductionContext |>.EstablishedBogoliubovParticleProductionAt framework.selectedBogoliubovProductionModel := framework.parker_bogoliubov_route_realizes_shared_particle_production setup calculation interpretation example : PhysicsLogic.Papers.CHSH1969.ExperimentalFramework.Countermodels.ObservedCountingRateViolation PhysicsLogic.Papers.CHSH1969.ExperimentalFramework.Countermodels.detectorDependenceWithLocalResponses ∧ ¬PhysicsLogic.Papers.CHSH1969.ExperimentalFramework.Countermodels.LocalResponseFactorizationFailed PhysicsLogic.Papers.CHSH1969.ExperimentalFramework.Countermodels.detectorDependenceWithLocalResponses := by have witness := PhysicsLogic.Papers.CHSH1969.ExperimentalFramework.Countermodels.observed_violation_does_not_identify_which_assumption_failed exact ⟨witness.1, witness.2.1⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.CHSH1969.Fidelity.countermodelMatrix := PhysicsLogic.Papers.CHSH1969.Fidelity.countermodel_matrix_is_realized example {Setting HiddenState Model Experiment Partition : Type} (framework : PhysicsLogic.Papers.CHSH1969.ExperimentalFramework Setting HiddenState Model Experiment Partition) (model : Model) (responses : PhysicsLogic.Papers.CHSH1969.LocalResponses Setting HiddenState) (factorizes : framework.LocalResponsesFactorize model responses) (domain : PhysicsLogic.Framework.DeterministicLocalResponse.NontrivialResponseDomain Setting HiddenState) (hiddenState : HiddenState) : (framework.responsePossibility model).FactorizesAt hiddenState := framework.local_responses_realize_generic_factorization model responses factorizes domain hiddenState open PhysicsLogic.Papers.KohnSham1965.Countermodels.OrbitalDiagnostics in example : NormalizedAt equalPhase ∧ diagonalDensityEnergy exactHamiltonian (density equalPhase) = diagonalDensityEnergy approximateHamiltonian (density equalPhase) ∧ relativePotential exactHamiltonian ≠ relativePotential approximateHamiltonian := PhysicsLogic.Papers.KohnSham1965.Countermodels.one_energy_readout_does_not_supply_a_functional_derivative example (mode : PhysicsLogic.Papers.KohnSham1965.ThermalPhysics.FermionMode) (physical : mode.PhysicalDomainAt) : mode.NonvacuousThermalFermionAt := mode.a_positive_temperature_fermion_realizes_the_thermal_gibbs_logic physical example {Density Mode Temperature : Type} (framework : PhysicsLogic.Papers.KohnSham1965.ThermalFramework Density Mode Temperature) (temperature : Temperature) (density : Density) (stationary : framework.thermalLocalStationaryAt temperature density) (calculation : framework.ThermalStationarityCalculation temperature density) : framework.ThermalSelfConsistentSolution temperature density := framework.finite_temperature_variational_route_yields_thermal_equations temperature density stationary calculation example {Density Mode Temperature : Type} (framework : PhysicsLogic.Papers.KohnSham1965.ThermalFramework Density Mode Temperature) (temperature : Temperature) (regime : framework.lowTemperatureRegime temperature) (calculation : framework.LowTemperatureSpecificHeatCalculation temperature) : framework.LowTemperatureSpecificHeatFormula temperature := framework.low_temperature_route_yields_specific_heat_formula temperature regime calculation example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.KohnSham1965.Fidelity.countermodelMatrix := PhysicsLogic.Papers.KohnSham1965.Fidelity.countermodel_matrix_is_realized example {Potential GroundState Density Regime : Type} (framework : PhysicsLogic.Papers.KohnSham1965.GroundStateFramework Potential GroundState Density Regime) (state : GroundState) (potential : Potential) (particleNumber : Nat) (ground : framework.groundStateFor state potential) (nondegenerate : framework.nondegenerateGroundStateFor state potential) (fixedParticleNumber : framework.particleNumber (framework.densityOf state) = particleNumber) (foundation : framework.HohenbergKohnVariationalFoundation) : (framework.toSharedDensityFunctionalContext state potential particleNumber) |>.EstablishedGroundDensityVariationalAt (framework.sharedGroundDensityModel state potential particleNumber) := framework.exact_ground_density_establishes_shared_variational_result state potential particleNumber ground nondegenerate fixedParticleNumber foundation example {Potential GroundState Density Regime : Type} (framework : PhysicsLogic.Papers.KohnSham1965.GroundStateFramework Potential GroundState Density Regime) (state : GroundState) (potential : Potential) (particleNumber : Nat) (setup : framework.SharedGroundDensityGroundingSetup state potential particleNumber) : (framework.sharedGroundDensityModel state potential particleNumber) |>.toAdmissibleDensityCompletenessContext.CompleteAt (framework.sharedGroundDensityModel state potential particleNumber) particleNumber := framework.exact_ground_density_realizes_scoped_descriptive_completeness state potential particleNumber setup example {T : PhysicsLogic.Papers.Lemaitre1927.LemaitreTypes} (framework : PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework T) (redshift : framework.EstablishedRedshiftAndVelocityDistanceRelation) : framework.periodValue framework.emittedProperPeriod < framework.periodValue framework.observedProperPeriod ∧ 0 < framework.velocityValue framework.equivalentDopplerVelocity := framework.established_redshift_has_positive_period_shift_and_recession_velocity redshift example : (PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.HasAdmissibleRadiusSignal PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.exactContractingRadiusSignalWithBlueshift ∧ PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.RetainsEquationTwentyTwoFormula PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.exactContractingRadiusSignalWithBlueshift) ∧ ¬PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.HasPositivePeriodShiftAndRecession PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.exactContractingRadiusSignalWithBlueshift := PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.exact_radius_ratio_without_growth_does_not_imply_positive_redshift example : PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.RetainsExactRedshiftAtSource PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.exactRedshiftWithoutNearbyVelocityApproximation ∧ ¬PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.RetainsEquationTwentyThreeNearbyApproximation PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.exactRedshiftWithoutNearbyVelocityApproximation := PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework.Countermodels.exact_equation_twenty_two_does_not_supply_nearby_equation_twenty_three example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Lemaitre1927.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Lemaitre1927.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Lemaitre1927.LemaitreTypes} (framework : PhysicsLogic.Papers.Lemaitre1927.ExpandingUniverseFramework T) (calibration : framework.EstablishedQualifiedObservationalCalibration) : framework.toNebularDistanceVelocityCalibrationContext |>.EstablishedQualifiedDistanceVelocityCalibrationAt framework.sharedCalibrationModel := framework.qualified_observational_calibration_realizes_shared_calibration calibration namespace LemaitreOriginalCalibrationIndependentReuse open PhysicsLogic.Papers.Lemaitre1927 open PhysicsLogic.Papers.Lemaitre1927.OriginalNebularCalibrationReadout open PhysicsLogic.Papers.Lemaitre1927.OriginalNebularCalibrationReadout.Countermodels example : SourceFormulaAt originalFortyThreeObjectCalibration := original_source_calibration_is_nonvacuously_realized example : originalFortyThreeObjectCalibration.radialVelocityCatalogCount = 43 ∧ originalFortyThreeObjectCalibration.selectedCommonNebulaCount = 42 ∧ originalFortyThreeObjectCalibration.excludedNGCIdentifier = 5194 ∧ originalFortyThreeObjectCalibration.associatedNGCIdentifier = 5195 := by obtain ⟨catalog, sample, _, excluded, associated⟩ := originalFortyThreeObjectCalibration |>.source_catalog_retains_forty_three_objects_and_associated_exclusion original_source_calibration_is_nonvacuously_realized exact ⟨catalog, sample, excluded, associated⟩ example (near far : ℝ) (nonnegative : 0 ≤ near) (ordered : near < far) : originalFortyThreeObjectCalibration.distanceWeightAt far < originalFortyThreeObjectCalibration.distanceWeightAt near := originalFortyThreeObjectCalibration |>.source_distance_weight_strictly_decreases_with_distance original_source_calibration_is_nonvacuously_realized near far nonnegative ordered example : originalFortyThreeObjectCalibration.weightedRoundedMeanVelocityResidualValue = (25 / 4 : ℝ) ∧ originalFortyThreeObjectCalibration.unweightedRoundedMeanVelocityResidualValue = 3 := by exact ⟨(originalFortyThreeObjectCalibration |>.source_weighted_rounded_means_do_not_exactly_realize_reported_fit original_source_calibration_is_nonvacuously_realized).2, (originalFortyThreeObjectCalibration |>.source_unweighted_rounded_means_do_not_exactly_realize_reported_fit original_source_calibration_is_nonvacuously_realized).2⟩ example : FortyTwoCommonNebulaeRetainedAt erroneousFortyEightObjectCatalogReadout ∧ ¬OriginalCatalogSelectionAt erroneousFortyEightObjectCatalogReadout := forty_two_selected_nebulae_do_not_identify_original_catalog_count example : ReportedWeightedExpansionRateRetainedAt erasedUnweightedFootnoteEstimateReadout ∧ ¬OriginalUnweightedCalibrationAt erasedUnweightedFootnoteEstimateReadout := weighted_estimate_does_not_supply_unweighted_footnote_estimate example : OriginalRadiusEstimatesAt originalFortyThreeObjectCalibration ∧ ¬ReportedRadiusRatioExactlyEqualsRoundedRadiiAt originalFortyThreeObjectCalibration := printed_radius_estimates_do_not_exactly_realize_printed_radius_ratio example {T : LemaitreTypes} (framework : ExpandingUniverseFramework T) (calibration : framework.EstablishedQualifiedObservationalCalibration) : framework.radialVelocityCatalogCountAt framework.nebularVelocityData = 43 ∧ framework.selectedCommonNebulaCountAt framework.nebulaSample = 42 := framework.established_calibration_retains_original_catalog_and_selected_sample calibration example {T : LemaitreTypes} (framework : ExpandingUniverseFramework T) (calibration : framework.EstablishedQualifiedObservationalCalibration) : framework.reportedWeightedExpansionRateKilometersPerSecondPerMegaparsecAt framework.expansionRateParameter = 625 ∧ framework.reportedUnweightedExpansionRateKilometersPerSecondPerMegaparsecAt framework.expansionRateParameter = 575 := by obtain ⟨weighted, unweighted, _⟩ := framework.established_calibration_retains_distinct_original_expansion_estimates calibration exact ⟨weighted, unweighted⟩ example {T : LemaitreTypes} (framework : ExpandingUniverseFramework T) (calibration : framework.EstablishedQualifiedObservationalCalibration) : framework.reportedInitialRadiusEstimateParsecAt framework.radiusEstimateParameter = 270000000 := (framework |>.established_calibration_distinguishes_initial_current_and_static_radii calibration).1 end LemaitreOriginalCalibrationIndependentReuse example : PhysicsLogic.Papers.Jarzynski1997.Countermodels.HasSufficientlyWeakInteraction PhysicsLogic.Papers.Jarzynski1997.Countermodels.weakButUndersizedNoncanonicalReservoir ∧ ¬PhysicsLogic.Papers.Jarzynski1997.Countermodels.HasLargeReservoirCanonicalSampling PhysicsLogic.Papers.Jarzynski1997.Countermodels.weakButUndersizedNoncanonicalReservoir := PhysicsLogic.Papers.Jarzynski1997.Countermodels.weak_coupling_does_not_supply_large_reservoir_sampling_scope example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Jarzynski1997.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Jarzynski1997.Fidelity.countermodel_matrix_is_realized example {Configuration Protocol Realization : Type} (framework : PhysicsLogic.Papers.Jarzynski1997.SwitchingFramework Configuration Protocol Realization) (protocol : Protocol) (betaPositive : 0 < framework.beta) (finite : framework.finiteClassicalSystem protocol) (admissible : framework.admissibleSwitchingProtocol protocol) (canonicalPreparation : framework.canonicalInitialPreparation protocol) (partition : framework.SystemPartitionRatioEncodesFreeEnergy protocol) (workDefinition : framework.workDefinedByGeneralizedForceIntegral protocol) (ensembleDomain : framework.NormalizedExponentialWorkEnsembleAt protocol) : (framework.toNonequilibriumWorkIdentityContext protocol) |>.SelectedSetupAt (framework.sharedNonequilibriumWorkModel protocol) := framework.jarzynski_inputs_realize_shared_nonequilibrium_work_setup protocol betaPositive finite admissible canonicalPreparation partition workDefinition ensembleDomain example : PhysicsLogic.Papers.Nordtvedt1968.EquationFortyCoefficientFormulaReadout.namedBransDickeCancellationFailure PhysicsLogic.Papers.Nordtvedt1968.EquationFortyCoefficientFormulaReadout.wrongNamedEquationFortyReadout ∧ ¬PhysicsLogic.Papers.Nordtvedt1968.EquationFortyCoefficientFormulaReadout.wrongNamedEquationFortyReadout.SourceFormulaAt := PhysicsLogic.Papers.Nordtvedt1968.EquationFortyCoefficientFormulaReadout.named_brans_dicke_failure_does_not_supply_equation_forty example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Nordtvedt1968.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Nordtvedt1968.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Nordtvedt1968.MassiveBodyTypes} (framework : PhysicsLogic.Papers.Nordtvedt1968.MassiveBodyFramework T) (setup : framework.BransDickeSelfGravityResponseSetup) (einsteinMetric : framework.coefficientsDescribeMetric framework.einsteinCoefficients framework.generalMetric framework.einsteinTheory framework.weakSlowRegime) (bransDickeMetric : framework.coefficientsDescribeMetric framework.bransDickeCoefficients framework.generalMetric framework.bransDickeTheory framework.weakSlowRegime) (bransDickeFailure : ¬framework.satisfiesStationaryCancellationA framework.bransDickeCoefficients) (stationaryCalculation : framework.StationaryTwoBodyAccelerationCalculation) (responseCalculation : framework.BransDickeBodyResponseCalculation) (interpretation : framework.BransDickeResponseInterpretation) : framework.toBodyDependentSelfGravityContext |>.BodyDependentSelfGravityRouteApplicableAt framework.sharedBodyDependentSelfGravityModel := framework.stationary_brans_dicke_result_makes_shared_response_route_applicable setup einsteinMetric bransDickeMetric bransDickeFailure stationaryCalculation responseCalculation interpretation example : PhysicsLogic.Papers.AndersonBergmann1951.CovariantConstraintFramework.Countermodels.primaryConstraintLostDespiteGeneratorClosure.HasInvariantGeneratorFunctionGroup ∧ PhysicsLogic.Papers.AndersonBergmann1951.CovariantConstraintFramework.Countermodels.primaryConstraintLostDespiteGeneratorClosure.HasHamiltonianBracketClosure ∧ ¬PhysicsLogic.Papers.AndersonBergmann1951.CovariantConstraintFramework.Countermodels.primaryConstraintLostDespiteGeneratorClosure.HasPreservedCompletedFamily := PhysicsLogic.Papers.AndersonBergmann1951.CovariantConstraintFramework.Countermodels.generator_function_group_does_not_by_itself_complete_constraints example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.AndersonBergmann1951.Fidelity.countermodelMatrix := PhysicsLogic.Papers.AndersonBergmann1951.Fidelity.countermodel_matrix_is_realized example {Theory PhasePoint Constraint Descriptor Generator : Type} (framework : PhysicsLogic.Papers.AndersonBergmann1951.CovariantConstraintFramework Theory PhasePoint Constraint Descriptor Generator) {theory : Theory} (setup : framework.CanonicalGeneratorSetup theory) (calculation : framework.InvariantGeneratorCalculation theory) : framework.toCanonicalGeneratorBracketClosureContext theory |>.EstablishedCanonicalGeneratorBracketClosureAt (framework.canonicalGeneratorBracketCalculationModel theory) := framework.invariant_generators_realize_shared_bracket_closure setup calculation namespace BHvKOriginalFewBodyIndependentReuse open PhysicsLogic.Papers.BedaqueHammerVanKolck1998 open PhysicsLogic.Papers.BedaqueHammerVanKolck1998.OriginalThreeBodySourceReadout open PhysicsLogic.Papers.BedaqueHammerVanKolck1998.OriginalThreeBodySourceReadout.Countermodels example : originalBosonicHeliumSourceReadout.SourceFormulaAt := original_bosonic_helium_source_is_nonvacuously_realized example : |3 * Real.sqrt 3 / (4 * Real.pi) - (827 / 2000 : Real)| < (1 / 100 : Real) := original_critical_threshold_agrees_with_reported_approximation example : originalBosonicHeliumSourceReadout.spinThreeHalfStatisticsWeightValue < originalBosonicHeliumSourceReadout.criticalStatisticsWeightValue ∧ originalBosonicHeliumSourceReadout.criticalStatisticsWeightValue < originalBosonicHeliumSourceReadout.bosonicStatisticsWeightValue ∧ originalBosonicHeliumSourceReadout.quartetLeadingThreeBodyForceRequired = false := source_boson_and_fermion_channels_lie_on_opposite_sides_of_threshold originalBosonicHeliumSourceReadout original_bosonic_helium_source_is_nonvacuously_realized example : originalBosonicHeliumSourceReadout.bosonicIntegralEquationCount = 1 ∧ originalBosonicHeliumSourceReadout.spinOneHalfIntegralEquationCount = 2 ∧ originalBosonicHeliumSourceReadout.bosonicIntegralEquationCount ≠ originalBosonicHeliumSourceReadout.spinOneHalfIntegralEquationCount := source_nuclear_doublet_is_not_the_single_bosonic_equation originalBosonicHeliumSourceReadout original_bosonic_helium_source_is_nonvacuously_realized example : 1 < discreteCutoffMultiplier (629 / 625 : Real) := positive_exponent_gives_nontrivial_discrete_cutoff_multiplier (629 / 625 : Real) (by norm_num) example : runningForceAtCutoff (629 / 625 : Real) (10 * discreteCutoffMultiplier (629 / 625 : Real)) 1 = runningForceAtCutoff (629 / 625 : Real) 10 1 := running_force_is_invariant_under_one_discrete_cutoff_cycle (629 / 625 : Real) 10 1 (by norm_num) (by norm_num) (by norm_num) example : runningForceAtPhase (629 / 625 : Real) 0 = 1 ∧ runningForceAtPhase (629 / 625 : Real) (runningPhaseShift (629 / 625 : Real)) = 0 ∧ runningForceAtPhase (629 / 625 : Real) Real.pi = 1 := positive_exponent_limit_cycle_returns_without_becoming_constant (629 / 625 : Real) (by norm_num) example : originalBosonicHeliumSourceReadout.fittedThreeBodyDatumCount = 1 ∧ originalBosonicHeliumSourceReadout.calibratedAtomDimerScatteringLengthValue ≠ originalBosonicHeliumSourceReadout.predictedAtomDimerEffectiveRangeValue := source_one_datum_does_not_identify_calibration_with_predicted_range originalBosonicHeliumSourceReadout original_bosonic_helium_source_is_nonvacuously_realized example : originalBosonicHeliumSourceReadout.heliumAtomDimerScatteringLengthAngstromValue ≠ (39 / 25 : Real) * originalBosonicHeliumSourceReadout.heliumTwoBodyScatteringLengthAngstromValue ∧ originalBosonicHeliumSourceReadout.heliumFittedRatioRoundingResidualAngstromValue = (117 / 250 : Real) := source_helium_scattering_lengths_do_not_exactly_realize_fitted_ratio originalBosonicHeliumSourceReadout original_bosonic_helium_source_is_nonvacuously_realized example : (57 / 100 : Real) * originalBosonicHeliumSourceReadout.heliumTwoBodyScatteringLengthAngstromValue ≠ originalBosonicHeliumSourceReadout.heliumEffectiveRangeAngstromValue ∧ originalBosonicHeliumSourceReadout.heliumEffectiveRangeRoundingResidualAngstromValue = (79 / 1000 : Real) := source_helium_effective_range_is_independently_rounded originalBosonicHeliumSourceReadout original_bosonic_helium_source_is_nonvacuously_realized example : originalBosonicHeliumSourceReadout.controlledShallowBoundStateCount = 1 ∧ originalBosonicHeliumSourceReadout.controlledShallowBoundStateCount < originalBosonicHeliumSourceReadout.candidateBoundStateCount := source_only_shallow_bound_state_has_controlled_eft_scope originalBosonicHeliumSourceReadout original_bosonic_helium_source_is_nonvacuously_realized example : frozenButPeriodicForceReadout.HasPositivePiPeriodicForce ∧ ¬frozenButPeriodicForceReadout.HasNonconstantOriginalForce := discrete_periodicity_alone_does_not_supply_nonconstant_running example : totalizedRunningForceAtPoleReadout.HasTotalizedSineRatioValue ∧ ¬totalizedRunningForceAtPoleReadout.IsAdmissibleNonpole := totalized_running_force_at_pole_is_not_physically_admissible example : reportedPairWithZeroCandidateRootReadout.RetainsReportedImaginaryPair ∧ ¬ReducedReportedRootReadout.HasExactPositiveCharacteristicRoot reportedPairWithZeroCandidateRootReadout := reported_imaginary_pair_does_not_supply_exact_characteristic_root example {T : ThreeBodyTypes} (framework : ThreeBodyRenormalizationFramework T) (renormalized : framework.EstablishedCutoffIndependentPredictions) (datum : framework.DatumFixesRunningParameterAndScale) (prediction : framework.OneDatumPredictionCalculation) : framework.originalFewBodySourceReadout.fittedThreeBodyDatumCount = 1 ∧ framework.datumRealValueAt framework.renormalizationDatum = (39 / 25 : Real) * framework.twoBodyScatteringLengthRealValue ∧ framework.observableRealValueAt framework.effectiveRangeObservable = (57 / 100 : Real) * framework.twoBodyScatteringLengthRealValue := framework.one_datum_prediction_retains_distinct_source_calibration_and_range renormalized datum prediction example {T : ThreeBodyTypes} (framework : ThreeBodyRenormalizationFramework T) (obstruction : framework.EstablishedThreeBodyCutoffObstruction) (renormalized : framework.EstablishedCutoffIndependentPredictions) (datum : framework.DatumFixesRunningParameterAndScale) (prediction : framework.OneDatumPredictionCalculation) (bound : framework.EstablishedQualifiedBoundStateResult) (helium : framework.HeliumApplicationInput) : framework.originalFewBodySourceReadout.SourceFormulaAt := framework.independent_nuclear_routes_reconstruct_complete_original_source_formula obstruction renormalized datum prediction bound helium end BHvKOriginalFewBodyIndependentReuse example : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework.Countermodels.formulaWithoutPhaseLock.RunningFormulaMatching ∧ ¬PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework.Countermodels.formulaWithoutPhaseLock.PhaseLockedTwoCutoffAgreement := PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework.Countermodels.running_formula_matching_does_not_supply_phase_locked_two_cutoff_agreement example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.BedaqueHammerVanKolck1998.Fidelity.countermodelMatrix := PhysicsLogic.Papers.BedaqueHammerVanKolck1998.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyTypes} (framework : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework T) (renormalized : framework.EstablishedCutoffIndependentPredictions) (datum : framework.DatumFixesRunningParameterAndScale) (prediction : framework.OneDatumPredictionCalculation) : framework.toPostCalibrationPredictionContext |>.EstablishedPostCalibrationPredictionAt framework.postCalibrationPredictionModel := framework.one_datum_prediction_realizes_shared_post_calibration_prediction renormalized datum prediction example {T : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyTypes} (framework : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework T) (predictions : framework.EstablishedCutoffIndependentPredictions) : framework.toRunningCountertermSolutionContext.SolvesAt framework.threeBodyForce framework () framework.selectedLowEnergyObservable := framework.cutoff_independent_predictions_realize_scoped_explanatory_success predictions example {T : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyTypes} (framework : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework T) (running : framework.EstablishedRunningCounterterm) (onShellSelected : framework.selectedLowEnergyObservable framework.onShellAmplitudeObservable) (matching : framework.AllSelectedPredictionsMatchTarget) : framework.selectedResummedRenormalizationHandoff |>.NonvacuousResummationGroundedRenormalizedPredictionAt := framework.dressed_two_body_cutoff_obstruction_realizes_composed_renormalized_prediction running onShellSelected matching example {T : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyTypes} (framework : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework T) (running : framework.EstablishedRunningCounterterm) (onShellSelected : framework.selectedLowEnergyObservable framework.onShellAmplitudeObservable) (matching : framework.AllSelectedPredictionsMatchTarget) : framework.selectedResummedRenormalizationHandoff |>.SourceSensitiveResummationGroundedRenormalizedPredictionAt framework.toLargeScatteringLengthResummationContext framework.toRenormalizedPredictionContext .dressedDibaryonSelfEnergyAndEffectiveRangeRecovery framework.BHvKLargeLengthResummationDerivationObtains .runningCountertermOnShellSelectionAndAllScaleMatching framework.bhvkRenormalizedPredictionDerivationObtains .dressedTwoBosonInputCutoffSensitiveThreeBodyEquationAndRunningForce framework.bhvkResummedPredictionHandoffDerivationObtains := framework |>.dressed_two_body_cutoff_obstruction_realizes_source_sensitive_renormalized_prediction running onShellSelected matching example : PhysicsLogic.Papers.FixsenEtAl1996.FirasFramework.Countermodels.satisfiesPlanckPlusResidualEquation PhysicsLogic.Papers.FixsenEtAl1996.FirasFramework.Countermodels.nonzeroSmallResidualReport ∧ PhysicsLogic.Papers.FixsenEtAl1996.FirasFramework.Countermodels.hasSubFiftyPpmResidual PhysicsLogic.Papers.FixsenEtAl1996.FirasFramework.Countermodels.nonzeroSmallResidualReport ∧ ¬PhysicsLogic.Papers.FixsenEtAl1996.FirasFramework.Countermodels.isExactBlackbodyAtReportedFrequency PhysicsLogic.Papers.FixsenEtAl1996.FirasFramework.Countermodels.nonzeroSmallResidualReport := PhysicsLogic.Papers.FixsenEtAl1996.FirasFramework.Countermodels.sub_fifty_ppm_residual_does_not_supply_exact_blackbody_equality example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.FixsenEtAl1996.Fidelity.countermodelMatrix := PhysicsLogic.Papers.FixsenEtAl1996.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.FixsenEtAl1996.FixsenTypes} (framework : PhysicsLogic.Papers.FixsenEtAl1996.FirasFramework T) (setup : framework.FirasObservationSetup) : framework.toCalibratedResidualBackgroundContext |>.SelectedCalibratedResidualBackgroundSetupAt framework.sharedBackgroundModel := framework.firas_setup_realizes_shared_selected_setup setup example : PhysicsLogic.Papers.CalderbankShor1996.QuantumCodeFramework.Countermodels.noiselessRecoveryWithoutFaultTolerance.HasNoiselessExactRecovery ∧ ¬PhysicsLogic.Papers.CalderbankShor1996.QuantumCodeFramework.Countermodels.noiselessRecoveryWithoutFaultTolerance.HasNoisyFaultTolerance := PhysicsLogic.Papers.CalderbankShor1996.QuantumCodeFramework.Countermodels.exact_supported_recovery_does_not_establish_noisy_fault_tolerance example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.CalderbankShor1996.Fidelity.countermodelMatrix := PhysicsLogic.Papers.CalderbankShor1996.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.CalderbankShor1996.CalderbankShorTypes} (framework : PhysicsLogic.Papers.CalderbankShor1996.QuantumCodeFramework T) (recovery : framework.EstablishedExactSupportedDecoherenceRecovery) : framework.toExactQuantumErrorRecoveryContext |>.EstablishedExactQuantumErrorRecoveryAt framework.toExactQuantumErrorRecoveryModel := framework.exact_decoding_realizes_shared_recovery_route recovery example : PhysicsLogic.Papers.Haldane1988.PaperFramework.Countermodels.ReducedHaldaneScopeReadout.HasZeroNetFluxAt PhysicsLogic.Papers.Haldane1988.PaperFramework.Countermodels.completeReducedHaldaneScopeReadout ∧ ¬PhysicsLogic.Papers.Haldane1988.PaperFramework.Countermodels.ReducedHaldaneScopeReadout.HasTimeReversalInvariantHoppingAt PhysicsLogic.Papers.Haldane1988.PaperFramework.Countermodels.completeReducedHaldaneScopeReadout := PhysicsLogic.Papers.Haldane1988.PaperFramework.Countermodels.zero_net_flux_does_not_imply_time_reversal_symmetry example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Haldane1988.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Haldane1988.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Haldane1988.HaldaneTypes} (framework : PhysicsLogic.Papers.Haldane1988.PaperFramework T) (phase : framework.EstablishedAnomalousHallPhase) : framework.toSharedContext.EstablishedGappedBandHallQuantizationAt framework.sharedGappedBandHallModel := framework.anomalous_phase_establishes_shared_gapped_band_hall_quantization phase example : PhysicsLogic.Papers.Choptuik1993.Countermodels.HasSelectedNumericalMassScaling PhysicsLogic.Papers.Choptuik1993.Countermodels.selectedScalarCriticalEvidence ∧ ¬PhysicsLogic.Papers.Choptuik1993.Countermodels.HasMathematicalMassScalingLaw PhysicsLogic.Papers.Choptuik1993.Countermodels.selectedScalarCriticalEvidence := PhysicsLogic.Papers.Choptuik1993.Countermodels.numerical_mass_scaling_evidence_does_not_become_mathematical_proof example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Choptuik1993.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Choptuik1993.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Choptuik1993.ChoptuikTypes} (context : PhysicsLogic.Papers.Choptuik1993.ScalarCollapseContext T) (setup : context.ThresholdSetup) (outcome : context.ThresholdOutcomeCalculation) (mass : context.MassScalingCalculation) (limit : context.VanishingMassBridge) : context.toSharedCriticalCollapseContext |>.EstablishedCriticalCollapseScalingAt context.selectedCriticalCollapseModel := context.scalar_threshold_and_mass_fit_establish_shared_scaling setup outcome mass limit example : PhysicsLogic.Papers.GRW1986.LocalizationFramework.Countermodels.ReducedGRWLocalizationReadout.HasStatisticalMixtureAt PhysicsLogic.Papers.GRW1986.LocalizationFramework.Countermodels.statisticalMixtureWithoutPointerRecord ∧ ¬PhysicsLogic.Papers.GRW1986.LocalizationFramework.Countermodels.ReducedGRWLocalizationReadout.HasDefinitePointerRecordAt PhysicsLogic.Papers.GRW1986.LocalizationFramework.Countermodels.statisticalMixtureWithoutPointerRecord := PhysicsLogic.Papers.GRW1986.LocalizationFramework.Countermodels.statistical_mixture_does_not_by_itself_supply_a_measurement_record example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.GRW1986.Fidelity.countermodelMatrix := PhysicsLogic.Papers.GRW1986.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.GRW1986.GRWTypes} (framework : PhysicsLogic.Papers.GRW1986.LocalizationFramework T) (branchSetup : framework.MacroscopicBranchPairSetup) (constituentSetup : framework.UniversalConstituentSetup) (calculation : framework.CompositeAmplificationCalculation) (interpretation : framework.BranchInterferenceInterpretation) : framework.toMacroscopicCoherenceSuppressionContext |>.EstablishedMacroscopicCoherenceSuppressionAt framework.macroscopicCoherenceSuppressionModel := framework.amplified_localization_realizes_shared_coherence_suppression branchSetup constituentSetup calculation interpretation example : PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.Countermodels.ParityViolationObserved PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.Countermodels.parityOnlyReadout ∧ ¬PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.Countermodels.ChargeConjugationViolationObserved PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.Countermodels.parityOnlyReadout ∧ ¬PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.Countermodels.TimeReversalViolationObserved PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.Countermodels.parityOnlyReadout := PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.Countermodels.parity_failure_alone_decides_neither_charge_conjugation_nor_time_reversal example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.GarwinLedermanWeinrich1957.Fidelity.countermodelMatrix := PhysicsLogic.Papers.GarwinLedermanWeinrich1957.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.GarwinLedermanWeinrich1957.Garwin1957Types} (framework : PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework T) (apparatus : framework.StoppedMuonApparatusResult) (precession : framework.PrecessionScanResult) (control : framework.StoppedPionControlResult) (setup : framework.ParityInferenceSetup) (prediction : framework.ParityConservationPredictionCalculation) (result : framework.ParityViolationResult) : framework.toControlledSymmetryFalsificationContext |>.EstablishedControlledSymmetryFalsificationAt framework.controlledSymmetryFalsificationModel := framework.glw_route_realizes_shared_controlled_symmetry_falsification apparatus precession control setup prediction result example : PhysicsLogic.Papers.Mori1965.PaperFramework.Countermodels.retainedModeWithoutEquationIdentity.HydrodynamicReductionAt PhysicsLogic.Papers.Mori1965.PaperFramework.Countermodels.exactHydrodynamicEquation PhysicsLogic.Papers.Mori1965.PaperFramework.Countermodels.localHydrodynamicEquation PhysicsLogic.Papers.Mori1965.PaperFramework.Countermodels.hydrodynamicRegime ∧ PhysicsLogic.Papers.Mori1965.PaperFramework.Countermodels.exactHydrodynamicEquation ≠ PhysicsLogic.Papers.Mori1965.PaperFramework.Countermodels.localHydrodynamicEquation := PhysicsLogic.Papers.Mori1965.PaperFramework.Countermodels.hydrodynamic_reduction_does_not_identify_exact_and_local_equations example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Mori1965.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Mori1965.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Mori1965.MoriTypes} (paper : PhysicsLogic.Papers.Mori1965.PaperFramework T) (correlation : paper.EstablishedForceCorrelation) : paper.selectedMemoryKernelContext.EstablishedExactMemoryKernelReductionAt paper.selectedMemoryKernelModel := paper.mori_routes_establish_shared_exact_memory_kernel_reduction correlation example : PhysicsLogic.Papers.Minkowski1908.Countermodels.nullResultWithoutWorldPostulate.HasMichelsonNullResult ∧ ¬PhysicsLogic.Papers.Minkowski1908.Countermodels.nullResultWithoutWorldPostulate.HasWorldPostulateSpeedBound := PhysicsLogic.Papers.Minkowski1908.Countermodels.michelson_null_result_does_not_deduce_the_world_postulate example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Minkowski1908.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Minkowski1908.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Minkowski1908.MinkowskiTypes} (context : PhysicsLogic.Papers.Minkowski1908.SpacetimeContext T) (setup : context.FiniteCSetup) (calculation : context.FiniteCGroupCalculation) (worldPostulate : context.WorldPostulate) : context.toSharedInertialFrameContext |>.EstablishedInertialFrameKinematicsAt context.selectedInertialFrameModel := context.minkowski_inputs_establish_shared_inertial_frame_kinematics setup calculation worldPostulate example : PhysicsLogic.Papers.Raychaudhuri1955.CosmologyFramework.Countermodels.finiteVolumeCollapseWithoutCurvatureDivergence.HasFinitePastVolumeCollapse ∧ ¬PhysicsLogic.Papers.Raychaudhuri1955.CosmologyFramework.Countermodels.finiteVolumeCollapseWithoutCurvatureDivergence.HasCurvatureDivergence := PhysicsLogic.Papers.Raychaudhuri1955.CosmologyFramework.Countermodels.finite_volume_collapse_does_not_establish_curvature_divergence example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Raychaudhuri1955.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Raychaudhuri1955.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Raychaudhuri1955.RaychaudhuriTypes} (framework : PhysicsLogic.Papers.Raychaudhuri1955.CosmologyFramework T) (setup : framework.SharedCongruenceSetup) (evolution : framework.SharedFocusingEvolution) (collapse : framework.SharedFiniteCollapseCalculation) (interpretation : framework.SharedCollapseInterpretation) : framework.toCongruenceFocusingContext |>.EstablishedCongruenceFocusingAt framework.sharedFocusingModel := framework.raychaudhuri_route_realizes_shared_congruence_focusing setup evolution collapse interpretation example : PhysicsLogic.Papers.Stanley1968.CriticalResponseControls.StableCriticalFreeEnergyConvergence ∧ Not (PhysicsLogic.Papers.Stanley1968.CriticalResponseControls.constructedCriticalFamily.AppendixAgreementAt (1 / 2) (1 / 4)) := PhysicsLogic.Papers.Stanley1968.CriticalResponseControls.uniform_stable_free_energy_convergence_does_not_establish_critical_index_agreement example : PhysicsLogic.Papers.Stanley1968.CriticalResponseControls.StableUniformFreeEnergyAndEntropyConvergence ∧ Not PhysicsLogic.Papers.Stanley1968.CriticalResponseControls.SpecificHeatTransportAtSelectedTemperature := PhysicsLogic.Papers.Stanley1968.CriticalResponseControls.uniform_free_energy_and_entropy_convergence_do_not_transport_specific_heat example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Stanley1968.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Stanley1968.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Stanley1968.StanleyTypes} (framework : PhysicsLogic.Papers.Stanley1968.SphericalLimitFramework T) (varies : framework.SpinDimensionFamilyVaries) (temperature : framework.SelectedNoncriticalTemperatureSetup) (remainder : framework.JointRemainderVanishes) : framework.toAsymptoticContributionContext |>.AsymptoticallyOrganizedAt () framework.NoncriticalFreeEnergySector := framework.normalized_remainder_realizes_shared_asymptotic_organization varies temperature remainder example : PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework.Countermodels.continuityEquationDerived PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework.Countermodels.conservedDensityWithoutActualConfiguration ∧ ¬PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework.Countermodels.actualConfigurationPostulated PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework.Countermodels.conservedDensityWithoutActualConfiguration := PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework.Countermodels.continuity_equation_does_not_supply_an_actual_configuration example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Bohm1952.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Bohm1952.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Bohm1952.BohmTypes} (framework : PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework T) (setup : framework.EquivarianceSetup) (calculation : framework.EquivarianceCalculation) : framework.toGuidedDensityEquivarianceContext |>.EstablishedGuidedDensityEquivarianceAt framework.sharedGuidedDensityEquivarianceModel := framework.bohm_route_realizes_shared_guided_density_equivariance setup calculation example : PhysicsLogic.Papers.Everett1957.RelativeStateFramework.Countermodels.calibratedBranchesWithoutSelection.containsBranch false false false ∧ PhysicsLogic.Papers.Everett1957.RelativeStateFramework.Countermodels.calibratedBranchesWithoutSelection.containsBranch false true false ∧ ¬PhysicsLogic.Papers.Everett1957.RelativeStateFramework.Countermodels.calibratedBranchesWithoutSelection.globallySelectsSingleOutcome false false := by have witness := PhysicsLogic.Papers.Everett1957.RelativeStateFramework.Countermodels.two_correlated_branches_do_not_select_one_global_outcome exact ⟨witness.2.2.1, witness.2.2.2.1, witness.2.2.2.2.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Everett1957.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Everett1957.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Everett1957.EverettTypes} (framework : PhysicsLogic.Papers.Everett1957.RelativeStateFramework T) (setup : framework.GoodMeasurementSetup) (calculation : framework.LinearMeasurementCalculation) (interpretation : framework.RelativeRecordInterpretation) : framework.toLinearRelativeStateMeasurementContext |>.EstablishedLinearRelativeStateMeasurementAt framework.relativeMeasurementModel := framework.everett_measurement_realizes_shared_relative_state_measurement setup calculation interpretation example : PhysicsLogic.Papers.Zeh1970.MeasurementFramework.Countermodels.zehDecoupledPureBoundaryReadout.HasOperationallySuppressedCoherenceAt ∧ PhysicsLogic.Papers.Zeh1970.MeasurementFramework.Countermodels.zehDecoupledPureBoundaryReadout.HasDiagonalReducedPointerDescriptionAt ∧ PhysicsLogic.Papers.Zeh1970.MeasurementFramework.Countermodels.zehDecoupledPureBoundaryReadout.HasGloballyPureSelectedStateAt ∧ ¬PhysicsLogic.Papers.Zeh1970.MeasurementFramework.Countermodels.zehDecoupledPureBoundaryReadout.HasDynamicallyProducedStatisticalMixtureAt ∧ ¬PhysicsLogic.Papers.Zeh1970.MeasurementFramework.Countermodels.zehDecoupledPureBoundaryReadout.HasOneActualizedBranchAt := PhysicsLogic.Papers.Zeh1970.MeasurementFramework.Countermodels.zeh_decoupling_realizes_shared_coherence_mixture_and_outcome_boundary example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Zeh1970.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Zeh1970.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Zeh1970.ZehTypes} (framework : PhysicsLogic.Papers.Zeh1970.MeasurementFramework T) (setup : framework.BranchDecouplingSetup) (calculation : framework.BranchDecouplingCalculation) : framework.toMacroscopicCoherenceSuppressionContext |>.EstablishedMacroscopicCoherenceSuppressionAt framework.macroscopicCoherenceSuppressionModel := framework.environmental_decoupling_realizes_shared_coherence_suppression setup calculation example : PhysicsLogic.Framework.SelfConsistentSpectralGapContext.Countermodels.setupWithoutSolution.GapEquationSetupAt PhysicsLogic.Framework.SelfConsistentSpectralGapContext.Countermodels.setupWithoutSolutionModel ∧ ¬PhysicsLogic.Framework.SelfConsistentSpectralGapContext.Countermodels.setupWithoutSolution.GapSolutionCalculationAt PhysicsLogic.Framework.SelfConsistentSpectralGapContext.Countermodels.setupWithoutSolutionModel := PhysicsLogic.Framework.SelfConsistentSpectralGapContext.Countermodels.self_consistency_setup_does_not_supply_nontrivial_solution example : PhysicsLogic.Framework.SelfConsistentSpectralGapContext.Countermodels.completeGapRoute.EstablishedSelfConsistentSpectralGapAt PhysicsLogic.Framework.SelfConsistentSpectralGapContext.Countermodels.completeGapModel ∧ ¬PhysicsLogic.Framework.TransverseMeissnerScreeningContext.Countermodels.gapEstablishedButResponseRejected.TransverseResponseSetupAt PhysicsLogic.Framework.TransverseMeissnerScreeningContext.Countermodels.gapRejectedModel := by have witness := PhysicsLogic.Framework.TransverseMeissnerScreeningContext.Countermodels.established_gap_without_admissibility_does_not_supply_transverse_response_setup exact ⟨witness.1, witness.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Nambu1960.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Nambu1960.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Nambu1960.NambuTypes} (framework : PhysicsLogic.Papers.Nambu1960.SuperconductivityFramework T) (setup : framework.SelfConsistentGapSetup) (calculation : framework.NonperturbativeGapCalculation) (interpretation : framework.GapInterpretationBridge) : framework.toSelfConsistentSpectralGapContext |>.EstablishedSelfConsistentSpectralGapAt framework.selfConsistentSpectralGapModel := framework.self_consistent_nonperturbative_solution_realizes_shared_gap setup calculation interpretation example : PhysicsLogic.Papers.LiebRobinson1972.PaperFramework.Countermodels.exponentiallySuppressedOutsideCone PhysicsLogic.Papers.LiebRobinson1972.PaperFramework.Countermodels.exponentiallySmallNonzeroCommutator ∧ ¬PhysicsLogic.Papers.LiebRobinson1972.PaperFramework.Countermodels.exactlyZeroOutsideCone PhysicsLogic.Papers.LiebRobinson1972.PaperFramework.Countermodels.exponentiallySmallNonzeroCommutator := PhysicsLogic.Papers.LiebRobinson1972.PaperFramework.Countermodels.exponentially_small_commutator_is_not_exact_zero example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.LiebRobinson1972.Fidelity.countermodelMatrix := PhysicsLogic.Papers.LiebRobinson1972.Fidelity.countermodel_matrix_is_realized example (framework : PhysicsLogic.Papers.LiebRobinson1972.PaperFramework) (finiteVelocity : framework.EstablishedFiniteGroupVelocity) : framework.toFiniteVelocityCommutatorConeContext |>.CommutatorConeCalculationAt framework.finiteVelocityCommutatorModel := framework.finite_velocity_result_realizes_shared_calculation finiteVelocity example : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.PlasmaWaveFramework.Countermodels.HasSelectedMomentAgreementAt PhysicsLogic.Papers.BernsteinGreeneKruskal1957.PlasmaWaveFramework.Countermodels.matchedMomentsWithDifferentPointValue ∧ ¬PhysicsLogic.Papers.BernsteinGreeneKruskal1957.PlasmaWaveFramework.Countermodels.HasPointwiseDistributionAgreementAt PhysicsLogic.Papers.BernsteinGreeneKruskal1957.PlasmaWaveFramework.Countermodels.matchedMomentsWithDifferentPointValue := PhysicsLogic.Papers.BernsteinGreeneKruskal1957.PlasmaWaveFramework.Countermodels.moment_agreement_does_not_make_surrogate_pointwise_exact example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.BernsteinGreeneKruskal1957.Fidelity.countermodelMatrix := PhysicsLogic.Papers.BernsteinGreeneKruskal1957.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.BGK1957Types} (framework : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.PlasmaWaveFramework T) (result : framework.SingularMomentSurrogateResult) (bridge : framework.MomentApproximationBridge) : framework.SingularSurrogateApproximatesExactMoments := framework.paper_moment_claim_realizes_generic_prediction_approximation result bridge example : PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework.Countermodels.HasGaussianTailCountedAt PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework.Countermodels.rawGaussianTailWithoutDoubling ∧ ¬PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework.Countermodels.HasFactorTwoMassNormalizationAt PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework.Countermodels.rawGaussianTailWithoutDoubling := PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework.Countermodels.gaussian_tail_counting_does_not_supply_the_factor_two_normalization example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.PressSchechter1974.Fidelity.countermodelMatrix := PhysicsLogic.Papers.PressSchechter1974.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.PressSchechter1974.PressSchechterTypes} (framework : PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework T) (analytic : framework.EstablishedAnalyticMassSpectrum) : framework.toSharedThresholdedAbundanceContext |>.EstablishedThresholdedCollapseAbundanceAt framework.sharedThresholdedAbundanceModel := framework.press_schechter_route_realizes_shared_thresholded_abundance analytic example {T : PhysicsLogic.Papers.PressSchechter1974.PressSchechterTypes} (framework : PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework T) (collapseSetup : framework.SphericalCollapseSetup) (collapseCalculation : framework.SphericalCollapseCalculation) (countingSetup : framework.MaximumVarianceCountingSetup) (tailCalculation : framework.GaussianTailCalculation) (correction : framework.UnderdenseAccretionCorrection) (spectrumCalculation : framework.AnalyticMassSpectrumCalculation) : framework.pressSchechterCollapseAbundanceHandoff |>.SourceSensitiveCollapseGroundedThresholdAbundanceAt framework.toSharedCollapseContext framework.toSharedThresholdedAbundanceContext .positiveOverdensitySphericalRecollapse framework.PressSchechterCollapseDerivationObtains .maximumVarianceGaussianTailAccretionAndSpectrum framework.pressSchechterAbundanceDerivationObtains .sphericalThresholdMaximumVarianceGaussianTailAndUnderdenseAccretion framework.pressSchechterCollapseAbundanceHandoffDerivationObtains := framework |>.press_schechter_spherical_gaussian_and_accretion_inputs_realize_source_sensitive_abundance_composition collapseSetup collapseCalculation countingSetup tailCalculation correction spectrumCalculation example {T : PhysicsLogic.Papers.PressSchechter1974.PressSchechterTypes} (framework : PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework T) (sourceSensitive : framework.pressSchechterCollapseAbundanceHandoff |>.SourceSensitiveCollapseGroundedThresholdAbundanceAt framework.toSharedCollapseContext framework.toSharedThresholdedAbundanceContext .positiveOverdensitySphericalRecollapse framework.PressSchechterCollapseDerivationObtains .maximumVarianceGaussianTailAccretionAndSpectrum framework.pressSchechterAbundanceDerivationObtains .sphericalThresholdMaximumVarianceGaussianTailAndUnderdenseAccretion framework.pressSchechterCollapseAbundanceHandoffDerivationObtains) : framework.fractionValueAt framework.selectedCompletedCollapsedFraction = 2 * framework.fractionValueAt framework.selectedRawCollapsedFraction /\ framework.EquationThirteenMassSpectrumFormulaAt := by obtain ⟨_formulae, _underdense, factorTwo, _equationTwelve, equationThirteen, _recollapse⟩ := framework |>.source_sensitive_press_schechter_composition_retains_gaussian_equations_factor_two_and_recollapse sourceSensitive exact ⟨factorTwo, equationThirteen⟩ example : PhysicsLogic.Papers.Einstein1911.LightGravityFramework.Countermodels.ReducedEquivalenceReadout.HasUniversalFreeFallAndMechanicalEquivalence PhysicsLogic.Papers.Einstein1911.LightGravityFramework.Countermodels.freeFallWithoutFullEquivalence ∧ ¬PhysicsLogic.Papers.Einstein1911.LightGravityFramework.Countermodels.freeFallWithoutFullEquivalence.HasFullProcessEquivalence := by have witness := PhysicsLogic.Papers.Einstein1911.LightGravityFramework.Countermodels.universal_free_fall_does_not_entail_full_process_equivalence exact ⟨witness.1, witness.2.1⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Einstein1911.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Einstein1911.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Einstein1911.LightGravityTypes} (framework : PhysicsLogic.Papers.Einstein1911.LightGravityFramework T) (setup : framework.EquivalenceSetup) (mechanics : framework.MechanicalEquivalenceCalculation) : framework.toAcceleratedFrameGravityContext |>.EstablishedAcceleratedFrameGravityAt framework.selectedAcceleratedFrameGravityModel := framework.einstein_1911_route_realizes_shared_accelerated_frame_gravity setup mechanics example : PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.qualifiedZKNumericalEvidenceReadout.HasProfileNearRecurrenceAt ∧ ¬PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.qualifiedZKNumericalEvidenceReadout.HasExactFullStateRecurrenceAt := PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.profile_reconstruction_does_not_establish_exact_state_recurrence example {T : PhysicsLogic.Papers.ZabuskyKruskal1965.ZKTypes} (framework : PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework T) (physical : framework.PhysicalKdVSetup) (position time : Real) : framework.evolutionTimeDerivativeValueAt position time + framework.evolutionFieldValueAt framework.selectedRun position time * framework.evolutionSpatialDerivativeValueAt position time + framework.dispersionParameter framework.selectedRun ^ 2 * framework.evolutionThirdSpatialDerivativeValueAt position time = 0 := (framework.physical_kdv_setup_retains_printed_equation_one physical).2.2 position time example {T : PhysicsLogic.Papers.ZabuskyKruskal1965.ZKTypes} (framework : PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework T) (protocol : framework.NumericalProtocol) (position : Real) : framework.evolutionFieldValueAt framework.selectedRun position 0 = Real.cos (Real.pi * position) := (framework.numerical_protocol_retains_printed_periodic_cosine_initial_data protocol).2 position example : PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.HasInviscidSteepeningBalanceAt PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.inviscidSteepeningWithoutKdV ∧ ¬PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.HasOriginalKdVBalanceAt PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.inviscidSteepeningWithoutKdV := by obtain ⟨_nonlinear, _dispersive, inviscid, notKdV⟩ := PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.inviscid_steepening_does_not_supply_original_dispersive_equation exact ⟨inviscid, notKdV⟩ example : PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.HasLinearDispersiveBalanceAt PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.linearDispersionWithoutKdV ∧ ¬PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.HasOriginalKdVBalanceAt PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.linearDispersionWithoutKdV := by obtain ⟨_nonlinear, _dispersive, linear, notKdV⟩ := PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.linear_dispersion_does_not_supply_original_nonlinear_equation exact ⟨linear, notKdV⟩ example : PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.HasPeriodTwoAt PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.phaseReversedPeriodicCosine ∧ ¬PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.HasSourceCosineInitialProfileAt PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.phaseReversedPeriodicCosine := by obtain ⟨periodic, _amplitude, notCosine⟩ := PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework.Countermodels.periodic_unit_amplitude_profile_does_not_fix_source_cosine_phase exact ⟨periodic, notCosine⟩ example {T : PhysicsLogic.Papers.ZabuskyKruskal1965.ZKTypes} (framework : PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework T) (physical : framework.PhysicalKdVSetup) (protocol : framework.NumericalProtocol) (validation : framework.NumericalValidationCalculation) (reliability : framework.NumericalReliabilityBridge) (nearRecurrence : framework.NearRecurrenceResult) : ∃ value, (framework.validatedNumericalModel value) |>.NonvacuousValidatedNumericalPredictionAt := framework.recurrence_route_exhibits_nonvacuous_shared_validated_numerical_prediction physical protocol validation reliability nearRecurrence example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.ZabuskyKruskal1965.Fidelity.countermodelMatrix := PhysicsLogic.Papers.ZabuskyKruskal1965.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.ZabuskyKruskal1965.ZKTypes} (framework : PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework T) (setup : framework.IsolatedSolitonSetup) (calculation : framework.IsolatedSolitonCalculation) (interpretation : framework.IsolatedSolitonInterpretationBridge) : framework.toSolitaryTravelingWaveContext |>.EstablishedSolitaryTravelingWaveAt framework.selectedSolitaryWaveModel := framework.isolated_pulse_realizes_shared_solitary_traveling_wave setup calculation interpretation example : PhysicsLogic.Papers.Haldane1983.HierarchyFramework.Countermodels.HasHomogeneousFiniteSphereStateAt PhysicsLogic.Papers.Haldane1983.HierarchyFramework.Countermodels.homogeneousFiniteSphereWithoutThermodynamicGap ∧ ¬PhysicsLogic.Papers.Haldane1983.HierarchyFramework.Countermodels.HasGappedThermodynamicFluidAt PhysicsLogic.Papers.Haldane1983.HierarchyFramework.Countermodels.homogeneousFiniteSphereWithoutThermodynamicGap := PhysicsLogic.Papers.Haldane1983.HierarchyFramework.Countermodels.finite_sphere_homogeneity_does_not_establish_a_thermodynamic_gap example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Haldane1983.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Haldane1983.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Haldane1983.HaldaneTypes} (framework : PhysicsLogic.Papers.Haldane1983.HierarchyFramework T) (fluid : framework.quantumFluidState framework.parentState framework.parentRegime) (gap : framework.noGaplessExcitations framework.parentState framework.parentRegime) (principle : framework.GapInterpretationPrinciple) : framework.toGappedIncompressibleFluidContext |>.EstablishedGappedIncompressibleFluidAt framework.sharedGappedFluidModel := framework.parent_fluid_realizes_shared_gapped_incompressible_fluid fluid gap principle example : PhysicsLogic.Papers.NiuThoulessWu1985.PaperFramework.Countermodels.finiteGapWithoutTwistIndependence.HasIntegerTwistAverageAt ∧ ¬PhysicsLogic.Papers.NiuThoulessWu1985.PaperFramework.Countermodels.finiteGapWithoutTwistIndependence.HasBoundaryTwistIndependenceAt := PhysicsLogic.Papers.NiuThoulessWu1985.PaperFramework.Countermodels.an_integer_twist_average_does_not_imply_pointwise_finite_volume_quantization example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.NiuThoulessWu1985.Fidelity.countermodelMatrix := PhysicsLogic.Papers.NiuThoulessWu1985.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.NiuThoulessWu1985.NTWTypes} (framework : PhysicsLogic.Papers.NiuThoulessWu1985.PaperFramework T) (setup : framework.TwistedBoundarySetup) : framework.SharedTopologicalResponseSetup := framework.twisted_boundary_inputs_select_shared_topological_response_setup setup example : PhysicsLogic.Papers.Thomas1935.ThomasFramework.Countermodels.thomasAlternativesWithoutSelectionReadout.HasCalibratedPairwiseObstructionAt ∧ ¬PhysicsLogic.Papers.Thomas1935.ThomasFramework.Countermodels.thomasAlternativesWithoutSelectionReadout.HasUniqueSelectedRemedyAt := by have witness := PhysicsLogic.Papers.Thomas1935.ThomasFramework.Countermodels.thomas_realizes_shared_three_body_remedy_boundary exact ⟨witness.1, witness.2.2.2.1⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Thomas1935.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Thomas1935.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Thomas1935.ThomasTypes} (framework : PhysicsLogic.Papers.Thomas1935.ThomasFramework T) (setup : framework.SingularPairwiseSetup) (calculation : framework.SingularVariationalCalculation) : framework.toSharedPairwiseInputContext |>.EstablishedThreeBodyPairwiseInputInsufficiencyAt framework.sharedPairwiseInputModel := framework.singular_pairwise_model_realizes_shared_input_insufficiency setup calculation example : PhysicsLogic.Papers.DurrGoldsteinZanghi1992.QuantumEquilibriumFramework.Countermodels.hasUniversalEquilibriumDistribution PhysicsLogic.Papers.DurrGoldsteinZanghi1992.QuantumEquilibriumFramework.Countermodels.universeDistributionWithoutSingleWorldFrequency ∧ ¬PhysicsLogic.Papers.DurrGoldsteinZanghi1992.QuantumEquilibriumFramework.Countermodels.hasSingleWorldEquilibriumFrequency PhysicsLogic.Papers.DurrGoldsteinZanghi1992.QuantumEquilibriumFramework.Countermodels.universeDistributionWithoutSingleWorldFrequency := PhysicsLogic.Papers.DurrGoldsteinZanghi1992.QuantumEquilibriumFramework.Countermodels.ensemble_of_universes_does_not_supply_single_world_empirical_frequencies example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.DurrGoldsteinZanghi1992.Fidelity.countermodelMatrix := PhysicsLogic.Papers.DurrGoldsteinZanghi1992.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.DurrGoldsteinZanghi1992.QuantumEquilibriumTypes} (framework : PhysicsLogic.Papers.DurrGoldsteinZanghi1992.QuantumEquilibriumFramework T) (setup : framework.EquivarianceSetup) (calculation : framework.EquivarianceCalculation) : framework.toGuidedDensityEquivarianceContext |>.EstablishedGuidedDensityEquivarianceAt framework.sharedGuidedDensityEquivarianceModel := framework.quantum_equilibrium_route_realizes_shared_guided_density_equivariance setup calculation namespace DGZOriginalGuidanceCurrentAndConditionalProbabilityReuseAudit open PhysicsLogic.Papers.DurrGoldsteinZanghi1992 open SourceFormulaCountermodels example : physicalTravelingWaveReadout.SourceFormulaAt /\ physicalTravelingWaveReadout.guidanceVelocityValue = 4 /\ physicalTravelingWaveReadout.schrodingerCurrentValue = 4 := traveling_wave_realizes_printed_guidance_current_and_continuity example : physicalTimeReversedWaveReadout.SourceFormulaAt /\ physicalTimeReversedWaveReadout.guidanceVelocityValue = -4 /\ physicalTimeReversedWaveReadout.schrodingerCurrentValue = -4 := time_reversed_wave_reverses_guidance_velocity_and_current example : physicalStationaryRealWaveReadout.SourceFormulaAt /\ physicalStationaryRealWaveReadout.bornDensityValue = 4 /\ physicalStationaryRealWaveReadout.guidanceVelocityValue = 0 := by have witness := real_stationary_wave_has_nonzero_born_density_without_motion exact ⟨witness.1, witness.2.1, witness.2.2.1⟩ example : physicalConditionalWaveReadout.SourceFormulaAt /\ physicalConditionalWaveReadout.conditionalProbabilityValueAt 0 = 9 / 25 /\ physicalConditionalWaveReadout.conditionalProbabilityValueAt 1 = 16 / 25 := by have witness := selected_environment_realizes_printed_conditional_wave_and_born_probabilities exact ⟨witness.1, witness.2.2.2.1, witness.2.2.2.2⟩ example : physicalConditionalWaveReadout.SourceFormulaAt /\ 0 < physicalConditionalWaveReadout.conditionalProbabilityValueAt 0 /\ 0 < physicalConditionalWaveReadout.conditionalProbabilityValueAt 1 /\ physicalConditionalWaveReadout.ActualSubsystemConfigurationAt 0 /\ ¬physicalConditionalWaveReadout.MultipleActualSubsystemConfigurationsAt := nontrivial_born_probabilities_coexist_with_one_actual_configuration example : physicalConditionalWaveReadout.SourceFormulaAt /\ ¬physicalConditionalWaveReadout.AutonomousConditionalSchrodingerAt := conditional_wave_source_does_not_supply_autonomous_schrodinger_evolution example : alternativeEnvironmentConditionalWaveReadout.SourceFormulaAt /\ alternativeEnvironmentConditionalWaveReadout.actualEnvironmentValue = 1 /\ alternativeEnvironmentConditionalWaveReadout.conditionalProbabilityValueAt 0 ≠ physicalConditionalWaveReadout.conditionalProbabilityValueAt 0 := by have witness := alternative_environment_changes_selected_conditional_wave_and_probability exact ⟨witness.1, witness.2.1, witness.2.2.2.2⟩ example : rescaledConditionalWaveReadout.SourceFormulaAt /\ rescaledConditionalWaveReadout.normalizationValue = 100 /\ ∀ subsystem, rescaledConditionalWaveReadout.conditionalProbabilityValueAt subsystem = physicalConditionalWaveReadout.conditionalProbabilityValueAt subsystem := by have witness := nonzero_wave_rescaling_preserves_normalized_conditional_probabilities exact ⟨witness.1, witness.2.2.1, witness.2.2.2⟩ example : ¬zeroAmplitudeNodeWithTotalizedVelocity.SourceFormulaAt := zero_wave_node_does_not_define_a_physical_guidance_velocity.2.2 example : ¬realGradientMistakenForBohmianVelocity.SourceFormulaAt := real_amplitude_gradient_does_not_supply_bohmian_velocity.2.2 example : ¬wrongMassCoefficientGuidance.SourceFormulaAt := wrong_mass_coefficient_does_not_supply_printed_guidance.2.2 example : ¬wrongSignedSchrodingerCurrent.SourceFormulaAt := reversing_only_the_current_breaks_born_guidance_identity.2 example : ¬linearAmplitudeMistakenForBornDensity.SourceFormulaAt := linear_amplitude_is_not_born_density.2.2 example : currentContinuityWithoutFluxSubstitution.WaveCurrentContinuityFormulaAt /\ ¬currentContinuityWithoutFluxSubstitution.SourceFormulaAt := schrodinger_current_continuity_does_not_supply_guidance_flux_substitution example : ¬otherEnvironmentMistakenForActualConditionalWave.SourceFormulaAt := alternative_environment_wave_is_not_actual_conditional_wave.2.2 example : normalizedLinearAmplitudesMistakenForBornProbabilities.conditionalProbabilityValueAt 0 + normalizedLinearAmplitudesMistakenForBornProbabilities.conditionalProbabilityValueAt 1 = 1 /\ ¬normalizedLinearAmplitudesMistakenForBornProbabilities.SourceFormulaAt := normalized_linear_weights_are_not_conditional_born_probabilities example : overlappingEnvironmentBranchReadout.EffectiveBranchDecompositionFormulaAt /\ overlappingEnvironmentBranchReadout.ConditionalWaveIdentityFormulaAt /\ ¬overlappingEnvironmentBranchReadout.SourceFormulaAt := by have witness := overlapping_environment_branch_does_not_supply_effective_wave_selection exact ⟨witness.1, witness.2.1, witness.2.2.2.2⟩ example {T : QuantumEquilibriumTypes} (framework : QuantumEquilibriumFramework T) (source : framework.dgzGuidedDensityEquivarianceDerivationObtains .guidanceCurrentIdentityAndContinuityEquations) : framework.guidanceCurrentFormulaReadout.schrodingerCurrentValue = framework.guidanceCurrentFormulaReadout.bornDensityValue * framework.guidanceCurrentFormulaReadout.guidanceVelocityValue := framework.source_derivation_yields_actual_born_guidance_current_identity source example {T : QuantumEquilibriumTypes} (framework : QuantumEquilibriumFramework T) (source : framework.dgzGuidedDensityEquivarianceDerivationObtains .guidanceCurrentIdentityAndContinuityEquations) : framework.guidanceCurrentFormulaReadout.waveDensityTimeDerivativeValue + framework.guidanceCurrentFormulaReadout.waveGuidanceFluxDivergenceValue = 0 := framework.source_derivation_yields_actual_wave_guidance_continuity source example {T : QuantumEquilibriumTypes} (context : SemanticContext T) (result : context.interpretAtom .guidedDynamicsResult) : context.framework.GuidanceSourceFormulaAt := context.guided_dynamics_atom_retains_printed_bohmian_velocity result example {T : QuantumEquilibriumTypes} (context : SemanticContext T) (result : context.interpretAtom .fundamentalConditionalProbability) : context.framework.ConditionalProbabilitySourceFormulaAt /\ context.framework.conditionalWaveFormulaReadout.conditionalProbabilityValueAt 0 + context.framework.conditionalWaveFormulaReadout.conditionalProbabilityValueAt 1 = 1 := context.conditional_probability_atom_retains_normalized_born_distribution result end DGZOriginalGuidanceCurrentAndConditionalProbabilityReuseAudit example : PhysicsLogic.Papers.Rovelli1996.RelationalFramework.Countermodels.recordingApparatusObserver.HasPhysicalObserverRole ∧ ¬PhysicsLogic.Papers.Rovelli1996.RelationalFramework.Countermodels.recordingApparatusObserver.IsConsciousObserver := PhysicsLogic.Papers.Rovelli1996.RelationalFramework.Countermodels.a_physical_observer_need_not_be_conscious example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Rovelli1996.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Rovelli1996.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Rovelli1996.RovelliTypes} (framework : PhysicsLogic.Papers.Rovelli1996.RelationalFramework T) (setup : framework.ThirdPersonMeasurementSetup) (calculation : framework.ThirdPersonLinearCalculation) (interpretation : framework.RelativeRecordInterpretation) : framework.toLinearRelativeStateMeasurementContext |>.EstablishedLinearRelativeStateMeasurementAt framework.relativeMeasurementModel := framework.rovelli_third_person_measurement_realizes_shared_relative_state_measurement setup calculation interpretation example : PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetFramework.Countermodels.HasSharpKMarkovIncrementsAt PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetFramework.Countermodels.sharpKMarkovWithCorrelatedGeneralFilter ∧ ¬PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetFramework.Countermodels.HasGeneralFilterMarkovIncrementsAt PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetFramework.Countermodels.sharpKMarkovWithCorrelatedGeneralFilter := PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetFramework.Countermodels.sharp_k_markov_exactness_does_not_transfer_to_general_filters example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.Fidelity.countermodelMatrix := PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetTypes} (framework : PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetFramework T) (established : framework.EstablishedSharpKMassSpectrum) : framework.toSharedThresholdedAbundanceContext |>.EstablishedThresholdedCollapseAbundanceAt framework.sharedThresholdedAbundanceModel := framework.bond_established_sharp_k_result_realizes_shared_thresholded_abundance established example : PhysicsLogic.Papers.Dirac1933.Framework.Countermodels.diracFiniteSmallActionPhaseReadout.HasFiniteSmallStationaryPhaseAt ∧ ¬PhysicsLogic.Papers.Dirac1933.Framework.Countermodels.diracFiniteSmallActionPhaseReadout.HasZeroParameterStationaryActionAt := by have witness := PhysicsLogic.Papers.Dirac1933.Framework.Countermodels.finite_small_planck_approximation_does_not_supply_zero_limit_action_principle exact ⟨witness.2.1, witness.2.2.1⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Dirac1933.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Dirac1933.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Dirac1933.DiracTypes} (framework : PhysicsLogic.Papers.Dirac1933.Framework T) (actionPhase : framework.EstablishedActionPhaseCorrespondence) (composition : framework.EstablishedTransformationComposition) (classical : framework.EstablishedClassicalActionLimit) : framework.selectedActionPhaseCompositionContext |>.EstablishedActionPhaseCompositionAt framework.selectedActionPhaseCompositionModel := framework.dirac_routes_establish_shared_action_phase_composition actionPhase composition classical example : PhysicsLogic.Papers.Bekenstein1973.BlackHoleEntropyFramework.Countermodels.characteristicTemperatureWithoutRadiation.HasCharacteristicTemperature ∧ ¬PhysicsLogic.Papers.Bekenstein1973.BlackHoleEntropyFramework.Countermodels.characteristicTemperatureWithoutRadiation.HasThermalRadiation := PhysicsLogic.Papers.Bekenstein1973.BlackHoleEntropyFramework.Countermodels.a_characteristic_temperature_is_not_a_radiation_temperature example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Bekenstein1973.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Bekenstein1973.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Bekenstein1973.BekensteinTypes} (framework : PhysicsLogic.Papers.Bekenstein1973.BlackHoleEntropyFramework T) (information : framework.InformationResult) (setup : framework.GeneralizedSecondLawSetup) (calculation : framework.InformationCompensationCalculation) (bridge : framework.SharedCompensationBridge) : framework.toGeneralizedEntropyBalanceContext |>.EstablishedGeneralizedEntropyBalanceAt framework.selectedGeneralizedEntropyModel := framework.bekenstein_route_establishes_shared_generalized_entropy_balance information setup calculation bridge example : PhysicsLogic.Papers.ArovasSchriefferWilczek1984.FractionalStatisticsFramework.Countermodels.HasNamedStatisticalPotentialAt PhysicsLogic.Papers.ArovasSchriefferWilczek1984.FractionalStatisticsFramework.Countermodels.wrongStatisticalPotentialFormulaReadout ∧ ¬PhysicsLogic.Papers.ArovasSchriefferWilczek1984.FractionalStatisticsFramework.Countermodels.wrongStatisticalPotentialFormulaReadout.SourceFormulaAt := by have witness := PhysicsLogic.Papers.ArovasSchriefferWilczek1984.FractionalStatisticsFramework.Countermodels.a_named_statistical_potential_does_not_supply_equations_thirteen_and_fourteen exact ⟨witness.1, witness.2.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.ArovasSchriefferWilczek1984.Fidelity.countermodelMatrix := PhysicsLogic.Papers.ArovasSchriefferWilczek1984.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.ArovasSchriefferWilczek1984.ArovasTypes} (framework : PhysicsLogic.Papers.ArovasSchriefferWilczek1984.FractionalStatisticsFramework T) (setup : framework.TwoQuasiholeStatisticsSetup) (calculation : framework.TwoQuasiholePhaseCalculation) (nonintegral : framework.nonintegralFilling framework.filling) (classification : framework.StatisticsClassificationBridge) : framework.toPlanarExchangeStatisticsContext |>.EstablishedPlanarExchangeStatisticsAt framework.planarExchangeStatisticsModel := framework.two_quasihole_route_realizes_shared_planar_exchange_statistics setup calculation nonintegral classification example : PhysicsLogic.Papers.ForsterNelsonStephen1977.EquationThreeNineFormulaReadout.Countermodels.NamedVelocityCorrelationScalingAt PhysicsLogic.Papers.ForsterNelsonStephen1977.EquationThreeNineFormulaReadout.Countermodels.wrongNamedVelocityCorrelationScaling ∧ ¬PhysicsLogic.Papers.ForsterNelsonStephen1977.EquationThreeNineFormulaReadout.Countermodels.wrongNamedVelocityCorrelationScaling.SourceFormulaAt := PhysicsLogic.Papers.ForsterNelsonStephen1977.EquationThreeNineFormulaReadout.Countermodels.named_velocity_scaling_does_not_supply_equation_three_nine example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.ForsterNelsonStephen1977.Fidelity.countermodelMatrix := PhysicsLogic.Papers.ForsterNelsonStephen1977.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.ForsterNelsonStephen1977.FNSTypes} (framework : PhysicsLogic.Papers.ForsterNelsonStephen1977.DynamicRGFramework T) (model : T.Model) (setup : framework.InfraredRGSetup model) (calculation : framework.MomentumShellCalculation model) : (framework.toCoarseGrainedScalingContext model).EstablishedSectorScalingAt model framework.selectedShell framework.infraredHydrodynamicRegime framework.VelocitySector := framework.shell_elimination_and_rescaling_establish_velocity_scaling model setup calculation example : PhysicsLogic.Papers.Zwanzig1973.PaperFramework.Countermodels.FixesInitialBathStatisticsAt PhysicsLogic.Papers.Zwanzig1973.PaperFramework.Countermodels.conditionalCanonicalLaw ∧ ¬PhysicsLogic.Papers.Zwanzig1973.PaperFramework.Countermodels.FixesInitialBathStatisticsAt PhysicsLogic.Papers.Zwanzig1973.PaperFramework.Countermodels.debyeFrequencyLaw := by have witness := PhysicsLogic.Papers.Zwanzig1973.PaperFramework.Countermodels.canonical_initial_data_and_debye_frequency_laws_are_not_identical exact ⟨witness.1, witness.2.2.1⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Zwanzig1973.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Zwanzig1973.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Zwanzig1973.ZwanzigTypes} (framework : PhysicsLogic.Papers.Zwanzig1973.PaperFramework T) (noiseRelation : framework.EstablishedCanonicalNoiseRelation) : framework.selectedMemoryKernelContext.EstablishedExactMemoryKernelReductionAt framework.selectedMemoryKernelModel := framework.zwanzig_routes_establish_shared_exact_memory_kernel_reduction noiseRelation example : PhysicsLogic.Papers.Weinberg1987.AnthropicBoundFramework.Countermodels.HasCalculatedAnthropicCapAt PhysicsLogic.Papers.Weinberg1987.AnthropicBoundFramework.Countermodels.structuredIntrinsicDistributionBelowCap ∧ ¬PhysicsLogic.Papers.Weinberg1987.AnthropicBoundFramework.Countermodels.IsSmoothFeaturelessBelowCapAt PhysicsLogic.Papers.Weinberg1987.AnthropicBoundFramework.Countermodels.structuredIntrinsicDistributionBelowCap := by have witness := PhysicsLogic.Papers.Weinberg1987.AnthropicBoundFramework.Countermodels.an_anthropic_cap_and_planck_scale_do_not_supply_featureless_intrinsic_distribution exact ⟨witness.1, witness.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Weinberg1987.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Weinberg1987.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Weinberg1987.CosmologicalConstantTypes} (framework : PhysicsLogic.Papers.Weinberg1987.AnthropicBoundFramework T) (collapse : framework.EstablishedNonlinearSphericalCollapse) : framework.toSharedCollapseContext.EstablishedCollapseCriterionAt framework.sharedCollapseModel := framework.weinberg_route_realizes_shared_collapse_criterion collapse example : PhysicsLogic.Papers.ReggeWheeler1957.Framework.Countermodels.reggeWheelerLinearWithoutNonlinearControlReadout.HasRegularLinearPacketsAt ∧ ¬PhysicsLogic.Papers.ReggeWheeler1957.Framework.Countermodels.reggeWheelerLinearWithoutNonlinearControlReadout.HasNonlinearStabilityAt := by have witness := PhysicsLogic.Papers.ReggeWheeler1957.Framework.Countermodels.regge_wheeler_realizes_shared_linear_stability_scope_boundary exact ⟨witness.2.1, witness.2.2.2.1⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.ReggeWheeler1957.Fidelity.countermodelMatrix := PhysicsLogic.Papers.ReggeWheeler1957.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.ReggeWheeler1957.ReggeWheelerTypes} (framework : PhysicsLogic.Papers.ReggeWheeler1957.Framework T) (scope : framework.EstablishedScopedStabilityConclusion) : framework.selectedSchwarzschildLinearModeStabilityContext |>.EstablishedScopedSchwarzschildLinearModeStabilityAt framework.selectedSchwarzschildLinearModeStabilityModel := framework.regge_wheeler_routes_establish_shared_linear_mode_stability scope example {T : PhysicsLogic.Papers.ReggeWheeler1957.ReggeWheelerTypes} (framework : PhysicsLogic.Papers.ReggeWheeler1957.Framework T) (scope : framework.EstablishedScopedStabilityConclusion) : framework.selectedOddMasterBoundaryHandoff |>.NonvacuousMasterGroundedScopedSchwarzschildLinearModeStabilityAt := framework.regge_wheeler_master_boundary_route_realizes_composed_scoped_stability scope example {T : PhysicsLogic.Papers.ReggeWheeler1957.ReggeWheelerTypes} (framework : PhysicsLogic.Papers.ReggeWheeler1957.Framework T) (masterSource : framework.ReggeWheelerOddMasterDerivationObtains .tensorHarmonicCanonicalGaugeOddRadialMasterPotential) (scope : framework.EstablishedScopedStabilityConclusion) : framework.selectedOddMasterBoundaryHandoff |>.SourceSensitiveMasterGroundedScopedSchwarzschildLinearModeStabilityAt framework.selectedSchwarzschildMasterEquationContext framework.selectedSchwarzschildLinearModeStabilityContext .independentlyDerived .tensorHarmonicCanonicalGaugeOddRadialMasterPotential framework.ReggeWheelerOddMasterDerivationObtains framework.ReggeWheelerOddMasterDerivationObtains .wormholeMismatchAndEvenBoundaryExtension framework.reggeWheelerGrowingModeExclusionDerivationObtains .originalOddMasterWormholeMismatchAndQualifiedEvenExtension framework.reggeWheelerMasterBoundaryHandoffDerivationObtains := framework |>.original_master_and_wormhole_boundary_realize_source_sensitive_scoped_stability masterSource scope example : PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework.Countermodels.identicalParametersForDistinctWells.SameLowEnergyParameters ∧ ¬PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework.Countermodels.identicalParametersForDistinctWells.SameWellShape := PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework.Countermodels.two_low_energy_parameters_do_not_determine_well_shape example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.BlattJackson1949.Fidelity.countermodelMatrix := PhysicsLogic.Papers.BlattJackson1949.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.BlattJackson1949.BlattJacksonTypes} (framework : PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework T) (law : framework.EstablishedShapeIndependentLaw) (higher : framework.EstablishedHigherOrderShapeScope) : framework.toLowEnergyEffectiveRangeContext |>.EstablishedEffectiveRangeReductionAt framework.effectiveRangeReductionModel := framework.established_local_routes_establish_shared_reduction law higher example : PhysicsLogic.Papers.Anderson1963.PlasmaPoleFormulaReadout.Countermodels.NamedFiniteFrequencyOutputsAt PhysicsLogic.Papers.Anderson1963.PlasmaPoleFormulaReadout.Countermodels.zeroResponseCoefficient ∧ ¬PhysicsLogic.Papers.Anderson1963.PlasmaPoleFormulaReadout.Countermodels.zeroResponseCoefficient.EquationsFourSevenAndEightAt := PhysicsLogic.Papers.Anderson1963.PlasmaPoleFormulaReadout.Countermodels.named_finite_frequency_outputs_do_not_supply_equations_four_seven_and_eight example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Anderson1963.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Anderson1963.AndersonTypes} (framework : PhysicsLogic.Papers.Anderson1963.PlasmonFramework T) (setup : framework.SuperconductingConversionSetup) (calculation : framework.CoulombModeConversionCalculation) : framework.toBrokenSymmetryModeConversionContext.EstablishedModeConversionAt framework.selectedMatterSymmetry framework.superconductingPhase framework.uncoupledGoldstoneMode framework.plasmonMode framework.uncoupledMatterSpectrum framework.interactingPhysicalSpectrum framework.electromagneticField framework.superconductor := framework.superconducting_conversion_establishes_shared_mode_conversion setup calculation example : PhysicsLogic.Papers.ThoulessTransport1983.ParticleTransportFramework.Countermodels.finiteTorusWithHalfUnitTransport.HasFiniteTorusAdiabaticCycle ∧ ¬PhysicsLogic.Papers.ThoulessTransport1983.ParticleTransportFramework.Countermodels.finiteTorusWithHalfUnitTransport.HasIntegerTransport := PhysicsLogic.Papers.ThoulessTransport1983.ParticleTransportFramework.Countermodels.finite_torus_adiabatic_transport_need_not_be_integer example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.ThoulessTransport1983.Fidelity.countermodelMatrix := PhysicsLogic.Papers.ThoulessTransport1983.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.ThoulessTransport1983.ThoulessTypes} (framework : PhysicsLogic.Papers.ThoulessTransport1983.ParticleTransportFramework T) (setup : framework.InfiniteFilledBandSetup) (following : framework.BandGapAdiabaticFollowingCalculation) (transport : framework.FilledBandIntegratedTransportCalculation) (index : framework.ChernIntegerIndexCalculation) (correspondence : framework.ParticleIndexCorrespondenceCalculation) : framework.toAdiabaticCycleTransportContext |>.EstablishedQuantizedAdiabaticCycleTransportAt framework.sharedTransportModel := framework.infinite_filled_band_cycle_establishes_quantized_particle_transport setup following transport index correspondence example : PhysicsLogic.Papers.Knox1995.CMBInferenceFramework.Countermodels.noiseFreeDistinctSkyEstimates.NoiseFreePossibleSkies ∧ ¬PhysicsLogic.Papers.Knox1995.CMBInferenceFramework.Countermodels.noiseFreeDistinctSkyEstimates.ExactRecoveryOnBothSkies := by have witness := PhysicsLogic.Papers.Knox1995.CMBInferenceFramework.Countermodels.noise_free_skies_need_not_identify_ensemble_target exact ⟨witness.1, witness.2.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Knox1995.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Knox1995.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Knox1995.CMBTypes} (framework : PhysicsLogic.Papers.Knox1995.CMBInferenceFramework T) (experiment : T.Experiment) (setup : framework.SingleSkyEstimatorSetup experiment) (calculation : framework.EstimatorVarianceCalculation experiment) : (framework.selectedSamplingVarianceContext experiment) |>.EstablishedSamplingVarianceAt (framework.samplingVarianceModel experiment) := framework.knox_estimator_variance_realizes_shared_route experiment setup calculation example {T : PhysicsLogic.Papers.Knox1995.CMBTypes} (framework : PhysicsLogic.Papers.Knox1995.CMBInferenceFramework T) (experiment : T.Experiment) (setup : framework.SingleSkyEstimatorSetup experiment) : framework.skyMultipoleAvailabilityContext |>.NonvacuousAvailableSectorAt framework.observedSky framework.selectedModel framework.selectedMultipoleSector := framework.observed_sky_realizes_nonvacuous_available_multipole_sector experiment setup example {T : PhysicsLogic.Papers.Knox1995.CMBTypes} (framework : PhysicsLogic.Papers.Knox1995.CMBInferenceFramework T) (experiment : T.Experiment) (setup : framework.SingleSkyEstimatorSetup experiment) : (framework.samplingVarianceModel experiment) |>.toObservableAvailabilityContext |>.NonvacuousAvailableSectorAt framework.observedSky framework.selectedModel (framework.samplingVarianceModel experiment).selectedObservableSector := (framework.samplingVarianceModel experiment) |>.single_realization_setup_realizes_nonvacuous_available_sector (framework.single_sky_setup_realizes_shared_sampling_variance_setup experiment setup).2 example : PhysicsLogic.Papers.RyuTakayanagi2006.EntanglementFramework.Countermodels.hasNamedStaticAreaProposal PhysicsLogic.Papers.RyuTakayanagi2006.EntanglementFramework.Countermodels.wrongPrintedAreaEntropyModel ∧ ¬PhysicsLogic.Papers.RyuTakayanagi2006.EntanglementFramework.Countermodels.realizesPrintedAreaEntropyFormula PhysicsLogic.Papers.RyuTakayanagi2006.EntanglementFramework.Countermodels.wrongPrintedAreaEntropyModel := PhysicsLogic.Papers.RyuTakayanagi2006.EntanglementFramework.Countermodels.named_static_area_proposal_does_not_supply_area_over_four_g example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.RyuTakayanagi2006.Fidelity.countermodelMatrix := PhysicsLogic.Papers.RyuTakayanagi2006.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.RyuTakayanagi2006.RTTypes} (framework : PhysicsLogic.Papers.RyuTakayanagi2006.EntanglementFramework T) (proposal : framework.StaticMinimalAreaProposal) {region : T.Region} {boundary : T.RegionBoundary} {surface : T.Surface} {regime : PhysicsLogic.Papers.RyuTakayanagi2006.Regime} {areaValue : T.AreaValue} {entropyValue : T.EntropyValue} (setup : framework.HolographicSetupAt region boundary surface regime) (areaCalculation : framework.SelectedAreaCalculation surface areaValue) (normalization : framework.SelectedAreaNormalization areaValue entropyValue) : (framework.toHolographicRegionEntropyContext region boundary surface regime areaValue entropyValue) |>.EstablishedHolographicRegionEntropyAt (framework.holographicEntropyModelAt region boundary surface regime areaValue entropyValue) := framework.rt_area_route_realizes_shared_holographic_region_entropy proposal setup areaCalculation normalization example : PhysicsLogic.Papers.EvansCohenMorriss1993.Framework.Countermodels.realizesOrdinaryTimeReverse PhysicsLogic.Papers.EvansCohenMorriss1993.Framework.Countermodels.finiteFixedShearReversalModel ∧ ¬PhysicsLogic.Papers.EvansCohenMorriss1993.Framework.Countermodels.realizesFixedShearOppositeStressPartner PhysicsLogic.Papers.EvansCohenMorriss1993.Framework.Countermodels.finiteFixedShearReversalModel PhysicsLogic.Papers.EvansCohenMorriss1993.Framework.Countermodels.finiteFixedShearReversalModel.ordinaryReversedShearRate PhysicsLogic.Papers.EvansCohenMorriss1993.Framework.Countermodels.finiteFixedShearReversalModel.ordinaryReversedStress := by have witness := PhysicsLogic.Papers.EvansCohenMorriss1993.Framework.Countermodels.ordinary_time_reverse_does_not_supply_fixed_shear_opposite_stress_pair exact ⟨witness.1, witness.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.EvansCohenMorriss1993.Fidelity.countermodelMatrix := PhysicsLogic.Papers.EvansCohenMorriss1993.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.EvansCohenMorriss1993.EvansCohenMorrissTypes} (framework : PhysicsLogic.Papers.EvansCohenMorriss1993.Framework T) (measure : framework.EstablishedSegmentMeasureProposal) (pairing : framework.EstablishedFixedShearPair) (ratio : framework.EstablishedProbabilityRatio) (secondLaw : framework.EstablishedSecondLawAssessment) (scope : framework.EstablishedScopeAssessment) : framework.selectedEntropyProductionFluctuationContext |>.EstablishedEntropyProductionFluctuationSymmetryAt framework.selectedEntropyProductionFluctuationModel := framework.ecm_routes_establish_shared_entropy_production_fluctuation_symmetry measure pairing ratio secondLaw scope example : PhysicsLogic.Papers.FuKane2009.QSHIJunctionFramework.Countermodels.localHamiltonianIsTwoPiPeriodic PhysicsLogic.Papers.FuKane2009.QSHIJunctionFramework.Countermodels.localReturnWithFractionalBranch ∧ ¬PhysicsLogic.Papers.FuKane2009.QSHIJunctionFramework.Countermodels.currentIsTwoPiPeriodic PhysicsLogic.Papers.FuKane2009.QSHIJunctionFramework.Countermodels.localReturnWithFractionalBranch := PhysicsLogic.Papers.FuKane2009.QSHIJunctionFramework.Countermodels.a_two_pi_periodic_local_hamiltonian_does_not_make_the_branch_current_two_pi_periodic example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.FuKane2009.Fidelity.countermodelMatrix := PhysicsLogic.Papers.FuKane2009.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.FuKane2009.FuKaneTypes} (framework : PhysicsLogic.Papers.FuKane2009.QSHIJunctionFramework T) (result : framework.EstablishedMajoranaInterface) : framework.toBoundaryMajoranaModeContext |>.EstablishedBoundaryMajoranaModeAt framework.boundaryMajoranaModel := framework.fu_kane_interface_establishes_shared_boundary_majorana result namespace FuKaneOriginalAndreevSwitchingNoiseReuseAudit open PhysicsLogic.Papers.FuKane2009 open SourceFormulaCountermodels example : physicalTransparentShortJunctionReadout.SourceFormulaAt /\ physicalTransparentShortJunctionReadout.transmissionProbabilityValue = 1 /\ physicalTransparentShortJunctionReadout.andreevEnergyValueAt 0 = 2 /\ physicalTransparentShortJunctionReadout.andreevEnergyValueAt Real.pi = 0 /\ physicalTransparentShortJunctionReadout.positiveBranchCurrentValueAt Real.pi = 1 := transparent_short_junction_realizes_original_equations_four_to_six example : physicalMagneticBarrierShortJunctionReadout.SourceFormulaAt /\ 0 < physicalMagneticBarrierShortJunctionReadout.transmissionProbabilityValue /\ physicalMagneticBarrierShortJunctionReadout.transmissionProbabilityValue < 1 /\ physicalMagneticBarrierShortJunctionReadout.andreevEnergyValueAt Real.pi = 0 /\ 0 < physicalMagneticBarrierShortJunctionReadout.positiveBranchCurrentValueAt Real.pi := magnetic_barrier_realizes_original_subunit_transmission_and_crossing example : physicalThermalQuasiparticleNoiseReadout.SourceFormulaAt := thermal_quasiparticle_source_realizes_original_rates_and_noise.1 example : physicalVariableRangeHoppingNoiseReadout.SourceFormulaAt := variable_range_hopping_source_realizes_distinct_activation_and_noise.1 example : physicalVariableRangeHoppingNoiseReadout.symmetricBathWeightValue = physicalThermalQuasiparticleNoiseReadout.symmetricBathWeightValue / 2 /\ physicalVariableRangeHoppingNoiseReadout.zeroFrequencyNoiseValue = 2 * physicalThermalQuasiparticleNoiseReadout.zeroFrequencyNoiseValue := slower_hopping_switching_doubles_zero_frequency_noise example : (∀ phase, physicalThermalQuasiparticleNoiseReadout.equilibriumCurrentValueAt (phase + 2 * Real.pi) = physicalThermalQuasiparticleNoiseReadout.equilibriumCurrentValueAt phase) /\ physicalTransparentShortJunctionReadout.positiveBranchCurrentValueAt (Real.pi + 2 * Real.pi) ≠ physicalTransparentShortJunctionReadout.positiveBranchCurrentValueAt Real.pi /\ physicalThermalQuasiparticleNoiseReadout.equilibriumCurrentValueAt Real.pi = 0 /\ physicalTransparentShortJunctionReadout.positiveBranchCurrentValueAt Real.pi = 1 := thermal_equilibrium_has_two_pi_period_while_fixed_parity_has_four_pi example : wrongFullPhaseAndreevSpectrum.andreevEnergyValueAt 0 = 2 /\ ¬wrongFullPhaseAndreevSpectrum.SpectrumSourceAt := full_phase_spectrum_matches_zero_phase_but_not_original_equation_four example : wrongMissingHalfCurrentFactor.SpectrumSourceAt /\ ¬wrongMissingHalfCurrentFactor.SourceFormulaAt := correct_spectrum_does_not_allow_missing_half_josephson_current_factor example : wrongUnshiftedOccupationHamiltonian.EquationFourAndreevEnergyAt /\ ¬wrongUnshiftedOccupationHamiltonian.SpectrumSourceAt := correct_andreev_energy_does_not_allow_unshifted_occupation_hamiltonian example : avoidedCrossingHalfPhaseReadout.andreevEnergyValueAt Real.pi = 1 /\ ¬avoidedCrossingHalfPhaseReadout.EquationFourAndreevEnergyAt := avoided_crossing_is_not_the_protected_original_andreev_spectrum example : wrongBarrierTransparencyReadout.BarrierDecayEquationAt /\ ¬wrongBarrierTransparencyReadout.TransmissionSourceAt := nonzero_magnetic_barrier_does_not_admit_perfect_transmission example : 0 < zeroTemperatureNoiseReadout.symmetricBathWeightValue /\ ¬zeroTemperatureNoiseReadout.SourceFormulaAt := positive_bath_weight_does_not_make_zero_temperature_switching_valid example : unsymmetricBathMistakenForSymmetricNoise.firstBathWeightValue ≠ unsymmetricBathMistakenForSymmetricNoise.secondBathWeightValue /\ ¬unsymmetricBathMistakenForSymmetricNoise.SourceFormulaAt := unequal_bath_weights_do_not_supply_the_symmetric_source_formula example : fixedBranchMistakenForThermalEquilibrium.equilibriumCurrentValueAt Real.pi = 1 /\ ¬fixedBranchMistakenForThermalEquilibrium.SourceFormulaAt := protected_branch_current_is_not_the_thermal_equilibrium_current example : flatSpectrumMistakenForLorentzianNoise.ZeroFrequencyReadoutAt /\ ¬flatSpectrumMistakenForLorentzianNoise.SourceFormulaAt := correct_zero_frequency_readout_does_not_allow_flat_noise_spectrum example : wrongZeroFrequencyNoiseReadout.EquationTwelveNoiseSpectrumAt /\ ¬wrongZeroFrequencyNoiseReadout.SourceFormulaAt := lorentzian_spectrum_does_not_allow_wrong_zero_frequency_readout example : unsuppressedQuasiparticleBathReadout.FormulaDomainAt /\ ¬unsuppressedQuasiparticleBathReadout.SourceFormulaAt := unsuppressed_bath_does_not_satisfy_quasiparticle_activation example {T : FuKaneTypes} (framework : QSHIJunctionFramework T) (result : framework.EstablishedJunctionSpectrum) : framework.andreevEnergyValueAt framework.andreevSpectrum Real.pi = 0 := framework.established_junction_spectrum_derives_actual_zero_energy_crossing result example {T : FuKaneTypes} (framework : QSHIJunctionFramework T) (result : framework.EstablishedFractionalJosephsonResponse) : (∀ phase, framework.branchCurrentValueAt framework.positiveBranchCurrent (phase + 4 * Real.pi) = framework.branchCurrentValueAt framework.positiveBranchCurrent phase) /\ framework.branchCurrentValueAt framework.positiveBranchCurrent (Real.pi + 2 * Real.pi) ≠ framework.branchCurrentValueAt framework.positiveBranchCurrent Real.pi := framework.established_fractional_response_derives_actual_four_pi_current result example {T : FuKaneTypes} (framework : QSHIJunctionFramework T) (result : framework.EstablishedParitySwitchingNoise) : framework.SelectedThermalSwitchingRateSourceFormulaAt /\ framework.SelectedThermalSwitchingNoiseSourceFormulaAt := framework.established_switching_noise_retains_printed_rate_and_noise_formulas result example {T : FuKaneTypes} (framework : QSHIJunctionFramework T) (result : framework.CompleteScopedResult) : framework.SelectedShortJunctionSpectrumSourceFormulaAt /\ framework.SelectedJosephsonCurrentSourceFormulaAt /\ framework.SelectedThermalSwitchingRateSourceFormulaAt /\ framework.SelectedThermalSwitchingNoiseSourceFormulaAt := framework.complete_result_retains_original_spectrum_current_and_noise_formulas result end FuKaneOriginalAndreevSwitchingNoiseReuseAudit example : PhysicsLogic.Papers.HastingsKoma2006.PaperFramework.Countermodels.correctedBosonicClustering PhysicsLogic.Papers.HastingsKoma2006.PaperFramework.Countermodels.correctedClusteringWithNondecayingGroundSector ∧ ¬PhysicsLogic.Papers.HastingsKoma2006.PaperFramework.Countermodels.rawBosonicCorrelationDecays PhysicsLogic.Papers.HastingsKoma2006.PaperFramework.Countermodels.correctedClusteringWithNondecayingGroundSector := PhysicsLogic.Papers.HastingsKoma2006.PaperFramework.Countermodels.corrected_bosonic_clustering_does_not_supply_raw_decay example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.HastingsKoma2006.Fidelity.countermodelMatrix := PhysicsLogic.Papers.HastingsKoma2006.Fidelity.countermodel_matrix_is_realized example (framework : PhysicsLogic.Papers.HastingsKoma2006.PaperFramework) (clustering : framework.EstablishedBosonicCorrectedClustering) : framework.toSpectralGapClusteringContext |>.EstablishedSpectralGapClusteringAt framework.selectedSpectralGapClusteringModel := framework.bosonic_clustering_establishes_shared_spectral_gap_clustering clustering example : PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.SelectionRuleCountermodels.compactChargedLoopWithoutWardIdentity.CompactChargedLoopSetup ∧ ¬PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.SelectionRuleCountermodels.compactChargedLoopWithoutWardIdentity.ChargedLoopExpectationVanishes := by have witness := PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.SelectionRuleCountermodels.compact_nontrivially_charged_loop_need_not_have_zero_expectation exact ⟨witness.1, witness.2.2.1⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.Fidelity.countermodelMatrix := PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.Fidelity.countermodel_matrix_is_realized example {TwoForm ThreeForm Loop Photon Background : Type} (framework : PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.MaxwellFramework TwoForm ThreeForm Loop Photon Background) {loop : Loop} (line : framework.loopSupportDimension loop = 1 ∧ framework.loopIsGenuine loop) : (framework.lineChargeContext loop).EstablishedFunctionalChargeAssignmentAt (framework.lineChargeModel loop) := framework.genuine_line_establishes_charge_assignment line example : PhysicsLogic.Papers.Dirac1950.ConstrainedHamiltonianFramework.Countermodels.diracFirstClassOrbitWithoutIdentificationBoundary.HasCompletedPreservedConstraintAt ∧ ¬PhysicsLogic.Papers.Dirac1950.ConstrainedHamiltonianFramework.Countermodels.diracFirstClassOrbitWithoutIdentificationBoundary.HasPhysicalGaugeIdentificationAt := by have witness := PhysicsLogic.Papers.Dirac1950.ConstrainedHamiltonianFramework.Countermodels.dirac_realizes_shared_constraint_gauge_boundary exact ⟨witness.1, witness.2.2.2.1⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Dirac1950.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Dirac1950.Fidelity.countermodel_matrix_is_realized example {System PhasePoint Constraint Multiplier Observable Value PhysicalState QuantumState Operator : Type} (framework : PhysicsLogic.Papers.Dirac1950.ConstrainedHamiltonianFramework System PhasePoint Constraint Multiplier Observable Value PhysicalState QuantumState Operator) (system : System) (primaryConstraint : Constraint) (constraintLinear : framework.totalHamiltonianLinearInConstraints system) (primary : framework.primaryConstraint system primaryConstraint) (setup : framework.ConsistencySetup system) (calculation : framework.ConsistencyClassificationCalculation system) (realization : framework.ConstraintCompletionRealizationCalculation system) : framework.toConstraintConsistencyCompletionContext system primaryConstraint |>.EstablishedConstraintConsistencyAt (framework.sharedConstraintConsistencyModel system primaryConstraint) := framework.dirac_consistency_route_realizes_shared_constraint_completion system primaryConstraint constraintLinear primary setup calculation realization example : PhysicsLogic.Papers.KaplanSavageWise1998.NuclearScatteringFramework.Countermodels.kswPostCalibrationPredictionModel.MatchesEmpiricalBoundaryAt PhysicsLogic.Papers.KaplanSavageWise1998.NuclearScatteringFramework.Countermodels.kswPostCalibrationEmpiricalReadout ∧ ¬PhysicsLogic.Papers.KaplanSavageWise1998.NuclearScatteringFramework.Countermodels.kswPostCalibrationEmpiricalReadout.HasEmpiricalAgreementAt := by have witness := PhysicsLogic.Papers.KaplanSavageWise1998.NuclearScatteringFramework.Countermodels.ksw_realizes_shared_post_calibration_empirical_boundary exact ⟨witness.1, witness.2.2.1⟩ namespace RenormalizationGroupPhysicalReuseAudit open PhysicsLogic.Paradigm.RenormalizationGroup.Countermodels example : runningButInvariant.effectiveAt .infrared = .infraredTheory /\ runningButInvariant.effectiveAt .ultraviolet = .ultravioletTheory /\ runningButInvariant.parameterAt .infrared = .infraredCoupling /\ runningButInvariant.parameterAt .ultraviolet = .ultravioletCoupling /\ runningButInvariant.targetPredicts () .protectedAmplitude .physicalValue /\ Not (runningButInvariant.targetPredicts () .protectedAmplitude .alternativeValue) /\ Not (runningButInvariant.observableIn .excludedAmplitude ()) := running_context_distinguishes_scales_couplings_and_predictions example : predictionlessAgreement.targetPredicts () .excludedAmplitude .alternativeValue /\ Not (predictionlessAgreement.TargetPredictionActiveAt () () (fun _ => True)) := excluded_prediction_does_not_activate_selected_scale_sector example : singleScaleAgreement.TargetPredictionActiveAt () () (fun _ => True) /\ Not (singleScaleAgreement.ActiveScaleComparisonAt () (fun _ => True)) := single_scale_has_prediction_but_no_second_comparison end RenormalizationGroupPhysicalReuseAudit example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.KaplanSavageWise1998.Fidelity.countermodelMatrix := PhysicsLogic.Papers.KaplanSavageWise1998.Fidelity.countermodel_matrix_is_realized example {Channel Scale Momentum Coupling Term Order Regime Fit : Type} (framework : PhysicsLogic.Papers.KaplanSavageWise1998.NuclearScatteringFramework Channel Scale Momentum Coupling Term Order Regime Fit) (channel : Channel) (momentum : Momentum) (rgControl : framework.pionInclusiveRGControlledCouplings) (parameterVariation : framework.SampledPionParameterVariation) (scaleIndependent : framework.fullAmplitudeScaleIndependent channel momentum) : framework.toRenormalizedPredictionContext channel momentum |>.EstablishedRenormalizedPredictionAt (framework.pionInclusiveRenormalizationModel channel momentum) := framework.pion_inclusive_renormalization_realizes_shared_prediction channel momentum rgControl parameterVariation scaleIndependent example {Channel Scale Momentum Coupling Term Order Regime Fit : Type} (framework : PhysicsLogic.Papers.KaplanSavageWise1998.NuclearScatteringFramework Channel Scale Momentum Coupling Term Order Regime Fit) (channel : Channel) (scale : Scale) (momentum : Momentum) (pionless : framework.pionlessMomentum momentum) (largeLength : framework.largeScatteringLength channel) (pds : framework.pdsSubtractionAt scale) (bubbleOrders : framework.pionlessLeadingContactOrderAt channel scale momentum = -1 ∧ framework.pionlessBubbleLoopOrderAt scale momentum = 1) (loopCalculation : framework.PDSLoopCalculation channel scale momentum) (resummationCalculation : framework.BubbleResummationCalculation channel scale momentum) (unitarityCalculation : framework.PionlessUnitarityCalculation channel scale momentum) (mixedExpansion : framework.KSWMixedExpansionAt channel scale momentum) (expansion : framework.amplitudeHasKSWExpansion channel scale momentum) (rgControl : framework.pionInclusiveRGControlledCouplings) (parameterVariation : framework.SampledPionParameterVariation) (scaleIndependent : framework.fullAmplitudeScaleIndependent channel momentum) : framework.selectedResummedRenormalizationHandoff channel scale momentum |>.NonvacuousResummationGroundedRenormalizedPredictionAt := framework.ksw_resummed_mixed_expansion_realizes_composed_renormalized_prediction channel scale momentum pionless largeLength pds bubbleOrders loopCalculation resummationCalculation unitarityCalculation mixedExpansion expansion rgControl parameterVariation scaleIndependent example {Channel Scale Momentum Coupling Term Order Regime Fit : Type} (framework : PhysicsLogic.Papers.KaplanSavageWise1998.NuclearScatteringFramework Channel Scale Momentum Coupling Term Order Regime Fit) (channel : Channel) (scale : Scale) (momentum : Momentum) (pionless : framework.pionlessMomentum momentum) (largeLength : framework.largeScatteringLength channel) (pds : framework.pdsSubtractionAt scale) (bubbleOrders : framework.pionlessLeadingContactOrderAt channel scale momentum = -1 ∧ framework.pionlessBubbleLoopOrderAt scale momentum = 1) (loopCalculation : framework.PDSLoopCalculation channel scale momentum) (resummationCalculation : framework.BubbleResummationCalculation channel scale momentum) (unitarityCalculation : framework.PionlessUnitarityCalculation channel scale momentum) (mixedExpansion : framework.KSWMixedExpansionAt channel scale momentum) (expansion : framework.amplitudeHasKSWExpansion channel scale momentum) (rgControl : framework.pionInclusiveRGControlledCouplings) (parameterVariation : framework.SampledPionParameterVariation) (scaleIndependent : framework.fullAmplitudeScaleIndependent channel momentum) : framework.selectedResummedRenormalizationHandoff channel scale momentum |>.SourceSensitiveResummationGroundedRenormalizedPredictionAt (framework.toLargeScatteringLengthResummationContext channel scale momentum) (framework.toRenormalizedPredictionContext channel momentum) .pdsLoopBubbleChainAndUnitarityRecovery (framework.KSWLargeLengthResummationDerivationObtains channel scale momentum) .pionInclusiveRunningCouplingsAndScaleIndependentFullAmplitude (framework.kswRenormalizedPredictionDerivationObtains channel momentum) .pionlessPdsBubbleAmplitudeFeedsPionInclusiveMixedExpansion (framework.kswResummedPredictionHandoffDerivationObtains channel scale momentum) := framework |>.ksw_resummed_mixed_expansion_realizes_source_sensitive_renormalized_prediction channel scale momentum pionless largeLength pds bubbleOrders loopCalculation resummationCalculation unitarityCalculation mixedExpansion expansion rgControl parameterVariation scaleIndependent namespace NuclearResummationPredictionSourceReuseAudit open PhysicsLogic.Framework.ResummedSectorRenormalizedPredictionHandoff.Countermodels example : distinctSystemHandoff.resummationModel.system ≠ distinctSystemHandoff.predictionModel.system ∧ distinctSystemHandoff.resummationModel.lowEnergyTarget ≠ distinctSystemHandoff.predictionModel.target ∧ ¬(distinctSystemHandoff |>.SourceSensitiveResummationGroundedRenormalizedPredictionAt distinctTwoBodyResummationContext PhysicsLogic.Framework.RenormalizedPredictionContext.Countermodels.directPredictionContext () availableResummedPredictionChildDerivationObtains () availableResummedPredictionChildDerivationObtains .absent missingResummedPredictionHandoffDerivationObtains) := by have witness := completed_resummed_prediction_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition exact ⟨witness.2.1, witness.2.2.2.1, witness.2.2.2.2.2.2⟩ end NuclearResummationPredictionSourceReuseAudit example : PhysicsLogic.Papers.THooft1974.PlanarGaugeFramework.Countermodels.HasPlanarLeadingOrderAt PhysicsLogic.Papers.THooft1974.PlanarGaugeFramework.Countermodels.completeReducedPlanarGaugeReadout ∧ ¬PhysicsLogic.Papers.THooft1974.PlanarGaugeFramework.Countermodels.HasFiniteRankLeadingExactnessAt PhysicsLogic.Papers.THooft1974.PlanarGaugeFramework.Countermodels.completeReducedPlanarGaugeReadout := by have witness := PhysicsLogic.Papers.THooft1974.PlanarGaugeFramework.Countermodels.large_rank_leading_order_need_not_be_exact_at_finite_rank exact ⟨witness.1, witness.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.THooft1974.Fidelity.countermodelMatrix := PhysicsLogic.Papers.THooft1974.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.THooft1974.THooftTypes} (framework : PhysicsLogic.Papers.THooft1974.PlanarGaugeFramework T) (setup : framework.GaugeInvariantCountingSetup framework.selectedSource framework.selectedPlanarDiagram) (indexCalculation : framework.IndexLineCalculation framework.selectedSource framework.selectedPlanarDiagram) (surface : framework.diagramHasIndexSurface framework.selectedPlanarDiagram) (normalized : framework.verticesNormalizedAtFixedRescaledCoupling framework.selectedPlanarDiagram) (observableIn : framework.observableIn framework.hadronicObservable framework.selectedRegime) (contributesAtEveryRank : ∀ rank, framework.diagramContributesTo framework.selectedPlanarDiagram (framework.rankIndexedTheory rank) framework.hadronicObservable) (trajectory : framework.THooftLargeRankTrajectory) (classification : framework.TopologicalClassification) (calculation : framework.PlanarDominanceCalculation) (nonplanarWitness : framework.NonleadingHadronicContributionWitness) (additionalBoundaryWitness : framework.AdditionalBoundaryHadronicContributionWitness) (planar : framework.planarDiagram framework.selectedPlanarDiagram) (oneBoundary : framework.quarkLoopCount framework.selectedPlanarDiagram = 1) (boundary : framework.quarksLieOnBoundary framework.selectedPlanarDiagram) : framework.toPlanarDiagramExpansionContext |>.EstablishedPlanarLargeRankSelectionAt framework.planarDiagramModel := framework.thooft_planar_route_realizes_shared_planar_selection setup indexCalculation surface normalized observableIn contributesAtEveryRank trajectory classification calculation nonplanarWitness additionalBoundaryWitness planar oneBoundary boundary example : PhysicsLogic.Papers.LaughlinHall1981.HallTransportFramework.Countermodels.mobilityGapFollowingWithHalfUnitTransfer.HasMobilityGapAndAdiabaticFollowing ∧ ¬PhysicsLogic.Papers.LaughlinHall1981.HallTransportFramework.Countermodels.mobilityGapFollowingWithHalfUnitTransfer.HasIntegerChargeTransfer := PhysicsLogic.Papers.LaughlinHall1981.HallTransportFramework.Countermodels.mobility_gap_setup_and_following_do_not_supply_integer_transfer example : PhysicsLogic.Papers.LaughlinHall1981.HallResistanceCorrectionFormulaReadout.Countermodels.originalEquationTwelveReadout.SourceFormulaAt ∧ PhysicsLogic.Papers.LaughlinHall1981.HallResistanceCorrectionFormulaReadout.Countermodels.originalEquationTwelveReadout.hallResistanceChangeValue ≠ 0 := PhysicsLogic.Papers.LaughlinHall1981.HallResistanceCorrectionFormulaReadout.Countermodels.original_equation_twelve_has_a_physical_nonzero_witness example : PhysicsLogic.Papers.LaughlinHall1981.HallTransportFramework.Countermodels.mobilityGapWithLocalizedStatesAtFermi.HasExtendedStateMobilityGap ∧ ¬PhysicsLogic.Papers.LaughlinHall1981.HallTransportFramework.Countermodels.mobilityGapWithLocalizedStatesAtFermi.HasDensityOfStatesGap := PhysicsLogic.Papers.LaughlinHall1981.HallTransportFramework.Countermodels.mobility_gap_does_not_require_a_density_of_states_gap example {T : PhysicsLogic.Papers.LaughlinHall1981.LaughlinTypes} (framework : PhysicsLogic.Papers.LaughlinHall1981.HallTransportFramework T) (assessment : framework.QualifiedErrorAssessment) : framework.selectedHallResistanceCorrectionFormulaReadout.EquationTwelveAt := (framework.qualified_error_assessment_retains_printed_signed_resistance_law assessment).1 example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.LaughlinHall1981.Fidelity.countermodelMatrix := PhysicsLogic.Papers.LaughlinHall1981.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.LaughlinHall1981.LaughlinTypes} (framework : PhysicsLogic.Papers.LaughlinHall1981.HallTransportFramework T) (setup : framework.LoopMobilityGapSetup) (following : framework.MobilityGapAdiabaticFollowingCalculation) (transport : framework.FluxIntegratedChargeTransferCalculation) (index : framework.GaugeReturnIntegerTransferCalculation) (correspondence : framework.ChargeIndexCorrespondenceCalculation) : framework.toAdiabaticCycleTransportContext |>.EstablishedQuantizedAdiabaticCycleTransportAt framework.sharedTransportModel := framework.dirty_interacting_flux_cycle_establishes_quantized_charge_transfer setup following transport index correspondence example : PhysicsLogic.Papers.Creutz1980.MonteCarloGaugeFramework.Countermodels.perturbativeScalingAgreement PhysicsLogic.Papers.Creutz1980.MonteCarloGaugeFramework.Countermodels.scalingAgreementWithoutContinuumConfinement ∧ ¬PhysicsLogic.Papers.Creutz1980.MonteCarloGaugeFramework.Countermodels.continuumConfinement PhysicsLogic.Papers.Creutz1980.MonteCarloGaugeFramework.Countermodels.scalingAgreementWithoutContinuumConfinement := PhysicsLogic.Papers.Creutz1980.MonteCarloGaugeFramework.Countermodels.perturbative_scaling_agreement_does_not_prove_continuum_confinement example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Creutz1980.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Creutz1980.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Creutz1980.CreutzTypes} (framework : PhysicsLogic.Papers.Creutz1980.MonteCarloGaugeFramework T) (latticeSetup : framework.LatticeMonteCarloSetup) (regulatedCalculation : framework.RegulatedGaugeCalculation) (setup : framework.LoopTensionSetup) (reliability : framework.ReliableLoopData) (calculation : framework.LoopFitCalculation) (potentialBridge : framework.LinearPotentialInterpretation) (confinementBridge : framework.ConfinementInterpretation) : framework.toLoopAreaConfinementContext |>.EstablishedLoopAreaConfinementAt framework.latticeDescription framework.finiteCutoffRegime framework.selectedLoop framework.selectedSurface framework.selectedSources framework.selectedStringTension := framework.monte_carlo_route_realizes_shared_loop_area_confinement latticeSetup regulatedCalculation setup reliability calculation potentialBridge confinementBridge example : PhysicsLogic.Papers.NachtergaeleSims2006.PaperFramework.Countermodels.exponentialClustering PhysicsLogic.Papers.NachtergaeleSims2006.PaperFramework.Countermodels.exponentiallyClusteredGaplessSystem ∧ ¬PhysicsLogic.Papers.NachtergaeleSims2006.PaperFramework.Countermodels.hasSpectralGap PhysicsLogic.Papers.NachtergaeleSims2006.PaperFramework.Countermodels.exponentiallyClusteredGaplessSystem := PhysicsLogic.Papers.NachtergaeleSims2006.PaperFramework.Countermodels.exponential_clustering_does_not_imply_spectral_gap example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.NachtergaeleSims2006.Fidelity.countermodelMatrix := PhysicsLogic.Papers.NachtergaeleSims2006.Fidelity.countermodel_matrix_is_realized example (framework : PhysicsLogic.Papers.NachtergaeleSims2006.PaperFramework) (cone : framework.EstablishedFiniteVelocityCone) : framework.toFiniteVelocityCommutatorConeContext |>.EstablishedFiniteVelocityCommutatorConeAt framework.finiteVelocityCommutatorModel := framework.finite_velocity_cone_establishes_shared_commutator_cone cone example : PhysicsLogic.Papers.ScottSrednickiWhite1993.SampleVarianceFramework.Countermodels.ensemblePredictionWithoutRealizedValue.NoiseFreePossibleEstimates ∧ ¬PhysicsLogic.Papers.ScottSrednickiWhite1993.SampleVarianceFramework.Countermodels.ensemblePredictionWithoutRealizedValue.ExactRecoveryOnBothSkies := by have witness := PhysicsLogic.Papers.ScottSrednickiWhite1993.SampleVarianceFramework.Countermodels.ensemble_target_does_not_identify_each_possible_sky_estimate exact ⟨witness.1, witness.2.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.ScottSrednickiWhite1993.Fidelity.countermodelMatrix := PhysicsLogic.Papers.ScottSrednickiWhite1993.Fidelity.countermodel_matrix_is_realized example (readout : PhysicsLogic.Papers.ScottSrednickiWhite1993.CircularGaussianVarianceFormulaReadout) (formula : readout.SourceFormulaAt) : readout.largeAreaApproximateVarianceValue = (4 * Real.pi / readout.capSolidAngleValue) * readout.fullSkyCosmicVarianceValue := readout.large_area_limit_is_four_pi_over_area_times_cosmic_variance formula example (readout : PhysicsLogic.Papers.ScottSrednickiWhite1993.HistoricalScanExperimentReadout) (values : readout.SourceValuesAt) : readout.maxLinearizedTemperatureAmplitudeUncertainty = (1 / 4 : Real) ∧ readout.sp91LinearizedTemperatureAmplitudeUncertainty = (3 / 10 : Real) := readout.max_and_sp91_amplitude_errors_are_one_quarter_and_three_tenths values example {T : PhysicsLogic.Papers.ScottSrednickiWhite1993.CMBTypes} (framework : PhysicsLogic.Papers.ScottSrednickiWhite1993.SampleVarianceFramework T) (result : framework.ExperimentUncertaintyConclusion) : framework.SelectedHistoricalScanExperimentValuesAt ∧ framework.SelectedMAXLinearScanFormulaAt ∧ framework.SelectedSP91LinearScanFormulaAt := framework.experiment_uncertainty_retains_equation_ten_and_historical_inputs result example {T : PhysicsLogic.Papers.ScottSrednickiWhite1993.CMBTypes} (framework : PhysicsLogic.Papers.ScottSrednickiWhite1993.SampleVarianceFramework T) (setup : framework.SingleRealizationSetup) (gaussian : framework.gaussianPrimordialFluctuations framework.selectedModel) (calculation : framework.GaussianMomentCalculation) : framework.toPowerSpectrumSamplingVarianceContext |>.EstablishedSamplingVarianceAt framework.samplingVarianceModel := framework.gaussian_moment_route_realizes_shared_sampling_variance setup gaussian calculation example : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework.Countermodels.changedPredictionConditionsWithoutRemoteImpulse.HasChangedPredictionConditions ∧ ¬PhysicsLogic.Papers.Bohr1935.ComplementarityFramework.Countermodels.changedPredictionConditionsWithoutRemoteImpulse.HasRemoteMechanicalDisturbance := PhysicsLogic.Papers.Bohr1935.ComplementarityFramework.Countermodels.contextual_influence_does_not_assert_remote_mechanical_disturbance example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Bohr1935.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Bohr1935.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (reply : framework.EPRReplyResult) : framework.toArrangementContext |>.ContextualInfluenceWithoutMechanicalDisturbanceAt framework.eprPositionArrangement framework.eprMomentumArrangement framework.selectedRegime := framework.epr_reply_realizes_shared_contextual_influence_without_mechanical_disturbance reply example : PhysicsLogic.Papers.Bohr1935.SlitDiffractionFormulaReadout.Countermodels.physicalSlitDiffractionFormulaReadout.SourceFormulaAt := PhysicsLogic.Papers.Bohr1935.SlitDiffractionFormulaReadout.Countermodels.physical_slit_values_realize_original_qualitative_planck_product example : PhysicsLogic.Papers.Bohr1935.SlitDiffractionFormulaReadout.Countermodels.lowerOrderUnityProductReadout.SourceFormulaAt ∧ PhysicsLogic.Papers.Bohr1935.SlitDiffractionFormulaReadout.Countermodels.upperOrderUnityProductReadout.SourceFormulaAt := by have distinctCoefficients := PhysicsLogic.Papers.Bohr1935.SlitDiffractionFormulaReadout.Countermodels.qualitative_planck_product_does_not_impose_an_exact_coefficient exact ⟨distinctCoefficients.1, distinctCoefficients.2.2.1⟩ example : PhysicsLogic.Papers.Bohr1935.EPRDoubleSlitPreparationReadout.Countermodels.physicalEPRDoubleSlitPreparationReadout.SourceFormulaAt := PhysicsLogic.Papers.Bohr1935.EPRDoubleSlitPreparationReadout.Countermodels.physical_double_slit_preparation_realizes_position_difference_and_momentum_sum example : PhysicsLogic.Papers.Bohr1935.EPRDoubleSlitPreparationReadout.Countermodels.wrongMeasuredDiaphragmMomentumReadout.FixedPositionDifferenceFormulaAt ∧ PhysicsLogic.Papers.Bohr1935.EPRDoubleSlitPreparationReadout.Countermodels.wrongMeasuredDiaphragmMomentumReadout.KnownOutgoingMomentumSumFormulaAt ∧ ¬PhysicsLogic.Papers.Bohr1935.EPRDoubleSlitPreparationReadout.Countermodels.wrongMeasuredDiaphragmMomentumReadout.SourceFormulaAt := PhysicsLogic.Papers.Bohr1935.EPRDoubleSlitPreparationReadout.Countermodels.fixed_correlation_values_do_not_supply_measured_diaphragm_momentum_balance example : PhysicsLogic.Papers.Bohr1935.EPRAlternativeRemotePredictionReadout.Countermodels.physicalAlternativeRemotePredictionReadout.SourceFormulaAt PhysicsLogic.Papers.Bohr1935.EPRDoubleSlitPreparationReadout.Countermodels.physicalEPRDoubleSlitPreparationReadout := PhysicsLogic.Papers.Bohr1935.EPRAlternativeRemotePredictionReadout.Countermodels.physical_alternative_predictions_realize_distinct_position_and_momentum_values example : PhysicsLogic.Papers.Bohr1935.EPRAlternativeRemotePredictionReadout.Countermodels.wrongRemoteMomentumPredictionReadout.PositionPredictionFormulaAt PhysicsLogic.Papers.Bohr1935.EPRDoubleSlitPreparationReadout.Countermodels.physicalEPRDoubleSlitPreparationReadout ∧ ¬PhysicsLogic.Papers.Bohr1935.EPRAlternativeRemotePredictionReadout.Countermodels.wrongRemoteMomentumPredictionReadout.SourceFormulaAt PhysicsLogic.Papers.Bohr1935.EPRDoubleSlitPreparationReadout.Countermodels.physicalEPRDoubleSlitPreparationReadout := PhysicsLogic.Papers.Bohr1935.EPRAlternativeRemotePredictionReadout.Countermodels.correct_position_prediction_does_not_supply_the_momentum_prediction example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (setup : framework.EPRPairSetup) : framework.eprDoubleSlitPreparationReadout.SourceFormulaAt ∧ framework.positionDifferenceFixedAt framework.measuredParticle framework.remoteParticle framework.selectedRegime ∧ framework.momentumSumFixedAt framework.measuredParticle framework.remoteParticle framework.selectedRegime := framework.epr_pair_setup_retains_original_double_slit_and_momentum_balance setup example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (setup : framework.EPRPairSetup) : framework.eprPositionSumUndeterminedAt framework.measuredParticle framework.remoteParticle framework.selectedRegime ∧ framework.eprMomentumDifferenceUndeterminedAt framework.measuredParticle framework.remoteParticle framework.selectedRegime ∧ framework.eprDoubleSlitPreparationReadout.positionSumDetermined = false ∧ framework.eprDoubleSlitPreparationReadout.momentumDifferenceDetermined = false := framework.epr_pair_setup_keeps_conjugate_position_sum_and_momentum_difference_open setup example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (setup : framework.EPRPairSetup) : framework.toRemoteCorrelationPredictionContext |>.SelectedRemoteCorrelationSetupAt framework.bohrRemoteCorrelationModel := framework.bohr_double_slit_realizes_selected_remote_correlation_setup setup example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (setup : framework.EPRPairSetup) (calculation : framework.EPRArrangementCalculation) : framework.bohrRemoteCorrelationModel |>.NonvacuousRemoteCorrelationPredictionAt := framework.bohr_double_slit_exhibits_nonvacuous_shared_remote_correlation setup calculation example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (setup : framework.EPRPairSetup) (calculation : framework.EPRArrangementCalculation) : framework.bohrRemoteCorrelationModel |>.DirectSourceStructuralRemoteCorrelationPredictionExplanationAt .doubleSlitPreparationAlternativePositionMomentumAndMechanicalNondisturbance framework.bohrRemoteCorrelationDerivationObtains := framework.double_slit_inputs_realize_source_structural_remote_correlation_explanation setup calculation example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (setup : framework.EPRPairSetup) (calculation : framework.EPRArrangementCalculation) (complementarity : framework.SlitComplementarityResult) (assessment : framework.EPRCriterionAssessment) : (framework.bohrRemoteCorrelationModel |>.NonvacuousRemoteCorrelationPredictionAt) ∧ (framework.toArrangementContext |>.ContextualInfluenceWithoutMechanicalDisturbanceAt framework.eprPositionArrangement framework.eprMomentumArrangement framework.selectedRegime) ∧ ¬(framework.toArrangementContext |>.PredictionConditionsPreservedAcrossAt framework.eprPositionArrangement framework.eprMomentumArrangement framework.selectedRegime) ∧ ¬framework.eprIncompletenessInferenceJustifiedAt framework.selectedRegime := framework.bohr_remote_prediction_and_contextual_reply_remain_logically_distinct setup calculation complementarity assessment example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (preparation : framework.FixedSlitOperationalPreparationAt) (formula : framework.SelectedSlitDiffractionFormulaAt) (stateChange : framework.SlitDiffractionStateChangeCalculation) : framework.toSlitStateChangeMeasurementContext.DisturbsAt framework.fixedDiaphragmArrangement framework.measuredParticle framework.selectedRegime := framework.fixed_slit_diffraction_realizes_local_measurement_disturbance preparation formula stateChange example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (setup : framework.FixedDiaphragmSetup) (calculation : framework.FixedDiaphragmCalculation) (preparation : framework.FixedSlitOperationalPreparationAt) (role : framework.SlitOperationalRoleAssessment) : framework.sharedOperationalErrorDisturbanceModel |>.NonvacuousOperationalErrorDisturbanceAt := framework.fixed_slit_realizes_nonvacuous_shared_operational_route setup calculation preparation role example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (setup : framework.FixedDiaphragmSetup) (calculation : framework.FixedDiaphragmCalculation) (preparation : framework.FixedSlitOperationalPreparationAt) (role : framework.SlitOperationalRoleAssessment) : framework.toOperationalErrorDisturbanceContext |>.SourceStructuralOperationalTradeoffExplanationAt framework.sharedOperationalErrorDisturbanceModel .fixedDiaphragmDeBroglieDiffractionLocalRecoilAndQualitativePlanckProduct framework.bohrSlitOperationalDerivationObtains := framework.fixed_slit_source_inputs_realize_source_structural_operational_tradeoff_explanation setup calculation preparation role example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (setup : framework.FixedDiaphragmSetup) (calculation : framework.FixedDiaphragmCalculation) (preparation : framework.FixedSlitOperationalPreparationAt) (role : framework.SlitOperationalRoleAssessment) (epr : framework.EPRArrangementResult) : (framework.sharedOperationalErrorDisturbanceModel |>.NonvacuousOperationalErrorDisturbanceAt) ∧ framework.slitDiffractionFormulaReadout.TransverseMomentumExchangeRelationAt ∧ framework.momentumExchangeUncontrolledAt framework.measuredParticle framework.diaphragm framework.fixedDiaphragmArrangement framework.selectedRegime ∧ ¬framework.mechanicallyDisturbsAt framework.eprPositionArrangement framework.remoteParticle framework.selectedRegime ∧ ¬framework.mechanicallyDisturbsAt framework.eprMomentumArrangement framework.remoteParticle framework.selectedRegime := framework.fixed_slit_local_exchange_preserves_epr_remote_mechanical_nondisturbance setup calculation preparation role epr example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (setup : framework.FixedDiaphragmSetup) (calculation : framework.FixedDiaphragmCalculation) (preparation : framework.FixedSlitOperationalPreparationAt) (role : framework.SlitOperationalRoleAssessment) (stateChange : framework.SlitDiffractionStateChangeCalculation) (epr : framework.EPRArrangementResult) : framework.toSlitStateChangeMeasurementContext.DisturbsAt framework.fixedDiaphragmArrangement framework.measuredParticle framework.selectedRegime ∧ framework.eprDoubleSlitPreparationReadout.SourceFormulaAt ∧ (framework.sharedOperationalErrorDisturbanceModel |>.NonvacuousOperationalErrorDisturbanceAt) ∧ ¬framework.mechanicallyDisturbsAt framework.eprPositionArrangement framework.remoteParticle framework.selectedRegime ∧ ¬framework.mechanicallyDisturbsAt framework.eprMomentumArrangement framework.remoteParticle framework.selectedRegime := framework.local_measurement_disturbance_and_epr_remote_nondisturbance_are_distinct setup calculation preparation role stateChange epr example {T : PhysicsLogic.Papers.Bohr1935.Bohr1935Types} (framework : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework T) (fixedSetup : framework.FixedDiaphragmSetup) (fixedCalculation : framework.FixedDiaphragmCalculation) (preparation : framework.FixedSlitOperationalPreparationAt) (role : framework.SlitOperationalRoleAssessment) (stateChange : framework.SlitDiffractionStateChangeCalculation) (eprSetup : framework.EPRPairSetup) (eprCalculation : framework.EPRArrangementCalculation) : framework.toSlitStateChangeMeasurementContext.DisturbsAt framework.fixedDiaphragmArrangement framework.measuredParticle framework.selectedRegime ∧ (framework.sharedOperationalErrorDisturbanceModel |>.NonvacuousOperationalErrorDisturbanceAt) ∧ (framework.bohrRemoteCorrelationModel |>.NonvacuousRemoteCorrelationPredictionAt) ∧ ¬framework.eprAlternativeArrangementsJointlyPerformedAt framework.eprPositionArrangement framework.eprMomentumArrangement framework.measuredParticle framework.selectedRegime ∧ ¬framework.mechanicallyDisturbsAt framework.eprPositionArrangement framework.remoteParticle framework.selectedRegime ∧ ¬framework.mechanicallyDisturbsAt framework.eprMomentumArrangement framework.remoteParticle framework.selectedRegime := framework.local_disturbance_and_counterfactual_remote_prediction_are_compatible fixedSetup fixedCalculation preparation role stateChange eprSetup eprCalculation example : PhysicsLogic.Papers.Bohr1935.ComplementarityFramework.Countermodels.localSlitStateChangeWithoutRemoteImpulse.HasLocalSlitStateChange ∧ ¬PhysicsLogic.Papers.Bohr1935.ComplementarityFramework.Countermodels.localSlitStateChangeWithoutRemoteImpulse.HasRemoteMechanicalDisturbance := PhysicsLogic.Papers.Bohr1935.ComplementarityFramework.Countermodels.local_slit_state_change_does_not_imply_remote_mechanical_disturbance example : (PhysicsLogic.Papers.Bohr1935.ComplementarityFramework.Countermodels.sourceValuedAlternativesWithoutJointMeasurementOrCommonReality |>.HasBothAlternativePredictions) ∧ ¬(PhysicsLogic.Papers.Bohr1935.ComplementarityFramework.Countermodels.sourceValuedAlternativesWithoutJointMeasurementOrCommonReality |>.HasJointlyPerformedProcedures) ∧ ¬(PhysicsLogic.Papers.Bohr1935.ComplementarityFramework.Countermodels.sourceValuedAlternativesWithoutJointMeasurementOrCommonReality |>.HasArrangementIndependentReality) := PhysicsLogic.Papers.Bohr1935.ComplementarityFramework.Countermodels.source_valued_alternative_predictions_do_not_imply_joint_measurement_or_common_reality example : PhysicsLogic.Papers.Dieks1982.DieksFramework.Countermodels.basisReplicationWithoutSuperpositionCrossTerm.ReplicatesSelectedBasisStates ∧ ¬PhysicsLogic.Papers.Dieks1982.DieksFramework.Countermodels.basisReplicationWithoutSuperpositionCrossTerm.ReplicatesArbitrarySuperposition := PhysicsLogic.Papers.Dieks1982.DieksFramework.Countermodels.x_basis_copying_is_retained_while_arbitrary_copying_is_excluded example (readout : PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout) (input : readout.InputZStateSuperpositionFormulaAt) (linearity : readout.MultiplierBranchLinearityFormulaAt) (orthogonality : readout.HalfSplitOccupationOrthogonalityAt) (bornRule : readout.BornProbabilityFromBranchAmplitudesAt) : readout.OutcomeProbabilitiesAt := readout.linear_branches_orthogonality_and_born_rule_determine_outcomes input linearity orthogonality bornRule example : PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.coherentBranchesWithoutBornProbabilities.NormalizedOccupationProbabilityAssignmentAt ∧ ¬PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.coherentBranchesWithoutBornProbabilities.BornProbabilityFromBranchAmplitudesAt := by obtain ⟨_occupation, _input, _linearity, _orthogonality, normalized, notBorn, _notOutcomes⟩ := PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.coherent_orthogonal_branches_do_not_determine_probabilities_without_born_rule exact ⟨normalized, notBorn⟩ example : PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.NamedCoherentLinearBurstAt PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.namedBranchWithoutPrintedAmplitudes ∧ ¬PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.namedBranchWithoutPrintedAmplitudes.SourceFormulaAt := PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.named_linear_burst_does_not_supply_printed_amplitudes example : PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.probabilitiesWithoutCoherentBranch.OutcomeProbabilitiesAt ∧ ¬PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.probabilitiesWithoutCoherentBranch.MultiplierBranchLinearityFormulaAt := PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.outcome_probabilities_do_not_reconstruct_coherent_multiplier_branch example : PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.physicalCoherentMultiplierReadout.SourceFormulaAt ∧ PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.physicalCoherentMultiplierReadout.jointBranchCoherenceValue = 1 / 2 := by obtain ⟨source, _distinctRecords, _orthogonal, coherence⟩ := PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.physical_coherent_multiplier_has_nonzero_joint_branch_interference exact ⟨source, coherence⟩ example : PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.coherentMultiplierWithEqualMachineRecords.SourceFormulaAt ∧ PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.coherentMultiplierWithEqualMachineRecords.OccupationSectorOrthogonalityAt := by obtain ⟨source, _equalRecords, orthogonal⟩ := PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.occupation_orthogonality_does_not_require_distinguishable_machine_records exact ⟨source, orthogonal⟩ example : PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.physicalCoherentMultiplierReadout.OutcomeProbabilitiesAt ∧ PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.coherentMultiplierWithNegativeRelativePhase.OutcomeProbabilitiesAt ∧ PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.physicalCoherentMultiplierReadout.jointBranchCoherenceValue ≠ PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.coherentMultiplierWithNegativeRelativePhase.jointBranchCoherenceValue := by obtain ⟨_positiveSource, _negativeSource, positiveOutcomes, negativeOutcomes, _positivePhase, _negativePhase, phasesDistinct⟩ := PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.opposite_signed_bursts_share_probabilities_but_not_joint_phase exact ⟨positiveOutcomes, negativeOutcomes, phasesDistinct⟩ example : PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.halfSplitAliasedWithAllPlusOccupation.OutcomeProbabilitiesAt ∧ PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.halfSplitAliasedWithAllPlusOccupation.CoherentBranchSuperpositionAt ∧ ¬PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.halfSplitAliasedWithAllPlusOccupation.OccupationSectorOrthogonalityAt := PhysicsLogic.Papers.Dieks1982.LinearMultiplierOccupationReadout.Countermodels.normalized_statistics_do_not_supply_distinct_occupation_sectors example {T : PhysicsLogic.Papers.Dieks1982.DieksTypes} (framework : PhysicsLogic.Papers.Dieks1982.DieksFramework T) (setup : framework.HerbertMultiplierSetup) (linearity : framework.ZStateLinearityCalculation) (statistics : framework.LinearBurstStatisticsCalculation) : framework.LinearMultiplierOccupationSourceFormulaAt := framework.actual_linear_statistics_retain_printed_multiplier_source_formula setup linearity statistics example {T : PhysicsLogic.Papers.Dieks1982.DieksTypes} (framework : PhysicsLogic.Papers.Dieks1982.DieksFramework T) (setup : framework.HerbertMultiplierSetup) (linearity : framework.ZStateLinearityCalculation) (statistics : framework.LinearBurstStatisticsCalculation) : framework.linearMultiplierOccupationReadout.OutcomeProbabilitiesAt := framework.linear_statistics_derive_exact_born_outcome_probabilities setup linearity statistics example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Dieks1982.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Dieks1982.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Dieks1982.DieksTypes} (framework : PhysicsLogic.Papers.Dieks1982.DieksFramework T) (setup : framework.XBasisMultiplierSetup) (unique : framework.UnitaryOutputUniqueness) (linearCalculation : framework.ZStateLinearityCalculation) (requiredCalculation : framework.RequiredZCloneCalculation) (mismatchCalculation : framework.LinearAndRequiredBurstMismatchCalculation) : framework.sharedReplicationModel |>.DirectSourceStructuralLinearReplicationObstructionExplanationAt .multiplierLinearityAndRequiredBurstMismatch framework.DieksReplicationDerivationObtains := framework.multiplier_source_inputs_realize_structural_replication_obstruction setup unique linearCalculation requiredCalculation mismatchCalculation example : PhysicsLogic.Papers.GallavottiCohen1995.Framework.Countermodels.realizesDissipativeReversibleDynamics PhysicsLogic.Papers.GallavottiCohen1995.Framework.Countermodels.dissipativeReversibleWithoutHyperbolicity ∧ ¬PhysicsLogic.Papers.GallavottiCohen1995.Framework.Countermodels.realizesUniformHyperbolicityData PhysicsLogic.Papers.GallavottiCohen1995.Framework.Countermodels.dissipativeReversibleWithoutHyperbolicity := PhysicsLogic.Papers.GallavottiCohen1995.Framework.Countermodels.dissipation_and_reversal_do_not_verify_hyperbolicity example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.GallavottiCohen1995.Fidelity.countermodelMatrix := PhysicsLogic.Papers.GallavottiCohen1995.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.GallavottiCohen1995.GallavottiCohenTypes} (framework : PhysicsLogic.Papers.GallavottiCohen1995.Framework T) (sourceSetup : framework.SRBWeightRatioSourceSetup) (sourceCalculation : framework.SRBWeightRatioSourceCalculation) (unproved : framework.chaoticHypothesisUnprovedForFluidAt framework.system .chaoticHypothesis) (leadingOnly : framework.ratioValidOnlyToLeadingLongTimeOrderAt framework.system .longObservation) (finiteCorrections : framework.finiteTimeCorrectionsRemainAt framework.correctionBound framework.system .longObservation) (proposedExtension : framework.extensionToReversibleFluidDynamicsProposedAt framework.dynamics framework.system .fluidExtension) : framework.selectedEntropyProductionFluctuationContext |>.QualifiedFluctuationRatioRouteApplicableAt framework.selectedEntropyProductionFluctuationModel := framework.gc_source_inputs_make_shared_qualified_ratio_route_applicable sourceSetup sourceCalculation unproved leadingOnly finiteCorrections proposedExtension example : PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.MagnetoRotonFramework.Countermodels.hasNamedProjectedEnergyRatio PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.MagnetoRotonFramework.Countermodels.wrongEquationFourNineteenModel ∧ ¬PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.MagnetoRotonFramework.Countermodels.realizesEquationFourNineteen PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.MagnetoRotonFramework.Countermodels.wrongEquationFourNineteenModel := PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.MagnetoRotonFramework.Countermodels.named_projected_energy_ratio_does_not_supply_equation_four_nineteen example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.Fidelity.countermodelMatrix := PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.Fidelity.countermodel_matrix_is_realized namespace PhysicsLogicProjectedHallSourceReuse open PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986 open PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.FiniteProjectedDensityAlgebraReadout open PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.FiniteProjectedSpectralReadout open PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.LowestLandauStructureProjectionReadout example : physicalProjectedDensityAlgebraReadout.SourceProjectedDensityAlgebraAt := physical_projected_density_matrices_realize_noncommuting_source_algebra example : projectedDensitySourceCoefficient 1 ((Real.pi : Complex) * Complex.I) = 2 * Complex.I := perpendicular_momenta_have_nonzero_source_commutator_coefficient example : physicalLowestLandauProjectionReadout.PositiveProjectedDensityResponseAt := physical_projection_subtracts_the_exact_cyclotron_structure_factor.1 example : physicalProjectedDensitySpectrum.SourceSpectralMomentRepresentationAt := physical_projected_spectral_measure_realizes_source_moments.1 example {T : GMPTypes} (framework : MagnetoRotonFramework T) (construction : framework.EstablishedProjectedSingleModeConstruction) : framework.sourceDensityAlgebraReadout.SourceProjectedDensityAlgebraAt := framework.established_projected_construction_retains_noncommuting_density_matrices construction example {T : GMPTypes} (framework : MagnetoRotonFramework T) (construction : framework.EstablishedProjectedSingleModeConstruction) : framework.SourceLowestLandauStructureProjectionAt := framework.established_projected_construction_retains_cyclotron_subtraction construction example {T : GMPTypes} (framework : MagnetoRotonFramework T) (bound : framework.EstablishedProjectedVariationalBound) : framework.SourceProjectedSpectralFirstMomentAt := framework.established_variational_bound_retains_actual_density_spectral_moments bound example {T : GMPTypes} (framework : MagnetoRotonFramework T) (bound : framework.EstablishedProjectedVariationalBound) : framework.sourceSpectralReadout.excitationEnergy framework.sourceSpectralReadout.lowestExcitationIndex ≤ framework.sourceSpectralReadout.reportedFirstMoment := framework.established_variational_bound_controls_lowest_spectral_excitation bound example {T : GMPTypes} (framework : MagnetoRotonFramework T) (dispersion : framework.EstablishedMagnetoRotonDispersion) : ¬framework.sourceSpectralReadout.SingleModeOscillatorStrengthSaturationAt := framework.established_dispersion_does_not_claim_single_mode_saturation dispersion example : ∀ proposedGap : Rat, 0 < proposedGap -> ∃ momentumMagnitude : Rat, 0 < momentumMagnitude ∧ momentumMagnitude ≤ 1 ∧ (gaplessDarkProjectedSpectrum momentumMagnitude).SourceSpectralMomentRepresentationAt ∧ (gaplessDarkProjectedSpectrum momentumMagnitude).SourceQuarticVisibleResponseAt momentumMagnitude ∧ (gaplessDarkProjectedSpectrum momentumMagnitude).reportedFirstMoment = 2 ∧ (gaplessDarkProjectedSpectrum momentumMagnitude).excitationEnergy (gaplessDarkProjectedSpectrum momentumMagnitude).lowestExcitationIndex < proposedGap := quartic_visible_response_does_not_imply_a_uniform_sector_gap example : Fidelity.countermodelMatrix .commutingDensityOperatorsNotProjectedSourceAlgebra := Fidelity.countermodel_matrix_is_realized .commutingDensityOperatorsNotProjectedSourceAlgebra example : Fidelity.countermodelMatrix .reversedDensityOrderNotSourceCommutator := Fidelity.countermodel_matrix_is_realized .reversedDensityOrderNotSourceCommutator example : Fidelity.countermodelMatrix .collinearMomentaNotNonzeroProjectedCommutator := Fidelity.countermodel_matrix_is_realized .collinearMomentaNotNonzeroProjectedCommutator example : Fidelity.countermodelMatrix .unprojectedStructureNotProjectedStructure := Fidelity.countermodel_matrix_is_realized .unprojectedStructureNotProjectedStructure example : Fidelity.countermodelMatrix .filledLandauLevelNotPositiveProjectedResponse := Fidelity.countermodel_matrix_is_realized .filledLandauLevelNotPositiveProjectedResponse example : Fidelity.countermodelMatrix .reportedMomentsNotNonnegativeSpectralWeights := Fidelity.countermodel_matrix_is_realized .reportedMomentsNotNonnegativeSpectralWeights example : Fidelity.countermodelMatrix .zeroStructureFactorNotVariationalQuotient := Fidelity.countermodel_matrix_is_realized .zeroStructureFactorNotVariationalQuotient example : Fidelity.countermodelMatrix .exactFirstMomentNotLowestExcitation := Fidelity.countermodel_matrix_is_realized .exactFirstMomentNotLowestExcitation example : Fidelity.countermodelMatrix .densityDarkLowestModeNotResponseCoupled := Fidelity.countermodel_matrix_is_realized .densityDarkLowestModeNotResponseCoupled example : Fidelity.countermodelMatrix .ninetyEightPercentStrengthNotSingleModeSaturation := Fidelity.countermodel_matrix_is_realized .ninetyEightPercentStrengthNotSingleModeSaturation example : Fidelity.countermodelMatrix .exactCollectiveEnergyNotSingleModeSaturation := Fidelity.countermodel_matrix_is_realized .exactCollectiveEnergyNotSingleModeSaturation example : Fidelity.countermodelMatrix .staticWeightFractionNotOscillatorStrengthFraction := Fidelity.countermodel_matrix_is_realized .staticWeightFractionNotOscillatorStrengthFraction example : Fidelity.countermodelMatrix .quarticVisibleResponseNotUniformSectorGap := Fidelity.countermodel_matrix_is_realized .quarticVisibleResponseNotUniformSectorGap end PhysicsLogicProjectedHallSourceReuse example {T : PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.GMPTypes} (framework : PhysicsLogic.Papers.GirvinMacDonaldPlatzman1986.MagnetoRotonFramework T) (setup : framework.FractionalHallSetup) (diagnosis : framework.UnprojectedSMADiagnosis) (construction : framework.ProjectedSingleModeConstruction) (energy : framework.ProjectedEnergyCalculation) : framework.toMomentumResolvedVariationalBoundContext |>.MomentumResolvedVariationalRouteApplicableAt framework.sharedVariationalModel := framework.gmp_source_inputs_make_shared_variational_route_applicable setup diagnosis construction energy example : PhysicsLogic.Papers.EvansColeman1994.Countermodels.HasNumericallyAgreeingMatterExponents PhysicsLogic.Papers.EvansColeman1994.Countermodels.radiationCriticalScopeBoundary ∧ ¬PhysicsLogic.Papers.EvansColeman1994.Countermodels.HasMathematicallyUniversalMatterExponent PhysicsLogic.Papers.EvansColeman1994.Countermodels.radiationCriticalScopeBoundary := PhysicsLogic.Papers.EvansColeman1994.Countermodels.numerical_exponent_agreement_does_not_prove_universal_exponent example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix := PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.EvansColeman1994.EvansColemanTypes} (context : PhysicsLogic.Papers.EvansColeman1994.RadiationFluidCollapseContext T) (setup : context.ThresholdSetup) (outcomeCalculation : context.ThresholdOutcomeCalculation) (massCalculation : context.MassScalingCalculation) (limitBridge : context.VanishingMassBridge) : context.toSharedCriticalCollapseContext |>.CriticalCollapseScalingRouteApplicableAt context.selectedCriticalCollapseModel := context.radiation_fluid_inputs_make_shared_route_applicable setup outcomeCalculation massCalculation limitBridge example : PhysicsLogic.Papers.JacobiPDEReduction1837.Countermodels.pdeSolutionWithoutCompleteParameterFamily.HasHamiltonJacobiSolution ∧ ¬PhysicsLogic.Papers.JacobiPDEReduction1837.Countermodels.pdeSolutionWithoutCompleteParameterFamily.HasCompleteIntegral := PhysicsLogic.Papers.JacobiPDEReduction1837.Countermodels.a_pde_solution_need_not_be_a_complete_integral example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.JacobiPDEReduction1837.Fidelity.countermodelMatrix := PhysicsLogic.Papers.JacobiPDEReduction1837.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.JacobiPDEReduction1837.JacobiTypes} (framework : PhysicsLogic.Papers.JacobiPDEReduction1837.JacobiFramework T) (timeSetup : framework.TimeDependentMechanicalSetup) (principalCalculation : framework.PrincipalVariationCalculation) (arbitraryFirstOrder : framework.timeDependentFirstOrderEquationAt framework.timeDependentEquation framework.arbitraryCompleteIntegral framework.mechanicalSystem framework.timeDependentRegime) (completeFamily : framework.parameterFamilyCompleteAt framework.arbitraryCompleteIntegral framework.timeDependentEquation framework.parameterFamily framework.timeDependentRegime) (recoveryCalculation : framework.CompleteIntegralRecoveryCalculation) : framework.toSharedCompleteIntegralDynamicsContext |>.CompleteIntegralDynamicsRouteApplicableAt framework.selectedCompleteIntegralDynamicsModel := framework.jacobi_source_inputs_make_shared_complete_integral_route_applicable timeSetup principalCalculation arbitraryFirstOrder completeFamily recoveryCalculation example : PhysicsLogic.Papers.PortegiesZwartBoekholt2014.PZBFramework.Countermodels.pzbDeterministicPointFailureWithEnsembleAgreement.HasFiniteResolutionDivergenceAt ∧ ¬PhysicsLogic.Papers.PortegiesZwartBoekholt2014.PZBFramework.Countermodels.pzbDeterministicPointFailureWithEnsembleAgreement.HasOnticIndeterminismAt := by have witness := PhysicsLogic.Papers.PortegiesZwartBoekholt2014.PZBFramework.Countermodels.pzb_realizes_shared_deterministic_point_and_ensemble_boundary exact ⟨witness.1, witness.2.2.1⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.PortegiesZwartBoekholt2014.Fidelity.countermodelMatrix := PhysicsLogic.Papers.PortegiesZwartBoekholt2014.Fidelity.countermodel_matrix_is_realized example (readout : PhysicsLogic.Papers.PortegiesZwartBoekholt2014.EnsembleProtocolReadout) (protocol : readout.SourceProtocolAt) : readout.totalRealizationCount = 40000 := readout.four_ensembles_contain_forty_thousand_initial_realizations protocol example (readout : PhysicsLogic.Papers.PortegiesZwartBoekholt2014.DissolutionCriterionReadout) (criterion : readout.SourceCriterionAt) : readout.EscaperUnboundAt readout.selectedIntegralStep ∧ readout.OutsideTwiceInitialVirialRadiusAt readout.selectedIntegralStep ∧ readout.RecedingFromCenterOfMassAt readout.selectedIntegralStep := readout.first_dissolution_requires_all_three_independent_escape_conditions criterion example {T : PhysicsLogic.Papers.PortegiesZwartBoekholt2014.PZBTypes} (framework : PhysicsLogic.Papers.PortegiesZwartBoekholt2014.PZBFramework T) (result : framework.EstablishedEnsembleStatisticalRecovery) : framework.SelectedDissolutionCriterionAt ∧ framework.SelectedEnsembleProtocolAt := framework.ensemble_recovery_retains_original_dissolution_and_protocol result example {T : PhysicsLogic.Papers.PortegiesZwartBoekholt2014.PZBTypes} (framework : PhysicsLogic.Papers.PortegiesZwartBoekholt2014.PZBFramework T) (setup : framework.IndividualComparisonSetup) (determinism : framework.ThreeBodyDynamicsDeterminismRealization) (calculation : framework.IndividualDivergenceCalculation) (bridge : framework.PointPredictionReliabilityBridge) : framework.toSharedTrajectoryContext |>.EstablishedFinitePrecisionPredictabilityLimitAt framework.selectedFinitePrecisionTrajectoryModel := framework.individual_comparison_realizes_shared_predictability_limit setup determinism calculation bridge example : PhysicsLogic.Papers.Zamolodchikov1986.CTheoremFramework.Countermodels.assumptionsWithoutCorrelationCalculation.CorrelatorCFunctionConstruction ∧ ¬PhysicsLogic.Papers.Zamolodchikov1986.CTheoremFramework.Countermodels.assumptionsWithoutCorrelationCalculation.StressTensorCorrelationCalculation := by have witness := PhysicsLogic.Papers.Zamolodchikov1986.CTheoremFramework.Countermodels.correlator_c_function_construction_does_not_supply_gradient_identity exact ⟨witness.2.1, witness.2.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Zamolodchikov1986.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Zamolodchikov1986.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Zamolodchikov1986.ZamolodchikovTypes} (framework : PhysicsLogic.Papers.Zamolodchikov1986.CTheoremFramework T) (paperSetup : framework.SelectedFlowSetup) (cFunction : framework.CorrelatorCFunctionConstruction) (runningTracksC : framework.RunningCTracksCorrelatorCFunction) (stressTensor : framework.StressTensorCorrelationCalculation) (positivity : framework.CouplingMetricPositivityCalculation) (integration : framework.RGIntegrationCalculation) (vanishingContractions : framework.VanishingBetaContractionCalculation) (fixedPointCoefficient : framework.FixedPointStressTensorCoefficientCalculation) : framework.toSharedContext.RunningMonotonicityCalculationAt framework.sharedModel := framework.c_theorem_realizes_shared_monotonicity_calculation paperSetup cFunction runningTracksC stressTensor positivity integration vanishingContractions fixedPointCoefficient example : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ChiralNuclearFramework.Countermodels.HasLeadingThreeBodyCancellationAt PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ChiralNuclearFramework.Countermodels.reducedUnitaryNuclearScope ∧ ¬PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ChiralNuclearFramework.Countermodels.HasNoHigherThreeBodyContributionAt PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ChiralNuclearFramework.Countermodels.reducedUnitaryNuclearScope := PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ChiralNuclearFramework.Countermodels.leading_three_body_cancellation_does_not_remove_all_higher_orders example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.Fidelity.countermodelMatrix := PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.Fidelity.countermodel_matrix_is_realized example {Sector Projector Operator Energy Potential Term Order Regime : Type} (framework : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ChiralNuclearFramework Sector Projector Operator Energy Potential Term Order Regime) (realization : framework.SharedPowerCountingRealization) (constructed : framework.potentialConstructedPerturbativelyThrough framework.lowEnergyRegime framework.secondOrder) (nonperturbative : ¬framework.fullFewNucleonDynamicsPerturbativeAt framework.selectedFullDynamics framework.lowEnergyRegime) (required : framework.potentialIterationRequiredForAt (framework.effectivePotentialAt 2 framework.secondOrder) framework.selectedFullDynamics framework.lowEnergyRegime) (generated : framework.fullFewNucleonDynamicsGeneratedByIterationOf (framework.effectivePotentialAt 2 framework.secondOrder) framework.selectedFullDynamics framework.lowEnergyRegime) : framework.toEffectiveInteractionDynamicsContext |>.MixedInteractionDynamicsRouteAt framework.sharedEffectiveInteractionModel := framework.low_energy_inputs_make_shared_mixed_dynamics_route_applicable realization constructed nonperturbative required generated example : PhysicsLogic.Papers.BognerFurnstahlPerry2007.PaperFramework.Countermodels.twoBodyPreservationWithoutInducedThreeBodyControl.HasUnitaryTwoBodyPreservation ∧ ¬PhysicsLogic.Papers.BognerFurnstahlPerry2007.PaperFramework.Countermodels.twoBodyPreservationWithoutInducedThreeBodyControl.HasInducedThreeBodyPreservation := PhysicsLogic.Papers.BognerFurnstahlPerry2007.PaperFramework.Countermodels.two_body_unitarity_does_not_control_omitted_induced_three_body_content example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.BognerFurnstahlPerry2007.Fidelity.countermodelMatrix := PhysicsLogic.Papers.BognerFurnstahlPerry2007.Fidelity.countermodel_matrix_is_realized namespace PhysicsLogicNuclearSrgSourceReuse open PhysicsLogic.Papers.BognerFurnstahlPerry2007 open PhysicsLogic.Papers.BognerFurnstahlPerry2007.FiniteNuclearSrgOperatorReadout open PhysicsLogic.Papers.BognerFurnstahlPerry2007.FiniteWeinbergBornReadout example : physicalSourceOperatorReadout.SourceOperatorFormulaAt := physical_source_operator_realizes_exact_double_commutator_flow example {T : BognerFurnstahlPerryTypes} (paper : PaperFramework T) (flow : paper.EstablishedUnitaryFlow) : paper.SourceOperatorFlowFormulaAt := paper.established_unitary_flow_retains_source_operator_equations flow example {T : BognerFurnstahlPerryTypes} (paper : PaperFramework T) (flow : paper.EstablishedUnitaryFlow) : paper.sourceOperatorReadout.resolutionLambda ^ 4 * paper.sourceOperatorReadout.flowParameter = 1 := paper.established_unitary_flow_retains_fourth_root_resolution_scale flow example {T : BognerFurnstahlPerryTypes} (paper : PaperFramework T) (convergence : paper.EstablishedConvergenceEvidence) : paper.SourceWeinbergBornDiagnosticAt := paper.established_convergence_retains_actual_weinberg_born_diagnostic convergence example {T : BognerFurnstahlPerryTypes} (paper : PaperFramework T) (convergence : paper.EstablishedConvergenceEvidence) : ¬paper.initialWeinbergReadout.BornSeriesConvergesAt /\ paper.evolvedWeinbergReadout.BornSeriesConvergesAt := paper.established_convergence_distinguishes_divergent_and_convergent_born_series convergence example (readout : FiniteWeinbergBornReadout) : readout.BornSeriesConvergesAt <-> |(readout.selectedEigenvalue : Real)| < 1 := born_series_converges_exactly_below_source_eigenvalue_threshold readout example : (repulsiveSourceEigenmode 2).selectedEigenvalue = -2 /\ ¬(repulsiveSourceEigenmode 2).BornSeriesConvergesAt := ⟨initial_repulsive_source_eigenmode_is_nonperturbative.2.1, initial_repulsive_source_eigenmode_is_nonperturbative.2.2.2⟩ example : (repulsiveSourceEigenmode (1 / 2)).selectedEigenvalue = -1 / 2 /\ (repulsiveSourceEigenmode (1 / 2)).BornSeriesConvergesAt := ⟨evolved_repulsive_source_eigenmode_has_convergent_born_series.2.1, evolved_repulsive_source_eigenmode_has_convergent_born_series.2.2.2⟩ example : Fidelity.countermodelMatrix .antiHermitianGeneratorNotSelectedKineticCommutator := Fidelity.countermodel_matrix_is_realized .antiHermitianGeneratorNotSelectedKineticCommutator example : Fidelity.countermodelMatrix .kineticGeneratorNotDoubleCommutatorEvolution := Fidelity.countermodel_matrix_is_realized .kineticGeneratorNotDoubleCommutatorEvolution example : Fidelity.countermodelMatrix .inverseSquareScaleNotFourthRootResolution := Fidelity.countermodel_matrix_is_realized .inverseSquareScaleNotFourthRootResolution example : Fidelity.countermodelMatrix .nonunitaryCongruenceNotSpectrumPreserving := Fidelity.countermodel_matrix_is_realized .nonunitaryCongruenceNotSpectrumPreserving example : Fidelity.countermodelMatrix .negativeWeinbergEigenvalueNotBornConvergence := Fidelity.countermodel_matrix_is_realized .negativeWeinbergEigenvalueNotBornConvergence example : Fidelity.countermodelMatrix .reducedWeinbergEigenvalueNotPerturbativeThreshold := Fidelity.countermodel_matrix_is_realized .reducedWeinbergEigenvalueNotPerturbativeThreshold example : Fidelity.countermodelMatrix .unitMagnitudeWeinbergModeNotConvergent := Fidelity.countermodel_matrix_is_realized .unitMagnitudeWeinbergModeNotConvergent example : Fidelity.countermodelMatrix .finiteBornTruncationNotInfiniteConvergence := Fidelity.countermodel_matrix_is_realized .finiteBornTruncationNotInfiniteConvergence example : Fidelity.countermodelMatrix .positiveEnergyNotSourceWeinbergDomain := Fidelity.countermodel_matrix_is_realized .positiveEnergyNotSourceWeinbergDomain example : Fidelity.countermodelMatrix .freeResolventPotentialOrderNotInterchangeable := Fidelity.countermodel_matrix_is_realized .freeResolventPotentialOrderNotInterchangeable example : Fidelity.countermodelMatrix .zeroVectorEquationNotWeinbergEigenmode := Fidelity.countermodel_matrix_is_realized .zeroVectorEquationNotWeinbergEigenmode end PhysicsLogicNuclearSrgSourceReuse example {T : PhysicsLogic.Papers.BognerFurnstahlPerry2007.BognerFurnstahlPerryTypes} (paper : PhysicsLogic.Papers.BognerFurnstahlPerry2007.PaperFramework T) (suppression : paper.EstablishedMomentumSuppression) (observables : paper.EstablishedObservablePreservation) : paper.selectedUnitarySimilarityContext |>.UnitarySimilarityDecouplingRouteApplicableAt paper.selectedUnitarySimilarityModel := paper.bogner_furnstahl_perry_routes_make_shared_unitary_similarity_route_applicable suppression observables example : PhysicsLogic.Papers.GlazekWilson1993.PaperFramework.Countermodels.finiteOrderRenormalizationWithHigherOrderResidual.HasSelectedOrderRenormalization ∧ ¬PhysicsLogic.Papers.GlazekWilson1993.PaperFramework.Countermodels.finiteOrderRenormalizationWithHigherOrderResidual.HasAllOrderCutoffIndependence := PhysicsLogic.Papers.GlazekWilson1993.PaperFramework.Countermodels.finite_order_counterterms_do_not_establish_all_order_cutoff_independence example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.GlazekWilson1993.Fidelity.countermodelMatrix := PhysicsLogic.Papers.GlazekWilson1993.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.GlazekWilson1993.GlazekWilsonTypes} (paper : PhysicsLogic.Papers.GlazekWilson1993.PaperFramework T) (diagnosisSetup : paper.DivergenceDiagnosisSetup) (diagnosisCalculation : paper.DivergenceDiagnosisCalculation) (localSetup : paper.SimilarityConstructionLocalSetup) (similarityCalculation : paper.SimilarityConstructionCalculation) : paper.selectedUnitarySimilarityContext |>.UnitarySimilarityDecouplingRouteApplicableAt paper.selectedUnitarySimilarityModel := paper.glazek_wilson_source_inputs_make_shared_unitary_similarity_route_applicable diagnosisSetup diagnosisCalculation localSetup similarityCalculation example : PhysicsLogic.Papers.OregRefaelVonOppen2010.HelicalWireFramework.Countermodels.zeroEnergyBoundState PhysicsLogic.Papers.OregRefaelVonOppen2010.HelicalWireFramework.Countermodels.zeroEnergyWithoutSelfConjugacy ∧ ¬PhysicsLogic.Papers.OregRefaelVonOppen2010.HelicalWireFramework.Countermodels.selfConjugateMode PhysicsLogic.Papers.OregRefaelVonOppen2010.HelicalWireFramework.Countermodels.zeroEnergyWithoutSelfConjugacy := PhysicsLogic.Papers.OregRefaelVonOppen2010.HelicalWireFramework.Countermodels.a_zero_energy_bound_state_is_not_automatically_majorana example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.OregRefaelVonOppen2010.Fidelity.countermodelMatrix := PhysicsLogic.Papers.OregRefaelVonOppen2010.Fidelity.countermodel_matrix_is_realized example (readout : PhysicsLogic.Papers.OregRefaelVonOppen2010.LinearInterfaceFormulaReadout) (formula : readout.SourceFormulaAt) : readout.minusOscillatorEnergySquaredAt 0 = 0 ∧ 0 < readout.plusOscillatorEnergySquaredAt 0 := readout.oscillator_has_one_zero_branch_and_one_gapped_branch formula example (readout : PhysicsLogic.Papers.OregRefaelVonOppen2010.TransportAndPairingSignatureReadout) (domain : readout.FormulaDomainAt) (periods : readout.FractionalJosephsonFluxPeriodsAt) : readout.fractionalJosephsonFluxPeriodValue = 2 * readout.conventionalJosephsonFluxPeriodValue := readout.source_fractional_flux_period_is_twice_conventional_period domain periods example {T : PhysicsLogic.Papers.OregRefaelVonOppen2010.OregTypes} (framework : PhysicsLogic.Papers.OregRefaelVonOppen2010.HelicalWireFramework T) (result : framework.CompleteScopedResult) : framework.SelectedBdGHamiltonianFormulaAt ∧ framework.SelectedCleanConductanceFormulaAt ∧ framework.SelectedLinearInterfaceFormulaAt .zeemanField framework.zeemanInterfaceHamiltonian framework.zeemanInterface framework.zeemanMode ∧ framework.SelectedLinearInterfaceFormulaAt .pairingGap framework.pairingInterfaceHamiltonian framework.pairingInterface framework.pairingMode ∧ framework.SelectedChemicalInterfaceMatchingFormulaAt ∧ framework.SelectedPairingSignDoubletFormulaAt ∧ framework.SelectedFractionalJosephsonFormulaAt := framework.complete_scoped_result_retains_all_original_source_formula_families result example {T : PhysicsLogic.Papers.OregRefaelVonOppen2010.OregTypes} (framework : PhysicsLogic.Papers.OregRefaelVonOppen2010.HelicalWireFramework T) (result : framework.ChemicalPotentialInterfaceResult) (descriptionValid : framework.descriptionValidInRegime framework.chemicalInterfaceHamiltonian framework.selectedRegime) (descriptionGoverns : framework.hamiltonianGovernsSystem framework.chemicalInterfaceHamiltonian framework.quantumWire framework.selectedRegime) (spinGapGoverns : framework.hamiltonianGovernsSystem framework.spinGapHamiltonian framework.quantumWire framework.selectedRegime) (superconductingGapGoverns : framework.hamiltonianGovernsSystem framework.superconductingGapHamiltonian framework.quantumWire framework.selectedRegime) : framework.toBoundaryMajoranaModeContext |>.BoundaryMajoranaRouteApplicableAt framework.boundaryMajoranaModel := framework.chemical_interface_inputs_make_shared_route_applicable result descriptionValid descriptionGoverns spinGapGoverns superconductingGapGoverns example : PhysicsLogic.Papers.Zerilli1970.Framework.Countermodels.realizesPrintedEvenMasterEquation PhysicsLogic.Papers.Zerilli1970.Framework.Countermodels.evenMasterEquationWithoutBoundaryAnalysis.masterEquation ∧ PhysicsLogic.Papers.Zerilli1970.Framework.Countermodels.evenMasterEquationWithoutBoundaryAnalysis.masterEquation.transformedK ≠ 0 ∧ ¬PhysicsLogic.Papers.Zerilli1970.Framework.Countermodels.evenBoundaryMismatchCalculatedAt PhysicsLogic.Papers.Zerilli1970.Framework.Countermodels.evenMasterEquationWithoutBoundaryAnalysis := PhysicsLogic.Papers.Zerilli1970.Framework.Countermodels.an_effective_potential_does_not_by_itself_prove_stability example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Zerilli1970.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Zerilli1970.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Zerilli1970.ZerilliTypes} (framework : PhysicsLogic.Papers.Zerilli1970.Framework T) (correctedSetup : framework.CorrectedEvenSystemSetup) (correctedCalculation : framework.CorrectedEvenSystemCalculation) (transformationValid : framework.validity.validAt framework.system .transformationConstruction) (transformationCalculation : framework.TransformationCalculation) (masterValid : framework.validity.validAt framework.system .evenMasterEquation) (masterCalculation : framework.EvenMasterCalculation) (applicationValid : framework.validity.validAt framework.system .applicationScope) (applicationAssessment : framework.ApplicationScopeAssessment) : framework.selectedSchwarzschildMasterEquationContext |>.SchwarzschildMasterEquationRouteApplicableAt framework.selectedSchwarzschildMasterEquationModel := framework.zerilli_source_inputs_make_shared_master_equation_route_applicable correctedSetup correctedCalculation transformationValid transformationCalculation masterValid masterCalculation applicationValid applicationAssessment example : PhysicsLogic.Papers.KomargodskiSchwimmer2011.ATheoremCountermodels.positiveCrossSectionWithoutAnomalySumRule.HasPositiveCrossSectionIntegral ∧ ¬PhysicsLogic.Papers.KomargodskiSchwimmer2011.ATheoremCountermodels.positiveCrossSectionWithoutAnomalySumRule.HasAnomalyDispersionSumRule := PhysicsLogic.Papers.KomargodskiSchwimmer2011.ATheoremCountermodels.positive_cross_section_does_not_supply_the_anomaly_dispersion_sum_rule example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.KomargodskiSchwimmer2011.Fidelity.countermodelMatrix := PhysicsLogic.Papers.KomargodskiSchwimmer2011.Fidelity.countermodel_matrix_is_realized example {Scale Mass : Type} (flow : PhysicsLogic.Papers.KomargodskiSchwimmer2011.RGFlowFramework Scale Mass) (lowEnergy : flow.DilatonLowEnergyTheorem) (dispersiveAssumptions : flow.HasDispersiveAssumptions) (positiveDecayConstant : flow.PositiveDecayConstant) (optical : flow.OpticalTheorem) (crossSection : flow.NontrivialPositiveCrossSection) (contour : flow.ContourDispersionCalculation) (conversion : flow.OpticalTheoremIntegralConversion) (integralPositivity : flow.PositiveCrossSectionIntegralCalculation) : flow.toForwardAmplitudePositivityContext |>.EstablishedForwardAmplitudePositivityAt () flow.forwardAmplitude flow.AnomalyDifference flow.totalCrossSection () := flow.dilaton_dispersion_establishes_shared_forward_amplitude_positivity lowEnergy dispersiveAssumptions positiveDecayConstant optical crossSection contour conversion integralPositivity example : PhysicsLogic.Papers.Miller1964.MillerFramework.Countermodels.millerDeterministicPointFailureWithoutEnsembleEvidence.HasFiniteResolutionDivergenceAt ∧ PhysicsLogic.Papers.Miller1964.MillerFramework.Countermodels.millerDeterministicPointFailureWithoutEnsembleEvidence.HasUniqueReferenceEvolutionAt ∧ ¬PhysicsLogic.Papers.Miller1964.MillerFramework.Countermodels.millerDeterministicPointFailureWithoutEnsembleEvidence.HasOnticIndeterminismAt ∧ ¬PhysicsLogic.Papers.Miller1964.MillerFramework.Countermodels.millerDeterministicPointFailureWithoutEnsembleEvidence.HasTestedEnsembleAgreementAt := PhysicsLogic.Papers.Miller1964.MillerFramework.Countermodels.miller_realizes_shared_deterministic_point_and_open_ensemble_boundary example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Miller1964.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Miller1964.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Miller1964.MillerTypes} (framework : PhysicsLogic.Papers.Miller1964.MillerFramework T) (setup : framework.ParallelExperimentSetup) (determinism : framework.StellarDynamicsDeterminismRealization) (calculation : framework.ParallelSeparationCalculation) (bridge : framework.PointPredictionReliabilityBridge) : framework.toSharedTrajectoryContext |>.EstablishedFinitePrecisionPredictabilityLimitAt framework.selectedFinitePrecisionTrajectoryModel := framework.miller_parallel_experiment_realizes_shared_predictability_limit setup determinism calculation bridge example : PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.HughesFramework.Countermodels.HasNuclearBoundAt PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.HughesFramework.Countermodels.reportedFiniteP2Experiment ∧ ¬PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.HughesFramework.Countermodels.EstablishesExactMassIsotropyAt PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.HughesFramework.Countermodels.reportedFiniteP2Experiment := by have witness := PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.HughesFramework.Countermodels.finite_nuclear_p2_bound_does_not_establish_unrestricted_isotropy exact ⟨witness.2.2.1, witness.2.2.2.2.1⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.Fidelity.countermodelMatrix := PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.HughesTypes} (framework : PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.HughesFramework T) (support : framework.toEvidenceSupportContext.SupportsModel framework.combinedNullEvidence framework.boundedSelectedAnisotropyModel framework.selectedRegime) (transfer : framework.toEvidenceSupportContext.ModelEvidenceTransfersToTheory framework.boundedSelectedAnisotropyModel framework.machianAnisotropyTheory framework.selectedRegime) : framework.toEvidenceSupportContext.SupportsTheory framework.combinedNullEvidence framework.machianAnisotropyTheory framework.selectedRegime := framework.model_support_reaches_theory_only_with_explicit_transfer support transfer example : PhysicsLogic.Papers.FeynmanCohen1956.CorrelatedExcitationFramework.Countermodels.controlledRotonEstimateWithoutExactSpectrum.HasVariationalRotonUpperBound ∧ ¬PhysicsLogic.Papers.FeynmanCohen1956.CorrelatedExcitationFramework.Countermodels.controlledRotonEstimateWithoutExactSpectrum.HasExactRotonEnergy := PhysicsLogic.Papers.FeynmanCohen1956.CorrelatedExcitationFramework.Countermodels.variational_upper_bound_does_not_supply_exact_sector_energy example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.FeynmanCohen1956.Fidelity.countermodelMatrix := PhysicsLogic.Papers.FeynmanCohen1956.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.FeynmanCohen1956.FeynmanCohenTypes} (framework : PhysicsLogic.Papers.FeynmanCohen1956.CorrelatedExcitationFramework T) (trial : framework.EstablishedBackflowTrial) (correlations : framework.EstablishedControlledCorrelationInputs) (normCalculation : framework.ImprovedNormCalculation) (numeratorCalculation : framework.ImprovedNumeratorCalculation) (variationalCalculation : framework.ImprovedVariationalCalculation) : framework.toMomentumResolvedVariationalBoundContext |>.MomentumResolvedVariationalRouteApplicableAt framework.sharedVariationalModel := framework.feynman_cohen_source_inputs_make_shared_variational_route_applicable trial correlations normCalculation numeratorCalculation variationalCalculation example : PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework.Countermodels.stiffnessRelationWithoutQuarterTransition.HasCalculatedStiffnessRelation ∧ ¬PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework.Countermodels.stiffnessRelationWithoutQuarterTransition.HasQuarterTransitionExponent := PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework.Countermodels.stiffness_relation_does_not_supply_quarter_transition_expectation example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.HalperinNelson1978.Fidelity.countermodelMatrix := PhysicsLogic.Papers.HalperinNelson1978.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.HalperinNelson1978.MeltingTypes} (framework : PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework T) (setup : framework.SmoothSolidSetup) (smoothCalculation : framework.SmoothSolidCalculation) (rg : framework.DislocationRGCalculation) (bridge : framework.DislocationUnbindingBridge) : framework.toDefectUnbindingTransitionContext |>.DefectUnbindingRouteApplicableAt framework.defectUnbindingModel := framework.dislocation_inputs_make_shared_unbinding_route_applicable setup smoothCalculation rg bridge example : PhysicsLogic.Papers.LindeLindeMezhlumian1993.StochasticInflationFramework.Countermodels.locallyStationaryWithGrowingPhysicalVolume.HasLocalStationarity ∧ ¬PhysicsLogic.Papers.LindeLindeMezhlumian1993.StochasticInflationFramework.Countermodels.locallyStationaryWithGrowingPhysicalVolume.HasGlobalVolumeStationarity := PhysicsLogic.Papers.LindeLindeMezhlumian1993.StochasticInflationFramework.Countermodels.local_stationarity_does_not_imply_global_stationarity example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.LindeLindeMezhlumian1993.Fidelity.countermodelMatrix := PhysicsLogic.Papers.LindeLindeMezhlumian1993.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.LindeLindeMezhlumian1993.LLMTypes} (framework : PhysicsLogic.Papers.LindeLindeMezhlumian1993.StochasticInflationFramework T) (setup : framework.OrdinaryDiffusionSetup) (spreadCalculation : framework.BrownianStepSpreadCalculation) (diffusionCalculation : framework.ComovingDiffusionCalculation) : framework.toRandomDisplacementDiffusionContext |>.RandomDisplacementDiffusionRouteApplicableAt framework.randomDisplacementModel := framework.llm_inputs_make_shared_random_displacement_route_applicable setup spreadCalculation diffusionCalculation example : (PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework.Countermodels.testBosonicEndpointAt PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework.Countermodels.discreteExchangePhasesWithoutContinuousFamily ∧ PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework.Countermodels.testFermionicEndpointAt PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework.Countermodels.discreteExchangePhasesWithoutContinuousFamily ∧ PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework.Countermodels.testIntermediateClassificationAt PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework.Countermodels.discreteExchangePhasesWithoutContinuousFamily ∧ PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework.Countermodels.testFractionalExchangePhaseAt PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework.Countermodels.discreteExchangePhasesWithoutContinuousFamily) ∧ ¬PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework.Countermodels.testContinuousStatisticsInterpolationAt PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework.Countermodels.discreteExchangePhasesWithoutContinuousFamily := by have witness := PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework.Countermodels.endpoint_intermediate_and_fractional_phase_do_not_supply_continuous_interpolation exact ⟨⟨witness.1, witness.2.1, witness.2.2.1, witness.2.2.2.1⟩, witness.2.2.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.WilczekFractionalSpin1982.Fidelity.countermodelMatrix := PhysicsLogic.Papers.WilczekFractionalSpin1982.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.WilczekFractionalSpin1982.WilczekTypes} (framework : PhysicsLogic.Papers.WilczekFractionalSpin1982.FractionalSpinFramework T) (setup : framework.TwoAnyonExchangeSetup) (calculation : framework.TwoAnyonPhaseCalculation) (intermediate : framework.IntermediateStatisticsParameter) (classification : framework.FractionalStatisticsBridge) : framework.toPlanarExchangeStatisticsContext |>.RouteApplicableAt framework.planarExchangeStatisticsModel := framework.two_anyon_inputs_make_shared_planar_exchange_route_applicable setup calculation intermediate classification example : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework.Countermodels.PeierlsSuppressionAvailable PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework.Countermodels.suppressedContourWithoutAnalyticity ∧ ¬PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework.Countermodels.MetastableAnalyticityAvailable PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework.Countermodels.suppressedContourWithoutAnalyticity := PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework.Countermodels.peierls_suppression_does_not_supply_metastable_analyticity example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.Fidelity.countermodelMatrix := PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroTypes} (framework : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework T) (phaseSum : framework.ControlledTorusPhaseSumResult) (physical : framework.physicalControl framework.selectedControl) (contained : framework.contains framework.selectedRegion framework.selectedControl) (control : framework.PartitionZeroControlResult) (bridge : framework.PhysicalSingularityBoundaryBridge) : framework.toPartitionZeroPhaseBoundaryContext |>.PartitionZeroPhaseBoundaryRouteApplicableAt framework.partitionZeroPhaseBoundaryModel := framework.bbckk_inputs_make_shared_partition_zero_route_applicable phaseSum physical contained control bridge example : PhysicsLogic.Papers.OrdonezRayVanKolck1995.ChiralPotentialFramework.Countermodels.HasIrreduciblePotentialAt PhysicsLogic.Papers.OrdonezRayVanKolck1995.ChiralPotentialFramework.Countermodels.reducedChiralPotentialScope ∧ ¬PhysicsLogic.Papers.OrdonezRayVanKolck1995.ChiralPotentialFramework.Countermodels.HasFullAmplitudeAt PhysicsLogic.Papers.OrdonezRayVanKolck1995.ChiralPotentialFramework.Countermodels.reducedChiralPotentialScope := PhysicsLogic.Papers.OrdonezRayVanKolck1995.ChiralPotentialFramework.Countermodels.irreducible_potential_without_iteration_does_not_supply_full_amplitude example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.OrdonezRayVanKolck1995.Fidelity.countermodelMatrix := PhysicsLogic.Papers.OrdonezRayVanKolck1995.Fidelity.countermodel_matrix_is_realized example {DegreeOfFreedom Contribution Potential Order Regime Regulator Channel Observable Value Fit : Type} (framework : PhysicsLogic.Papers.OrdonezRayVanKolck1995.ChiralPotentialFramework DegreeOfFreedom Contribution Potential Order Regime Regulator Channel Observable Value Fit) (counting : framework.ChiralPowerCountingCalculation) (organization : framework.IrreduciblePotentialOrganization) : framework.toEffectiveInteractionDynamicsContext |>.MixedInteractionDynamicsRouteAt framework.sharedEffectiveInteractionModel := framework.chiral_potential_inputs_make_shared_mixed_dynamics_route_applicable counting organization example : PhysicsLogic.Papers.Eardley1975.BinaryPulsarFramework.Countermodels.calculatedDipoleWithoutObservedExcess.HasCompletedScalarDipoleCalculation ∧ ¬PhysicsLogic.Papers.Eardley1975.BinaryPulsarFramework.Countermodels.calculatedDipoleWithoutObservedExcess.HasObservedExcessRadiation := PhysicsLogic.Papers.Eardley1975.BinaryPulsarFramework.Countermodels.calculated_scalar_dipole_does_not_establish_an_observed_binary_excess example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Eardley1975.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Eardley1975.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Eardley1975.BinaryPulsarTypes} (framework : PhysicsLogic.Papers.Eardley1975.BinaryPulsarFramework T) (generalSetup : framework.GeneralizedEIHSetup) (massCalculation : framework.FieldDependentMassCalculation) (sensitivitySetup : framework.CompactSensitivitySetup) (neutronCalculation : framework.NeutronStarSensitivityCalculation) (blackHoleCalculation : framework.BlackHoleAndOrdinarySensitivityCalculation) (dipoleSetup : framework.DipoleRadiationSetup) (dipoleCalculation : framework.DipoleRadiationCalculation) : framework.toBodyDependentSelfGravityContext |>.BodyDependentSelfGravityRouteApplicableAt framework.sharedBodyDependentSelfGravityModel := framework.compact_sensitivity_and_dipole_routes_make_shared_route_applicable generalSetup massCalculation sensitivitySetup neutronCalculation blackHoleCalculation dipoleSetup dipoleCalculation namespace EardleyOriginalScalarDipoleClockReuseAudit open PhysicsLogic.Papers.Eardley1975 open PhysicsLogic.Papers.Eardley1975.SourceFormulaCountermodels example : physicalOrdinaryMatterPair.SourceFormulaAt := ordinary_matter_source_scalar_dipole_is_nonvacuous example : physicalUnequalCompactPair.SourceFormulaAt := unequal_compact_source_scalar_dipole_is_nonvacuous example : physicalIdenticalCompactPair.SourceFormulaAt := identical_compact_source_scalar_dipole_is_nonvacuous example : physicalBlackHoleCompanionPair.SourceFormulaAt := black_hole_companion_source_scalar_dipole_is_nonvacuous example : physicalEccentricCompactPair.SourceFormulaAt := eccentric_compact_source_scalar_dipole_is_nonvacuous example : physicalPsr1913Estimate.SourceFormulaAt := original_psr_observations_and_rounded_decay_are_nonvacuous example : physicalNonblackHolePulsarClock.SourceFormulaAt := nonblack_hole_companion_pulsar_clock_is_nonvacuous example : physicalBlackHolePulsarClock.SourceFormulaAt := black_hole_companion_pulsar_clock_is_nonvacuous example : physicalBlackHoleCompanionPair.secondScalarMassSolarValue = 0 /\ physicalBlackHoleCompanionPair.fractionalOrbitalDecayPerYearValue < 0 /\ physicalBlackHolePulsarClock.fractionalLocalGravityVariationValue = 0 /\ physicalBlackHolePulsarClock.fractionalPulsePeriodVariationValue = 0 := by obtain ⟨_, _, _, scalar, _, decay, gravity, pulse⟩ := black_hole_companion_suppresses_clock_without_suppressing_dipole exact ⟨scalar, decay, gravity, pulse⟩ example : ¬namedScalarChargeIgnoringCompactSensitivity.SourceFormulaAt := named_scalar_mass_does_not_supply_sensitivity_weighted_scalar_charge.2.2 example : ¬pairForceMissingSensitivityCrossTerm.SourceFormulaAt := sensitivity_labels_do_not_supply_the_force_cross_term.2.2.2 example : ¬identicalSensitivitiesWithInventedDipole.SourceFormulaAt := identical_compact_sensitivities_do_not_support_a_nonzero_dipole.2.2 example : ¬eccentricOrbitWithoutPrintedEnhancement.SourceFormulaAt := eccentric_binary_decay_requires_the_printed_eccentricity_factor.2.2 example : ¬observedPsrMassFunctionMistakenForReducedMass.SourceFormulaAt := measured_psr_mass_function_is_not_the_binary_reduced_mass.2.2 example : ¬blackHoleCompanionWithInventedClockSignal.SourceFormulaAt := black_hole_companion_cannot_support_a_nonzero_local_clock_signal.2.2 example : ¬localGravityWithWrongSignedPulseResponse.SourceFormulaAt := local_gravity_variation_does_not_allow_the_wrong_pulse_response_sign.2.2 example (row : NeutronStarTableRow) : 0 < row.massSolarValue /\ 0 < row.sensitivityValue /\ row.sensitivityValue < 1 := row.every_printed_neutron_star_has_positive_subunit_sensitivity example : NeutronStarTableRow.massSolarValue .maximumMass141 = 141 / 100 /\ NeutronStarTableRow.sensitivityValue .maximumMass141 = 78 / 100 := NeutronStarTableRow.source_table_retains_selected_and_maximum_mass_rows.2.2 example (readout : ScalarDipoleFormulaReadout) (source : readout.SourceFormulaAt) (equal : readout.firstSensitivityValue = readout.secondSensitivityValue) : readout.scalarDipoleMomentValue = 0 /\ readout.fractionalOrbitalDecayPerYearValue = 0 := ⟨readout.equal_sensitivities_cancel_center_of_mass_scalar_dipole source equal, readout.equal_sensitivities_cancel_scalar_dipole_period_decay source equal⟩ example (readout : ScalarDipoleFormulaReadout) (source : readout.SourceFormulaAt) (different : readout.firstSensitivityValue ≠ readout.secondSensitivityValue) : readout.scalarDipoleMomentValue ≠ 0 := readout.unequal_sensitivities_give_nonzero_center_of_mass_scalar_dipole source different example (readout : ScalarDipoleFormulaReadout) (source : readout.SourceFormulaAt) (different : readout.firstSensitivityValue ≠ readout.secondSensitivityValue) : readout.fractionalOrbitalDecayPerYearValue < 0 := readout.unequal_sensitivities_make_scalar_dipole_period_decay_negative source different example (readout : PulsarClockFormulaReadout) (source : readout.SourceFormulaAt) : readout.fractionalLocalGravityVariationValue / 2 ≤ -readout.fractionalPulsePeriodVariationValue := readout.equation_thirteen_and_stellar_bound_give_equation_fourteen source example (readout : PulsarClockFormulaReadout) (source : readout.SourceFormulaAt) (blackHole : readout.companionSensitivityValue = 1) : readout.fractionalLocalGravityVariationValue = 0 /\ readout.fractionalPulsePeriodVariationValue = 0 := readout.black_hole_companion_eliminates_local_gravity_and_clock_variation source blackHole example {T : BinaryPulsarTypes} (framework : BinaryPulsarFramework T) (result : framework.DipoleDecayResult) : framework.ScalarDipoleSourceFormulaAt /\ framework.PsrDipoleSourceFormulaAt := framework.dipole_decay_result_retains_scalar_charge_eccentric_decay_and_psr_estimate result example {T : BinaryPulsarTypes} (framework : BinaryPulsarFramework T) (result : framework.DipoleDecayResult) : framework.scalarDipoleFormulaReadout.scalarDipoleMomentValue ≠ 0 /\ framework.scalarDipoleFormulaReadout.fractionalOrbitalDecayPerYearValue < 0 := framework.selected_dipole_result_establishes_nonzero_dipole_and_negative_decay result example {T : BinaryPulsarTypes} (framework : BinaryPulsarFramework T) (result : framework.DipoleDecayResult) (identical : framework.bodySensitivityValueAt framework.pulsar = framework.bodySensitivityValueAt framework.companion) : framework.psrDipoleEstimateFormulaReadout.fractionalOrbitalDecayPerYearValue = 0 := framework.identical_psr_compact_sensitivities_cancel_selected_decay result identical example {T : BinaryPulsarTypes} (framework : BinaryPulsarFramework T) (observed : framework.PsrObservedParameters) : framework.psrDipoleEstimateFormulaReadout.observedOrbitalPeriodHoursValue = 31 / 4 /\ framework.psrDipoleEstimateFormulaReadout.observedMassFunctionSolarValue = 13 / 100 /\ framework.psrDipoleEstimateFormulaReadout.observedOrbitalEccentricityValue = 123 / 200 := framework.psr_observed_parameters_retain_original_period_mass_function_and_eccentricity observed example {T : BinaryPulsarTypes} (framework : BinaryPulsarFramework T) (result : framework.VaryingGClockResult) : framework.PulsarClockSourceFormulaAt /\ framework.pulsarClockFormulaReadout.fractionalLocalGravityVariationValue / 2 ≤ -framework.pulsarClockFormulaReadout.fractionalPulsePeriodVariationValue := ⟨framework.clock_result_retains_printed_gravity_inertia_and_pulse_equations result, framework.clock_source_result_derives_printed_half_variation_bound result⟩ example {T : BinaryPulsarTypes} (framework : BinaryPulsarFramework T) (dipole : framework.DipoleDecayResult) (clock : framework.VaryingGClockResult) (blackHole : framework.bodySensitivityValueAt framework.companion = 1) (pulsarDifferent : framework.bodySensitivityValueAt framework.pulsar ≠ 1) : framework.psrDipoleEstimateFormulaReadout.fractionalOrbitalDecayPerYearValue < 0 /\ framework.pulsarClockFormulaReadout.fractionalLocalGravityVariationValue = 0 /\ framework.pulsarClockFormulaReadout.fractionalPulsePeriodVariationValue = 0 := framework.black_hole_psr_companion_suppresses_clock_but_preserves_negative_decay dipole clock blackHole pulsarDifferent example {T : BinaryPulsarTypes} (context : SemanticContext T) (dipole : context.interpretAtom .dipoleDecayResult) : context.framework.scalarDipoleFormulaReadout.scalarDipoleMomentValue ≠ 0 /\ context.framework.scalarDipoleFormulaReadout.fractionalOrbitalDecayPerYearValue < 0 := context.dipole_result_atom_proves_nonzero_scalar_dipole_and_negative_decay dipole example {T : BinaryPulsarTypes} (context : SemanticContext T) (dipole : context.interpretAtom .dipoleDecayResult) (clock : context.interpretAtom .varyingGClockResult) (blackHole : context.framework.bodySensitivityValueAt context.framework.companion = 1) (pulsarDifferent : context.framework.bodySensitivityValueAt context.framework.pulsar ≠ 1) : context.framework.psrDipoleEstimateFormulaReadout.fractionalOrbitalDecayPerYearValue < 0 /\ context.framework.pulsarClockFormulaReadout.fractionalLocalGravityVariationValue = 0 /\ context.framework.pulsarClockFormulaReadout.fractionalPulsePeriodVariationValue = 0 := context.black_hole_psr_atoms_suppress_clock_without_suppressing_orbital_decay dipole clock blackHole pulsarDifferent end EardleyOriginalScalarDipoleClockReuseAudit example : PhysicsLogic.Papers.AdamsEtAl2006.PaperFramework.Countermodels.HasNonGravitationalForwardResultAt PhysicsLogic.Papers.AdamsEtAl2006.PaperFramework.Countermodels.nonGravitationalResultWithMasslessGravitonPole ∧ PhysicsLogic.Papers.AdamsEtAl2006.PaperFramework.Countermodels.HasMasslessGravitonForwardPoleAt PhysicsLogic.Papers.AdamsEtAl2006.PaperFramework.Countermodels.nonGravitationalResultWithMasslessGravitonPole ∧ ¬PhysicsLogic.Papers.AdamsEtAl2006.PaperFramework.Countermodels.HasGravitationalForwardExtensionAt PhysicsLogic.Papers.AdamsEtAl2006.PaperFramework.Countermodels.nonGravitationalResultWithMasslessGravitonPole := PhysicsLogic.Papers.AdamsEtAl2006.PaperFramework.Countermodels.the_non_gravitational_forward_argument_does_not_cover_massless_gravity example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.AdamsEtAl2006.Fidelity.countermodelMatrix := PhysicsLogic.Papers.AdamsEtAl2006.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.AdamsEtAl2006.AdamsTypes} (framework : PhysicsLogic.Papers.AdamsEtAl2006.PaperFramework T) (setup : framework.RegulatedForwardAmplitudeSetup) (dispersion : framework.AnalyticDispersionCalculation) (optical : framework.OpticalTheoremCalculation) (integral : framework.PositiveWeightedIntegralCalculation) (sign : framework.SumRuleSignCalculation) : framework.toSharedContext.EstablishedForwardAmplitudePositivityAt framework.forwardChannel framework.forwardAmplitude framework.leadingDerivativeCoefficient framework.regulatedSpectralData framework.regulatedDispersiveRegime := framework.paper_calculations_establish_shared_forward_amplitude_positivity setup dispersion optical integral sign example : PhysicsLogic.Papers.AbrahamsAndersonLicciardelloRamakrishnan1979.Fidelity.countermodelMatrix .continuityNotMonotonicity := PhysicsLogic.Papers.AbrahamsAndersonLicciardelloRamakrishnan1979.Framework.Countermodels.continuity_does_not_force_monotonicity example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.AbrahamsAndersonLicciardelloRamakrishnan1979.Fidelity.countermodelMatrix := PhysicsLogic.Papers.AbrahamsAndersonLicciardelloRamakrishnan1979.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.MacKinnonKramer1981.Fidelity.countermodelMatrix .finiteSizeCollapseNotMicroscopicDerivation := PhysicsLogic.Papers.MacKinnonKramer1981.Framework.Countermodels.finite_size_collapse_does_not_prove_one_parameter_scaling example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.MacKinnonKramer1981.Fidelity.countermodelMatrix := PhysicsLogic.Papers.MacKinnonKramer1981.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.Young1979.Fidelity.countermodelMatrix .selectedRecursionNotFullSubstrateEquivalence := PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework.DependencyCountermodels.single_recursion_coefficient_does_not_establish_full_substrate_equivalence example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Young1979.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Young1979.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.HawkingOpenUniverses1965.Fidelity.countermodelMatrix .nonnegativeConvergenceNotStrictMargin := PhysicsLogic.Papers.HawkingOpenUniverses1965.RobertsonWalkerGeometry.the_marginal_flat_sphere_has_zero_outward_convergence example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.HawkingOpenUniverses1965.Fidelity.countermodelMatrix := PhysicsLogic.Papers.HawkingOpenUniverses1965.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.Vishveshwara1970.Fidelity.countermodelMatrix .linearWavePacketsNotNonlinearStability := PhysicsLogic.Papers.Vishveshwara1970.Framework.Countermodels.regular_linear_wave_packets_do_not_prove_nonlinear_stability example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Vishveshwara1970.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Vishveshwara1970.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.EnglertBrout1964.Fidelity.countermodelMatrix .brokenGeneratorMassNotEveryGaugeMass := by have witness := PhysicsLogic.Papers.EnglertBrout1964.Framework.Countermodels.one_broken_generator_does_not_make_every_gauge_field_massive exact ⟨⟨witness.1, witness.2.1, witness.2.2.1⟩, witness.2.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.EnglertBrout1964.Fidelity.countermodelMatrix := PhysicsLogic.Papers.EnglertBrout1964.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.LiuSinclairSrivastava2018.Fidelity.countermodelMatrix .nonzeroPinnedContributionsNotParentNoncancellation := PhysicsLogic.Papers.LiuSinclairSrivastava2018.FisherZeroFramework.Countermodels.nonzero_pinned_contributions_do_not_exclude_parent_cancellation example : PhysicsLogic.Papers.LiuSinclairSrivastava2018.Fidelity.countermodelMatrix .disagreementWeightedSpinSumNotUnweightedCount := PhysicsLogic.Papers.LiuSinclairSrivastava2018.FisherZeroFramework.Countermodels.ising_configuration_sum_is_not_unweighted_spin_count example : PhysicsLogic.Papers.LiuSinclairSrivastava2018.Fidelity.countermodelMatrix .pinnedConfigurationSumNotUnpinnedSum := PhysicsLogic.Papers.LiuSinclairSrivastava2018.FisherZeroFramework.Countermodels.pinned_ising_configuration_sum_is_not_unpinned_sum example : PhysicsLogic.Papers.LiuSinclairSrivastava2018.Fidelity.countermodelMatrix .unpinnedNonvanishingNotArbitraryPinnedNonvanishing := PhysicsLogic.Papers.LiuSinclairSrivastava2018.FisherZeroFramework.Countermodels.unpinned_configuration_sum_does_not_imply_arbitrary_pinned_nonvanishing example : PhysicsLogic.Papers.LiuSinclairSrivastava2018.Fidelity.countermodelMatrix .pinnedNonvanishingNotUnpinnedNonvanishing := PhysicsLogic.Papers.LiuSinclairSrivastava2018.FisherZeroFramework.Countermodels.nonzero_pinned_configuration_does_not_prevent_unpinned_cancellation example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.LiuSinclairSrivastava2018.Fidelity.countermodelMatrix := PhysicsLogic.Papers.LiuSinclairSrivastava2018.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.LiuSinclairSrivastava2018.FisherZeroTypes} (framework : PhysicsLogic.Papers.LiuSinclairSrivastava2018.FisherZeroFramework T) (result : framework.FisherZeroFreeNeighborhood) (interaction : T.Interaction) (graph : T.Graph) (nearby : framework.withinComplexRadius interaction framework.selectedRealInteraction framework.selectedRadius) (setup : framework.IsingGraphSetupAt graph) : (framework.finiteConfigurationGraphAt graph).pinnedIsingPartitionValue (framework.pinnedSpinAt graph) (framework.complexInteractionValue interaction) ≠ 0 := framework.fisher_neighborhood_establishes_actual_pinned_ising_sum_nonvanishing result interaction graph nearby setup example : PhysicsLogic.Papers.PetersRegts2017.Fidelity.countermodelMatrix .univariateDomainNotMultivariateProductNonvanishing := by have witness := PhysicsLogic.Papers.PetersRegts2017.HardCoreZeroFramework.Countermodels.univariate_domain_does_not_imply_literal_multivariate_product_domain exact ⟨⟨witness.1, witness.2.1⟩, witness.2.2⟩ example : PhysicsLogic.Papers.PetersRegts2017.Fidelity.countermodelMatrix .independentSetSumNotUnconstrainedOccupationSum := PhysicsLogic.Papers.PetersRegts2017.HardCoreZeroFramework.Countermodels.independent_set_sum_is_not_unrestricted_binary_configuration_sum example : PhysicsLogic.Papers.PetersRegts2017.Fidelity.countermodelMatrix .constantActivityNonvanishingNotMultivariateNonvanishing := PhysicsLogic.Papers.PetersRegts2017.HardCoreZeroFramework.Countermodels.constant_activity_sum_does_not_control_arbitrary_multivariate_sum example : PhysicsLogic.Papers.PetersRegts2017.Fidelity.countermodelMatrix .finiteConfigurationSumNotUncontrolledNonvanishing := PhysicsLogic.Papers.PetersRegts2017.HardCoreZeroFramework.Countermodels.finite_independent_set_sum_does_not_imply_uncontrolled_nonvanishing example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.PetersRegts2017.Fidelity.countermodelMatrix := PhysicsLogic.Papers.PetersRegts2017.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.PetersRegts2017.HardCoreZeroTypes} (framework : PhysicsLogic.Papers.PetersRegts2017.HardCoreZeroFramework T) (result : framework.SokalConjectureResult) (activity : T.Activity) (graph : T.Graph) (inDomain : framework.activityBelongsToDomain activity framework.selectedDomain) (setup : framework.GraphSetupAt graph) : (framework.finiteConfigurationGraphAt graph).hardCorePartitionValue (fun _ => framework.complexActivityValue activity) ≠ 0 := framework.sokal_domain_establishes_actual_univariate_independence_polynomial_nonvanishing result activity graph inDomain setup example {T : PhysicsLogic.Papers.AbrahamsAndersonLicciardelloRamakrishnan1979.ScalingTheoryTypes} (framework : PhysicsLogic.Papers.AbrahamsAndersonLicciardelloRamakrishnan1979.Framework T) (identification : framework.EstablishedFiniteScaleConductanceIdentification) (flow : framework.EstablishedOneParameterFlow) (asymptotics : framework.EstablishedLocalizedAndDiffusiveAsymptotics) (topology : framework.EstablishedRegularFlowTopology) (threeDimensional : framework.EstablishedThreeDimensionalCriticality) (twoDimensional : framework.EstablishedTwoDimensionalNoTrueMetal) (scope : framework.EstablishedQualifiedScope) (distinct : framework.DistinctDimensionalSectors) : framework.toOneParameterLocalizationScalingContext |>.RouteApplicableAt framework.sharedLocalizationScalingModel := framework.source_inputs_make_shared_scaling_route_applicable identification flow asymptotics topology threeDimensional twoDimensional scope distinct example {T : PhysicsLogic.Papers.MacKinnonKramer1981.NumericalScalingTypes} (framework : PhysicsLogic.Papers.MacKinnonKramer1981.Framework T) (source : framework.macKinnonKramerResolventDecayDerivationObtains .stabilizedRecursiveResolventAndSampledFiniteSizeLocalization) : framework.toResolventDecayLocalizationContext |>.ResolventDecayLocalizationRouteApplicableAt framework.sharedResolventDecayLocalizationModel := framework.macKinnonKramer_source_inputs_make_shared_resolvent_decay_route_applicable source example {T : PhysicsLogic.Papers.Young1979.MeltingTypes} (framework : PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework T) (solidSetup : framework.SolidSetup) (recursionSetup : framework.RecursionSetup) (recursionCalculation : framework.RecursionCalculation) (flowCalculation : framework.FlowCalculation) (bridge : framework.UnbindingBridge) : framework.toDefectUnbindingTransitionContext |>.DefectUnbindingRouteApplicableAt framework.defectUnbindingModel := framework.young_inputs_make_shared_unbinding_route_applicable solidSetup recursionSetup recursionCalculation flowCalculation bridge example {T : PhysicsLogic.Papers.HawkingOpenUniverses1965.HawkingTypes} (framework : PhysicsLogic.Papers.HawkingOpenUniverses1965.OpenUniverseFramework T) (setup : framework.OpenUniverseModelSetup) (calculation : framework.LargeSpherePastConvergenceCalculation) (global : framework.GlobalCausalSetupAt framework.homogeneousSpacetime framework.homogeneousCauchySurface framework.homogeneousLargeSphere framework.homogeneousPastBoundary) (focusing : framework.ImportedFocusingCalculationAt framework.homogeneousSpacetime framework.homogeneousCauchySurface framework.homogeneousLargeSphere framework.homogeneousPastBoundary) (compactBoundary : framework.ImportedCompactBoundaryCalculationAt framework.homogeneousSpacetime framework.homogeneousCauchySurface framework.homogeneousLargeSphere framework.homogeneousPastBoundary) (topology : framework.ImportedTopologyObstructionAt framework.homogeneousSpacetime framework.homogeneousCauchySurface framework.homogeneousLargeSphere framework.homogeneousPastBoundary) : framework.toTrappedSurfaceIncompletenessContext |>.TrappedSurfaceIncompletenessRouteApplicableAt framework.incompletenessModel := framework.hawking_large_sphere_source_inputs_make_shared_incompleteness_route_applicable setup calculation global focusing compactBoundary topology example {T : PhysicsLogic.Papers.Vishveshwara1970.VishveshwaraTypes} (framework : PhysicsLogic.Papers.Vishveshwara1970.Framework T) (setup : framework.OddHigherModeSetup) : framework.selectedSchwarzschildMasterEquationContext |>.SchwarzschildMasterEquationRouteApplicableAt framework.selectedSchwarzschildMasterEquationModel := framework.vishveshwara_odd_setup_makes_shared_master_equation_route_applicable setup example {T : PhysicsLogic.Papers.Vishveshwara1970.VishveshwaraTypes} (framework : PhysicsLogic.Papers.Vishveshwara1970.Framework T) (kruskal : framework.EstablishedKruskalProblem) (oddHigher : framework.EstablishedOddHigherModeStability) (oddDipole : framework.EstablishedOddDipoleGaugeRemoval) (even : framework.EstablishedEvenModeStability) (realPackets : framework.EstablishedRealFrequencyWavePackets) (scope : framework.EstablishedCompletenessScope) : framework.selectedOddMasterBoundaryHandoff |>.NonvacuousMasterGroundedScopedSchwarzschildLinearModeStabilityAt := framework.vishveshwara_master_boundary_route_realizes_composed_scoped_stability kruskal oddHigher oddDipole even realPackets scope example {T : PhysicsLogic.Papers.Vishveshwara1970.VishveshwaraTypes} (framework : PhysicsLogic.Papers.Vishveshwara1970.Framework T) (kruskal : framework.EstablishedKruskalProblem) (oddHigher : framework.EstablishedOddHigherModeStability) (oddDipole : framework.EstablishedOddDipoleGaugeRemoval) (even : framework.EstablishedEvenModeStability) (realPackets : framework.EstablishedRealFrequencyWavePackets) (scope : framework.EstablishedCompletenessScope) : framework.selectedOddMasterBoundaryHandoff |>.SourceSensitiveMasterGroundedScopedSchwarzschildLinearModeStabilityAt framework.selectedSchwarzschildMasterEquationContext framework.selectedSchwarzschildLinearModeStabilityContext .adoptedPriorResult .adoptedReggeWheelerModesCorrectedRadialAndPrintedOddEquations framework.vishveshwaraAdoptedOddMasterSourceObtains framework.vishveshwaraIndependentOddMasterDerivationObtains .kruskalRegularityOddDipoleGaugeAndEvenBoundaryDivergence framework.vishveshwaraGrowingModeExclusionDerivationObtains .adoptedOddMasterCorrectedKruskalDivergenceAndIncompleteEvenScope framework.vishveshwaraMasterBoundaryHandoffDerivationObtains := framework |>.adopted_master_and_kruskal_boundary_realize_source_sensitive_scoped_stability kruskal oddHigher oddDipole even realPackets scope namespace SchwarzschildMasterSourceOrderReuseAudit open PhysicsLogic.Framework.SchwarzschildOddMasterBoundaryHandoff.Countermodels example : (distinctStageHandoff |>.SelectedOddMasterSourceEvidenceAt distinctStageMasterContext .adoptedPriorResult () availableMasterBoundaryChildDerivationObtains absentIndependentMasterDerivationObtains) ∧ ¬(distinctStageHandoff |>.SelectedOddMasterSourceEvidenceAt distinctStageMasterContext .independentlyDerived () availableMasterBoundaryChildDerivationObtains absentIndependentMasterDerivationObtains) := original_and_adopted_master_sources_cannot_be_identified.2 example : distinctStageHandoff.masterModel.frequency ≠ distinctStageHandoff.stabilityModel.growingFrequency ∧ ¬(distinctStageHandoff |>.SourceSensitiveMasterGroundedScopedSchwarzschildLinearModeStabilityAt distinctStageMasterContext distinctStageStabilityContext .independentlyDerived () availableMasterBoundaryChildDerivationObtains availableMasterBoundaryChildDerivationObtains () availableMasterBoundaryChildDerivationObtains .absent missingMasterBoundaryHandoffDerivationObtains) := by have witness := completed_master_stability_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition exact ⟨witness.2.1, witness.2.2.2.2.2⟩ end SchwarzschildMasterSourceOrderReuseAudit example {T : PhysicsLogic.Papers.EnglertBrout1964.EnglertBroutTypes} (framework : PhysicsLogic.Papers.EnglertBrout1964.Framework T) (vacuumSetup : framework.BrokenVacuumSetup) (vacuumAssessment : framework.BrokenVacuumAssessment) (abelianLocalSetup : framework.AbelianMassLocalSetup) (abelianCalculation : framework.AbelianPolarizationCalculation) (compactLocalSetup : framework.CompactGroupMassLocalSetup) (compactCalculation : framework.CompactGroupMassCalculation) (chiralLocalSetup : framework.ChiralModelLocalSetup) (chiralCalculation : framework.ChiralWardCalculation) (rotatedLocalSetup : framework.RotatedVacuumInterpretationLocalSetup) (rotatedBridge : framework.RotatedVacuumInterpretationBridge) (scopeCalculation : framework.ScopeAssessmentCalculation) : framework.selectedGaugeFieldMassGenerationContext |>.GaugeFieldMassGenerationRouteApplicableAt framework.selectedGaugeFieldMassGenerationModel := framework.englert_brout_source_inputs_make_shared_gauge_mass_route_applicable vacuumSetup vacuumAssessment abelianLocalSetup abelianCalculation compactLocalSetup compactCalculation chiralLocalSetup chiralCalculation rotatedLocalSetup rotatedBridge scopeCalculation example {T : PhysicsLogic.Papers.LiuSinclairSrivastava2018.FisherZeroTypes} (framework : PhysicsLogic.Papers.LiuSinclairSrivastava2018.FisherZeroFramework T) (saw : framework.SAWTreeRecurrenceResult) (domain : framework.ComplexInvariantDomainResult) (controls : framework.RecurrenceDomainControlsPinnedRatio) (cancellation : framework.PinnedCancellationCalculation) (base : framework.PinnedBaseCaseCalculation) (step : framework.PinnedInductionStepCalculation) (interaction : T.Interaction) (nearby : framework.withinComplexRadius interaction framework.selectedRealInteraction framework.selectedRadius) : framework.toFinitePartitionNonvanishingContext interaction |>.FinitePartitionNonvanishingRouteApplicableAt (framework.finitePartitionNonvanishingModelAt interaction) := framework.pinned_base_and_step_make_shared_nonvanishing_route_applicable saw domain controls cancellation base step interaction nearby example {T : PhysicsLogic.Papers.PetersRegts2017.HardCoreZeroTypes} (framework : PhysicsLogic.Papers.PetersRegts2017.HardCoreZeroFramework T) (ratioIdentity : framework.RatioIdentityCalculation) (recurrence : framework.WeitzOccupationRatioResult) (domain : framework.ComplexInvariantDomainResult) (controls : framework.RecurrenceDomainControlsOccupationRatio) (cancellation : framework.PartitionCancellationCalculation) (base : framework.EmptyGraphBaseCalculation) (step : framework.VertexSubsetInductionStepCalculation) (assignment : T.Assignment) : framework.toFinitePartitionNonvanishingContext assignment |>.FinitePartitionNonvanishingRouteApplicableAt (framework.finitePartitionNonvanishingModelAt assignment) := framework.base_and_vertex_subset_step_make_shared_nonvanishing_route_applicable ratioIdentity recurrence domain controls cancellation base step assignment example : PhysicsLogic.Papers.Adler1969.Fidelity.countermodelMatrix .namedAnomalousTermNotPrintedDivergence := PhysicsLogic.Papers.Adler1969.AxialVertexFramework.Countermodels.anomalous_term_label_does_not_supply_equation_thirty example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Adler1969.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Adler1969.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.Rosen1940.Fidelity.countermodelMatrix .formalAuxiliaryMetricNotPhysicalGeometry := PhysicsLogic.Papers.Rosen1940.RosenFramework.Countermodels.formal_auxiliary_status_does_not_supply_physical_geometry example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Rosen1940.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Rosen1940.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.Rosen1940.Fidelity.countermodelMatrix .equalMassLabelsNotPrintedRestFrameFormula := PhysicsLogic.Papers.Rosen1940.RosenFramework.Countermodels.equal_mass_labels_do_not_supply_printed_rest_frame_mass_formula example : PhysicsLogic.Papers.Rosen1940.Fidelity.countermodelMatrix .equalInertialGravitationalMassNotConstantMass := by obtain ⟨massLaw, equalMasses, notConstant⟩ := PhysicsLogic.Papers.Rosen1940.RosenFramework.Countermodels.equal_inertial_and_gravitational_mass_does_not_imply_constant_mass exact ⟨⟨massLaw, equalMasses⟩, notConstant⟩ example {T : PhysicsLogic.Papers.Rosen1940.Rosen1940Types} (framework : PhysicsLogic.Papers.Rosen1940.RosenFramework T) (motion : framework.FlatSpaceMotionResult) : framework.SelectedPartIIMotionFormulaAt := framework.variable_mass_motion_retains_part_ii_printed_mass_formulas motion namespace OriginalGravitationalTensorFormulaReuseAudit open PhysicsLogic.Papers.Einstein1916.RelativityFramework.Countermodels open PhysicsLogic.Papers.Rosen1940.PartITensorFormulaReadout.Countermodels example {T : PhysicsLogic.Papers.Einstein1916.RelativityTypes} (framework : PhysicsLogic.Papers.Einstein1916.RelativityFramework T) (metric : T.Metric) (situation : T.Situation) (geometry : framework.metricDeterminesSpacetimeGeometry metric situation) (physical : framework.physicalMetric metric) (calculation : framework.CurvatureTensorCalculation metric situation) : framework.EquationFortyThreeRiemannChristoffelFormulaAt metric := framework.constructed_curvature_retains_equation_forty_three metric situation geometry physical calculation example {T : PhysicsLogic.Papers.Einstein1916.RelativityTypes} (framework : PhysicsLogic.Papers.Einstein1916.RelativityFramework T) (metric : T.Metric) (situation : T.Situation) (coordinates : T.CoordinateSystem) (point : PhysicsLogic.Papers.Einstein1916.CoordinatePoint) (mu nu : Fin 4) (geometry : framework.metricDeterminesSpacetimeGeometry metric situation) (physical : framework.physicalMetric metric) (admissible : framework.coordinatePointAdmissibleIn point coordinates) (nondegenerate : framework.squareRootNegativeDeterminantValueAt metric point ≠ 0) (connection : framework.EquationFortyFiveConnectionSignFormulaAt metric) (tensor : framework.CurvatureTensorCalculation metric situation) (contraction : framework.CurvatureContractionCalculation metric situation) : ((framework.toMetricConnectionCurvatureContext metric situation coordinates point mu nu) |>.NonvacuousMetricConnectionCurvatureAt) ∧ ((framework.toMetricConnectionCurvatureContext metric situation coordinates point mu nu) |>.SourceStructuralMetricConnectionCurvatureExplanationAt .directChristoffelRiemannAndSeparateContraction (framework.einsteinMetricCurvatureDerivationObtains metric connection (tensor geometry physical).2 (contraction (tensor geometry physical).1).2)) := ⟨framework.direct_metric_calculations_realize_nonvacuous_metric_connection_curvature metric situation coordinates point mu nu geometry physical admissible nondegenerate connection tensor contraction, framework.direct_metric_source_realizes_source_curvature_explanation metric situation coordinates point mu nu geometry physical admissible nondegenerate connection tensor contraction⟩ example : HasSymmetricLowerChristoffelIndices originalIndexedChristoffelReadout ∧ RetainsEquationFortyFiveOriginalIndexOrder originalIndexedChristoffelReadout ∧ ¬RetainsUnpermutedChristoffelIndexOrder originalIndexedChristoffelReadout := symmetric_christoffel_bracket_rejects_equation_forty_five_index_permutation example {Metric Tensor : Type} (readout : PhysicsLogic.Papers.Rosen1940.PartITensorFormulaReadout Metric Tensor) (source : readout.SourceFormulaAt) : 0 < readout.kappaValue := readout.kappa_is_positive_for_nondegenerate_metrics source example {T : PhysicsLogic.Papers.Rosen1940.Rosen1940Types} (framework : PhysicsLogic.Papers.Rosen1940.RosenFramework T) (mu nu : Fin 4) (formal : framework.FormalTwoMetricResult) (calculation : framework.TensorReformulationCalculation) : ((framework.toMetricConnectionCurvatureContext mu nu) |>.NonvacuousMetricConnectionCurvatureAt) ∧ ((framework.toMetricConnectionCurvatureContext mu nu) |>.SourceStructuralMetricConnectionCurvatureExplanationAt .formalFlatDifferenceAndRewrittenRicci (framework.rosenMetricCurvatureDerivationObtains formal.1 (calculation formal).1 (calculation formal).2.1 (calculation formal).2.2.2.2)) ∧ framework.flatMetricIntroducedAsFormalAuxiliary framework.flatMetric ∧ framework.flatMetricAddsNoPhysicalPostulateInPartI framework.flatMetric := by have retained := framework.shared_rosen_metric_curvature_retains_formal_background_and_original_formulas mu nu formal calculation exact ⟨retained.2.2.2, framework.formal_flat_source_realizes_source_metric_curvature_explanation mu nu formal calculation, retained.1, retained.2.1⟩ example : HasOnlySquareRootDeterminantDomain degenerateGravitationalMetricReadout ∧ ¬degenerateGravitationalMetricReadout.FormulaDomainAt := by have boundary := square_root_domain_does_not_supply_nondegenerate_gravitational_metric exact ⟨boundary.1, boundary.2.2.2.2.2⟩ end OriginalGravitationalTensorFormulaReuseAudit section MetricConnectionCurvature open PhysicsLogic.Framework.MetricConnectionCurvatureContext.Countermodels example : PhysicalEinsteinDirectMetricRiemannContractionAt physicalEinstein1916MetricCurvatureReadout ∧ PhysicalRosenFlatDifferenceRicciRewriteAt physicalRosen1940MetricCurvatureReadout ∧ physicalRosen1940MetricCurvatureReadout.flatConnectionValue ≠ 0 ∧ ¬(physicalMetricCurvatureEstablishment.establishedAt .backgroundIndependenceEstablished .rosen1940) := by have sources := metric_curvature_sources_distinguish_einstein_riemann_contraction_rosen_flat_difference_and_open_background_scope exact ⟨sources.1, sources.2.1, sources.2.2.2.1, sources.2.2.2.2.2.2⟩ example : ¬(rosenWithPhysicallyAssignedFlatBackground |>.NonvacuousMetricConnectionCurvatureAt) := formal_auxiliary_curvature_does_not_assign_physical_flat_geometry example : ¬(einsteinWithDegenerateMetric |>.NonvacuousMetricConnectionCurvatureAt) := degenerate_metric_does_not_realize_contracted_curvature end MetricConnectionCurvature example : PhysicsLogic.Papers.CallanMaldacena1996.Fidelity.countermodelMatrix .extremalAgreementNotNearExtremalControl := PhysicsLogic.Papers.CallanMaldacena1996.Framework.Countermodels.extremal_agreement_does_not_supply_near_extremal_control example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.CallanMaldacena1996.Fidelity.countermodelMatrix := PhysicsLogic.Papers.CallanMaldacena1996.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.DeLaOssaQuevedo1992.Fidelity.countermodelMatrix .originalNonAbelianIsometryNotDualIsometry := PhysicsLogic.Papers.DeLaOssaQuevedo1992.Countermodels.nonabelian_isometry_does_not_force_dual_isometry example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.DeLaOssaQuevedo1992.Fidelity.countermodelMatrix := PhysicsLogic.Papers.DeLaOssaQuevedo1992.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.RocekVerlinde1991.Fidelity.countermodelMatrix .localReductionNotGlobalHolonomyEquivalence := PhysicsLogic.Papers.RocekVerlinde1991.Countermodels.local_dual_reduction_does_not_supply_global_cft_equivalence example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.RocekVerlinde1991.Fidelity.countermodelMatrix := PhysicsLogic.Papers.RocekVerlinde1991.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.SeibergDuality1994.Fidelity.countermodelMatrix .anomalyMatchingNotDualityProof := PhysicsLogic.Papers.SeibergDuality1994.Framework.Countermodels.anomaly_matching_does_not_prove_duality example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.SeibergDuality1994.Fidelity.countermodelMatrix := PhysicsLogic.Papers.SeibergDuality1994.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.WittenBaryons1979.Fidelity.countermodelMatrix .weakPairForceNotWeakTotalBaryonInteraction := PhysicsLogic.Papers.WittenBaryons1979.LargeNBaryonFramework.Countermodels.weak_pair_force_does_not_make_total_baryon_interaction_weak example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.WittenBaryons1979.Fidelity.countermodelMatrix := PhysicsLogic.Papers.WittenBaryons1979.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.WittenDynamicalBreaking1981.Fidelity.countermodelMatrix .fourZeroModesNotClosedThreeFermionLoophole := PhysicsLogic.Papers.WittenDynamicalBreaking1981.DynamicalBreakingFramework.FourDimensionalZeroModeData.four_zero_mode_count_does_not_close_three_fermion_loophole example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.WittenDynamicalBreaking1981.Fidelity.countermodelMatrix := PhysicsLogic.Papers.WittenDynamicalBreaking1981.Fidelity.countermodel_matrix_is_realized example {T : PhysicsLogic.Papers.Adler1969.AdlerTypes} (framework : PhysicsLogic.Papers.Adler1969.AxialVertexFramework T) (calculation : framework.MasslessAnomalyCalculation) (criterion : framework.CurrentAnomalyIdentification) : framework.toQuantumCurrentAnomalyContext |>.CurrentAnomalyRouteApplicableAt framework.masslessQuantumCurrentAnomalyModel := framework.massless_anomaly_inputs_make_shared_route_applicable calculation criterion example {T : PhysicsLogic.Papers.Rosen1940.Rosen1940Types} (framework : PhysicsLogic.Papers.Rosen1940.RosenFramework T) (formal : framework.FormalTwoMetricResult) (physical : framework.PhysicalReinterpretationResult) : framework.CovariantFixedBackgroundResult := framework.physical_flat_metric_and_dynamical_gravitational_metric_realize_covariant_fixed_background formal physical example (framework : PhysicsLogic.Papers.CallanMaldacena1996.Framework) {sector : PhysicsLogic.Papers.CallanMaldacena1996.ChargeSector} (macroscopic : framework.EstablishedExtremalMacroscopicEntropyAt sector) (microscopic : framework.EstablishedExtremalMicroscopicEntropyAt sector) (sameSector : framework.sameD1D5MomentumSector sector) (controlled : framework.extremalComparisonControlled sector) (agreement : framework.extremalEntropyAgreement sector (framework.macroscopicEntropy .extremal sector) (framework.microscopicEntropy .extremal sector)) : (framework.microstateEntropyContext .extremal sector) |>.BlackHoleMicrostateEntropyRouteApplicableAt (framework.sharedMicrostateEntropyModel .extremal sector) := framework.paper_inputs_make_shared_extremal_entropy_route_applicable macroscopic microscopic sameSector controlled agreement example {T : PhysicsLogic.Papers.DeLaOssaQuevedo1992.DeLaOssaQuevedoTypes} (framework : PhysicsLogic.Papers.DeLaOssaQuevedo1992.DeLaOssaQuevedoFramework T) (setup : framework.NonAbelianParentSetup) (construction : framework.NonAbelianParentConstruction) (calculation : framework.NonAbelianParentCalculation) : framework.toSharedParentActionDualReductionContext |>.ParentActionDualReductionRouteApplicableAt framework.selectedParentActionDualReductionModel := framework.nonabelian_source_inputs_make_shared_parent_action_route_applicable setup construction calculation example {T : PhysicsLogic.Papers.RocekVerlinde1991.RocekVerlindeTypes} (framework : PhysicsLogic.Papers.RocekVerlinde1991.RocekVerlindeFramework T) (setup : framework.BosonicDualReductionSetup) (calculation : framework.BosonicDualReductionCalculation) : framework.toSharedParentActionDualReductionContext |>.ParentActionDualReductionRouteApplicableAt framework.selectedParentActionDualReductionModel := framework.bosonic_source_inputs_make_shared_parent_action_route_applicable setup calculation example {FieldValue : Type} [Field FieldValue] (framework : PhysicsLogic.Papers.SeibergDuality1994.Framework) (data : PhysicsLogic.Papers.SeibergDuality1994.MassDeformationData FieldValue) (coherence : data.parameters = framework.parameters) (proposal : framework.SeibergDualityProposal) (dictionary : framework.EstablishedDualDictionary) (route : data.MassDeformationRouteApplicable) (support : framework.massDeformationSupportsProposalAt data.parameters .oneFlavorMagneticFlow) (notProof : ¬framework.massDeformationProvesDualityAt data.parameters .oneFlavorMagneticFlow) : framework.toMagneticMassDeformationContext data |>.MagneticMassDeformationRouteApplicableAt (framework.sharedMagneticMassDeformationModel data) := framework.one_flavor_mass_source_inputs_make_shared_route_applicable data coherence proposal dictionary route support notProof example {T : PhysicsLogic.Papers.WittenBaryons1979.WittenBaryonTypes} (framework : PhysicsLogic.Papers.WittenBaryons1979.LargeNBaryonFramework T) (setup : framework.ReviewedPlanarSetup) (calculation : framework.ReviewedPlanarCountingCalculation) (fixed : ∀ rank, framework.rescaledCouplingAt rank = framework.fixedRescaledCoupling) (nonplanarWitness : framework.SelectedNonplanarContributionWitness) (additionalBoundaryWitness : framework.SelectedAdditionalBoundaryContributionWitness) (planar : framework.planarDiagram framework.selectedPlanarDiagram) (minimum : framework.minimumQuarkLoopsFor framework.selectedPlanarDiagram framework.selectedObservable) : framework.toPlanarDiagramExpansionContext |>.PlanarSelectionRouteApplicableAt framework.planarDiagramModel := framework.witten_source_inputs_make_shared_planar_route_applicable setup calculation fixed nonplanarWitness additionalBoundaryWitness planar minimum example {State Regime Supercharge Fermion Auxiliary Instanton EffectiveOperator : Type} (framework : PhysicsLogic.Papers.WittenDynamicalBreaking1981.DynamicalBreakingFramework State Regime Supercharge Fermion Auxiliary Instanton EffectiveOperator) (regime : Regime) (hamiltonian : framework.HamiltonianSquareCalculation regime) (ground : framework.GroundStateCalculation regime) (zeroState : framework.HasZeroEnergyStateAt regime) : framework.toSupersymmetricVacuumEnergyContext regime |>.UnbrokenVacuumRouteApplicableAt regime := framework.zero_state_and_calculations_make_shared_unbroken_route_applicable regime hamiltonian ground zeroState example : PhysicsLogic.Papers.GubserKlebanovPolyakov1998.Fidelity.countermodelMatrix .sharedBoundaryProfileNotBulkEquation := by have witness := PhysicsLogic.Papers.GubserKlebanovPolyakov1998.CorrelatorFramework.Countermodels.common_boundary_profile_does_not_select_a_physical_bulk_solution exact ⟨⟨witness.1, witness.2.1⟩, witness.2.2⟩ example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.GubserKlebanovPolyakov1998.Fidelity.countermodelMatrix := PhysicsLogic.Papers.GubserKlebanovPolyakov1998.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.boundaryDictionaryWithoutScalarContext.BoundaryDictionaryResultAt PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.boundaryDictionaryWithoutScalarModel ∧ PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.boundaryDictionaryWithoutScalarModel.scalarModel.dimensionlessMassSquared PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.boundaryDictionaryWithoutScalarModel.scalarModel.field = 1 ∧ ¬PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.boundaryDictionaryWithoutScalarContext.ScalarDictionaryResultAt PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.boundaryDictionaryWithoutScalarModel := by obtain ⟨boundary, _boundaryDimension, _operatorDimension, mass, noScalar⟩ := PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.boundary_dictionary_does_not_supply_scalar_dictionary exact ⟨boundary, mass, noScalar⟩ example : PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.scalarDictionaryWithoutCompatibleSourceContext.ScalarDictionaryResultAt PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.scalarDictionaryWithoutCompatibleSourceModel ∧ ¬PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.scalarDictionaryWithoutCompatibleSourceModel.sourceSelectsField PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.scalarDictionaryWithoutCompatibleSourceModel.boundaryModel.source PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.scalarDictionaryWithoutCompatibleSourceModel.scalarModel.field ∧ ¬PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.scalarDictionaryWithoutCompatibleSourceContext.LinkedScalarSourceSetupAt PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.scalarDictionaryWithoutCompatibleSourceModel := by obtain ⟨scalar, _source, _field, noSource, noLinked⟩ := PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.scalar_dictionary_does_not_supply_source_compatibility exact ⟨scalar, noSource, noLinked⟩ example : PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.childDictionariesWithoutScalingContext.BoundaryDictionaryResultAt PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.childDictionariesWithoutScalingModel ∧ PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.childDictionariesWithoutScalingContext.ScalarDictionaryResultAt PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.childDictionariesWithoutScalingModel ∧ PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.childDictionariesWithoutScalingContext.LinkedScalarSourceSetupAt PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.childDictionariesWithoutScalingModel ∧ ¬PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.childDictionariesWithoutScalingContext.ScalarTwoPointCalculationBridgeAt PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.childDictionariesWithoutScalingModel := by obtain ⟨boundary, scalar, linked, _correlator, _dimension, _noScaling, noBridge⟩ := PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels.child_dictionaries_do_not_supply_two_point_scaling exact ⟨boundary, scalar, linked, noBridge⟩ example : PhysicsLogic.Papers.HorowitzStrominger1996.Fidelity.countermodelMatrix .leadingEntropyMatchNotNoPhaseTransition := PhysicsLogic.Papers.HorowitzStrominger1996.Framework.Countermodels.matching_leading_entropies_does_not_prove_a_phase_transition_absent example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.HorowitzStrominger1996.Fidelity.countermodelMatrix := PhysicsLogic.Papers.HorowitzStrominger1996.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.ItzhakiMaldacenaSonnenscheinYankielowicz1998.Fidelity.countermodelMatrix .massiveEscapeNotThermalRadiation := PhysicsLogic.Papers.ItzhakiMaldacenaSonnenscheinYankielowicz1998.PaperFramework.Countermodels.massive_escape_does_not_logically_supply_hawking_radiation example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.ItzhakiMaldacenaSonnenscheinYankielowicz1998.Fidelity.countermodelMatrix := PhysicsLogic.Papers.ItzhakiMaldacenaSonnenscheinYankielowicz1998.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.KutasovSchwimmer1995.Fidelity.countermodelMatrix .anomalyMatchingNotDualityProof := PhysicsLogic.Papers.KutasovSchwimmer1995.Framework.Countermodels.anomaly_matching_does_not_prove_duality example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.KutasovSchwimmer1995.Fidelity.countermodelMatrix := PhysicsLogic.Papers.KutasovSchwimmer1995.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.LewkowyczMaldacena2013.Fidelity.countermodelMatrix .integerReplicasNotAnalyticContinuation := PhysicsLogic.Papers.LewkowyczMaldacena2013.ReplicaGravityFramework.Countermodels.integer_replica_traces_do_not_supply_analytic_continuation example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.LewkowyczMaldacena2013.Fidelity.countermodelMatrix := PhysicsLogic.Papers.LewkowyczMaldacena2013.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.Polchinski1995.Fidelity.countermodelMatrix .bpsEvidenceNotDirectRRSource := PhysicsLogic.Papers.Polchinski1995.PaperFramework.Countermodels.bps_scaling_evidence_does_not_by_itself_establish_rr_source example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.Polchinski1995.Fidelity.countermodelMatrix := PhysicsLogic.Papers.Polchinski1995.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.StromingerVafa1996.Fidelity.countermodelMatrix .protectedIndexNotActualDegeneracy := PhysicsLogic.Papers.StromingerVafa1996.Framework.Countermodels.moduli_independent_index_does_not_make_actual_degeneracy_invariant example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.StromingerVafa1996.Fidelity.countermodelMatrix := PhysicsLogic.Papers.StromingerVafa1996.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.TownsendSupermembrane1995.Fidelity.countermodelMatrix .equalChargeMassNotStateIdentity := PhysicsLogic.Papers.TownsendSupermembrane1995.PaperFramework.Countermodels.identical_quantum_numbers_do_not_definitionally_identify_states example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.TownsendSupermembrane1995.Fidelity.countermodelMatrix := PhysicsLogic.Papers.TownsendSupermembrane1995.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.WittenHolography1998.Fidelity.countermodelMatrix .actionCrossingNotContributingSaddles := PhysicsLogic.Papers.WittenHolography1998.Countermodels.action_crossing_does_not_supply_saddle_contributions example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.WittenHolography1998.Fidelity.countermodelMatrix := PhysicsLogic.Papers.WittenHolography1998.Fidelity.countermodel_matrix_is_realized example : PhysicsLogic.Papers.WittenStringDynamics1995.Fidelity.countermodelMatrix .towerIdentityNotDecompactification := PhysicsLogic.Papers.WittenStringDynamics1995.PaperFramework.Countermodels.tower_identity_does_not_supply_the_coupling_radius_limit example : PhysicsLogic.Logic.PaperFidelity.CountermodelMatrixRealized PhysicsLogic.Papers.WittenStringDynamics1995.Fidelity.countermodelMatrix := PhysicsLogic.Papers.WittenStringDynamics1995.Fidelity.countermodel_matrix_is_realized example {Source Operator BulkField BulkSolution Variation BoundaryPoint AsymptoticVector : Type} (framework : PhysicsLogic.Papers.GubserKlebanovPolyakov1998.CorrelatorFramework Source Operator BulkField BulkSolution Variation BoundaryPoint AsymptoticVector) (source : Source) (solution : BulkSolution) (variationScope : Nonempty Variation) (proposal : framework.GeneratingFunctionalProposalAt source solution) (generates : framework.SourceGeneratesConnectedCorrelators source) (differentiation : framework.FunctionalDifferentiationCalculation) : (framework.toBoundaryGeneratingFunctionalContext source solution) |>.BoundaryCorrelatorRouteApplicableAt (framework.boundaryGeneratingFunctionalModel source solution) := framework.gkp_source_inputs_make_shared_correlator_route_applicable source solution variationScope proposal generates differentiation example {Source Operator BulkField BulkSolution Variation BoundaryPoint AsymptoticVector : Type} (framework : PhysicsLogic.Papers.GubserKlebanovPolyakov1998.CorrelatorFramework Source Operator BulkField BulkSolution Variation BoundaryPoint AsymptoticVector) (source : Source) (solution : BulkSolution) (scope : framework.LeadingLargeNSupergravityScope) (computed : framework.toBoundaryGeneratingFunctionalContext source solution |>.BulkComputesBoundaryCorrelatorsAt (framework.boundaryGeneratingFunctionalModel source solution)) (selectedVariation : Variation) : (framework.correlatorClassicalRecoveryContext source solution) |>.toClassicalRecoveryContext.EstablishedAt .strongCouplingGaugeTheory .leadingLargeNStrongCoupling .leadingLargeNStrongCoupling := framework.boundary_action_realizes_prediction_grounded_classical_recovery source solution scope computed selectedVariation example (framework : PhysicsLogic.Papers.HorowitzStrominger1996.Framework) {sector : PhysicsLogic.Papers.HorowitzStrominger1996.ChargeSector} (macroscopic : framework.EstablishedMacroscopicEntropyAt sector) (microscopic : framework.EstablishedMicroscopicEntropyAt sector) (sameSector : framework.sameMassAndChargesSelected sector) (comparison : framework.WeakStrongComparisonControlAt sector) (agreement : framework.leadingNearExtremalAgreement sector (framework.blackStringEntropy sector) (framework.statisticalEntropy sector)) : (framework.microstateEntropyContext sector) |>.BlackHoleMicrostateEntropyRouteApplicableAt (framework.sharedMicrostateEntropyModel sector) := framework.paper_inputs_make_shared_microstate_entropy_route_applicable macroscopic microscopic sameSector comparison agreement example (framework : PhysicsLogic.Papers.ItzhakiMaldacenaSonnenscheinYankielowicz1998.PaperFramework) (setup : framework.D2Setup) (limit : framework.FieldTheoryLimitAt .d2) (calculation : framework.D2FieldTheoryLimitCalculation) : framework.toBraneWorldvolumeDecouplingContext |>.BraneWorldvolumeDecouplingRouteApplicableAt framework.selectedD2DecouplingModel := framework.d2_inputs_make_shared_decoupling_route_applicable setup limit calculation example {T : PhysicsLogic.Papers.KutasovSchwimmer1995.KutasovSchwimmerTypes} (framework : PhysicsLogic.Papers.KutasovSchwimmer1995.Framework T) (setup : framework.OneFlavorMassSetup) (calculation : framework.OneFlavorMassCalculation) : framework.toMagneticMassDeformationContext |>.MagneticMassDeformationRouteApplicableAt framework.sharedMagneticMassDeformationModel := framework.one_flavor_mass_source_inputs_make_shared_route_applicable setup calculation example {T : PhysicsLogic.Papers.LewkowyczMaldacena2013.LMTypes} (framework : PhysicsLogic.Papers.LewkowyczMaldacena2013.ReplicaGravityFramework T) (subregion : framework.AdSCFTSubregionSetup) (minimal : framework.MinimalSurfaceCondition) (gravityArea : framework.GeneralizedGravityAreaEntropy) (adsCft : framework.AdSCFTEntropyInterpretationBridge) (areaCalculation : framework.surfaceArea framework.selectedReplicaSurface = framework.selectedAreaValue) (normalization : framework.entropyFromArea framework.selectedAreaValue = framework.selectedEntropyValue) : framework.toHolographicRegionEntropyContext |>.HolographicEntropyCalculationApplicableAt framework.holographicEntropyModel := framework.replica_outputs_make_shared_holographic_entropy_calculation_applicable subregion minimal gravityArea adsCft areaCalculation normalization example (framework : PhysicsLogic.Papers.Polchinski1995.PaperFramework) (source : framework.EstablishedBoundaryStateRRSource) : framework.rrChargeContext.FunctionalAssignmentRouteApplicableAt framework.rrChargeModel := framework.selected_rr_source_makes_shared_charge_route_applicable source example (framework : PhysicsLogic.Papers.StromingerVafa1996.Framework) {sector : PhysicsLogic.Papers.StromingerVafa1996.ChargeSector} (microscopic : framework.EstablishedMicroscopicIndexGrowthAt sector) (macroscopicReliability : sector.LargeChargeRegime ∧ framework.semiclassicalBlackHoleReliable sector) (connection : framework.weakToStrongCouplingConnection sector) (protection : framework.ellipticGenusProtectedAcrossConnection sector) (growthEquality : framework.indexedGrowthAt .weakDbrane sector = framework.indexedGrowthAt .strongBlackHole sector) : (framework.protectedIndexContext sector) |>.ProtectedIndexRouteApplicableAt (framework.protectedIndexModel sector) := framework.paper_inputs_make_shared_protected_index_route_applicable microscopic macroscopicReliability connection protection growthEquality example (framework : PhysicsLogic.Papers.TownsendSupermembrane1995.PaperFramework) (lowEnergySetup : framework.LowEnergyReductionSetup) (lowEnergyCalculation : framework.LowEnergyReductionCalculation) (discreteSpectrum : framework.nonperturbativeSupermembraneSpectrumDiscrete .compactifiedSupermembrane) (magneticSixbrane : framework.magneticSixbraneIncluded framework.sixbraneState) (towerCalculation : framework.ChargedTowerCalculation) (matchingCalculation : framework.KaluzaKleinTowerMatchingCalculation) (identificationBridge : framework.KaluzaKleinStateIdentificationBridge) (originBridge : framework.CompactifiedSupermembraneOriginBridge) : framework.selectedKaluzaKleinTowerContext |>.KaluzaKleinTowerRouteApplicableAt framework.selectedKaluzaKleinTowerModel := framework.townsend_source_inputs_make_shared_kaluza_klein_tower_route_applicable lowEnergySetup lowEnergyCalculation discreteSpectrum magneticSixbrane towerCalculation matchingCalculation identificationBridge originBridge example {Source Operator BulkField BulkSolution BoundaryPoint : Type} (framework : PhysicsLogic.Papers.WittenHolography1998.Framework Source Operator BulkField BulkSolution BoundaryPoint) (source : Source) (solution : BulkSolution) (variationScope : Nonempty framework.SourceVariation) (proposal : framework.GeneratingFunctionalProposal) (regime : framework.ClassicalSupergravityRegime source) (admissible : framework.sourceAdmissible source) (boundaryValue : framework.sourceIsBoundaryValueOf source solution) (equations : framework.solutionSolvesBulkEquations solution) (generates : framework.SourceGeneratesCFTCorrelators source) (evaluation : framework.ClassicalSaddleEvaluation) (extraction : framework.OnShellActionCorrelatorPrinciple) : framework.toBoundaryGeneratingFunctionalContext source solution |>.BoundaryCorrelatorRouteApplicableAt (framework.boundaryGeneratingFunctionalModel source solution) := framework.witten_source_inputs_make_shared_correlator_route_applicable source solution variationScope proposal regime admissible boundaryValue equations generates evaluation extraction example (framework : PhysicsLogic.Papers.WittenStringDynamics1995.PaperFramework) (rrSetup : framework.RRTowerSetup) (seed : framework.RRSeedExistenceAssumption) (quantizationCalculation : framework.RRChargeQuantizationCalculation) (completion : framework.RRAllChargeTowerExpectation) (towerCalculation : framework.RRTowerCalculation) (spectrumCalculation : framework.StrongCouplingSpectrumCalculation) (spectrumInterpretation : framework.StrongCouplingSpectrumInterpretation) (localSetup : framework.KaluzaKleinIdentificationLocalSetup) (matchingCalculation : framework.KaluzaKleinTowerMatchingCalculation) (identificationBridge : framework.KaluzaKleinStateIdentificationBridge) (originBridge : framework.ElevenDimensionalOriginInterpretationBridge) : framework.selectedKaluzaKleinTowerContext |>.KaluzaKleinTowerRouteApplicableAt framework.selectedKaluzaKleinTowerModel := framework.witten_source_inputs_make_shared_kaluza_klein_tower_route_applicable rrSetup seed quantizationCalculation completion towerCalculation spectrumCalculation spectrumInterpretation localSetup matchingCalculation identificationBridge originBridge example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature : Real) (contour : Contour) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (circulation : PhysicsLogic.Papers.KosterlitzThouless1973.QuantizedCirculation model) (irrotational : PhysicsLogic.Papers.KosterlitzThouless1973.LocallyIrrotationalAwayFromVortices model) (winding : model.winding contour ≠ 0) : (PhysicsLogic.Papers.KosterlitzThouless1973.toPhaseWindingCirculationContext model temperature contour) |>.NonvacuousPhaseWindingCirculationAt := PhysicsLogic.Papers.KosterlitzThouless1973.nonzero_angular_contour_realizes_nonvacuous_phase_winding model temperature contour scope circulation irrotational winding example {T : PhysicsLogic.Papers.FeynmanTwoFluid1954.FeynmanTypes} (framework : PhysicsLogic.Papers.FeynmanTwoFluid1954.AtomicTwoFluidFramework T) (result : framework.TwoFluidDescriptionResult) : framework.toPhaseWindingCirculationContext |>.NonvacuousPhaseWindingCirculationAt := framework.two_fluid_result_realizes_nonvacuous_phase_winding_circulation result example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature : Real) (contour : Contour) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (circulation : PhysicsLogic.Papers.KosterlitzThouless1973.QuantizedCirculation model) (irrotational : PhysicsLogic.Papers.KosterlitzThouless1973.LocallyIrrotationalAwayFromVortices model) (winding : model.winding contour ≠ 0) : (PhysicsLogic.Papers.KosterlitzThouless1973.toPhaseWindingCirculationContext model temperature contour) |>.SourceStructuralPhaseWindingCirculationExplanationAt .quantizedAngularContourAwayFromVortexCore (PhysicsLogic.Papers.KosterlitzThouless1973.kosterlitzThoulessPhaseWindingDerivationObtains model temperature contour) := PhysicsLogic.Papers.KosterlitzThouless1973.nonzero_angular_contour_realizes_source_structural_phase_winding_circulation_explanation model temperature contour scope circulation irrotational winding example {T : PhysicsLogic.Papers.FeynmanTwoFluid1954.FeynmanTypes} (framework : PhysicsLogic.Papers.FeynmanTwoFluid1954.AtomicTwoFluidFramework T) (source : framework.PermanentRingPhaseSourceAt) : framework.toPhaseWindingCirculationContext |>.SourceStructuralPhaseWindingCirculationExplanationAt .manyBodyPhaseGradientAndSingleValuedRing (framework.feynmanPhaseWindingDerivationObtains) := framework.permanent_ring_source_inputs_realize_source_structural_phase_winding_circulation_explanation source example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (neutral : PhysicsLogic.Papers.KosterlitzThouless1973.NeutralVortexGas model) (unitVortices : PhysicsLogic.Papers.KosterlitzThouless1973.UnitVorticesPresent model) : (PhysicsLogic.Papers.KosterlitzThouless1973.toNeutralDefectConfigurationContext model temperature) |>.NonvacuousNeutralDefectConfigurationAt := PhysicsLogic.Papers.KosterlitzThouless1973.neutral_unit_vortices_realize_nonvacuous_defect_configuration model temperature scope neutral unitVortices example {T : PhysicsLogic.Papers.HalperinNelson1978.MeltingTypes} (framework : PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework T) (setup : framework.SmoothSolidSetup) (result : framework.SmoothSolidResult) : framework.toNeutralDefectConfigurationContext |>.NonvacuousNeutralDefectConfigurationAt := framework.smooth_solid_result_realizes_nonvacuous_neutral_dislocations setup result example {T : PhysicsLogic.Papers.Young1979.MeltingTypes} (framework : PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework T) (setup : framework.SolidSetup) : framework.toNeutralDefectConfigurationContext |>.NonvacuousNeutralDefectConfigurationAt := framework.solid_setup_realizes_nonvacuous_neutral_dislocations setup example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (neutral : PhysicsLogic.Papers.KosterlitzThouless1973.NeutralVortexGas model) (unitVortices : PhysicsLogic.Papers.KosterlitzThouless1973.UnitVorticesPresent model) : (PhysicsLogic.Papers.KosterlitzThouless1973.toNeutralDefectConfigurationContext model temperature) |>.SourceStructuralNeutralDefectConfigurationExplanationAt .scopedNeutralUnitVortexGas (PhysicsLogic.Papers.KosterlitzThouless1973.kosterlitzThoulessNeutralDefectDerivationObtains model temperature) := PhysicsLogic.Papers.KosterlitzThouless1973.neutral_unit_vortices_realize_source_structural_neutral_defect_configuration_explanation model temperature scope neutral unitVortices example {T : PhysicsLogic.Papers.HalperinNelson1978.MeltingTypes} (framework : PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework T) (setup : framework.SmoothSolidSetup) (result : framework.SmoothSolidResult) : framework.toNeutralDefectConfigurationContext |>.SourceStructuralNeutralDefectConfigurationExplanationAt .smoothSolidBurgersPair (PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework.halperinNelsonNeutralDefectDerivationObtains framework) := framework.smooth_solid_result_realizes_source_structural_neutral_defect_configuration_explanation setup result example {T : PhysicsLogic.Papers.Young1979.MeltingTypes} (framework : PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework T) (setup : framework.SolidSetup) : framework.toNeutralDefectConfigurationContext |>.SourceStructuralNeutralDefectConfigurationExplanationAt .solidVectorBurgersPair (PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework.youngNeutralDefectDerivationObtains framework) := framework.solid_setup_realizes_source_structural_neutral_defect_configuration_explanation setup example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature area : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (large : model.latticeSpacing ^ (2 : Nat) < area) (screened : PhysicsLogic.Papers.KosterlitzThouless1973.ScreenedStiffnessAt model temperature) : (PhysicsLogic.Papers.KosterlitzThouless1973.toLogarithmicDefectThermodynamicsContext model temperature area) |>.NonvacuousLogarithmicDefectThermodynamicsAt := PhysicsLogic.Papers.KosterlitzThouless1973.xy_vortex_thermodynamics_realizes_nonvacuous_logarithmic_competition model temperature area scope large screened example {T : PhysicsLogic.Papers.HalperinNelson1978.MeltingTypes} (framework : PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework T) (setup : framework.SmoothSolidSetup) (result : framework.SmoothSolidResult) : framework.toLogarithmicDefectThermodynamicsContext |>.NonvacuousLogarithmicDefectThermodynamicsAt := framework.smooth_solid_result_realizes_nonvacuous_logarithmic_thermodynamics setup result example {T : PhysicsLogic.Papers.Young1979.MeltingTypes} (framework : PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework T) (setup : framework.SolidSetup) : framework.toLogarithmicDefectThermodynamicsContext |>.NonvacuousLogarithmicDefectThermodynamicsAt := framework.solid_setup_realizes_nonvacuous_logarithmic_thermodynamics setup example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature area : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (large : model.latticeSpacing ^ (2 : Nat) < area) (screened : PhysicsLogic.Papers.KosterlitzThouless1973.ScreenedStiffnessAt model temperature) : (PhysicsLogic.Papers.KosterlitzThouless1973.toLogarithmicDefectThermodynamicsContext model temperature area) |>.SourceStructuralLogarithmicDefectThermodynamicsExplanationAt .finiteAreaScreenedVortexCompetition (PhysicsLogic.Papers.KosterlitzThouless1973.kosterlitzThoulessLogarithmicDefectDerivationObtains model temperature area) := PhysicsLogic.Papers.KosterlitzThouless1973.xy_vortex_thermodynamics_realizes_source_structural_logarithmic_defect_thermodynamics_explanation model temperature area scope large screened example {T : PhysicsLogic.Papers.HalperinNelson1978.MeltingTypes} (framework : PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework T) (setup : framework.SmoothSolidSetup) (result : framework.SmoothSolidResult) : framework.toLogarithmicDefectThermodynamicsContext |>.SourceStructuralLogarithmicDefectThermodynamicsExplanationAt .smoothSolidFiniteCoreCompetition (PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework.halperinNelsonLogarithmicDefectDerivationObtains framework) := framework.smooth_solid_result_realizes_source_structural_logarithmic_defect_thermodynamics_explanation setup result example {T : PhysicsLogic.Papers.Young1979.MeltingTypes} (framework : PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework T) (setup : framework.SolidSetup) : framework.toLogarithmicDefectThermodynamicsContext |>.SourceStructuralLogarithmicDefectThermodynamicsExplanationAt .solidFiniteCoreCompetition (PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework.youngLogarithmicDefectDerivationObtains framework) := framework.solid_setup_realizes_source_structural_logarithmic_defect_thermodynamics_explanation setup example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (screening : PhysicsLogic.Papers.KosterlitzThouless1973.DipoleScreeningResponseAt model temperature) (measure : 0 < model.shellMeasure) (density : 0 < model.shellPairDensity temperature) : (PhysicsLogic.Papers.KosterlitzThouless1973.toDipoleScreeningResponseContext model temperature) |>.NonvacuousDipoleScreeningResponseAt := PhysicsLogic.Papers.KosterlitzThouless1973.positive_original_pair_shell_realizes_nonvacuous_dipole_response model temperature scope screening measure density example {T : PhysicsLogic.Papers.Young1979.MeltingTypes} (framework : PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework T) (setup : framework.SolidSetup) (measure : 0 < framework.dipoleShellMeasureValue) (density : 0 < framework.dipoleShellPairDensityValue) (polarizability : 0 < framework.dipoleShellPolarizabilityValue) : framework.toDipoleScreeningResponseContext |>.NonvacuousDipoleScreeningResponseAt := framework.positive_solid_shell_realizes_nonvacuous_dipole_screening setup measure density polarizability example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (screening : PhysicsLogic.Papers.KosterlitzThouless1973.DipoleScreeningResponseAt model temperature) (measure : 0 < model.shellMeasure) (density : 0 < model.shellPairDensity temperature) : (PhysicsLogic.Papers.KosterlitzThouless1973.toDipoleScreeningResponseContext model temperature) |>.SourceStructuralDipoleScreeningResponseExplanationAt .finiteCoreDilutePairIteratedDielectric (PhysicsLogic.Papers.KosterlitzThouless1973.kosterlitzThoulessDipoleScreeningDerivationObtains model temperature) := PhysicsLogic.Papers.KosterlitzThouless1973.positive_original_pair_shell_realizes_source_structural_dipole_screening_response_explanation model temperature scope screening measure density example {T : PhysicsLogic.Papers.Young1979.MeltingTypes} (framework : PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework T) (setup : framework.SolidSetup) (measure : 0 < framework.dipoleShellMeasureValue) (density : 0 < framework.dipoleShellPairDensityValue) (polarizability : 0 < framework.dipoleShellPolarizabilityValue) : framework.toDipoleScreeningResponseContext |>.SourceStructuralDipoleScreeningResponseExplanationAt .triangularBoundDislocationShell (PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework.youngDipoleScreeningDerivationObtains framework) := framework.positive_solid_shell_realizes_source_structural_dipole_screening_response_explanation setup measure density polarizability example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (powerLaw : PhysicsLogic.Papers.KosterlitzThouless1973.SpinWavePowerLaw model temperature) : (PhysicsLogic.Papers.KosterlitzThouless1973.toCorrelationDecayChannelContext model temperature) |>.NonvacuousAlgebraicCorrelationDecayAt := PhysicsLogic.Papers.KosterlitzThouless1973.spin_wave_power_law_realizes_nonvacuous_correlation_decay model temperature scope powerLaw example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (powerLaw : PhysicsLogic.Papers.KosterlitzThouless1973.SpinWavePowerLaw model temperature) : (PhysicsLogic.Papers.KosterlitzThouless1973.toCorrelationDecayChannelContext model temperature) |>.SourceStructuralAlgebraicCorrelationDecayExplanationAt .scopedSpinWavePowerLawBeyondCore (PhysicsLogic.Papers.KosterlitzThouless1973.kosterlitzThoulessSpinWaveCorrelationDerivationObtains model temperature) := PhysicsLogic.Papers.KosterlitzThouless1973.spin_wave_power_law_realizes_source_structural_correlation_decay_explanation model temperature scope powerLaw example (readout : PhysicsLogic.Papers.HastingsKoma2006.CorrectedBosonicClusteringFormulaReadout) (formula : readout.SourceFormulaAt) (nonzero : norm (readout.CorrectedCorrelationValueAt readout.selectedDistanceValue) ≠ 0) : readout.NonvacuousCorrectedCorrelationDecayAt := readout.nonzero_source_formula_realizes_nonvacuous_corrected_decay formula nonzero example (readout : PhysicsLogic.Papers.HastingsKoma2006.CorrectedBosonicClusteringFormulaReadout) (formula : readout.SourceFormulaAt) (nonzero : norm (readout.CorrectedCorrelationValueAt readout.selectedDistanceValue) ≠ 0) : readout.SourceStructuralCorrectedCorrelationDecayExplanationAt := readout |>.nonzero_source_formula_realizes_source_structural_corrected_correlation_decay_explanation formula nonzero example : PhysicsLogic.Framework.PhaseWindingCirculationContext.Countermodels.irrotationalZeroWinding.ScopedPhaseWindingCirculationAt ∧ ¬PhysicsLogic.Framework.PhaseWindingCirculationContext.Countermodels.irrotationalZeroWinding.NonvacuousPhaseWindingCirculationAt := PhysicsLogic.Framework.PhaseWindingCirculationContext.Countermodels.scoped_zero_winding_does_not_establish_nonvacuous_circulation example : PhysicsLogic.Framework.PhaseWindingCirculationContext.Countermodels.irrotationalWindingRing.NonvacuousPhaseWindingCirculationAt ∧ PhysicsLogic.Framework.PhaseWindingCirculationContext.Countermodels.irrotationalWindingRing.DirectStructuralPhaseWindingCirculationExplanationAt ∧ ¬PhysicsLogic.Framework.PhaseWindingCirculationContext.Countermodels.irrotationalWindingRing.SourceStructuralPhaseWindingCirculationExplanationAt .absent PhysicsLogic.Framework.PhaseWindingCirculationContext.Countermodels.missingPhaseWindingDerivationObtains := PhysicsLogic.Framework.PhaseWindingCirculationContext.Countermodels.completed_phase_winding_route_and_ordinary_explanation_without_source_derivation_have_no_source_explanation example : PhysicsLogic.Framework.NeutralDefectConfigurationContext.Countermodels.emptyNeutralGas.ChargeNeutralAt ∧ ¬PhysicsLogic.Framework.NeutralDefectConfigurationContext.Countermodels.emptyNeutralGas.NonvacuousNeutralDefectConfigurationAt := PhysicsLogic.Framework.NeutralDefectConfigurationContext.Countermodels.empty_charge_neutral_gas_is_not_a_nonvacuous_configuration example : PhysicsLogic.Framework.NeutralDefectConfigurationContext.Countermodels.scalarPair.NonvacuousNeutralDefectConfigurationAt ∧ PhysicsLogic.Framework.NeutralDefectConfigurationContext.Countermodels.scalarPair.DirectStructuralNeutralDefectConfigurationExplanationAt ∧ ¬PhysicsLogic.Framework.NeutralDefectConfigurationContext.Countermodels.scalarPair.SourceStructuralNeutralDefectConfigurationExplanationAt .absent PhysicsLogic.Framework.NeutralDefectConfigurationContext.Countermodels.missingNeutralDefectDerivationObtains := PhysicsLogic.Framework.NeutralDefectConfigurationContext.Countermodels.completed_neutral_configuration_and_ordinary_explanation_without_source_derivation_have_no_source_explanation example : 0 < PhysicsLogic.Framework.LogarithmicDefectThermodynamicsContext.Countermodels.entropyFreeLogarithmicDefect.logarithmicScaleFactor ∧ PhysicsLogic.Framework.LogarithmicDefectThermodynamicsContext.Countermodels.entropyFreeLogarithmicDefect.positionalEntropy = 0 ∧ ¬PhysicsLogic.Framework.LogarithmicDefectThermodynamicsContext.Countermodels.entropyFreeLogarithmicDefect.NonvacuousLogarithmicDefectThermodynamicsAt := PhysicsLogic.Framework.LogarithmicDefectThermodynamicsContext.Countermodels.logarithmic_scale_without_entropy_is_not_nonvacuous_thermodynamics example : PhysicsLogic.Framework.LogarithmicDefectThermodynamicsContext.Countermodels.screenedBoundPair.NonvacuousLogarithmicDefectThermodynamicsAt ∧ PhysicsLogic.Framework.LogarithmicDefectThermodynamicsContext.Countermodels.screenedBoundPair.DirectStructuralLogarithmicDefectThermodynamicsExplanationAt ∧ ¬PhysicsLogic.Framework.LogarithmicDefectThermodynamicsContext.Countermodels.screenedBoundPair.SourceStructuralLogarithmicDefectThermodynamicsExplanationAt .absent PhysicsLogic.Framework.LogarithmicDefectThermodynamicsContext.Countermodels.missingLogarithmicDefectDerivationObtains := PhysicsLogic.Framework.LogarithmicDefectThermodynamicsContext.Countermodels.completed_logarithmic_thermodynamics_and_ordinary_explanation_without_source_derivation_have_no_source_explanation example : PhysicsLogic.Framework.DipoleScreeningResponseContext.Countermodels.nonpolarizableDenseShell.ScopedDipoleScreeningResponseAt ∧ 0 < PhysicsLogic.Framework.DipoleScreeningResponseContext.Countermodels.nonpolarizableDenseShell.shellPairDensity ∧ ¬PhysicsLogic.Framework.DipoleScreeningResponseContext.Countermodels.nonpolarizableDenseShell.NonvacuousDipoleScreeningResponseAt := PhysicsLogic.Framework.DipoleScreeningResponseContext.Countermodels.a_dense_nonpolarizable_shell_is_not_nonvacuous_screening example : PhysicsLogic.Framework.DipoleScreeningResponseContext.Countermodels.screenedDipoleShell.NonvacuousDipoleScreeningResponseAt ∧ PhysicsLogic.Framework.DipoleScreeningResponseContext.Countermodels.screenedDipoleShell.DirectStructuralDipoleScreeningResponseExplanationAt ∧ ¬PhysicsLogic.Framework.DipoleScreeningResponseContext.Countermodels.screenedDipoleShell.SourceStructuralDipoleScreeningResponseExplanationAt .absent PhysicsLogic.Framework.DipoleScreeningResponseContext.Countermodels.missingDipoleScreeningDerivationObtains := PhysicsLogic.Framework.DipoleScreeningResponseContext.Countermodels.completed_dipole_screening_route_and_ordinary_explanation_without_source_derivation_have_no_source_explanation example : PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels.zeroAlgebraicChannel.ScopedAlgebraicCorrelationDecayAt ∧ ¬PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels.zeroAlgebraicChannel.NonvacuousAlgebraicCorrelationDecayAt := PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels.a_zero_algebraic_channel_can_decay_vacuously example : PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels.zeroExponentialChannel.ScopedExponentialCorrelationDecayAt ∧ ¬PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels.zeroExponentialChannel.NonvacuousExponentialCorrelationDecayAt := PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels.a_zero_exponential_channel_can_decay_vacuously example : PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels.algebraicDecayChannel.NonvacuousAlgebraicCorrelationDecayAt ∧ PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels.algebraicDecayChannel.DirectStructuralAlgebraicCorrelationDecayExplanationAt ∧ ¬PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels.algebraicDecayChannel.SourceStructuralAlgebraicCorrelationDecayExplanationAt .absent PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels.missingCorrelationDecayDerivationObtains := PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels.completed_correlation_decay_route_and_ordinary_explanation_without_source_derivation_have_no_source_explanation example {T : PhysicsLogic.Papers.WuEtAl1957.Wu1957Types} (framework : PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework T) (setup : framework.ExperimentAndDecaySetup) (prediction : framework.ParityConservationPredictionCalculation) (result : framework.EstablishedParityViolation) : framework.controlledSymmetryFalsificationModel |>.NonvacuousControlledSymmetryFalsificationAt := framework.wu_route_exhibits_nonvacuous_controlled_symmetry_falsification setup prediction result example {T : PhysicsLogic.Papers.GarwinLedermanWeinrich1957.Garwin1957Types} (framework : PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework T) (apparatus : framework.StoppedMuonApparatusResult) (precession : framework.PrecessionScanResult) (control : framework.StoppedPionControlResult) (setup : framework.ParityInferenceSetup) (prediction : framework.ParityConservationPredictionCalculation) (result : framework.ParityViolationResult) : framework.controlledSymmetryFalsificationModel |>.NonvacuousControlledSymmetryFalsificationAt := framework.glw_route_exhibits_nonvacuous_controlled_symmetry_falsification apparatus precession control setup prediction result example {T : PhysicsLogic.Papers.CoffmanKunduWootters2000.CKWTypes} (framework : PhysicsLogic.Papers.CoffmanKunduWootters2000.DistributedEntanglementFramework T) (opening : framework.OpeningScope) (pure : framework.PureStateSelection) (definition : framework.TangleDefinitionCalculation) (identification : framework.SelectedABTangleReadoutAt) (positive : 0 < framework.tangleAB framework.selectedPureState) (bridge : framework.SelectedABSeparabilityTangleBridgeAt) : framework.sharedBipartiteEntanglementModel |>.NonvacuousBipartiteEntanglementAt := framework.selected_pair_tangle_exhibits_nonvacuous_bipartite_entanglement opening pure definition identification positive bridge example {Mode Observer : Type} (framework : PhysicsLogic.Papers.AMPS2012.Framework Mode Observer) (old : framework.IsOldBlackHole) (postulate2 : framework.Postulate2) (postulate4 : framework.Postulate4) (accessible : framework.SelectedModeIsAccessibleLateRadiation) (thermality : framework.SemiclassicalExteriorThermality) (noDrama : framework.SemiclassicalNoDramaInteriorPairPrinciple) (bridge : framework.PureInteriorPairSeparabilityEntropyBridgeAt) : framework.sharedBipartiteEntanglementModel |>.NonvacuousBipartiteEntanglementAt := framework.pure_thermal_interior_pair_exhibits_nonvacuous_bipartite_entanglement old postulate2 postulate4 accessible thermality noDrama bridge example (framework : PhysicsLogic.Papers.Polchinski1995.PaperFramework) (source : framework.polchinskiQuantizedRRFluxDerivationObtains .boundaryStateAnnulusFactorizationAndMinimalDiracPairing) : framework.quantizedRRFluxChargeContext |>.NonvacuousQuantizedBraneRRFluxChargeAt framework.quantizedRRFluxChargeModel := framework.polchinski_source_derivation_exhibits_nonvacuous_quantized_rr_flux_charge source example {Boundary Observable Value Symmetry Coupling Effect : Type} (framework : PhysicsLogic.Papers.Maldacena1997.D3Framework Boundary Observable Value Symmetry Coupling Effect) (source : framework.maldacenaD3RRFluxDerivationObtains .chargedSupergravitySolutionAndQuantizedFiveFormFlux) : framework.quantizedD3RRFluxChargeContext |>.NonvacuousQuantizedBraneRRFluxChargeAt framework.quantizedD3RRFluxChargeModel := framework.maldacena_source_derivation_exhibits_nonvacuous_quantized_d3_rr_flux_charge source example (model : PhysicsLogic.Papers.MerminWagner1966.Model) (channel : PhysicsLogic.Papers.MerminWagner1966.OrderChannel) (scope : PhysicsLogic.Papers.MerminWagner1966.SourceScope model) (ordering : PhysicsLogic.Papers.MerminWagner1966.OrderingChannelSetupAt model) (finitePackage : forall field, PhysicsLogic.Papers.MerminWagner1966.SmallNonzeroField field -> PhysicsLogic.Papers.MerminWagner1966.FiniteVolumeBogoliubovPackageAt model channel field) (transfer : PhysicsLogic.Papers.MerminWagner1966.ThermodynamicEquationTransferAt model channel) (infrared : PhysicsLogic.Papers.MerminWagner1966.LowDimensionalInfraredCalculationAt model channel) (spontaneous : PhysicsLogic.Papers.MerminWagner1966.SpontaneousLimitAt model channel) : (PhysicsLogic.Papers.MerminWagner1966.sharedInfraredModel model channel) |>.NonvacuousInfraredOrderSuppressionAt := PhysicsLogic.Papers.MerminWagner1966.mermin_wagner_route_exhibits_nonvacuous_infrared_order_suppression model channel scope ordering finitePackage transfer infrared spontaneous example (model : PhysicsLogic.Papers.MerminWagner1966.Model) (channel : PhysicsLogic.Papers.MerminWagner1966.OrderChannel) (scope : PhysicsLogic.Papers.MerminWagner1966.SourceScope model) (ordering : PhysicsLogic.Papers.MerminWagner1966.OrderingChannelSetupAt model) (finitePackage : forall field, PhysicsLogic.Papers.MerminWagner1966.SmallNonzeroField field -> PhysicsLogic.Papers.MerminWagner1966.FiniteVolumeBogoliubovPackageAt model channel field) (transfer : PhysicsLogic.Papers.MerminWagner1966.ThermodynamicEquationTransferAt model channel) (infrared : PhysicsLogic.Papers.MerminWagner1966.LowDimensionalInfraredCalculationAt model channel) (spontaneous : PhysicsLogic.Papers.MerminWagner1966.SpontaneousLimitAt model channel) : (PhysicsLogic.Papers.MerminWagner1966.toInfraredOrderSuppressionContext model channel) |>.SourceStructuralInfraredOrderSuppressionExplanationAt (PhysicsLogic.Papers.MerminWagner1966.sharedInfraredModel model channel) .finiteVolumeBogoliubovAndOrderedThermodynamicLimit (PhysicsLogic.Papers.MerminWagner1966.merminWagnerInfraredDerivationObtains model channel) := PhysicsLogic.Papers.MerminWagner1966.mermin_wagner_source_inputs_realize_source_structural_infrared_order_suppression_explanation model channel scope ordering finitePackage transfer infrared spontaneous example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (xy : PhysicsLogic.Papers.KosterlitzThouless1973.NearestNeighborXYHamiltonian model) (calculation : PhysicsLogic.Papers.KosterlitzThouless1973.OrdinaryOrderSuppressionCalculation model temperature) (spontaneous : PhysicsLogic.Papers.KosterlitzThouless1973.SpontaneousMagnetizationLimit model temperature) : (PhysicsLogic.Papers.KosterlitzThouless1973.infraredModel model temperature) |>.NonvacuousInfraredOrderSuppressionAt := PhysicsLogic.Papers.KosterlitzThouless1973.ordinary_order_input_exhibits_nonvacuous_infrared_order_suppression model temperature scope xy calculation spontaneous example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (temperature : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model temperature) (xy : PhysicsLogic.Papers.KosterlitzThouless1973.NearestNeighborXYHamiltonian model) (calculation : PhysicsLogic.Papers.KosterlitzThouless1973.OrdinaryOrderSuppressionCalculation model temperature) (spontaneous : PhysicsLogic.Papers.KosterlitzThouless1973.SpontaneousMagnetizationLimit model temperature) : (PhysicsLogic.Papers.KosterlitzThouless1973.infraredContext model temperature) |>.SourceStructuralInfraredOrderSuppressionExplanationAt (PhysicsLogic.Papers.KosterlitzThouless1973.infraredModel model temperature) .quotedOrdinaryOrderSuppressionAndSpontaneousLimit (PhysicsLogic.Papers.KosterlitzThouless1973.kosterlitzThoulessInfraredDerivationObtains model temperature) := PhysicsLogic.Papers.KosterlitzThouless1973.kosterlitz_thouless_source_inputs_realize_source_structural_infrared_order_suppression_explanation model temperature scope xy calculation spontaneous example {Symmetry OrderParameter : Type} (framework : PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.InfraredHigherFormFramework Symmetry OrderParameter) (scope : framework.HasLowDimensionalNoOrderScopeAt) (charged : framework.HasChargedOrderParameterAt) (divergence : framework.InfraredPotentialDivergesAt) (law : framework.HasCoulombExpectationLawAt) (criterion : framework.SuppressedChargedOrderImpliesUnbrokenAt) : framework.sharedInfraredModel |>.NonvacuousInfraredOrderSuppressionAt := framework.higher_form_route_exhibits_nonvacuous_infrared_order_suppression scope charged divergence law criterion example {Symmetry OrderParameter : Type} (framework : PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.InfraredHigherFormFramework Symmetry OrderParameter) (scope : framework.HasLowDimensionalNoOrderScopeAt) (charged : framework.HasChargedOrderParameterAt) (divergence : framework.InfraredPotentialDivergesAt) (law : framework.HasCoulombExpectationLawAt) (criterion : framework.SuppressedChargedOrderImpliesUnbrokenAt) : framework.toInfraredOrderSuppressionContext |>.SourceStructuralInfraredOrderSuppressionExplanationAt framework.sharedInfraredModel .dimensionCoulombPotentialAndWardSelection (PhysicsLogic.Papers.GeneralizedGlobalSymmetries2014.InfraredHigherFormFramework.higherFormInfraredDerivationObtains framework) := framework.higher_form_source_inputs_realize_source_structural_infrared_order_suppression_explanation scope charged divergence law criterion example {T : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyTypes} (framework : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework T) (predictions : framework.EstablishedCutoffIndependentPredictions) : framework.toSelectedObservableCompletenessContext.CompleteAt framework framework.lowEnergyRegime := framework.cutoff_independent_predictions_realize_scoped_descriptive_completeness predictions example {Mode Observer : Type} (framework : PhysicsLogic.Papers.AMPS2012.Framework Mode Observer) (infall : framework.NoDramaAtHorizon) (mass : framework.SelectedObserverHasNonzeroMass) : framework.toInfallingObserverRoleContext |>.NonvacuousAllObserversPhysicalAt framework.horizon := framework.massive_infalling_observer_realizes_nonvacuous_physical_observer_scope infall mass example {Mode Observer : Type} (framework : PhysicsLogic.Papers.AMPS2012.Framework Mode Observer) (accessible : framework.SelectedModeIsAccessibleLateRadiation) : framework.lateModeAvailabilityContext |>.NonvacuousAvailableSectorAt framework.horizon framework.age (fun mode => mode = framework.selectedMode) := framework.accessible_late_mode_realizes_nonvacuous_available_sector framework.selectedMode accessible example {T : PhysicsLogic.Papers.Metropolis1953.MetropolisTypes} (framework : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework T) (oldConfiguration proposedConfiguration : T.Configuration) (canonical : framework.EquationTwoCanonicalConfigurationFormulaAt) (proposal : framework.PositiveSymmetricProposalPairAt oldConfiguration proposedConfiguration) (transitions : framework.UphillPairTransitionFormulaAt oldConfiguration proposedConfiguration) : framework.PairwisePopulationFluxBalanceAt oldConfiguration proposedConfiguration := framework.canonical_equation_two_and_transition_formulas_balance_population_flux oldConfiguration proposedConfiguration canonical proposal transitions example {T : PhysicsLogic.Papers.Metropolis1953.MetropolisTypes} (framework : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework T) (balance : framework.PairwiseBalanceResult) : framework.CanonicalPairwisePopulationFluxAt := framework.established_metropolis_balance_retains_population_flux_equation balance example {T : PhysicsLogic.Papers.Metropolis1953.MetropolisTypes} (framework : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework T) (sampling : framework.CanonicalSamplingResult) : framework.CanonicalPairwisePopulationFluxAt := framework.canonical_sampling_retains_population_flux_equation sampling example : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.rejectionDiscardingPopulationFlow.HasBoltzmannPopulationRatio /\ Not PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.rejectionDiscardingPopulationFlow.BalancesPopulationFlux := by obtain ⟨⟨_domain, _symmetric, canonical, _downhill⟩, _notCounted, notBalanced⟩ := PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.discarding_rejected_attempts_destroys_canonical_population_balance exact ⟨canonical, notBalanced⟩ example : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.noncanonicalEqualPopulationFlow.RetainsUphillRejectedAttempts /\ Not PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.noncanonicalEqualPopulationFlow.BalancesPopulationFlux := by obtain ⟨⟨_domain, _symmetric, accepted, _downhill⟩, _notCanonical, notBalanced⟩ := PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework.Countermodels.correct_acceptance_without_canonical_population_does_not_balance_flux exact ⟨accepted, notBalanced⟩ example {T : PhysicsLogic.Papers.Metropolis1953.MetropolisTypes} (framework : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework T) (target : framework.CanonicalTargetResult) (balance : framework.PairwiseBalanceResult) (valid : framework.calculationValidAt framework.finitePeriodicDescription framework.finiteModelRegime) (calculation : framework.CanonicalEquilibrationCalculation) (interpretation : framework.CanonicalEnsembleInterpretation) : framework.equilibriumSamplingModel.toAsymptoticSamplingSolutionContext |>.SolvesAt framework.equilibriumSamplingModel.update framework.equilibriumSamplingModel () (fun ensemble => ensemble = framework.equilibriumSamplingModel.ensemble) := framework.metropolis_route_realizes_scoped_asymptotic_sampling_solution target balance valid calculation interpretation example {T : PhysicsLogic.Papers.Creutz1980.CreutzTypes} (framework : PhysicsLogic.Papers.Creutz1980.MonteCarloGaugeFramework T) (gauge : framework.RegulatedGaugeResult) (setup : framework.HeatBathSetup) (valid : framework.calculationValidAt framework.latticeDescription framework.finiteCutoffRegime) (calculation : framework.HeatBathCalculation) : framework.heatBathEquilibriumSamplingModel.toAsymptoticSamplingSolutionContext |>.SolvesAt framework.heatBathEquilibriumSamplingModel.update framework.heatBathEquilibriumSamplingModel () (fun ensemble => ensemble = framework.heatBathEquilibriumSamplingModel.ensemble) := framework.heat_bath_route_realizes_scoped_asymptotic_sampling_solution gauge setup valid calculation example {T : PhysicsLogic.Papers.PenziasWilson1965.PenziasWilsonTypes} (framework : PhysicsLogic.Papers.PenziasWilson1965.HornRadiometerFramework T) (setup : framework.HornRadiometerSetup) (calibration : framework.ReferenceCalibrationCalculation) (accounting : framework.ForegroundAccountingCalculation) (uncertainty : framework.ResidualUncertaintyCalculation) (robustness : framework.InstrumentAndSkyRobustnessCalculation) (interpretation : framework.ResidualDetectionInterpretationCalculation) (bridge : framework.ResidualDetectionBridgeCalculation) : framework.sharedBackgroundModel.toReferenceCalibrationContext.CalibratedBy framework.sharedBackgroundModel.observation framework.sharedBackgroundModel.calibrationReference framework.sharedBackgroundModel.calibrationReference framework.sharedBackgroundModel.rawSignal framework.sharedBackgroundModel.instrument framework.sharedBackgroundModel.regime := framework.horn_measurement_realizes_selected_reference_calibration setup calibration accounting uncertainty robustness interpretation bridge example {T : PhysicsLogic.Papers.FixsenEtAl1996.FixsenTypes} (framework : PhysicsLogic.Papers.FixsenEtAl1996.FirasFramework T) (reduction : framework.EstablishedFullDataReduction) (separation : framework.MonopoleComponentSeparationCalculation) (uncertainty : framework.MonopoleUncertaintyCalculation) (robustness : framework.AlternateAnalysisRobustnessCalculation) (interpretation : framework.MonopoleDetectionInterpretationCalculation) (bridge : framework.MonopoleDetectionBridgeCalculation) : framework.sharedBackgroundModel.toReferenceCalibrationContext.CalibratedBy framework.sharedBackgroundModel.observation framework.sharedBackgroundModel.calibrationReference framework.sharedBackgroundModel.calibrationReference framework.sharedBackgroundModel.rawSignal framework.sharedBackgroundModel.instrument framework.sharedBackgroundModel.regime := framework.firas_measurement_realizes_selected_reference_calibration reduction separation uncertainty robustness interpretation bridge example {T : PhysicsLogic.Papers.AharonovBohm1959.AharonovBohmTypes} (framework : PhysicsLogic.Papers.AharonovBohm1959.ElectromagneticPotentialFramework T) (setup : framework.MagneticInterferenceSetup) (calculation : framework.MagneticPhaseCalculation) (observation : framework.MagneticObservationCalculation) (interpretation : framework.MagneticInterpretationBridge) (distinctPathWitness : framework.upperPath ≠ framework.lowerPath) (selectedGaugeWitness : framework.electromagneticGaugeRelated framework.vectorPotential framework.vectorPotential) : framework.fieldFreeMagneticInterferenceModel |>.toSelectedGaugePhaseCovarianceContext.NonvacuousLawCovariant framework.fieldFreeMagneticInterferenceModel.relativePhase := framework.magnetic_route_realizes_nonvacuous_gauge_phase_covariance setup calculation observation interpretation distinctPathWitness selectedGaugeWitness example {T : PhysicsLogic.Papers.Berry1984.BerryTypes} (framework : PhysicsLogic.Papers.Berry1984.GeometricalPhaseFramework T) (setup : framework.AharonovBohmSetup) (calculation : framework.AharonovBohmCalculation) (observation : framework.AharonovBohmObservationBridge) (distinctPathWitness : framework.transportedBoxPath ≠ framework.referenceBoxPath) (selectedGaugeWitness : framework.electromagneticGaugeRelated framework.electromagneticVectorPotential framework.electromagneticVectorPotential) : framework.fieldFreeMagneticInterferenceModel |>.toSelectedGaugePhaseCovarianceContext.NonvacuousLawCovariant framework.fieldFreeMagneticInterferenceModel.relativePhase := framework.transported_box_realizes_nonvacuous_gauge_phase_covariance setup calculation observation distinctPathWitness selectedGaugeWitness example {T : PhysicsLogic.Papers.Friedmann1922.FriedmannTypes} (framework : PhysicsLogic.Papers.Friedmann1922.CurvatureWorldFramework T) (monotone : framework.EstablishedMonotoneWorldAssessment) (periodic : framework.EstablishedPeriodicWorldAssessment) : (framework.toSelectedWorldBranchDeterminationContext |>.AdmissibleDomainInhabitedAt framework.regime) ∧ ¬(framework.toSelectedWorldBranchDeterminationContext |>.ReducedDeterminesFullAt framework.regime) := framework.distinct_world_branches_refute_reduced_cosmological_determination monotone periodic example {T : PhysicsLogic.Papers.Heisenberg1927.HeisenbergTypes} (framework : PhysicsLogic.Papers.Heisenberg1927.HeisenbergFramework T) (setup : framework.GammaMicroscopeSetup) : framework.toPostMeasurementOperationalValueContext |>.OperationallyEstablishedValueAt framework.gammaApparatus framework.firstRecord framework.firstLocalizedState framework.position framework.position framework.firstPositionValue framework.firstPositionValue framework.electronSystem framework.quantumRegime := framework.gamma_microscope_establishes_postmeasurement_operational_position setup example {T : PhysicsLogic.Papers.Gamow1928.GamowTypes} (framework : PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework T) (result : framework.EstablishedGeigerNuttallComparison) : (framework.finiteBarrierDecayModel.toMetastableDecayProbabilityDescription |>.PredictivelyBranchesAt framework.finiteBarrierDecayModel framework.finiteBarrierDecayModel.lowerEnergySelection) ∧ (framework.finiteBarrierDecayModel.toMetastableDecayPossibilityContext |>.PhysicallyBranchesAt framework.finiteBarrierDecayModel.lowerEnergySelection) := framework.gamow_decay_realizes_physical_probability_branching result example {T : PhysicsLogic.Papers.GurneyCondon1928.GurneyCondonTypes} (framework : PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework T) (result : framework.EstablishedOverlapRateComparison) : (framework.finiteBarrierDecayModel.toMetastableDecayProbabilityDescription |>.PredictivelyBranchesAt framework.finiteBarrierDecayModel framework.finiteBarrierDecayModel.lowerEnergySelection) ∧ (framework.finiteBarrierDecayModel.toMetastableDecayPossibilityContext |>.PhysicallyBranchesAt framework.finiteBarrierDecayModel.lowerEnergySelection) := framework.gurney_condon_decay_realizes_physical_probability_branching result example {T : PhysicsLogic.Papers.GarwinLedermanWeinrich1957.Garwin1957Types} (framework : PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework T) (result : framework.MagneticMomentResult) : (framework.toMuonMomentUncertaintyAssessmentContext |>.ScopedUncertaintyAssessmentAt PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.selectedMuonMomentUncertaintySource framework.precessionScanRecord 0.10 framework.positiveMuonMoment framework.experimentalRegime) ∧ ¬(framework.toMuonMomentUncertaintyAssessmentContext |>.StatisticallyLicensedUncertaintyAssessmentAt PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.selectedMuonMomentUncertaintySource framework.precessionScanRecord 0.10 framework.positiveMuonMoment framework.experimentalRegime) ∧ ¬framework.exactGTwoEstablishedAt framework.positiveMuonMoment framework.experimentalRegime ∧ framework.precisionMagneticResonanceExperimentProposedAt framework.experimentalRegime ∧ ¬framework.precisionMagneticResonanceExperimentPerformedAt framework.experimentalRegime := framework.finite_accuracy_moment_realizes_scoped_carbon_field_uncertainty result example (framework : PhysicsLogic.Papers.LiebRobinson1972.PaperFramework) (velocity : framework.EstablishedFiniteGroupVelocity) : framework.EstablishedFiniteGroupVelocity ∧ ¬(PhysicsLogic.Framework.FiniteVelocityCommutatorConeContext.Countermodels.approximateConeContext |>.ExactSignalExclusionBridgeForConeAt PhysicsLogic.Framework.FiniteVelocityCommutatorConeContext.Countermodels.controlledModel PhysicsLogic.Framework.FiniteVelocityCommutatorConeContext.Countermodels.exactSignallingPropagation) := framework.finite_group_velocity_does_not_supply_exact_signal_exclusion_bridge velocity example (framework : PhysicsLogic.Papers.NachtergaeleSims2006.PaperFramework) (cone : framework.EstablishedFiniteVelocityCone) : framework.EstablishedFiniteVelocityCone ∧ ¬(PhysicsLogic.Framework.FiniteVelocityCommutatorConeContext.Countermodels.approximateConeContext |>.ExactSignalExclusionBridgeForConeAt PhysicsLogic.Framework.FiniteVelocityCommutatorConeContext.Countermodels.controlledModel PhysicsLogic.Framework.FiniteVelocityCommutatorConeContext.Countermodels.exactSignallingPropagation) := framework.finite_velocity_cone_does_not_supply_exact_signal_exclusion_bridge cone example {T : PhysicsLogic.Papers.EPR1935.EPRTypes} (framework : PhysicsLogic.Papers.EPR1935.CompletenessFramework T) (setup : framework.SeparatedSystemsSetup) (correlated : framework.conditionalWavefunctionFormulaReadout |>.EquationNineCorrelatedContinuumStateAt) (calculation : framework.CorrelatedContinuumNonfactorizationCalculation) : framework.toCorrelatedPairQuantumNonseparabilityContext |>.NonseparableCompositeAt framework.wavefunctionDescription framework.combinedSystem framework.combinedState framework.firstSystem framework.secondSystem framework.selectedRegime := framework.correlated_joint_state_realizes_nonseparable_quantum_composition setup correlated calculation example {T : PhysicsLogic.Papers.EPR1935.EPRTypes} (framework : PhysicsLogic.Papers.EPR1935.CompletenessFramework T) (setup : framework.SeparatedSystemsSetup) (correlated : framework.conditionalWavefunctionFormulaReadout |>.EquationNineCorrelatedContinuumStateAt) (calculation : framework.CorrelatedContinuumNonfactorizationCalculation) : framework.toCorrelatedPairQuantumNonseparabilityContext |>.AdmitsQuantumNonseparabilityAt framework.wavefunctionDescription framework.combinedSystem framework.firstSystem framework.secondSystem framework.selectedRegime := framework.correlated_joint_state_exhibits_inhabited_quantum_nonseparability setup correlated calculation example : PhysicsLogic.Papers.EPR1935.CompletenessFramework.Countermodels.wrongNamedConditionalWavefunctionReadout.FormulaDomainAt ∧ (PhysicsLogic.Papers.EPR1935.CompletenessFramework.Countermodels.wrongNamedConditionalWavefunctionReadout |>.EquationNineCorrelatedContinuumStateAt) ∧ ∃ firstWavefunction secondWavefunction : Real → Complex, ∀ firstPosition secondPosition, PhysicsLogic.Papers.EPR1935.CompletenessFramework.Countermodels.wrongNamedConditionalWavefunctionReadout.jointWavefunctionValueAt firstPosition secondPosition = firstWavefunction firstPosition * secondWavefunction secondPosition := PhysicsLogic.Papers.EPR1935.CompletenessFramework.Countermodels.correlated_continuum_formula_without_nonfactorization_can_factor example {T : PhysicsLogic.Papers.Kibble1976.Kibble1976Types} (framework : PhysicsLogic.Papers.Kibble1976.CosmicDomainFramework T) (setup : framework.SymmetryBreakingSetup) (orbit : framework.SelectedVacuumOrbitRealizationBridge) : (framework.toCosmologicalVacuumBreakingContext |>.SpontaneouslyBrokenIn framework.symmetryGroup framework.orderedPhase) ∧ framework.symmetryRestoredInPhaseAt framework.symmetryGroup framework.normalPhase framework.aboveCriticalTemperature framework.formationRegime ∧ framework.vacuumManifoldIsCosetAt framework.vacuumManifold framework.symmetryGroup framework.unbrokenSubgroup framework.formationRegime ∧ framework.phaseTransitionAt framework.cosmos framework.transition framework.normalPhase framework.orderedPhase framework.criticalTemperature framework.formationRegime := framework.cosmological_transition_realizes_continuous_vacuum_orbit_breaking setup orbit example {T : PhysicsLogic.Papers.Kibble1976.Kibble1976Types} (framework : PhysicsLogic.Papers.Kibble1976.CosmicDomainFramework T) (setup : framework.SymmetryBreakingSetup) (orbit : framework.SelectedVacuumOrbitRealizationBridge) : framework.toCosmologicalVacuumBreakingContext.continuousSymmetry framework.symmetryGroup ∧ ∃ state, framework.toCosmologicalVacuumBreakingContext.toUnclassifiedSymmetryContext |>.LawSymmetricSelectedMovedStateAt () framework.symmetryGroup state framework.orderedPhase := framework.the_cosmological_vacuum_orbit_retains_law_specific_state_selection setup orbit namespace PhysicsLogicKibbleSourceReuse open PhysicsLogic.Papers.Kibble1976 open PhysicsLogic.Papers.Kibble1976.VacuumOrientationChamberEnsemble open PhysicsLogic.Papers.Kibble1976.VacuumOrientationChamberEnsemble.RationalVacuumSimplex example {T : Kibble1976Types} (framework : CosmicDomainFramework T) (assumptions : framework.ProductionEstimateAssumptions) : framework.OriginalInitialProductionFractionFormulaAt := framework.independent_shortest_path_chambers_compute_original_defect_fractions assumptions example {T : Kibble1976Types} (framework : CosmicDomainFramework T) (assumptions : framework.ProductionEstimateAssumptions) : framework.oneInFourInitialEdgesFormStringsAt framework.stringEstimateDomainFamily framework.circleVacuumManifold framework.string framework.stringEdge framework.formationRegime /\ framework.oneInEightInitialVerticesFormMonopolesAt framework.monopoleEstimateDomainFamily framework.twoSphereVacuumManifold framework.monopole framework.monopoleVertex framework.formationRegime := framework.source_chamber_assumptions_establish_actual_defect_production_claims assumptions example : (equallyCountedShortestPathEnsemble 3).totalConfigurationWeight = 8 /\ (equallyCountedShortestPathEnsemble 3).trappedConfigurationWeight = 2 /\ (equallyCountedShortestPathEnsemble 3).initialDefectProbability = 1 / 4 /\ (equallyCountedShortestPathEnsemble 4).totalConfigurationWeight = 16 /\ (equallyCountedShortestPathEnsemble 4).trappedConfigurationWeight = 2 /\ (equallyCountedShortestPathEnsemble 4).initialDefectProbability = 1 / 8 := actual_chamber_counts_distinguish_strings_from_monopoles example : circularVacuaEnclosingOrigin.vacuaOnUnitManifold = true /\ circularVacuaWithinSemicircle.vacuaOnUnitManifold = true /\ circularVacuaEnclosingOrigin.circularVacuumTrapsString = true /\ circularVacuaWithinSemicircle.circularVacuumTrapsString = false := actual_circle_vacuum_vectors_distinguish_string_obstruction example : sphericalVacuaEnclosingOrigin.vacuaOnUnitManifold = true /\ sphericalVacuaWithinHemisphere.vacuaOnUnitManifold = true /\ sphericalVacuaEnclosingOrigin.sphericalVacuumTrapsMonopole = true /\ sphericalVacuaWithinHemisphere.sphericalVacuumTrapsMonopole = false := actual_sphere_vacuum_vectors_distinguish_monopole_obstruction example : forall coordinate : Fin 2, Finset.univ.sum (fun domain : Fin 3 => circularVacuaEnclosingOrigin.circleOrientationCoefficient domain * circularVacuaEnclosingOrigin.vacuumAt domain coordinate) = 0 := circle_signed_coefficients_cancel_actual_vacuum_vectors example : forall coordinate : Fin 3, Finset.univ.sum (fun domain : Fin 4 => sphericalVacuaEnclosingOrigin.sphereOrientationCoefficient domain * sphericalVacuaEnclosingOrigin.vacuumAt domain coordinate) = 0 := sphere_signed_coefficients_cancel_actual_vacuum_vectors example : Fidelity.countermodelMatrix .biasedIndependentCircleNotUnbiasedStringFraction := Fidelity.countermodel_matrix_is_realized .biasedIndependentCircleNotUnbiasedStringFraction example : Fidelity.countermodelMatrix .biasedIndependentSphereNotUnbiasedMonopoleFraction := Fidelity.countermodel_matrix_is_realized .biasedIndependentSphereNotUnbiasedMonopoleFraction example : Fidelity.countermodelMatrix .unbiasedCorrelatedChoicesNotIndependentStringFraction := Fidelity.countermodel_matrix_is_realized .unbiasedCorrelatedChoicesNotIndependentStringFraction example : Fidelity.countermodelMatrix .independentUnbiasedChoicesNotShortestPathStringFraction := Fidelity.countermodel_matrix_is_realized .independentUnbiasedChoicesNotShortestPathStringFraction example : Fidelity.countermodelMatrix .threeDomainStringFractionNotFourDomainMonopoleFraction := Fidelity.countermodel_matrix_is_realized .threeDomainStringFractionNotFourDomainMonopoleFraction example : Fidelity.countermodelMatrix .rawFavorableWeightNotNormalizedProbability := Fidelity.countermodel_matrix_is_realized .rawFavorableWeightNotNormalizedProbability example : Fidelity.countermodelMatrix .unitCircleVacuaNotStringObstruction := Fidelity.countermodel_matrix_is_realized .unitCircleVacuaNotStringObstruction example : Fidelity.countermodelMatrix .unitSphereVacuaNotMonopoleObstruction := Fidelity.countermodel_matrix_is_realized .unitSphereVacuaNotMonopoleObstruction end PhysicsLogicKibbleSourceReuse example {Volume Regime Energy : Type} (system : PhysicsLogic.Papers.WittenIndex1982.IndexSystem Volume Regime Energy) (volume : Volume) (regulators : system.PeriodicFiniteVolumeIndexSetup) (finiteZero : ∀ selectedVolume, system.groundStateEnergy (system.periodic selectedVolume) = system.zeroEnergy) (converges : system.periodicGroundEnergyConvergesToInfiniteVolumeAt (fun selectedVolume => system.groundStateEnergy (system.periodic selectedVolume)) (system.infiniteVolumeGroundStateEnergyFor .periodic)) : system.toPeriodicZeroEnergyThermodynamicLimitContext |>.ThermodynamicTransferSetupAt system.zeroEnergy := system.periodic_zero_energy_family_realizes_nonvacuous_thermodynamic_setup volume regulators finiteZero converges example {Volume Regime Energy : Type} (system : PhysicsLogic.Papers.WittenIndex1982.IndexSystem Volume Regime Energy) (limit : system.PeriodicZeroEnergyLimitPrinciple) : system.toPeriodicZeroEnergyThermodynamicLimitContext |>.UniformPropertyTransferBridge := system.periodic_zero_energy_limit_realizes_thermodynamic_transfer_bridge limit example {Volume Regime Energy : Type} (system : PhysicsLogic.Papers.WittenIndex1982.IndexSystem Volume Regime Energy) (volume : Volume) (regulators : system.PeriodicFiniteVolumeIndexSetup) (finiteZero : ∀ selectedVolume, system.groundStateEnergy (system.periodic selectedVolume) = system.zeroEnergy) (converges : system.periodicGroundEnergyConvergesToInfiniteVolumeAt (fun selectedVolume => system.groundStateEnergy (system.periodic selectedVolume)) (system.infiniteVolumeGroundStateEnergyFor .periodic)) (limit : system.PeriodicZeroEnergyLimitPrinciple) : system.toPeriodicZeroEnergyThermodynamicLimitContext |>.EstablishedThermodynamicPropertyAt system.zeroEnergy := system.periodic_zero_energy_realizes_shared_thermodynamic_limit volume regulators finiteZero converges limit example : (PhysicsLogic.Papers.WittenIndex1982.IndexSystem.ThermodynamicCountermodels.finiteBrokenInfiniteRestored |>.ThermodynamicTransferSetupAt true) ∧ (PhysicsLogic.Papers.WittenIndex1982.IndexSystem.ThermodynamicCountermodels.finiteBrokenInfiniteRestored |>.DirectInfinitePropertyAt false) ∧ ¬(PhysicsLogic.Papers.WittenIndex1982.IndexSystem.ThermodynamicCountermodels.finiteBrokenInfiniteRestored |>.DirectInfinitePropertyAt true) ∧ ¬(PhysicsLogic.Papers.WittenIndex1982.IndexSystem.ThermodynamicCountermodels.finiteBrokenInfiniteRestored |>.UniformPropertyTransferBridge) := PhysicsLogic.Papers.WittenIndex1982.IndexSystem.ThermodynamicCountermodels.finite_volume_breaking_can_be_restored_at_infinite_volume example {T : PhysicsLogic.Papers.Ramsey1950.RamseyTypes} (framework : PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework T) (transition : framework.EstablishedSeparatedFieldTransition) : framework.toSeparatedFieldResponseCausalityContext |>.RetardedResponseAt framework.separatedFieldProcedure framework.firstOscillatingField framework.singleVelocityProbability framework.molecularBeam framework.selectedRegime := framework.separated_field_evolution_realizes_retarded_transition_response transition example {T : PhysicsLogic.Papers.Ramsey1950.RamseyTypes} (framework : PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework T) (transition : framework.EstablishedSeparatedFieldTransition) : ¬(framework.toSeparatedFieldResponseCausalityContext |>.FutureDependenceAt framework.separatedFieldProcedure framework.firstOscillatingField framework.singleVelocityProbability framework.molecularBeam framework.selectedRegime) := framework.separated_field_transition_excludes_future_pulse_dependence transition example : (∃ sourceRegion responseRegion, PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.orderedPulseAndReadoutWithoutEvolution.stimulusPresentAt () () () sourceRegion () ∧ PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.orderedPulseAndReadoutWithoutEvolution.responsePresentAt () () responseRegion () ∧ PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.orderedPulseAndReadoutWithoutEvolution.notLaterThan sourceRegion responseRegion) ∧ ¬(PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.orderedPulseAndReadoutWithoutEvolution |>.ResponseDependenceWitnessAt () () () () ()) ∧ ¬(PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.orderedPulseAndReadoutWithoutEvolution |>.RetardedResponseAt () () () () ()) := PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.Countermodels.ordered_pulse_and_readout_without_evolution_do_not_establish_dependence example (framework : PhysicsLogic.Papers.StromingerVafa1996.Framework) (origin : framework.EstablishedMicroscopicOriginAt framework.selectedSector) : framework.toElectricChargeLargeParameterContext |>.LargeParameterSetupAt .strongBlackHole (fun observable => observable = .ellipticGenusGrowth) := framework.electric_charge_family_realizes_nonvacuous_large_parameter_setup origin example (framework : PhysicsLogic.Papers.StromingerVafa1996.Framework) (origin : framework.EstablishedMicroscopicOriginAt framework.selectedSector) : framework.toElectricChargeAsymptoticContributionContext |>.HasRelevantNonleadingContributionAt .strongBlackHole (fun observable => observable = .ellipticGenusGrowth) := framework.symmetric_product_offset_realizes_nonleading_entropy_contribution origin example (framework : PhysicsLogic.Papers.StromingerVafa1996.Framework) (origin : framework.EstablishedMicroscopicOriginAt framework.selectedSector) : framework.toElectricChargeAsymptoticContributionContext |>.AsymptoticallyOrganizedAt .strongBlackHole (fun observable => observable = .ellipticGenusGrowth) := framework.leading_entropy_match_realizes_shared_large_charge_organization origin example : PhysicsLogic.Papers.StromingerVafa1996.Framework.Countermodels.ReducedStromingerVafaEntropyReadout.HasLeadingLargeChargeAgreementAt PhysicsLogic.Papers.StromingerVafa1996.Framework.Countermodels.completeReducedStromingerVafaEntropyReadout ∧ ¬PhysicsLogic.Papers.StromingerVafa1996.Framework.Countermodels.ReducedStromingerVafaEntropyReadout.HasExactFiniteChargeEqualityAt PhysicsLogic.Papers.StromingerVafa1996.Framework.Countermodels.completeReducedStromingerVafaEntropyReadout := PhysicsLogic.Papers.StromingerVafa1996.Framework.Countermodels.leading_large_charge_agreement_does_not_imply_finite_charge_equality example {Boundary Observable Value Symmetry Coupling Effect : Type} (framework : PhysicsLogic.Papers.Maldacena1997.D3Framework Boundary Observable Value Symmetry Coupling Effect) (boundary : Boundary) (observable : Observable) (admissible : framework.admissibleBoundaryCondition boundary) (window : framework.LocalSupergravityProcessEnergyWindow observable) (reliable : framework.SupergravityReliableAtLargeN) (computes : framework.localSupergravityComputes observable) (correct : framework.ReliableSupergravityCalculationsAreCorrect) : framework.toLocalSupergravityClassicalRecoveryContext |>.EstablishedAt .nearHorizonTypeIIB (boundary, observable) (boundary, observable) := framework.local_supergravity_window_realizes_controlled_classical_bulk_recovery boundary observable admissible window reliable computes correct example {Boundary Observable Value Symmetry Coupling Effect : Type} (framework : PhysicsLogic.Papers.Maldacena1997.D3Framework Boundary Observable Value Symmetry Coupling Effect) (boundary : Boundary) (observable : Observable) (admissible : framework.admissibleBoundaryCondition boundary) (window : framework.LocalSupergravityProcessEnergyWindow observable) (reliable : framework.SupergravityReliableAtLargeN) (computes : framework.localSupergravityComputes observable) (correct : framework.ReliableSupergravityCalculationsAreCorrect) : (framework.localSupergravityPredictionRecoveryContext boundary observable) |>.toClassicalRecoveryContext.EstablishedAt .nearHorizonTypeIIB .largeNCurvatureControlled .largeNCurvatureControlled := framework.local_supergravity_prediction_match_realizes_classical_recovery boundary observable admissible window reliable computes correct example : (PhysicsLogic.Paradigm.ClassicalQuantum.Countermodels.matchedPredictionWithoutClassicalInterpretation |>.ControlledClassicalApproximationAt .selected) ∧ ¬(PhysicsLogic.Paradigm.ClassicalQuantum.Countermodels.matchedPredictionWithoutClassicalInterpretation |>.ClassicalInterpretationBridgeAt .selected) ∧ ¬(PhysicsLogic.Paradigm.ClassicalQuantum.Countermodels.matchedPredictionWithoutClassicalInterpretation |>.toClassicalRecoveryContext.EstablishedAt .quantum .selected .selected) := PhysicsLogic.Paradigm.ClassicalQuantum.Countermodels.controlled_prediction_match_does_not_by_itself_establish_classical_recovery example : PhysicsLogic.Papers.Maldacena1997.D3Framework.Countermodels.energyWindowCountermodel.SupergravityReliableAtLargeN ∧ ¬(PhysicsLogic.Papers.Maldacena1997.D3Framework.Countermodels.energyWindowCountermodel |>.toLocalSupergravityClassicalRecoveryContext.EstablishedAt .nearHorizonTypeIIB ((), ()) ((), ())) := PhysicsLogic.Papers.Maldacena1997.D3Framework.Countermodels.large_n_reliability_without_energy_window_does_not_establish_classical_recovery example {Sector Projector Operator Energy Potential Term Order Regime : Type} (framework : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ChiralNuclearFramework Sector Projector Operator Energy Potential Term Order Regime) (lowEnergy : framework.LowEnergyChiralSetup) (counting : framework.SharedPowerCountingRealization) (scales : framework.ChiralDimensionalScaleSelection) : framework.toChiralDimensionalFormContext |>.ScopedDimensionalFormAt .regulatedLowEnergyExpansion PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.selectedChiralDimensionalInputs PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.selectedChiralDimensionalConstraints framework.leadingFourNucleonContact framework.lowEnergyRegime := framework.low_energy_scales_realize_scoped_chiral_dimensional_form lowEnergy counting scales example {Sector Projector Operator Energy Potential Term Order Regime : Type} (framework : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ChiralNuclearFramework Sector Projector Operator Energy Potential Term Order Regime) (lowEnergy : framework.LowEnergyChiralSetup) (counting : framework.SharedPowerCountingRealization) (scales : framework.ChiralDimensionalScaleSelection) (leadingContact : framework.potentialContains (framework.effectivePotentialAt 2 framework.leadingOrder) framework.leadingFourNucleonContact) (independent : framework.lowEnergyConstantNotFixedByDimensionalCountingAt framework.leadingFourNucleonContact framework.lowEnergyRegime) : framework.toChiralDimensionalFormContext |>.OpenCoefficientDimensionalFormAt .regulatedLowEnergyExpansion framework.leadingFourNucleonContact PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.selectedChiralDimensionalInputs PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.selectedChiralDimensionalConstraints framework.leadingFourNucleonContact framework.lowEnergyRegime := framework.leading_contact_realizes_open_chiral_dimensional_coefficient lowEnergy counting scales leadingContact independent example {Sector Projector Operator Energy Potential Term Order Regime : Type} (framework : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ChiralNuclearFramework Sector Projector Operator Energy Potential Term Order Regime) (lowEnergy : framework.LowEnergyChiralSetup) (counting : framework.SharedPowerCountingRealization) (scales : framework.ChiralDimensionalScaleSelection) (leadingContact : framework.potentialContains (framework.effectivePotentialAt 2 framework.leadingOrder) framework.leadingFourNucleonContact) (determined : framework.lowEnergyConstantPhysicallyDeterminedAt framework.leadingFourNucleonContact framework.lowEnergyRegime) : framework.toChiralDimensionalFormContext |>.PhysicallyEstablishedDimensionalFormAt .regulatedLowEnergyExpansion framework.leadingFourNucleonContact PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.selectedChiralDimensionalInputs PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.selectedChiralDimensionalConstraints framework.leadingFourNucleonContact framework.lowEnergyRegime := framework.independently_determined_contact_establishes_normalized_chiral_form lowEnergy counting scales leadingContact determined example {Sector Projector Operator Energy Potential Term Order Regime : Type} (framework : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ChiralNuclearFramework Sector Projector Operator Energy Potential Term Order Regime) (lowEnergy : framework.LowEnergyChiralSetup) (counting : framework.SharedPowerCountingRealization) (scales : framework.ChiralDimensionalScaleSelection) (undetermined : ¬framework.lowEnergyConstantPhysicallyDeterminedAt framework.leadingFourNucleonContact framework.lowEnergyRegime) : (framework.toChiralDimensionalFormContext |>.ScopedDimensionalFormAt .regulatedLowEnergyExpansion PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.selectedChiralDimensionalInputs PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.selectedChiralDimensionalConstraints framework.leadingFourNucleonContact framework.lowEnergyRegime) ∧ ¬(framework.toChiralDimensionalFormContext |>.PhysicallyEstablishedDimensionalFormAt .regulatedLowEnergyExpansion framework.leadingFourNucleonContact PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.selectedChiralDimensionalInputs PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.selectedChiralDimensionalConstraints framework.leadingFourNucleonContact framework.lowEnergyRegime) := framework.chiral_dimensional_counting_does_not_establish_normalized_potential lowEnergy counting scales undetermined example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (cold hot area : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model cold) (neutral : PhysicsLogic.Papers.KosterlitzThouless1973.NeutralVortexGas model) (unitVortices : PhysicsLogic.Papers.KosterlitzThouless1973.UnitVorticesPresent model) (large : model.latticeSpacing ^ (2 : Nat) < area) (screened : PhysicsLogic.Papers.KosterlitzThouless1973.ScreenedStiffnessAt model cold) (bound : PhysicsLogic.Papers.KosterlitzThouless1973.BoundVortexPhase model cold) (crossing : PhysicsLogic.Papers.KosterlitzThouless1973.CrossesDissociationScale model hot) (dissociation : PhysicsLogic.Papers.KosterlitzThouless1973.DissociationBridge model hot) : (PhysicsLogic.Papers.KosterlitzThouless1973.selectedNeutralLogarithmicUnbindingHandoff model cold hot area) |>.NonvacuousNeutralThermodynamicsGroundedDefectUnbindingAt := PhysicsLogic.Papers.KosterlitzThouless1973.neutral_vortex_dielectric_breakdown_realizes_composed_defect_unbinding model cold hot area scope neutral unitVortices large screened bound crossing dissociation example {Site Contour : Type} (model : PhysicsLogic.Papers.KosterlitzThouless1973.KTModel Site Contour) (cold hot area : Real) (scope : PhysicsLogic.Papers.KosterlitzThouless1973.SourceScope model cold) (neutral : PhysicsLogic.Papers.KosterlitzThouless1973.NeutralVortexGas model) (unitVortices : PhysicsLogic.Papers.KosterlitzThouless1973.UnitVorticesPresent model) (large : model.latticeSpacing ^ (2 : Nat) < area) (screened : PhysicsLogic.Papers.KosterlitzThouless1973.ScreenedStiffnessAt model cold) (bound : PhysicsLogic.Papers.KosterlitzThouless1973.BoundVortexPhase model cold) (crossing : PhysicsLogic.Papers.KosterlitzThouless1973.CrossesDissociationScale model hot) (dissociation : PhysicsLogic.Papers.KosterlitzThouless1973.DissociationBridge model hot) : (PhysicsLogic.Papers.KosterlitzThouless1973.selectedNeutralLogarithmicUnbindingHandoff model cold hot area) |>.SourceSensitiveNeutralThermodynamicsGroundedDefectUnbindingAt .scopedNeutralUnitVortexGas (PhysicsLogic.Papers.KosterlitzThouless1973.kosterlitzThoulessNeutralDefectDerivationObtains model cold) .finiteAreaScreenedVortexCompetition (PhysicsLogic.Papers.KosterlitzThouless1973.kosterlitzThoulessLogarithmicDefectDerivationObtains model cold area) .dielectricBreakdownAndDissociation (PhysicsLogic.Papers.KosterlitzThouless1973.kosterlitzThoulessDefectUnbindingDerivationObtains model cold hot) .neutralScreeningAndDielectricDissociation (PhysicsLogic.Papers.KosterlitzThouless1973.kosterlitzThoulessDefectCompositionDerivationObtains model cold hot area) := PhysicsLogic.Papers.KosterlitzThouless1973.neutral_vortex_dielectric_breakdown_realizes_source_sensitive_composed_defect_unbinding model cold hot area scope neutral unitVortices large screened bound crossing dissociation example {T : PhysicsLogic.Papers.HalperinNelson1978.MeltingTypes} (framework : PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework T) (setup : framework.SmoothSolidSetup) (smoothCalculation : framework.SmoothSolidCalculation) (rg : framework.DislocationRGCalculation) (bridge : framework.DislocationUnbindingBridge) : framework.selectedNeutralLogarithmicDislocationHandoff |>.NonvacuousNeutralThermodynamicsGroundedDefectUnbindingAt := framework.quadratic_fugacity_dislocations_realize_composed_defect_unbinding setup smoothCalculation rg bridge example {T : PhysicsLogic.Papers.HalperinNelson1978.MeltingTypes} (framework : PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework T) (setup : framework.SmoothSolidSetup) (smoothCalculation : framework.SmoothSolidCalculation) (rg : framework.DislocationRGCalculation) (bridge : framework.DislocationUnbindingBridge) (perturbativeScope : framework.PerturbativeScopeSetup) : framework.selectedNeutralLogarithmicDislocationHandoff |>.SourceSensitiveNeutralThermodynamicsGroundedDefectUnbindingAt .smoothSolidBurgersPair (framework.halperinNelsonNeutralDefectDerivationObtains) .smoothSolidFiniteCoreCompetition (framework.halperinNelsonLogarithmicDefectDerivationObtains) .smoothElasticDislocationDecompositionAndQuadraticFugacityRecursion (framework.halperinNelsonDefectUnbindingDerivationObtains) .neutralLogarithmicQuadraticFugacityHandoff (framework.halperinNelsonDefectCompositionDerivationObtains) := framework.quadratic_fugacity_dislocations_realize_source_sensitive_composed_defect_unbinding setup smoothCalculation rg bridge perturbativeScope example {T : PhysicsLogic.Papers.Young1979.MeltingTypes} (framework : PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework T) (solidSetup : framework.SolidSetup) (recursionSetup : framework.RecursionSetup) (recursionCalculation : framework.RecursionCalculation) (flowCalculation : framework.FlowCalculation) (bridge : framework.UnbindingBridge) : framework.selectedNeutralLogarithmicDislocationHandoff |>.NonvacuousNeutralThermodynamicsGroundedDefectUnbindingAt := framework.self_consistent_vector_gas_realizes_composed_defect_unbinding solidSetup recursionSetup recursionCalculation flowCalculation bridge example {T : PhysicsLogic.Papers.Young1979.MeltingTypes} (framework : PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework T) (solidSetup : framework.SolidSetup) (solidCalculation : framework.SolidCalculation) (recursionSetup : framework.RecursionSetup) (recursionCalculation : framework.RecursionCalculation) (periodicSetup : framework.PeriodicSubstrateSetup) (periodicCalculation : framework.PeriodicSubstrateCalculation) (consistencyBridge : framework.PeriodicSubstrateConsistencyBridge) (flowCalculation : framework.FlowCalculation) (bridge : framework.UnbindingBridge) : framework.selectedNeutralLogarithmicDislocationHandoff |>.SourceSensitiveNeutralThermodynamicsGroundedDefectUnbindingAt .solidVectorBurgersPair (framework.youngNeutralDefectDerivationObtains) .solidFiniteCoreCompetition (framework.youngLogarithmicDefectDerivationObtains) .selfConsistentVectorGasRecursionAndPeriodicSubstrateConsistencyCheck (framework.youngDefectUnbindingDerivationObtains) .neutralLogarithmicSelfConsistentVectorGasHandoff (framework.youngDefectCompositionDerivationObtains) := framework.self_consistent_vector_gas_realizes_source_sensitive_composed_defect_unbinding solidSetup solidCalculation recursionSetup recursionCalculation periodicSetup periodicCalculation consistencyBridge flowCalculation bridge open PhysicsLogic.Framework.NeutralLogarithmicDefectUnbindingHandoff.Countermodels example : (completedDefectHandoff |>.NonvacuousNeutralThermodynamicsGroundedDefectUnbindingAt) ∧ completedDefectHandoff.neutralConfiguration.chargeRank = 2 ∧ completedDefectHandoff.thermodynamics.screenedBindingCoefficient < completedDefectHandoff.thermodynamics.bareBindingCoefficient ∧ completedDefectHandoff.transitionModel.orderedRegime ≠ completedDefectHandoff.transitionModel.unboundRegime := composed_defect_route_keeps_screened_binding_and_unbound_regime_distinct example : (completedEndpointsWithoutCalculation.neutralConfiguration |>.NonvacuousNeutralDefectConfigurationAt) ∧ (completedEndpointsWithoutCalculation.thermodynamics |>.NonvacuousLogarithmicDefectThermodynamicsAt) ∧ (completedEndpointsWithoutCalculation.transitionModel |>.NonvacuousDefectUnbindingTransitionAt) ∧ ¬(completedEndpointsWithoutCalculation |>.SelectedNeutralLogarithmicUnbindingHandoffAt) := completed_neutral_thermodynamics_and_unbinding_do_not_supply_flow_handoff example : (completedDefectHandoff |>.NonvacuousNeutralThermodynamicsGroundedDefectUnbindingAt) ∧ (selectedNeutralPair |>.SourceStructuralNeutralDefectConfigurationExplanationAt () availableChildDerivationObtains) ∧ (selectedLogarithmicCompetition |>.SourceStructuralLogarithmicDefectThermodynamicsExplanationAt () availableChildDerivationObtains) ∧ (PhysicsLogic.Framework.DefectUnbindingTransitionContext.Countermodels.completedTechnicalModel |>.DirectSourceMechanisticDefectUnbindingExplanationAt () availableChildDerivationObtains) ∧ ¬(completedDefectHandoff |>.SourceSensitiveNeutralThermodynamicsGroundedDefectUnbindingAt () availableChildDerivationObtains () availableChildDerivationObtains () availableChildDerivationObtains .absent missingNeutralLogarithmicUnbindingCompositionDerivationObtains) := completed_composition_and_child_source_explanations_without_handoff_derivation_have_no_source_sensitive_composition example {T : PhysicsLogic.Papers.CoffmanKunduWootters2000.CKWTypes} (framework : PhysicsLogic.Papers.CoffmanKunduWootters2000.DistributedEntanglementFramework T) (opening : framework.OpeningScope) (pure : framework.PureStateSelection) (definition : framework.TangleDefinitionCalculation) (identification : framework.SelectedABTangleReadoutAt) (positive : 0 < framework.tangleAB framework.selectedPureState) (bridge : framework.SelectedABSeparabilityTangleBridgeAt) (monogamy : framework.PureMonogamyBound) (saturation : framework.tangleAWithBC framework.selectedPureState ≤ framework.tangleAB framework.selectedPureState) : framework.selectedEntangledDominantPairSharingHandoff |>.NonvacuousEntanglementGroundedTripartiteSharingAt := framework.selected_ab_tangle_realizes_composed_entanglement_sharing opening pure definition identification positive bridge monogamy saturation example {T : PhysicsLogic.Papers.CoffmanKunduWootters2000.CKWTypes} (framework : PhysicsLogic.Papers.CoffmanKunduWootters2000.DistributedEntanglementFramework T) (opening : framework.OpeningScope) (pure : framework.PureStateSelection) (definition : framework.TangleDefinitionCalculation) (identification : framework.SelectedABTangleReadoutAt) (positive : 0 < framework.tangleAB framework.selectedPureState) (bridge : framework.SelectedABSeparabilityTangleBridgeAt) (formula : framework.SpinFlipTraceFormulaAt) (composite : framework.CompositeTangleIdentification) (saturation : framework.SelectedFirstPairSaturationCalculation) : framework.selectedEntangledDominantPairSharingHandoff |>.SourceSensitiveEntanglementGroundedTripartiteSharingAt .selectedABSpinFlipTangleWitness framework.CoffmanKunduWoottersBipartiteEntanglementDerivationObtains .spinFlipTraceDeterminantCompositeTangleAndSelectedPairSaturation framework.CoffmanKunduWoottersTripartiteSharingDerivationObtains .selectedABTangleMonogamySaturationHandoff framework.CoffmanKunduWoottersEntanglementSharingHandoffDerivationObtains := framework.selected_ab_tangle_realizes_source_sensitive_composed_entanglement_sharing opening pure definition identification positive bridge formula composite saturation example {Mode Observer : Type} (framework : PhysicsLogic.Papers.AMPS2012.Framework Mode Observer) (old : framework.IsOldBlackHole) (postulate1 : framework.Postulate1) (postulate2 : framework.Postulate2) (postulate4 : framework.Postulate4) (accessible : framework.SelectedModeIsAccessibleLateRadiation) (strongSubadditivity : framework.entropy.StrongSubadditivity) (unitarity : framework.OldUnitaryRadiationEntropyPrinciple) (thermality : framework.SemiclassicalExteriorThermality) (noDrama : framework.SemiclassicalNoDramaInteriorPairPrinciple) (factorization : framework.PureInteriorPairFactorization) (bridge : framework.PureInteriorPairSeparabilityEntropyBridgeAt) : False := framework.old_hole_interior_pair_contradicts_composed_entanglement_sharing old postulate1 postulate2 postulate4 accessible strongSubadditivity unitarity thermality noDrama factorization bridge example {Mode Observer : Type} (framework : PhysicsLogic.Papers.AMPS2012.Framework Mode Observer) (old : framework.IsOldBlackHole) (postulate1 : framework.Postulate1) (postulate2 : framework.Postulate2) (postulate4 : framework.Postulate4) (accessible : framework.SelectedModeIsAccessibleLateRadiation) (strongSubadditivity : framework.entropy.StrongSubadditivity) (unitarity : framework.OldUnitaryRadiationEntropyPrinciple) (thermality : framework.SemiclassicalExteriorThermality) (noDrama : framework.SemiclassicalNoDramaInteriorPairPrinciple) (factorization : framework.PureInteriorPairFactorization) (bridge : framework.PureInteriorPairSeparabilityEntropyBridgeAt) : framework.selectedEntangledCompetingPairSharingHandoff |>.SourceSensitiveEntangledPairSharingHandoffAt .oldHoleThermalPureInteriorPair framework.AMPSBipartiteEntanglementDerivationObtains .oldHoleStrongSubadditivityNoDramaEntropyRoute framework.AMPSTripartiteSharingDerivationObtains .oldHoleThermalInteriorPairStrongSubadditivityHandoff framework.AMPSEntanglementSharingHandoffDerivationObtains := framework.old_hole_interior_pair_realizes_source_sensitive_entanglement_sharing_handoff old postulate1 postulate2 postulate4 accessible strongSubadditivity unitarity thermality noDrama factorization bridge example {Mode Observer : Type} (framework : PhysicsLogic.Papers.AMPS2012.Framework Mode Observer) (old : framework.IsOldBlackHole) (postulate1 : framework.Postulate1) (postulate2 : framework.Postulate2) (postulate4 : framework.Postulate4) (accessible : framework.SelectedModeIsAccessibleLateRadiation) (strongSubadditivity : framework.entropy.StrongSubadditivity) (unitarity : framework.OldUnitaryRadiationEntropyPrinciple) (thermality : framework.SemiclassicalExteriorThermality) (noDrama : framework.SemiclassicalNoDramaInteriorPairPrinciple) (factorization : framework.PureInteriorPairFactorization) (bridge : framework.PureInteriorPairSeparabilityEntropyBridgeAt) : False := framework.old_hole_source_sensitive_interior_pair_contradicts_entanglement_sharing old postulate1 postulate2 postulate4 accessible strongSubadditivity unitarity thermality noDrama factorization bridge open PhysicsLogic.Framework.EntangledPairTripartiteSharingHandoff.Countermodels example : completedDominantPairSharing |>.NonvacuousEntanglementGroundedTripartiteSharingAt := dominant_entangled_pair_and_completed_sharing_really_coexist example : (completedDominantPairSharing |>.NonvacuousEntanglementGroundedTripartiteSharingAt) ∧ (dominantEntangledPairContext |>.SourceStructuralBipartiteEntanglementExplanationAt dominantEntangledPair () availablePairSharingChildDerivationObtains) ∧ (PhysicsLogic.Framework.TripartiteSharingContext.Countermodels.completedTripartiteSharing |>.SourceStructuralTripartiteSharingExclusionExplanationAt PhysicsLogic.Framework.TripartiteSharingContext.Countermodels.completedTripartiteSharing.selectedModel () availablePairSharingChildDerivationObtains) ∧ ¬(completedDominantPairSharing |>.SourceSensitiveEntanglementGroundedTripartiteSharingAt () availablePairSharingChildDerivationObtains () availablePairSharingChildDerivationObtains .absent missingEntangledPairSharingHandoffDerivationObtains) := completed_sharing_composition_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition example : PhysicsLogic.Framework.BipartiteEntanglementModel.NonvacuousBipartiteEntanglementAt completedPairAndSharingWithoutCalculation.entanglementModel /\ PhysicsLogic.Framework.TripartiteSharingContext.NonvacuousTripartiteSharingConstraintAt completedPairAndSharingWithoutCalculation.sharingContext completedPairAndSharingWithoutCalculation.sharingModel /\ Not (completedPairAndSharingWithoutCalculation |>.SelectedEntangledPairSharingHandoffAt) := completed_pair_and_sharing_do_not_supply_paper_local_handoff example : PhysicsLogic.Framework.BipartiteEntanglementModel.NonvacuousBipartiteEntanglementAt competingPairWithoutSharingBound.entanglementModel /\ competingPairWithoutSharingBound.SelectedEntangledPairSharingHandoffAt /\ Not (PhysicsLogic.Framework.TripartiteSharingContext.SharingBoundAt competingPairWithoutSharingBound.sharingContext competingPairWithoutSharingBound.sharingModel) := competing_pair_handoff_does_not_supply_the_sharing_bound example : let changed := { completedDominantPairSharing with pairDemandRole := .competing } Not changed.SelectedEntangledPairSharingAlignmentAt := dominant_pair_cannot_be_reclassified_as_competing example {T : PhysicsLogic.Papers.Guth1981.CosmologyTypes} (framework : PhysicsLogic.Papers.Guth1981.InflationFramework T) (initial : T.InitialCondition) (region : T.Region) (rate : T.Rate) (phase : T.Phase) (chaotic : framework.chaoticGlobalInitialStateAllowed initial) (hot : framework.hotLocalRegion region initial) (thermalization : framework.localThermalizationExpected region) (anisotropy : framework.anisotropyDampingImported region) (inhomogeneity : framework.inhomogeneityDampingExpected region) (expansion : framework.exponentialExpansion initial) (localCalculation : framework.LocalPatchCalculation initial region) (admissible : framework.admissibleInitial framework.standardModel initial) (applies : framework.mechanismApplies framework.inflationaryMechanism framework.standardModel initial) (solutionRule : framework.InflationarySolutionRule) (distinctTargets : framework.horizonTarget ≠ framework.flatnessTarget) (firstOrder : framework.firstOrderPhaseTransition phase) (tunneling : framework.bubbleNucleationIsTunneling phase) (hugeExpansion : framework.hugeExpansionBeforeTransitionPlausible initial phase) (slow : framework.slowNucleationRelativeToExpansion rate initial) (random : framework.randomBubbleFormation initial phase) (walls : framework.latentHeatInitiallyInBubbleWalls phase) (collisions : framework.thermalizationRequiresManyWallCollisions phase) (range : framework.enormousBubbleSizeRange initial phase) (regimeAssessment : framework.PercolationRegimeAssessment initial phase rate) (percolationCalculation : framework.PercolationCalculation initial phase rate) (thermalizationCalculation : framework.ClusterThermalizationCalculation initial phase) (exitAssessment : framework.GracefulExitAssessment initial phase) : (framework.selectedSameMechanismInflationBubbleHandoff initial region rate phase) |>.NonvacuousInflationaryPatchBubbleObstructionAt := framework.local_patch_and_slow_nucleation_realize_composed_inflation_bubble_failure initial region rate phase chaotic hot thermalization anisotropy inhomogeneity expansion localCalculation admissible applies solutionRule distinctTargets firstOrder tunneling hugeExpansion slow random walls collisions range regimeAssessment percolationCalculation thermalizationCalculation exitAssessment example {T : PhysicsLogic.Papers.Guth1981.CosmologyTypes} (framework : PhysicsLogic.Papers.Guth1981.InflationFramework T) (initial : T.InitialCondition) (region : T.Region) (rate : T.Rate) (phase : T.Phase) (chaotic : framework.chaoticGlobalInitialStateAllowed initial) (hot : framework.hotLocalRegion region initial) (thermalization : framework.localThermalizationExpected region) (anisotropy : framework.anisotropyDampingImported region) (inhomogeneity : framework.inhomogeneityDampingExpected region) (expansion : framework.exponentialExpansion initial) (localCalculation : framework.LocalPatchCalculation initial region) (admissible : framework.admissibleInitial framework.standardModel initial) (applies : framework.mechanismApplies framework.inflationaryMechanism framework.standardModel initial) (solutionRule : framework.InflationarySolutionRule) (distinctTargets : framework.horizonTarget ≠ framework.flatnessTarget) (firstOrder : framework.firstOrderPhaseTransition phase) (tunneling : framework.bubbleNucleationIsTunneling phase) (hugeExpansion : framework.hugeExpansionBeforeTransitionPlausible initial phase) (slow : framework.slowNucleationRelativeToExpansion rate initial) (random : framework.randomBubbleFormation initial phase) (walls : framework.latentHeatInitiallyInBubbleWalls phase) (collisions : framework.thermalizationRequiresManyWallCollisions phase) (range : framework.enormousBubbleSizeRange initial phase) (regimeAssessment : framework.PercolationRegimeAssessment initial phase rate) (percolationCalculation : framework.PercolationCalculation initial phase rate) (thermalizationCalculation : framework.ClusterThermalizationCalculation initial phase) (exitAssessment : framework.GracefulExitAssessment initial phase) : (framework.selectedSameMechanismInflationBubbleHandoff initial region rate phase) |>.SourceSensitiveInflationaryPatchBubbleObstructionAt (framework.toInflationaryPatchContext initial region) (framework.toBubbleTransitionCompletionContext initial rate phase) .localSubhorizonPatchExpansionAndProblemTransfer (framework.guthInflationaryPatchDerivationObtains initial region) .slowNucleationPercolationAndThermalizationFailure (framework.guthBubbleCompletionDerivationObtains initial phase rate) .localPatchSlowNucleationSameMechanismComparison (framework.guthInflationBubbleHandoffDerivationObtains initial region rate phase) := framework |>.local_patch_and_slow_nucleation_realize_source_sensitive_composed_inflation_bubble_failure initial region rate phase chaotic hot thermalization anisotropy inhomogeneity expansion localCalculation admissible applies solutionRule distinctTargets firstOrder tunneling hugeExpansion slow random walls collisions range regimeAssessment percolationCalculation thermalizationCalculation exitAssessment example {T : PhysicsLogic.Papers.AlbrechtSteinhardt1982.CosmologyTypes} (framework : PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework T) (initial : T.InitialCondition) (region : T.Region) (admissible : framework.admissibleInitial framework.selectedModel initial) (applies : framework.mechanismApplies framework.fluctuationExpansionMechanism framework.selectedModel initial) (homogeneous : framework.fluctuationRegionHomogeneous region initial) (expansion : framework.exponentialExpansionContinuesIn region initial) (coverage : framework.regionContainsObservedUniverse region initial) (homogeneityBridge : framework.HomogeneityInterpretationBridge region initial) (flatnessBridge : framework.FlatnessInterpretationBridge region initial) (distinctTargets : framework.homogeneityTarget ≠ framework.flatnessTarget) (distinctMechanisms : framework.fluctuationExpansionMechanism ≠ framework.standardBubbleMechanism) (firstOrder : framework.firstOrderTransition framework.symmetricPhase framework.brokenPhase) (enoughExpansion : framework.sufficientExpansionForCosmologicalPuzzles initial) (outpaced : framework.expansionDominatesBubbleProductionAndGrowth initial) (failureCalculation : framework.OldInflationFailureCalculation initial) : (framework.selectedReplacementInflationBubbleHandoff initial region) |>.NonvacuousInflationaryPatchBubbleObstructionAt := framework.fluctuation_patch_and_old_bubbles_realize_composed_inflation_comparison initial region admissible applies homogeneous expansion coverage homogeneityBridge flatnessBridge distinctTargets distinctMechanisms firstOrder enoughExpansion outpaced failureCalculation example {T : PhysicsLogic.Papers.AlbrechtSteinhardt1982.CosmologyTypes} (framework : PhysicsLogic.Papers.AlbrechtSteinhardt1982.NewInflationFramework T) (initial : T.InitialCondition) (region : T.Region) (admissible : framework.admissibleInitial framework.selectedModel initial) (applies : framework.mechanismApplies framework.fluctuationExpansionMechanism framework.selectedModel initial) (homogeneous : framework.fluctuationRegionHomogeneous region initial) (expansion : framework.exponentialExpansionContinuesIn region initial) (coverage : framework.regionContainsObservedUniverse region initial) (homogeneityBridge : framework.HomogeneityInterpretationBridge region initial) (flatnessBridge : framework.FlatnessInterpretationBridge region initial) (distinctTargets : framework.homogeneityTarget ≠ framework.flatnessTarget) (distinctMechanisms : framework.fluctuationExpansionMechanism ≠ framework.standardBubbleMechanism) (firstOrder : framework.firstOrderTransition framework.symmetricPhase framework.brokenPhase) (enoughExpansion : framework.sufficientExpansionForCosmologicalPuzzles initial) (outpaced : framework.expansionDominatesBubbleProductionAndGrowth initial) (failureCalculation : framework.OldInflationFailureCalculation initial) : (framework.selectedReplacementInflationBubbleHandoff initial region) |>.SourceSensitiveInflationaryPatchBubbleObstructionAt (framework.toInflationaryPatchContext initial region) (framework.toBubbleTransitionCompletionContext initial) .homogeneousFluctuationRegionExpansionAndProblemTransfer (framework.albrechtSteinhardtInflationaryPatchDerivationObtains initial region) .standardBubbleFailureUnderSufficientExpansion (framework.albrechtSteinhardtBubbleCompletionDerivationObtains initial) .homogeneousFluctuationReplacesFailedStandardBubble (framework |>.albrechtSteinhardtInflationBubbleHandoffDerivationObtains initial region) := framework |>.fluctuation_patch_and_old_bubbles_realize_source_sensitive_composed_inflation_comparison initial region admissible applies homogeneous expansion coverage homogeneityBridge flatnessBridge distinctTargets distinctMechanisms firstOrder enoughExpansion outpaced failureCalculation open PhysicsLogic.Framework.InflationaryPatchBubbleObstructionHandoff.Countermodels example : sameMechanismPatchAndObstruction |>.NonvacuousInflationaryPatchBubbleObstructionAt := successful_patch_and_same_mechanism_bubble_failure_really_coexist example : (sameMechanismPatchAndObstruction |>.NonvacuousInflationaryPatchBubbleObstructionAt) ∧ (successfulSameMechanismPatchContext |>.DirectSourceStructuralInflationaryPatchExplanationAt (successfulPatch false) () availableInflationBubbleChildDerivationObtains) ∧ (obstructedStandardBubbleContext |>.DirectSourceStructuralBubbleCompletionObstructionExplanationAt () () () () availableInflationBubbleChildDerivationObtains) ∧ ¬(sameMechanismPatchAndObstruction |>.SourceSensitiveInflationaryPatchBubbleObstructionAt successfulSameMechanismPatchContext obstructedStandardBubbleContext () availableInflationBubbleChildDerivationObtains () availableInflationBubbleChildDerivationObtains .absent missingInflationBubbleHandoffDerivationObtains) := completed_patch_bubble_composition_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition example : PhysicsLogic.Framework.InflationaryPatchBubbleObstructionHandoff.NonvacuousInflationaryPatchBubbleObstructionAt replacementPatchAndOldBubbleObstruction /\ replacementPatchAndOldBubbleObstruction.patchCalculation.mechanism ≠ replacementPatchAndOldBubbleObstruction.bubbleCalculation.mechanism /\ Not (PhysicsLogic.Framework.BubbleTransitionCalculationModel.sideEffect replacementPatchAndOldBubbleObstruction.bubbleCalculation replacementPatchAndOldBubbleObstruction.patchCalculation.mechanism replacementPatchAndOldBubbleObstruction.bubbleCalculation.model replacementPatchAndOldBubbleObstruction.bubbleCalculation.initial replacementPatchAndOldBubbleObstruction.bubbleCalculation.consequence) := replacement_patch_and_old_bubble_failure_keep_mechanisms_distinct example : PhysicsLogic.Framework.InflationaryPatchCalculationModel.NonvacuousInflationaryPatchAt completedPatchAndObstructionWithoutHandoff.patchCalculation /\ PhysicsLogic.Framework.BubbleTransitionCalculationModel.NonvacuousBubbleCompletionObstructionAt completedPatchAndObstructionWithoutHandoff.bubbleCalculation /\ Not (completedPatchAndObstructionWithoutHandoff |>.SelectedPatchBubbleObstructionHandoffAt) := completed_patch_and_bubble_failure_do_not_supply_source_handoff example : let changed := { sameMechanismPatchAndObstruction with mechanismRelation := .replacementMechanism } Not changed.SelectedPatchBubbleAlignmentAt := same_mechanism_patch_cannot_be_reclassified_as_replacement example {T : PhysicsLogic.Papers.Einstein1911.LightGravityTypes} (framework : PhysicsLogic.Papers.Einstein1911.LightGravityFramework T) (equivalence : framework.physicallyEquivalentForAllProcesses framework.gravitationalFrame framework.acceleratedFrame framework.firstOrderRegime) (comparison : framework.FrequencyComparisonSetup) (frequencyCalculation : framework.FrequencyShiftCalculation) (coordinateSpeedCalculation : framework.CoordinateLightSpeedCalculation) (valid : framework.validity.validAt framework.firstOrderDescription framework.firstOrderRegime) (variableSpeed : framework.coordinateLightSpeedDependsOnPotential framework.firstOrderRegime) (transverse : framework.transverseLightRayInField framework.selectedRay framework.firstOrderRegime) (huygens : framework.huygensConstructionApplied framework.selectedRay framework.firstOrderRegime) (deflectionCalculation : framework.LightDeflectionCalculation) : framework.firstOrderOpticalFrequencyRayHandoff |>.NonvacuousGravitationalFrequencyAndRayDeflectionAt := framework.first_order_clock_and_huygens_routes_realize_composed_solar_optics equivalence comparison frequencyCalculation coordinateSpeedCalculation valid variableSpeed transverse huygens deflectionCalculation example {T : PhysicsLogic.Papers.Einstein1911.LightGravityTypes} (framework : PhysicsLogic.Papers.Einstein1911.LightGravityFramework T) (equivalence : framework.physicallyEquivalentForAllProcesses framework.gravitationalFrame framework.acceleratedFrame framework.firstOrderRegime) (comparison : framework.FrequencyComparisonSetup) (frequencyCalculation : framework.FrequencyShiftCalculation) (coordinateSpeedCalculation : framework.CoordinateLightSpeedCalculation) (valid : framework.validity.validAt framework.firstOrderDescription framework.firstOrderRegime) (variableSpeed : framework.coordinateLightSpeedDependsOnPotential framework.firstOrderRegime) (transverse : framework.transverseLightRayInField framework.selectedRay framework.firstOrderRegime) (huygens : framework.huygensConstructionApplied framework.selectedRay framework.firstOrderRegime) (deflectionCalculation : framework.LightDeflectionCalculation) : framework.firstOrderOpticalFrequencyRayHandoff |>.SourceSensitiveGravitationalFrequencyAndRayDeflectionAt framework.toGravitationalFrequencyShiftContext framework.toGravitationalRayDeflectionContext .acceleratedFrameDopplerClockComparison framework.einstein1911FrequencyShiftDerivationObtains .coordinateLightSpeedHuygensSolarDeflection framework.einstein1911RayDeflectionDerivationObtains .acceleratedClockCoordinateSpeedHuygensSolarOptics framework.einstein1911OpticalHandoffDerivationObtains := framework |>.first_order_clock_and_huygens_realize_source_sensitive_composed_solar_optics equivalence comparison frequencyCalculation coordinateSpeedCalculation valid variableSpeed transverse huygens deflectionCalculation example {T : PhysicsLogic.Papers.Einstein1916.RelativityTypes} (framework : PhysicsLogic.Papers.Einstein1916.RelativityFramework T) (approximation : T.Approximation) (metric : T.Metric) (source : T.Source) (situation : T.Situation) (redshift deflection : T.Phenomenon) (pointMass : framework.pointMassSource source situation) (rayPasses : framework.lightRayPassesPointMassSource framework.selectedLightRay source situation) (solution : framework.staticFirstOrderMetricSolution approximation metric source situation) (comparison : framework.StaticPointMassComparisonSetup approximation metric source situation) (frequencyCalculation : framework.StaticFrequencyShiftCalculation approximation metric source situation redshift) (deflectionCalculation : framework.StaticLightDeflectionCalculation approximation metric source situation deflection) (radialContraction : framework.radialRodCoordinateContraction metric situation) (tangentialUnchanged : framework.tangentialRodCoordinateLengthUnchanged metric situation) : (framework.staticMetricOpticalFrequencyRayHandoff approximation metric source situation redshift deflection) |>.NonvacuousGravitationalFrequencyAndRayDeflectionAt := framework.static_metric_clock_and_null_ray_realize_composed_solar_optics approximation metric source situation redshift deflection pointMass rayPasses solution comparison frequencyCalculation deflectionCalculation radialContraction tangentialUnchanged example {T : PhysicsLogic.Papers.Einstein1916.RelativityTypes} (framework : PhysicsLogic.Papers.Einstein1916.RelativityFramework T) (approximation : T.Approximation) (metric : T.Metric) (source : T.Source) (situation : T.Situation) (redshift deflection : T.Phenomenon) (pointMass : framework.pointMassSource source situation) (rayPasses : framework.lightRayPassesPointMassSource framework.selectedLightRay source situation) (solution : framework.staticFirstOrderMetricSolution approximation metric source situation) (comparison : framework.StaticPointMassComparisonSetup approximation metric source situation) (frequencyCalculation : framework.StaticFrequencyShiftCalculation approximation metric source situation redshift) (deflectionCalculation : framework.StaticLightDeflectionCalculation approximation metric source situation deflection) (radialContraction : framework.radialRodCoordinateContraction metric situation) (tangentialUnchanged : framework.tangentialRodCoordinateLengthUnchanged metric situation) : (framework.staticMetricOpticalFrequencyRayHandoff approximation metric source situation redshift deflection) |>.SourceSensitiveGravitationalFrequencyAndRayDeflectionAt (framework.toGravitationalFrequencyShiftContext approximation metric source situation redshift) (framework.toGravitationalRayDeflectionContext approximation metric source situation deflection) .staticMetricLocalClockComparison (framework.einstein1916FrequencyShiftDerivationObtains approximation metric source situation redshift) .staticMetricNullRaySolarDeflection (framework.einstein1916RayDeflectionDerivationObtains approximation metric source situation deflection) .staticMetricClockNullRaySpatialRodSolarOptics (framework.einstein1916OpticalHandoffDerivationObtains approximation metric source situation redshift deflection) := framework |>.static_metric_clock_and_null_ray_realize_source_sensitive_composed_solar_optics approximation metric source situation redshift deflection pointMass rayPasses solution comparison frequencyCalculation deflectionCalculation radialContraction tangentialUnchanged open PhysicsLogic.Framework.GravitationalFrequencyRayHandoff.Countermodels example : PhysicsLogic.Framework.GravitationalFrequencyRayHandoff.NonvacuousGravitationalFrequencyAndRayDeflectionAt firstOrderOpticalHandoff /\ PhysicsLogic.Framework.GravitationalFrequencyRayHandoff.NonvacuousGravitationalFrequencyAndRayDeflectionAt staticMetricOpticalHandoff /\ firstOrderOpticalHandoff.reportedSolarDeflectionHundredths = 83 /\ staticMetricOpticalHandoff.reportedSolarDeflectionHundredths = 170 /\ firstOrderOpticalHandoff.reportedSolarDeflectionHundredths < staticMetricOpticalHandoff.reportedSolarDeflectionHundredths := first_order_and_static_metric_optics_keep_source_angles_distinct example : (firstOrderOpticalHandoff |>.NonvacuousGravitationalFrequencyAndRayDeflectionAt) ∧ firstOrderOpticalHandoff.reportedSolarDeflectionHundredths = 83 ∧ (completedFrequencyShiftContext |>.DirectSourceStructuralGravitationalFrequencyShiftExplanationAt completedFrequencyShift () availableOpticalChildDerivationObtains) ∧ (firstOrderSolarRayContext |>.DirectSourceStructuralGravitationalRayDeflectionExplanationAt firstOrderSolarRay () availableOpticalChildDerivationObtains) ∧ ¬(firstOrderOpticalHandoff |>.SourceSensitiveGravitationalFrequencyAndRayDeflectionAt completedFrequencyShiftContext firstOrderSolarRayContext () availableOpticalChildDerivationObtains () availableOpticalChildDerivationObtains .absent missingOpticalHandoffDerivationObtains) := completed_optical_composition_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition example : PhysicsLogic.Framework.GravitationalFrequencyShiftModel.NonvacuousGravitationalFrequencyShiftAt completedPredictionsWithoutOpticalHandoff.frequencyModel /\ PhysicsLogic.Framework.GravitationalRayDeflectionModel.NonvacuousGravitationalRayDeflectionAt completedPredictionsWithoutOpticalHandoff.rayModel /\ Not (completedPredictionsWithoutOpticalHandoff |>.SelectedGravitationalFrequencyRayHandoffAt) := frequency_and_deflection_outputs_do_not_supply_source_handoff example : PhysicsLogic.Framework.GravitationalFrequencyShiftModel.NonvacuousGravitationalFrequencyShiftAt frequencyShiftWithoutDeflectionCalculation.frequencyModel /\ Not (PhysicsLogic.Framework.GravitationalRayDeflectionModel.NonvacuousGravitationalRayDeflectionAt frequencyShiftWithoutDeflectionCalculation.rayModel) := gravitational_frequency_shift_does_not_supply_ray_calculation example : PhysicsLogic.Framework.GravitationalRayDeflectionModel.NonvacuousGravitationalRayDeflectionAt deflectionWithoutFrequencyShift.rayModel /\ Not (PhysicsLogic.Framework.GravitationalFrequencyShiftModel.NonvacuousGravitationalFrequencyShiftAt deflectionWithoutFrequencyShift.frequencyModel) := ray_deflection_does_not_supply_local_clock_frequency_shift example : let changed := { firstOrderOpticalHandoff with reportedSolarDeflectionHundredths := 170 } PhysicsLogic.Framework.GravitationalFrequencyShiftModel.NonvacuousGravitationalFrequencyShiftAt changed.frequencyModel /\ PhysicsLogic.Framework.GravitationalRayDeflectionModel.NonvacuousGravitationalRayDeflectionAt changed.rayModel /\ Not changed.SelectedGravitationalFrequencyRayAlignmentAt := incorrectly_reported_angle_fails_exact_optical_calibration example : let changed := { firstOrderOpticalHandoff with derivationMode := .staticMetricNullRay } Not changed.SelectedGravitationalFrequencyRayHandoffAt := first_order_source_cannot_be_reclassified_as_static_metric example {System PhasePoint Constraint Multiplier Observable Value PhysicalState QuantumState Operator : Type} (framework : PhysicsLogic.Papers.Dirac1950.ConstrainedHamiltonianFramework System PhasePoint Constraint Multiplier Observable Value PhysicalState QuantumState Operator) (system : System) (primaryConstraint selectedGenerator : Constraint) (constraintLinear : framework.totalHamiltonianLinearInConstraints system) (primary : framework.primaryConstraint system primaryConstraint) (consistencySetup : framework.ConsistencySetup system) (consistencyCalculation : framework.ConsistencyClassificationCalculation system) (completionCalculation : framework.ConstraintCompletionRealizationCalculation system) (generatorSetup : framework.HamiltonianStartingSetup system) (poissonIdentityCalculation : framework.FirstClassPoissonIdentityClosureCalculation system) (selectedGeneratorCompleted : framework.weakConstraintEquation system selectedGenerator) (selectedGeneratorFirstClass : framework.firstClass system selectedGenerator) : (framework.completedConstraintFirstClassGeneratorHandoff system primaryConstraint selectedGenerator) |>.NonvacuousCompletedConstraintGeneratorClosureAt := framework.completed_weak_constraints_and_poisson_identity_realize_composed_closure system primaryConstraint selectedGenerator constraintLinear primary consistencySetup consistencyCalculation completionCalculation generatorSetup poissonIdentityCalculation selectedGeneratorCompleted selectedGeneratorFirstClass example {System PhasePoint Constraint Multiplier Observable Value PhysicalState QuantumState Operator : Type} (framework : PhysicsLogic.Papers.Dirac1950.ConstrainedHamiltonianFramework System PhasePoint Constraint Multiplier Observable Value PhysicalState QuantumState Operator) (system : System) (primaryConstraint selectedGenerator : Constraint) (constraintLinear : framework.totalHamiltonianLinearInConstraints system) (primary : framework.primaryConstraint system primaryConstraint) (consistencySetup : framework.ConsistencySetup system) (consistencyCalculation : framework.ConsistencyClassificationCalculation system) (completionCalculation : framework.ConstraintCompletionRealizationCalculation system) (generatorSetup : framework.HamiltonianStartingSetup system) (poissonIdentityCalculation : framework.FirstClassPoissonIdentityClosureCalculation system) (selectedGeneratorCompleted : framework.weakConstraintEquation system selectedGenerator) (selectedGeneratorFirstClass : framework.firstClass system selectedGenerator) : (framework.completedConstraintFirstClassGeneratorHandoff system primaryConstraint selectedGenerator) |>.SourceSensitiveCompletedConstraintGeneratorClosureAt (framework.toConstraintConsistencyCompletionContext system primaryConstraint) (framework.toCanonicalGeneratorBracketClosureContext system) .iterativeWeakConstraintCompletion (framework.diracConstraintConsistencyDerivationObtains system primaryConstraint) .poissonIdentityAndWeakConstraintRepresentative (framework.DiracCanonicalGeneratorBracketDerivationObtains system) .iterativeWeakConstraintCompletionThenPoissonIdentityFirstClassClosure (framework.diracConstraintGeneratorHandoffDerivationObtains system primaryConstraint selectedGenerator) := framework |>.completed_weak_constraints_and_poisson_identity_realize_source_sensitive_composed_closure system primaryConstraint selectedGenerator constraintLinear primary consistencySetup consistencyCalculation completionCalculation generatorSetup poissonIdentityCalculation selectedGeneratorCompleted selectedGeneratorFirstClass example {Theory PhasePoint Constraint Descriptor Generator : Type} (framework : PhysicsLogic.Papers.AndersonBergmann1951.CovariantConstraintFramework Theory PhasePoint Constraint Descriptor Generator) {theory : Theory} (primaryConstraint : Constraint) (selectedGenerator : Generator) (selectedDescriptor : Descriptor) (hamiltonian : framework.canonicalHamiltonianConstructed theory) (primary : framework.primaryConstraint theory primaryConstraint) (noContradiction : ¬framework.contradictoryConstraintGenerated theory) (generatorSetup : framework.CanonicalGeneratorSetup theory) (generatorCalculation : framework.InvariantGeneratorFunctionGroupCalculation theory) (selectedInvariant : framework.invariantCanonicalGenerator theory selectedGenerator) (selectedExtension : framework.generatorExtendsInvariantTransformation theory selectedGenerator selectedDescriptor) (selectedAmbiguity : framework.generatorAmbiguousByQuadraticPrimaryTerms theory selectedGenerator) (hamiltonianClosure : framework.generatorBracketsWithHamiltonianClose theory) (completionCalculation : framework.ConsistencyCompletionCalculation theory) : (framework.invariantGeneratorConstraintCompletionHandoff theory primaryConstraint selectedGenerator selectedDescriptor) |>.NonvacuousCompletedConstraintGeneratorClosureAt := framework.invariant_generators_and_hamiltonian_brackets_realize_composed_completion primaryConstraint selectedGenerator selectedDescriptor hamiltonian primary noContradiction generatorSetup generatorCalculation selectedInvariant selectedExtension selectedAmbiguity hamiltonianClosure completionCalculation example {Theory PhasePoint Constraint Descriptor Generator : Type} (framework : PhysicsLogic.Papers.AndersonBergmann1951.CovariantConstraintFramework Theory PhasePoint Constraint Descriptor Generator) {theory : Theory} (primaryConstraint : Constraint) (selectedGenerator : Generator) (selectedDescriptor : Descriptor) (hamiltonian : framework.canonicalHamiltonianConstructed theory) (primary : framework.primaryConstraint theory primaryConstraint) (noContradiction : ¬framework.contradictoryConstraintGenerated theory) (generatorSetup : framework.CanonicalGeneratorSetup theory) (generatorCalculation : framework.InvariantGeneratorFunctionGroupCalculation theory) (selectedInvariant : framework.invariantCanonicalGenerator theory selectedGenerator) (selectedExtension : framework.generatorExtendsInvariantTransformation theory selectedGenerator selectedDescriptor) (selectedAmbiguity : framework.generatorAmbiguousByQuadraticPrimaryTerms theory selectedGenerator) (hamiltonianClosure : framework.generatorBracketsWithHamiltonianClose theory) (completionCalculation : framework.ConsistencyCompletionCalculation theory) : (framework.invariantGeneratorConstraintCompletionHandoff theory primaryConstraint selectedGenerator selectedDescriptor) |>.SourceSensitiveCompletedConstraintGeneratorClosureAt (framework.toConstraintConsistencyCompletionContext theory primaryConstraint) (framework.toCanonicalGeneratorBracketClosureContext theory) .invariantGeneratorBracketCompletion (framework.andersonBergmannConstraintConsistencyDerivationObtains theory primaryConstraint) .invariantTransformationCommutatorCanonicalExpansionAndEquationFiveSix (framework.AndersonBergmannCanonicalGeneratorBracketDerivationObtains theory) .invariantGeneratorFunctionGroupHamiltonianBracketThenConstraintCompletion (framework.andersonBergmannConstraintGeneratorHandoffDerivationObtains theory primaryConstraint selectedGenerator selectedDescriptor) := framework |>.invariant_generators_and_hamiltonian_brackets_realize_source_sensitive_composed_completion primaryConstraint selectedGenerator selectedDescriptor hamiltonian primary noContradiction generatorSetup generatorCalculation selectedInvariant selectedExtension selectedAmbiguity hamiltonianClosure completionCalculation open PhysicsLogic.Framework.CompletedConstraintGeneratorHandoff.Countermodels example : PhysicsLogic.Framework.CompletedConstraintGeneratorHandoff.NonvacuousCompletedConstraintGeneratorClosureAt completedConstraintAndClosedGeneratorHandoff /\ PhysicsLogic.Framework.CompletedConstraintGeneratorHandoff.NonvacuousCompletedConstraintGeneratorClosureAt invariantGeneratorThenCompletionHandoff /\ completedConstraintAndClosedGeneratorHandoff.derivationOrder ≠ invariantGeneratorThenCompletionHandoff.derivationOrder := completed_constraints_and_invariant_generators_keep_source_orders_distinct example : (completedConstraintAndClosedGeneratorHandoff |>.NonvacuousCompletedConstraintGeneratorClosureAt) /\ completedConstraintAndClosedGeneratorHandoff.derivationOrder = .completedConstraintsThenFirstClassClosure /\ ¬(completedConstraintAndClosedGeneratorHandoff |>.paperLocalHamiltonianBracketDerivationAt false .admissible) /\ (PhysicsLogic.Framework.ConstraintConsistencyCompletionContext.Countermodels.directConstraintConsistencyContext |>.DirectSourceStructuralConstraintConsistencyCompletionExplanationAt PhysicsLogic.Framework.ConstraintConsistencyCompletionContext.Countermodels.directConstraintConsistencyModel () availableConstraintGeneratorChildDerivationObtains) /\ (PhysicsLogic.Framework.CanonicalGeneratorBracketClosureContext.Countermodels.selectedDataContext |>.SourceStructuralCanonicalGeneratorBracketClosureExplanationAt PhysicsLogic.Framework.CanonicalGeneratorBracketClosureContext.Countermodels.selectedDataModel () availableConstraintGeneratorChildDerivationObtains) /\ ¬(completedConstraintAndClosedGeneratorHandoff |>.SourceSensitiveCompletedConstraintGeneratorClosureAt PhysicsLogic.Framework.ConstraintConsistencyCompletionContext.Countermodels.directConstraintConsistencyContext PhysicsLogic.Framework.CanonicalGeneratorBracketClosureContext.Countermodels.selectedDataContext () availableConstraintGeneratorChildDerivationObtains () availableConstraintGeneratorChildDerivationObtains .absent missingConstraintGeneratorHandoffDerivationObtains) := completed_constraint_generator_composition_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition example : PhysicsLogic.Framework.ConstraintConsistencyCalculationModel.NonvacuousConstraintConsistencyCompletionAt completedOutputsWithoutSourceHandoff.consistencyModel /\ PhysicsLogic.Framework.CanonicalGeneratorBracketCalculationModel.NonvacuousCanonicalGeneratorBracketClosureAt completedOutputsWithoutSourceHandoff.generatorModel /\ Not (completedOutputsWithoutSourceHandoff |>.SelectedCompletedConstraintGeneratorHandoffAt) := completed_constraint_and_generator_outputs_do_not_supply_source_handoff example : PhysicsLogic.Framework.ConstraintConsistencyCalculationModel.NonvacuousConstraintConsistencyCompletionAt completedConstraintWithoutClosedGenerator.consistencyModel /\ Not (PhysicsLogic.Framework.CanonicalGeneratorBracketCalculationModel.NonvacuousCanonicalGeneratorBracketClosureAt completedConstraintWithoutClosedGenerator.generatorModel) := completed_constraint_preservation_does_not_supply_generator_closure example : PhysicsLogic.Framework.CanonicalGeneratorBracketCalculationModel.NonvacuousCanonicalGeneratorBracketClosureAt closedGeneratorWithoutConstraintPreservation.generatorModel /\ Not (PhysicsLogic.Framework.ConstraintConsistencyCalculationModel.NonvacuousConstraintConsistencyCompletionAt closedGeneratorWithoutConstraintPreservation.consistencyModel) := closed_generators_do_not_supply_constraint_consistency_completion example : PhysicsLogic.Framework.ConstraintConsistencyCalculationModel.NonvacuousConstraintConsistencyCompletionAt invariantGeneratorWithoutHamiltonianBracketHandoff.consistencyModel /\ PhysicsLogic.Framework.CanonicalGeneratorBracketCalculationModel.NonvacuousCanonicalGeneratorBracketClosureAt invariantGeneratorWithoutHamiltonianBracketHandoff.generatorModel /\ Not (invariantGeneratorWithoutHamiltonianBracketHandoff |>.SelectedCompletedConstraintGeneratorHandoffAt) := invariant_generator_closure_does_not_supply_hamiltonian_bracket_handoff example : let changed := { completedConstraintAndClosedGeneratorHandoff with derivationOrder := .invariantGeneratorClosureThenCompletion } Not changed.SelectedCompletedConstraintGeneratorHandoffAt := completed_first_source_cannot_be_reclassified_as_generator_first example : PhysicsLogic.Framework.CompletedConstraintGeneratorHandoff.NonvacuousCompletedConstraintGeneratorClosureAt completedConstraintAndClosedGeneratorHandoff /\ PhysicsLogic.Framework.ConstraintGaugeBoundaryReadout.HasCompletedPreservedConstraintAt completedConstraintGeneratorWithoutPhysicalIdentification /\ PhysicsLogic.Framework.ConstraintGaugeBoundaryReadout.HasConstraintSurfaceGaugeOrbitAt completedConstraintGeneratorWithoutPhysicalIdentification /\ Not (PhysicsLogic.Framework.ConstraintGaugeBoundaryReadout.HasPhysicalGaugeIdentificationAt completedConstraintGeneratorWithoutPhysicalIdentification) := completed_constraints_and_closed_generators_do_not_identify_gauge_states namespace NuclearShellPhysicalReuseAudit open PhysicsLogic.Framework.SpinOrbitShellClosureContext.Countermodels example : splitWithoutHigherJReorderingModel.centralScheme ≠ splitWithoutHigherJReorderingModel.spinOrbitScheme /\ Not (splitWithoutHigherJReorderingModel.higherJPartnersReorderedIntoPrecedingShellAt .central .split .neutron .selected) /\ successfulControlReordering.EstablishedSpinOrbitShellClosureAt successfulControlReorderingModel := by obtain ⟨schemes, _patterns, _split, notReordered, _controlReordering, controlClosure, _controlSchemes⟩ := shell_reordering_distinguishes_species_schemes_patterns_and_control_routes exact ⟨schemes, notReordered, controlClosure⟩ end NuclearShellPhysicalReuseAudit namespace RenormalizationFlowPhysicalReuseAudit open PhysicsLogic.Framework.ExactRegulatedScaleFlowContext.Countermodels example : emptySetupModel.descriptionAt .initial = .ultraviolet /\ emptySetupModel.descriptionAt .selected = .infrared /\ Not flowWithoutProjectionModel.ProjectionSetup /\ completedPhysicalFlowContext.EstablishedExactProjectedScaleFlowAt completedPhysicalFlowModel := by obtain ⟨ultraviolet, infrared, _high, _low, _missingSetup, _controlSetup, _fullFlow, missingProjection, projected⟩ := regulated_flow_roles_distinguish_scales_families_equations_sectors_and_control_routes exact ⟨ultraviolet, infrared, missingProjection, projected⟩ end RenormalizationFlowPhysicalReuseAudit namespace FrequencyNoisePhysicalReuseAudit open PhysicsLogic.Framework.EquilibriumNoiseDissipationContext.Countermodels example : oneFrequencyOnlyModel.dissipationValueAt .target .low = .low /\ oneFrequencyOnlyModel.dissipationValueAt .target .high = .high /\ oneFrequencyOnlyModel.noiseValueAt .target .drive .high = .quiet /\ controlledFrequencyNoise.EstablishedEquilibriumNoiseDissipationAt controlledFrequencyNoiseModel := by obtain ⟨_system, _temperature, _lowFrequency, _highFrequency, _excludedControl, low, high, observed, _prediction, established⟩ := frequency_noise_roles_distinguish_systems_temperatures_frequencies_values_and_controls exact ⟨low, high, observed, established⟩ example : PhysicalNyquistPlanckReplacementAt physicalNyquist1928PlanckReplacementReadout ∧ PhysicalNyquistCurrentTransferAndCarrierScopeAt physicalNyquist1928CurrentTransferReadout ∧ PhysicalCallenWeltonClassicalLimitAndApplicationsAt physicalCallenWelton1951ClassicalApplicationsReadout ∧ PhysicalCallenWeltonCoherenceInterpretationAt physicalCallenWelton1951CoherenceReadout ∧ PhysicalKuboClassicalLimitAndScopeAt physicalKubo1957ClassicalScopeReadout ∧ physicalNyquist1928PlanckReplacementReadout.zeroPointEnergyValue = 0 ∧ 0 < physicalCallenWelton1951QuantumNoiseReadout.zeroPointEnergyValue ∧ ¬PhysicalNyquistPlanckReplacementAt physicalNyquistHalfQuantumControl ∧ ¬PhysicalCallenWeltonClassicalLimitAndApplicationsAt physicalCallenWeltonExactClassicalLimitControl ∧ ¬PhysicalKuboSymmetrizedConductivityNoiseAt physicalKuboOrderedEqualsSymmetrizedControl ∧ ¬PhysicalKuboClassicalLimitAndScopeAt physicalKuboProjectedBandSumRuleControl := by obtain ⟨nyquistPlanck, nyquistCurrent, callenApplications, callenCoherence, kubo, _nyquistCallen, _callenKubo, zeroPointFree, halfQuantum, noNyquistHalf, noExactClassical, _noCompleted, noOrdered, _noJoint, noProjected⟩ := noise_sources_distinguish_nyquist_circuit_callen_welton_quantum_force_and_kubo_symmetrized_correlation exact ⟨nyquistPlanck, nyquistCurrent, callenApplications, callenCoherence, kubo, zeroPointFree, halfQuantum, noNyquistHalf, noExactClassical, noOrdered, noProjected⟩ example : physicalNoiseScopePossibility.TentativeAt physicalNoiseScopeEstablishment .nyquistPlanckReplacementIncludesHalfQuantum .nyquist1928 ∧ physicalNoiseScopePossibility.TentativeAt physicalNoiseScopeEstablishment .callenWeltonGeneralIrreversibilityTheoryCompleted .callenWelton1951 ∧ physicalNoiseScopePossibility.TentativeAt physicalNoiseScopeEstablishment .kuboSymmetrizedCorrelationIsJointObservedCorrelation .kubo1957 ∧ physicalNoiseScopePossibility.TentativeAt physicalNoiseScopeEstablishment .kuboProjectedBandInheritsFullSumRule .kubo1957 := by obtain ⟨nyquistHalf, _carriers, _gas, _classical, _vacuum, callenTheory, kuboObserved, _static, projected, _einstein⟩ := noise_source_overclaims_remain_epistemically_open exact ⟨nyquistHalf, callenTheory, kuboObserved, projected⟩ end FrequencyNoisePhysicalReuseAudit namespace CosmologicalInferencePhysicalReuseAudit open PhysicsLogic.Framework.StandardCandleAccelerationInferenceCountermodels example : physicalStandardCandleModel.sample = .highRedshift /\ physicalStandardCandleModel.modelFamily ≠ physicalStandardCandleModel.referenceModel /\ Not (physicallyObservedWithoutFitContext.CosmologicalFitCalculationAt physicallyObservedWithoutFitModel) /\ physicallyFitWithoutAccelerationContext.EstablishedCosmologicalFitAt physicallyFitWithoutAccelerationModel /\ Not (physicallyFitWithoutAccelerationContext.AccelerationInterpretationCalculationAt physicallyFitWithoutAccelerationModel) /\ controlStandardCandleContext.EstablishedStandardCandleAccelerationInferenceAt controlStandardCandleModel := by obtain ⟨sample, _calibration, _observation, distinctModels, _acceleration, _distinctInterpretations, _observations, noFit, completedFit, noAcceleration, _selectedInference, controlInference, _openAlternatives⟩ := standard_candle_roles_distinguish_samples_calibrations_observations_models_fits_and_control_routes exact ⟨sample, distinctModels, noFit, completedFit, noAcceleration, controlInference⟩ end CosmologicalInferencePhysicalReuseAudit namespace CriticalCollapsePhysicalReuseAudit open PhysicsLogic.Framework.CriticalCollapseScalingCountermodels example : physicalCriticalCollapseModel.subcriticalParameter ≠ physicalCriticalCollapseModel.threshold ∧ physicalCriticalCollapseModel.subcriticalEvolution ≠ physicalCriticalCollapseModel.supercriticalEvolution ∧ Not (physicalCriticalCollapseModel.formsBlackHoleAt .dispersal .nearCritical) ∧ Not (physicalOutcomesWithoutScalingContext.MassScalingCalculationAt physicalOutcomesWithoutScalingModel) ∧ radiationFluidCollapseControlContext.EstablishedCriticalCollapseScalingAt radiationFluidCollapseControlModel := by obtain ⟨belowThreshold, _aboveThreshold, distinctEvolutions, _disperses, noBlackHole, _blackHole, _doesNotDisperse, _setup, _outcomes, noScaling, _selectedScaling, controlScaling⟩ := critical_collapse_distinguishes_threshold_sides_evolutions_families_and_controls exact ⟨belowThreshold, distinctEvolutions, noBlackHole, noScaling, controlScaling⟩ end CriticalCollapsePhysicalReuseAudit namespace BrownianTransportPhysicalReuseAudit open PhysicsLogic.Framework.ThermalDragBrownianTransportContext.Countermodels example : physicalBrownianModel.stochasticModel.regime ≠ physicalBrownianModel.dragRegime ∧ Not (physicalBrownianWithoutThermalDrive.ThermalDragSetupAt physicalBrownianWithoutThermalDriveModel) ∧ physicalBrownianWithoutHydrodynamicDrag.ThermalDragSetupAt physicalBrownianWithoutHydrodynamicDragModel ∧ controlBrownianContext.EstablishedThermalDragBrownianTransportAt controlBrownianModel := by obtain ⟨distinctRegimes, _carrier, _medium, _drag, _coefficient, _distinctSystems, _diffusion, noDrive, dragSetup, _noDrag, _selectedTransport, controlTransport⟩ := brownian_transport_distinguishes_diffusion_drag_regimes_media_and_controls exact ⟨distinctRegimes, noDrive, dragSetup, controlTransport⟩ example : PhysicalEinsteinOsmoticThermodynamicBalanceAt physicalEinsteinOsmoticReadout ∧ physicalEinsteinOsmoticReadout.osmoticPressureValue = 12 * Real.pi := physical_einstein_osmotic_balance_retains_molecular_pressure_and_force example : PhysicalEinsteinStokesFluxAndExactCoefficientAt physicalEinsteinStokesCoefficientReadout ∧ physicalEinsteinStokesCoefficientReadout.diffusionCoefficientValue = 1 := physical_einstein_stokes_flux_balance_retains_the_exact_six_pi_coefficient example : PhysicalEinsteinGaussianRootTimeSpreadAt physicalEinsteinGaussianSpreadReadout ∧ physicalEinsteinGaussianSpreadReadout.rmsDisplacementValue = 2 := physical_einstein_gaussian_retains_normalization_exponent_and_root_time example : PhysicalSmoluchowskiPersistentSmallParticleAt physicalSmoluchowskiPersistentFlightReadout ∧ physicalSmoluchowskiPersistentFlightReadout.smallParticleDisplacementValue = 1 / 2 := physical_smoluchowski_persistent_flight_retains_original_eight_thirds_prefactor example : ¬PhysicalEinsteinStokesFluxAndExactCoefficientAt physicalEinsteinStokesSixPiFactorErasedControl := by obtain ⟨_, _, _, _, erasedFactor, _, _⟩ := einstein_osmotic_pressure_stokes_prefactor_flux_and_exact_coefficient_are_independent exact erasedFactor example : ¬PhysicalSmoluchowskiApproximateCoefficientComparisonAt physicalSmoluchowskiCoefficientEqualityAssertedControl := by obtain ⟨_, _, _, _, _, _, _, _, _, distinctCoefficients, _, _, _, _⟩ := smoluchowski_collision_excess_persistence_relaxation_comparison_and_scope_are_independent exact distinctCoefficients example : ¬physicalBrownianEstablishment.establishedAt .originalSmoluchowskiPaperCompletesDirectExperimentalVerification .smoluchowski1906 := brownian_source_overclaims_remain_epistemically_unestablished.2.2.2.2.1 end BrownianTransportPhysicalReuseAudit namespace NuclearScatteringPhysicalReuseAudit open PhysicsLogic.Framework.LowEnergyEffectiveRangeContext.Countermodels example : physicalEffectiveRangeModel.zeroEnergy ≠ physicalEffectiveRangeModel.selectedEnergy ∧ Not (physicalRangeWithoutReferenceModel.zeroEnergyReferenceControlledAt .neutronProton .zero .finiteLow .lowMomentum) ∧ physicalRangeWithoutSmallCorrectionContext.EstablishedTwoParameterEffectiveRangeLawAt physicalRangeWithoutSmallCorrectionModel ∧ Not (physicalRangeWithoutSmallCorrectionContext.EstablishedEffectiveRangeReductionAt physicalRangeWithoutSmallCorrectionModel) ∧ controlledScatteringChannelContext.EstablishedEffectiveRangeReductionAt controlledScatteringChannelModel := by obtain ⟨distinctEnergies, _distinctChannels, _setup, noReference, twoParameterLaw, noControlledReduction, _unresolvedInteraction, _selectedReduction, controlReduction⟩ := effective_range_distinguishes_zero_energy_reference_channels_laws_and_controls exact ⟨distinctEnergies, noReference, twoParameterLaw, noControlledReduction, controlReduction⟩ end NuclearScatteringPhysicalReuseAudit namespace GappedHallPhysicalReuseAudit open PhysicsLogic.Framework.GappedBandHallContext.Countermodels example : physicalHallModel.index = 1 ∧ physicalHallControlModel.index = 2 ∧ physicalHallModel.band ≠ physicalHallControlModel.band ∧ Not (physicalHallWithoutIndexModel.integerHallIndexAt 1 .occupied .source .selected .selected) ∧ physicalHallControlContext.EstablishedGappedBandHallQuantizationAt physicalHallControlModel := by obtain ⟨selectedIndex, controlIndex, _distinctDescriptions, distinctBands, _distinctConductance, _selectedGap, _excludedGap, _setup, noIndex, _calculatedIndex, _missingConductance, _selectedQuantization, controlQuantization⟩ := gapped_hall_roles_distinguish_bands_indices_conductance_and_controls exact ⟨selectedIndex, controlIndex, distinctBands, noIndex, controlQuantization⟩ example : PhysicalTKNNMagneticBlochBoundaryWindingAt physicalTKNN1982MagneticBlochReadout := physical_tknn_magnetic_bloch_boundary_winding_is_nonvacuous example : PhysicalTKNNDiophantineSubbandFormulaAt physicalTKNN1982MagneticBlochReadout := physical_tknn_original_diophantine_subband_formula_is_nonvacuous example : PhysicalTKNNConditionalGapAndLinearScopeAt physicalTKNN1982MagneticBlochReadout := physical_tknn_conditional_gap_and_linear_response_scope_is_nonvacuous example : ¬PhysicalTKNNMagneticBlochBoundaryWindingAt physicalTKNNZeroUniformFluxControl := tknn_magnetic_flux_boundary_winding_and_diophantine_branch_are_independent.2.1 example : ¬PhysicalTKNNMagneticBlochBoundaryWindingAt physicalTKNNMissingMagneticBlochBoundaryControl := tknn_magnetic_flux_boundary_winding_and_diophantine_branch_are_independent.2.2.1 example : ¬PhysicalTKNNMagneticBlochBoundaryWindingAt physicalTKNNMissingKuboIdentityControl := tknn_magnetic_flux_boundary_winding_and_diophantine_branch_are_independent.2.2.2.1 example : ¬PhysicalTKNNMagneticBlochBoundaryWindingAt physicalTKNNMissingBoundaryPhasePatchingControl := tknn_magnetic_flux_boundary_winding_and_diophantine_branch_are_independent.2.2.2.2.1 example : ¬PhysicalTKNNDiophantineSubbandFormulaAt physicalTKNNWrongDiophantineGapControl := tknn_magnetic_flux_boundary_winding_and_diophantine_branch_are_independent.2.2.2.2.2.2.1 example : ¬PhysicalTKNNDiophantineSubbandFormulaAt physicalTKNNEvenFluxMissingGapControl := tknn_magnetic_flux_boundary_winding_and_diophantine_branch_are_independent.2.2.2.2.2.2.2.1 example : ¬PhysicalTKNNDiophantineSubbandFormulaAt physicalTKNNEqualFractionalSubbandControl := tknn_magnetic_flux_boundary_winding_and_diophantine_branch_are_independent.2.2.2.2.2.2.2.2 example : PhysicalTKNNConditionalGapAndLinearScopeAt physicalTKNNBelievedInterpolationGapClosureControl := tknn_conditional_gap_stability_tunneling_and_clean_scope_are_independent.2.1 example : physicalTKNNBelievedInterpolationGapClosureControl.minimumInterpolationGapValue = 0 ∧ physicalTKNNBelievedInterpolationGapClosureControl.initialInterpolationIndexValue ≠ physicalTKNNBelievedInterpolationGapClosureControl.finalInterpolationIndexValue := ⟨tknn_conditional_gap_stability_tunneling_and_clean_scope_are_independent.2.2.1, tknn_conditional_gap_stability_tunneling_and_clean_scope_are_independent.2.2.2.1⟩ example : ¬PhysicalTKNNConditionalGapAndLinearScopeAt physicalTKNNClaimedGapNonclosureProofControl := tknn_conditional_gap_stability_tunneling_and_clean_scope_are_independent.2.2.2.2.1 example : ¬PhysicalTKNNConditionalGapAndLinearScopeAt physicalTKNNNoStrongFieldTunnelingControl := tknn_conditional_gap_stability_tunneling_and_clean_scope_are_independent.2.2.2.2.2.1 example : ¬PhysicalTKNNConditionalGapAndLinearScopeAt physicalTKNNInteractingDisorderedSystemControl := tknn_conditional_gap_stability_tunneling_and_clean_scope_are_independent.2.2.2.2.2.2 example : PhysicalHaldaneZeroFluxHoneycombMassFormulaAt physicalHaldane1988ZeroFluxReadout := physical_haldane_zero_flux_honeycomb_mass_formula_is_nonvacuous example : PhysicalHaldaneWeakProbeStredaTransportAt physicalHaldane1988ZeroFluxReadout := physical_haldane_weak_probe_streda_transport_is_nonvacuous example : PhysicalHaldaneContinuumLatticeCompletionAt physicalHaldane1988ZeroFluxReadout := physical_haldane_continuum_lattice_completion_is_nonvacuous example : physicalHaldane1988ZeroFluxReadout.inducedNumberDensityDifferenceValue ≠ physicalHaldane1988ZeroFluxReadout.inducedChargeDensityDifferenceValue := physical_haldane_charge_density_direct_streda_and_dimensionless_halves_are_distinct |>.2.1 example : physicalHaldane1988ZeroFluxReadout.numberDensityDerivativeValue ≠ physicalHaldane1988ZeroFluxReadout.densityDerivativeValue := physical_haldane_charge_density_direct_streda_and_dimensionless_halves_are_distinct |>.2.2.1 example : ¬PhysicalHaldaneWeakProbeStredaTransportAt physicalHaldaneNumberDensityUsedAsChargeControl := physical_haldane_charge_density_direct_streda_and_dimensionless_halves_are_distinct |>.2.2.2.1 example : ¬PhysicalHaldaneWeakProbeStredaTransportAt physicalHaldaneExtraChargeInStredaControl := physical_haldane_charge_density_direct_streda_and_dimensionless_halves_are_distinct |>.2.2.2.2.1 example : ¬PhysicalHaldaneContinuumLatticeCompletionAt physicalHaldaneUnnormalizedContinuumHalvesControl := physical_haldane_charge_density_direct_streda_and_dimensionless_halves_are_distinct |>.2.2.2.2.2.2 example : ¬PhysicalHaldaneZeroFluxHoneycombMassFormulaAt physicalHaldaneNonzeroNetFluxControl := haldane_zero_flux_mass_sign_streda_and_lattice_halves_are_independent.2.1 example : ¬PhysicalHaldaneZeroFluxHoneycombMassFormulaAt physicalHaldaneTimeReversalInvariantHoppingControl := haldane_zero_flux_mass_sign_streda_and_lattice_halves_are_independent.2.2.1 example : ¬PhysicalHaldaneZeroFluxHoneycombMassFormulaAt physicalHaldaneWrongCornerMassCoefficientControl := haldane_zero_flux_mass_sign_streda_and_lattice_halves_are_independent.2.2.2.1 example : ¬PhysicalHaldaneZeroFluxHoneycombMassFormulaAt physicalHaldaneSameMassTrivialReadout := haldane_zero_flux_mass_sign_streda_and_lattice_halves_are_independent.2.2.2.2.1 example : ¬PhysicalHaldaneWeakProbeStredaTransportAt physicalHaldaneMissingWeakProbeControl := haldane_zero_flux_mass_sign_streda_and_lattice_halves_are_independent.2.2.2.2.2.2.1 example : ¬PhysicalHaldaneWeakProbeStredaTransportAt physicalHaldaneWrongStredaDensityControl := haldane_zero_flux_mass_sign_streda_and_lattice_halves_are_independent.2.2.2.2.2.2.2.1 example : ¬PhysicalHaldaneWeakProbeStredaTransportAt physicalHaldaneBerryCalculationSubstitutionControl := haldane_zero_flux_mass_sign_streda_and_lattice_halves_are_independent.2.2.2.2.2.2.2.2.1 example : ¬PhysicalHaldaneContinuumLatticeCompletionAt physicalHaldaneMissingLatticeHalfControl := haldane_zero_flux_mass_sign_streda_and_lattice_halves_are_independent.2.2.2.2.2.2.2.2.2.2 example : PhysicalHaldaneSameMassTrivialPhaseAt physicalHaldaneSameMassTrivialReadout := physical_haldane_same_mass_trivial_phase_is_nonvacuous example : PhysicalHaldaneCriticalUndoubledConeAt physicalHaldaneCriticalUndoubledConeReadout := physical_haldane_critical_undoubled_cone_is_nonvacuous example : PhysicalHaldaneSpinDoubledProtectedResponseAt physicalHaldaneSpinDoubledProtectedReadout := physical_haldane_spin_doubled_protected_response_is_nonvacuous example : PhysicalHaldaneSpinfulZeemanMetalAt physicalHaldaneSpinfulZeemanMetalReadout := physical_haldane_spinful_zeeman_metal_is_nonvacuous example : ¬PhysicalHaldaneSpinDoubledProtectedResponseAt physicalHaldaneSpinfulZeemanMetalReadout := haldane_trivial_critical_spin_doubled_and_zeeman_metal_phases_are_distinct.2.2.2.2.1 example : physicalTKNN1982MagneticBlochReadout.uniformBackgroundMagneticFieldValue ≠ physicalHaldane1988ZeroFluxReadout.uniformBackgroundMagneticFieldValue := by norm_num [physicalTKNN1982MagneticBlochReadout, physicalHaldane1988ZeroFluxReadout] example : PhysicalTKNNDiophantineSubbandFormulaAt physicalTKNN1982MagneticBlochReadout ∧ PhysicsLogic.Papers.TKNN1982.PaperFramework.Countermodels.completeReducedTKNNReadout.RetainsEquationNineFormulaAt := ⟨physical_tknn_original_diophantine_subband_formula_is_nonvacuous, PhysicsLogic.Papers.TKNN1982.PaperFramework.Countermodels.complete_reduced_TKNN_readout_retains_weak_potential_formulas.2.1⟩ example : PhysicalHaldaneZeroFluxHoneycombMassFormulaAt physicalHaldane1988ZeroFluxReadout ∧ (PhysicsLogic.Papers.Haldane1988.PaperFramework.Countermodels.hasNamedStredaCalculation PhysicsLogic.Papers.Haldane1988.PaperFramework.Countermodels.wrongPrintedOppositeMassStredaModel ∧ ¬PhysicsLogic.Papers.Haldane1988.PaperFramework.Countermodels.retainsPrintedMassAndResponseFormula PhysicsLogic.Papers.Haldane1988.PaperFramework.Countermodels.wrongPrintedOppositeMassStredaModel) := ⟨physical_haldane_zero_flux_honeycomb_mass_formula_is_nonvacuous, PhysicsLogic.Papers.Haldane1988.PaperFramework.Countermodels.named_streda_calculation_does_not_supply_printed_mass_and_response_formula⟩ end GappedHallPhysicalReuseAudit namespace OperationalMeasurementPhysicalReuseAudit open PhysicsLogic.Framework.OperationalErrorDisturbanceContext.Countermodels example : physicalOperationalDisturbanceModel.preparedState ≠ physicalOperationalDisturbanceModel.interactionInputState ∧ physicalOperationalDisturbanceModel.measuredObservable ≠ physicalOperationalDisturbanceModel.disturbedObservable ∧ Not (operationalPreparationWithoutInputBridgeModel.preparationSuppliesInteractionInputAt .objectPreparation .object .preparedObject .objectProbeInput .selected) ∧ representedOperationalMeasurementWithoutTradeoffModel.ErrorDisturbanceRepresentation ∧ operationalDisturbanceControlContext.EstablishedOperationalErrorDisturbanceAt operationalDisturbanceControlModel := by obtain ⟨distinctInput, _distinctOutput, distinctObservables, _prepared, _performed, noBridge, representations, _noTradeoff, _notDispersion, _selectedTradeoff, controlTradeoff⟩ := operational_disturbance_distinguishes_states_observables_apparatus_and_controls exact ⟨distinctInput, distinctObservables, noBridge, representations, controlTradeoff⟩ end OperationalMeasurementPhysicalReuseAudit namespace ViscousBoundaryLayerPhysicalReuseAudit open PhysicsLogic.Framework.ThinViscousBoundaryLayerReductionContext.Countermodels example : physicalBoundaryLayerModel.fullFlow ≠ physicalBoundaryLayerModel.layerFlow ∧ physicalBoundaryLayerModel.wallVelocity ≠ physicalBoundaryLayerModel.scaledNormalVelocity ∧ Not physicalMomentumWithoutContinuityModel.ContinuityReduction ∧ Not physicalUncoupledBoundaryLayerModel.CoupledEquations ∧ controlBoundaryLayerContext.EstablishedReductionAt controlBoundaryLayerModel := by obtain ⟨distinctFlows, _distinctMomentum, _distinctContinuity, distinctVelocities, _setup, _momentum, noContinuity, _uncoupledMomentum, _uncoupledContinuity, noCoupling, _selectedReduction, controlReduction⟩ := boundary_layer_distinguishes_full_flow_reduced_equations_velocities_and_controls exact ⟨distinctFlows, distinctVelocities, noContinuity, noCoupling, controlReduction⟩ example : PhysicalPrandtlSteadyPlanarReductionAt physicalPrandtlSteadyPlanarReadout ∧ PhysicalPrandtlQualifiedSeparationAt physicalPrandtlSeparationReadout ∧ PhysicalPrandtlLocalSuctionAt physicalPrandtlSuctionReadout ∧ physicalPrandtlSuctionReadout.cylinderWallShearAfterSuctionValue = 1 ∧ physicalPrandtlSuctionReadout.outerWallShearAfterSuctionValue = 0 := by exact ⟨physical_prandtl_steady_planar_wall_layer_and_equations_are_nonvacuous.1, physical_prandtl_separation_retains_vorticity_and_singular_global_effect.1, physical_prandtl_suction_suppresses_one_boundary_but_not_the_other.1, physical_prandtl_suction_suppresses_one_boundary_but_not_the_other.2.2.1, physical_prandtl_suction_suppresses_one_boundary_but_not_the_other.2.2.2⟩ example : PhysicalBlasiusThinLayerOrderCountingAt physicalBlasiusOrderCountingReadout ∧ PhysicalBlasiusFlatPlateSimilarityAt physicalBlasiusSimilarityReadout ∧ physicalBlasiusSimilarityReadout.farFieldProfileDerivativeValue = 2 ∧ PhysicalBlasiusMatchedFlatPlateDragAt physicalBlasiusDragReadout ∧ physicalBlasiusDragReadout.exactTwoFaceDragValue < physicalBlasiusDragReadout.reportedRoundedTwoFaceDragValue ∧ physicalBlasiusDragReadout.exactTwoFaceDragValue ≠ physicalBlasiusAlternateViscosityDragReadout.exactTwoFaceDragValue := by exact ⟨physical_blasius_imported_program_and_retained_curvature_are_nonvacuous.1, physical_blasius_similarity_retains_its_original_far_field_normalization.1, physical_blasius_similarity_retains_its_original_far_field_normalization.2.2.2, physical_blasius_matched_drag_distinguishes_exact_and_rounded_two_faces.1, physical_blasius_matched_drag_distinguishes_exact_and_rounded_two_faces.2.2.2, physical_blasius_reduced_onset_does_not_erase_viscosity_dependent_drag.2.2.2.2⟩ example : ¬physicalBoundaryLayerEstablishment.establishedAt .adversePressureSufficientForSeparation .prandtl1904 ∧ ¬physicalBoundaryLayerEstablishment.establishedAt .roundedDragCoefficientIsExactProfileConstant .blasius1908 ∧ physicalBoundaryLayerPossibility.TentativeAt physicalBoundaryLayerEstablishment .improvedSeparationApproximation .blasius1908 := by exact ⟨boundary_layer_source_overclaims_remain_epistemically_unestablished.2.1, boundary_layer_source_overclaims_remain_epistemically_unestablished.2.2.2.2.1, boundary_layer_source_overclaims_remain_epistemically_unestablished.2.2.2.2.2.2.2.2.2.2.1⟩ end ViscousBoundaryLayerPhysicalReuseAudit namespace StellarEquilibriumPhysicalReuseAudit open PhysicsLogic.Framework.SelfGravitatingMassLimitContext.Countermodels example : physicalStellarMassLimitModel.body ≠ neutronStarMassLimitControlModel.body ∧ physicalStellarMassLimitModel.gravity ≠ neutronStarMassLimitControlModel.gravity ∧ stellarFiniteMassWithoutInterpretationModel.finiteEquilibriumMassLimitCalculatedAt .chandrasekharLimit .whiteDwarf .whiteDwarfModel ∧ Not (stellarFiniteMassWithoutInterpretationModel.maximumEquilibriumMassInterpretedAt .chandrasekharLimit .whiteDwarf .whiteDwarfModel) ∧ neutronStarMassLimitControl.EstablishedSelfGravitatingEquilibriumMassLimitAt neutronStarMassLimitControlModel := by obtain ⟨distinctBodies, _distinctMatter, distinctGravity, _distinctMasses, _setup, _missingEquationOfState, finiteMass, noMaximumInterpretation, _selectedLimit, controlLimit⟩ := stellar_mass_limits_distinguish_bodies_matter_gravity_and_scoped_controls exact ⟨distinctBodies, distinctGravity, finiteMass, noMaximumInterpretation, controlLimit⟩ end StellarEquilibriumPhysicalReuseAudit namespace SuperconductingScreeningPhysicalReuseAudit open PhysicsLogic.Framework.TransverseMeissnerScreeningContext.Countermodels open PhysicsLogic.Framework.BrokenSymmetryModeConversionContext.Countermodels example : physicalMeissnerScreeningModel.field ≠ .residualBulk ∧ physicalMeissnerScreeningModel.current ≠ .aligned ∧ physicalAlignedNonscreeningModel.inducedCurrentCalculatedAt .transverseKernel .aligned .appliedTransverse .longWavelength ∧ Not (physicalAlignedNonscreeningModel.longWavelengthCurrentOpposesFieldAt .aligned .appliedTransverse .longWavelength) ∧ controlMeissnerScreeningContext.EstablishedTransverseMeissnerScreeningAt controlMeissnerScreeningModel := by obtain ⟨_field, distinctFields, _current, distinctCurrents, _distinctSystems, alignedCurrent, notOpposing, _opposing, _noExclusion, _selectedScreening, controlScreening⟩ := meissner_screening_distinguishes_fields_currents_systems_and_controls exact ⟨distinctFields, distinctCurrents, alignedCurrent, notOpposing, controlScreening⟩ example : PhysicalBCSPippardPenetrationAndScopeAt physicalBCS1957LondonScreeningReadout ∧ PhysicalNambuNeutralCollectiveScreeningAt physicalNambu1960NeutralScreeningReadout ∧ physicalBCS1957LondonScreeningReadout.caseIICoherenceFactorUsed = true ∧ physicalNambu1960NeutralScreeningReadout.longitudinalGaugeRestorationEstablished = true ∧ physicalNambu1960NeutralScreeningReadout.chargeWardVertexMatrix 0 1 = -4 ∧ physicalNambu1960NeutralCollectivePairReadout.chargeWardVertexZeroOne = -4 := by refine ⟨physical_bcs_nonlocal_penetration_and_unresolved_scope_are_nonvacuous, physical_nambu_neutral_pair_is_gapless_only_without_coulomb, ?_, ?_, ?_, ?_⟩ all_goals norm_num [physicalBCS1957LondonScreeningReadout, physicalNambu1960NeutralScreeningReadout, physicalMeissnerMatrix, physicalNambu1960NeutralCollectivePairReadout] example : PhysicalNambuNeutralCollectiveScreeningAt physicalNambu1960NeutralScreeningReadout ∧ PhysicalNambuCoulombPlasmaScreeningAt physicalNambu1960CoulombScreeningReadout ∧ physicalNambu1960NeutralScreeningReadout.zeroMomentumPairFrequencySquaredValue = 0 ∧ physicalNambu1960CoulombScreeningReadout.plasmaFrequencySquaredValue = 4 ∧ physicalNambu1960CoulombConvertedPlasmaReadout.nambuPlasmaFrequencySquared = 4 ∧ ¬PhysicalNambuCoulombPlasmaScreeningAt physicalNambuZeroModeRetainedInCoulombSpectrumControl := by obtain ⟨neutral, _wrongNeutral, _neutralCoulomb, coulomb, _wrongPlasma, _missingPolarization, noInteractingZeroMode, _masslessCoulomb⟩ := nambu_neutral_collective_pair_and_coulomb_plasma_have_distinct_spectra refine ⟨neutral, coulomb, ?_, ?_, ?_, noInteractingZeroMode⟩ all_goals norm_num [physicalNambu1960CoulombScreeningReadout, physicalNambu1960NeutralScreeningReadout, physicalBCS1957LondonScreeningReadout, physicalNambu1960CoulombConvertedPlasmaReadout, physicalNambu1960NeutralCollectivePairReadout] example : physicalMeissnerScopePossibility.TentativeAt physicalMeissnerScopeEstablishment .bcsOriginalCalculationEstablishesLongitudinalGaugeInvariance .bardeenCooperSchrieffer1957 ∧ physicalMeissnerScopePossibility.TentativeAt physicalMeissnerScopeEstablishment .nambuNeutralGaplessPairRemainsInCoulombSpectrum .nambu1960 ∧ physicalMeissnerScopePossibility.TentativeAt physicalMeissnerScopeEstablishment .nambuWardIdentityEstablishesAllExternalFieldOrders .nambu1960 := by obtain ⟨bcsGaugeRestoration, _bcsTransport, _bcsStrongCoupling, _nambuBareResponse, _nambuTransverseErasure, nambuCoulombZeroMode, nambuAllOrders, _nambuBrokenGauge, _nambuCoreConservation⟩ := meissner_source_overclaims_remain_epistemically_open exact ⟨bcsGaugeRestoration, nambuCoulombZeroMode, nambuAllOrders⟩ end SuperconductingScreeningPhysicalReuseAudit namespace QuantumRecoveryPhysicalReuseAudit open PhysicsLogic.Framework.ExactQuantumErrorRecoveryContext.Countermodels example : physicalExactRecoveryModel.amplitudeSyndrome ≠ physicalExactRecoveryModel.phaseSyndrome ∧ physicalExactRecoveryModel.amplitudeRecovery ≠ physicalExactRecoveryModel.phaseRecovery ∧ physicalAmplitudeRecoveryOnlyContext.EstablishedAmplitudeRecoveryAt physicalAmplitudeRecoveryOnlyModel ∧ ¬physicalAmplitudeRecoveryOnlyContext.EstablishedTwoBasisRecoveryAt physicalAmplitudeRecoveryOnlyModel ∧ physicalLogicalStateRevealedModel.amplitudeSyndromeRevealsLogicalStateAt .amplitude .logicalProtected .selected ∧ cosetRecoveryControlContext.EstablishedExactQuantumErrorRecoveryAt cosetRecoveryControlModel := by obtain ⟨distinctSyndromes, distinctRepairs, _distinctEncodings, amplitudeRecovery, noComplementaryRecovery, _identifies, reveals, _notPrivate, _selectedRecovery, controlRecovery⟩ := exact_recovery_distinguishes_syndromes_repairs_encodings_and_nonrevealing_controls exact ⟨distinctSyndromes, distinctRepairs, amplitudeRecovery, noComplementaryRecovery, reveals, controlRecovery⟩ end QuantumRecoveryPhysicalReuseAudit namespace StochasticDiffusionPhysicalReuseAudit open PhysicsLogic.Framework.RandomDisplacementDiffusionContext.Countermodels example : physicalBrownianDiffusionModel.system ≠ inflationaryDiffusionControlModel.system ∧ physicalBrownianDiffusionModel.time ≠ .initial ∧ ¬physicalBiasedImpulseModel.directionalBiasControlled .suspendedParticle .biasedImpulse .diluteSuspension ∧ physicalSpreadWithoutTransportModel.meanSquareDisplacementLinear .suspendedParticle .molecularImpulse .observed .diluteSuspension ∧ ¬physicalSpreadWithoutTransportModel.diffusiveTransport .brownianSuspension .suspendedParticle .diluteSuspension ∧ inflationaryDiffusionControlContext.EstablishedRandomDisplacementDiffusionAt inflationaryDiffusionControlModel := by obtain ⟨distinctSystems, _distinctSteps, distinctTimes, _dependence, biased, _invalidSetup, spreading, _rootSpread, noTransport, _brownian, inflation⟩ := stochastic_diffusion_distinguishes_systems_steps_times_bias_and_transport_controls exact ⟨distinctSystems, distinctTimes, biased, spreading, noTransport, inflation⟩ end StochasticDiffusionPhysicalReuseAudit namespace GravitationalRedshiftPhysicalReuseAudit open PhysicsLogic.Framework.GravitationalFrequencyShiftContext.Countermodels example : physicalFirstOrderShiftModel.emitter ≠ physicalFirstOrderShiftModel.receiver ∧ physicalFirstOrderShiftModel.description ≠ staticMetricShiftControlModel.description ∧ physicalFrequencyWithoutClockModel.frequencyShiftRelativeToLocalClocks .upwardLight .lowerEmitter .upperReceiver .equivalenceApproximation ∧ ¬physicalFrequencyWithoutClockModel.localClockRatesDiffer .lowerEmitter .upperReceiver .equivalenceApproximation ∧ ¬physicalClockWithoutSpectrumModel.spectralShiftPredicted .firstOrderRedshift .upwardLight .lowerEmitter .upperReceiver .equivalenceApproximation ∧ staticMetricShiftControlContext.EstablishedGravitationalFrequencyShiftAt .stationaryMetric .metricEmitter .metricReceiver .metricLight .metricRedshift .staticWeakField := by obtain ⟨distinctLocations, distinctDescriptions, _distinctSignals, _distinctPhenomena, frequency, noClock, _clock, noSpectrum, _selectedShift, controlShift⟩ := gravitational_redshift_distinguishes_locations_signals_clock_bridges_and_metric_controls exact ⟨distinctLocations, distinctDescriptions, frequency, noClock, noSpectrum, controlShift⟩ example (readout : PhysicsLogic.Papers.Einstein1911.FirstOrderOpticalFormulaReadout) (domain : readout.FormulaDomainAt) (equationTwoA : readout.EquationTwoAAt) : (readout.emittedFrequencyValue - readout.receivedFrequencyValue) / readout.emittedFrequencyValue = -(readout.potentialDifferenceValue / readout.referenceLightSpeedValue ^ 2) := readout.equation_two_a_derives_fractional_gravitational_redshift domain equationTwoA example (readout : PhysicsLogic.Papers.Einstein1911.FirstOrderOpticalFormulaReadout) (domain : readout.FormulaDomainAt) (equationTwoA : readout.EquationTwoAAt) : 0 < (readout.emittedFrequencyValue - readout.receivedFrequencyValue) / readout.emittedFrequencyValue ↔ readout.potentialDifferenceValue < 0 := readout.equation_two_a_has_positive_redshift_iff_negative_potential domain equationTwoA example : PhysicsLogic.Papers.Einstein1911.FirstOrderOpticalFormulaReadout.Countermodels.positivePotentialBlueShift.EquationTwoAAt ∧ 0 < PhysicsLogic.Papers.Einstein1911.FirstOrderOpticalFormulaReadout.Countermodels.positivePotentialBlueShift.potentialDifferenceValue ∧ ¬(0 < (PhysicsLogic.Papers.Einstein1911.FirstOrderOpticalFormulaReadout.Countermodels.positivePotentialBlueShift.emittedFrequencyValue - PhysicsLogic.Papers.Einstein1911.FirstOrderOpticalFormulaReadout.Countermodels.positivePotentialBlueShift.receivedFrequencyValue) / PhysicsLogic.Papers.Einstein1911.FirstOrderOpticalFormulaReadout.Countermodels.positivePotentialBlueShift.emittedFrequencyValue) := by obtain ⟨_domain, frequencyFormula, positivePotential, notRedshift⟩ := PhysicsLogic.Papers.Einstein1911.FirstOrderOpticalFormulaReadout.Countermodels.equation_two_a_with_positive_potential_does_not_predict_redshift exact ⟨frequencyFormula, positivePotential, notRedshift⟩ example : PhysicalEinstein1911AcceleratedPotentialFrequencyAt physicalEinstein1911FrequencyReadout := physical_einstein_1911_accelerated_potential_frequency_shift_is_nonvacuous example : PhysicalEinstein1911UnresolvedSolarSpectralEvidenceAt physicalEinstein1911FrequencyReadout := physical_einstein_1911_solar_redshift_evidence_remains_inconclusive example : PhysicalEinstein1916StaticMetricClockFrequencyAt physicalEinstein1916FrequencyReadout := physical_einstein_1916_static_metric_clock_frequency_shift_is_nonvacuous example : PhysicalEinstein1916OpenRedshiftTestAt physicalEinstein1916FrequencyReadout := physical_einstein_1916_predicted_redshift_test_remains_open example : ¬PhysicalEinstein1911AcceleratedPotentialFrequencyAt physical1911WrongPotentialSignControl := einstein_1911_acceleration_potential_coordinate_speed_and_spectral_scope_are_independent.2.1 example : ¬PhysicalEinstein1911AcceleratedPotentialFrequencyAt physical1911LocalSpeedSubstitutionControl := einstein_1911_acceleration_potential_coordinate_speed_and_spectral_scope_are_independent.2.2.1 example : ¬PhysicalEinstein1911AcceleratedPotentialFrequencyAt physical1911MissingDopplerCalculationControl := einstein_1911_acceleration_potential_coordinate_speed_and_spectral_scope_are_independent.2.2.2.1 example : ¬PhysicalEinstein1911AcceleratedPotentialFrequencyAt physical1911ClaimedNonuniformDerivationControl := einstein_1911_acceleration_potential_coordinate_speed_and_spectral_scope_are_independent.2.2.2.2.1 example : ¬PhysicalEinstein1911UnresolvedSolarSpectralEvidenceAt physical1911IgnoredSpectralSystematicsControl := einstein_1911_acceleration_potential_coordinate_speed_and_spectral_scope_are_independent.2.2.2.2.2.2.1 example : ¬PhysicalEinstein1911UnresolvedSolarSpectralEvidenceAt physical1911ClaimedConclusiveRedshiftControl := einstein_1911_acceleration_potential_coordinate_speed_and_spectral_scope_are_independent.2.2.2.2.2.2.2 example : ¬PhysicalEinstein1916StaticMetricClockFrequencyAt physical1916WrongTemporalMetricControl := einstein_1916_metric_normalization_clock_rates_and_open_test_are_independent.2.1 example : ¬PhysicalEinstein1916StaticMetricClockFrequencyAt physical1916WrongClockNormalizationControl := einstein_1916_metric_normalization_clock_rates_and_open_test_are_independent.2.2.1 example : ¬PhysicalEinstein1916StaticMetricClockFrequencyAt physical1916NoPointMassMetricControl := einstein_1916_metric_normalization_clock_rates_and_open_test_are_independent.2.2.2.1 example : ¬PhysicalEinstein1916OpenRedshiftTestAt physical1916ClaimedConclusiveRedshiftControl := einstein_1916_metric_normalization_clock_rates_and_open_test_are_independent.2.2.2.2.2.1 example : ¬PhysicalEinstein1916OpenRedshiftTestAt physical1916ClaimedExactStrongFieldControl := einstein_1916_metric_normalization_clock_rates_and_open_test_are_independent.2.2.2.2.2.2 example : PhysicalEinstein1911UnresolvedSolarSpectralEvidenceAt physicalEinstein1911FrequencyReadout ∧ PhysicsLogic.Papers.Einstein1911.LightGravityFramework.Countermodels.ReducedSpectralEvidenceReadout.HasPredictedSolarRedshift PhysicsLogic.Papers.Einstein1911.LightGravityFramework.Countermodels.predictedShiftWithoutConclusiveEvidence ∧ ¬PhysicsLogic.Papers.Einstein1911.LightGravityFramework.Countermodels.ReducedSpectralEvidenceReadout.HasConclusiveSpectralTest PhysicsLogic.Papers.Einstein1911.LightGravityFramework.Countermodels.predictedShiftWithoutConclusiveEvidence := ⟨physical_einstein_1911_solar_redshift_evidence_remains_inconclusive, PhysicsLogic.Papers.Einstein1911.LightGravityFramework.Countermodels.predicted_spectral_shift_need_not_be_empirically_conclusive.1, PhysicsLogic.Papers.Einstein1911.LightGravityFramework.Countermodels.predicted_spectral_shift_need_not_be_empirically_conclusive.2.2⟩ example : PhysicalEinstein1911AcceleratedPotentialFrequencyAt physicalEinstein1911FrequencyReadout ∧ PhysicsLogic.Papers.Einstein1911.FirstOrderOpticalFormulaReadout.Countermodels.HasNamedOpticalCalculation PhysicsLogic.Papers.Einstein1911.FirstOrderOpticalFormulaReadout.Countermodels.namedOpticalCalculationWithoutPrintedFormulas := ⟨physical_einstein_1911_accelerated_potential_frequency_shift_is_nonvacuous, PhysicsLogic.Papers.Einstein1911.FirstOrderOpticalFormulaReadout.Countermodels.named_optical_calculation_does_not_supply_printed_equations_two_two_a_and_three.1⟩ end GravitationalRedshiftPhysicalReuseAudit namespace CongruenceFocusingPhysicalReuseAudit open PhysicsLogic.Framework.CongruenceFocusing.Countermodels example : physicalTimelikeFocusingModel.congruence ≠ nullFocusingControlModel.congruence ∧ physicalTimelikeFocusingModel.transverseProfile ≠ nullFocusingControlModel.transverseProfile ∧ ¬physicalContractionWithoutCurvatureModel.FocusingEvolution ∧ physicalCollapseWithoutInterpretationModel.profileVanishesAt .timelikeVolumeScale .timelikeGeodesics .timelikeFocus .timelikeMatterConvergence ∧ ¬physicalCollapseWithoutInterpretationModel.congruenceFocusesAt .timelikeGeodesics .past .timelikeFocus .timelikeMatterConvergence ∧ nullFocusingControlContext.EstablishedCongruenceFocusingAt nullFocusingControlModel := by obtain ⟨distinctCongruences, distinctProfiles, _distinctDirections, _distinctParameters, _setup, _contraction, noEvolution, collapse, noFocus, _timelike, nullFocus⟩ := geodesic_focusing_distinguishes_timelike_null_profiles_directions_and_controls exact ⟨distinctCongruences, distinctProfiles, noEvolution, collapse, noFocus, nullFocus⟩ end CongruenceFocusingPhysicalReuseAudit namespace BipartiteEntanglementPhysicalReuseAudit open PhysicsLogic.Framework.BipartiteEntanglementContext.Countermodels example : physicalBipartiteEntanglementModel.firstSubsystem ≠ physicalBipartiteEntanglementModel.secondSubsystem ∧ physicalBipartiteEntanglementModel.parentState ≠ horizonEntanglementControlModel.parentState ∧ physicalClassicalMixtureModel.PositiveEntanglementWitnessAt ∧ physicalClassicalMixtureModel.SeparableAt ∧ ¬physicalClassicalMixtureModel.EntangledAt ∧ ¬physicalParentWithoutReductionModel.SelectedBipartiteSetupAt ∧ horizonEntanglementControlContext.EstablishedBipartiteEntanglementAt horizonEntanglementControlModel := by obtain ⟨distinctSubsystems, distinctParents, _distinctJointStates, positiveWitness, _classicalCorrelation, separable, _uncalibrated, notEntangled, _prepared, missingReduction, _laboratory, horizon⟩ := bipartite_entanglement_distinguishes_parents_pairs_subsystems_and_separable_controls exact ⟨distinctSubsystems, distinctParents, positiveWitness, separable, notEntangled, missingReduction, horizon⟩ example : let context := selectedReadoutContext nonseparableWithoutWitnessCalibration let model := context.selectedModel model.NonvacuousBipartiteEntanglementAt ∧ ¬model.SeparableWitnessUpperBoundAt ∧ ¬context.BipartiteEntanglementRouteApplicableAt model ∧ ¬context.EstablishedBipartiteEntanglementAt model := nonseparable_positive_state_without_witness_calibration_does_not_complete_route example : let context := selectedReadoutContext certifiedEntangledPair let model := context.selectedModel context.EstablishedBipartiteEntanglementAt model ∧ context.DirectStructuralBipartiteEntanglementExplanationAt model ∧ ¬context.SourceStructuralBipartiteEntanglementExplanationAt model .absent missingEntanglementDerivationObtains := completed_entanglement_and_ordinary_explanation_without_source_derivation_have_no_source_explanation end BipartiteEntanglementPhysicalReuseAudit namespace LinearTransportPhysicalReuseAudit open PhysicsLogic.Framework.LinearResponseTransportContext.Countermodels example : physicalElectricTransportModel.force ≠ thermalTransportControlModel.force ∧ physicalElectricTransportModel.flux ≠ thermalTransportControlModel.flux ∧ ¬physicalTransportWithoutLateKernelModel.EquilibriumKernelRepresentation ∧ physicalResponseWithoutTransportModel.EquilibriumKernelRepresentation ∧ ¬physicalResponseWithoutTransportModel.TransportReadout ∧ thermalTransportControlContext.EstablishedLinearResponseTransportAt thermalTransportControlModel := by obtain ⟨distinctForces, distinctFluxes, _distinctKernelValues, _earlyKernel, _noLateKernel, incompleteKernel, _causal, responseKernel, noTransport, _electric, thermal⟩ := linear_transport_distinguishes_forces_fluxes_kernel_times_values_and_controls exact ⟨distinctForces, distinctFluxes, incompleteKernel, responseKernel, noTransport, thermal⟩ end LinearTransportPhysicalReuseAudit namespace HomogeneousExpansionPhysicalReuseAudit open PhysicsLogic.Framework.HomogeneousCosmicExpansionContext.Countermodels example : physicalFriedmannExpansionModel.earlierTime ≠ physicalFriedmannExpansionModel.laterTime ∧ physicalFriedmannExpansionModel.earlierRadius ≠ physicalFriedmannExpansionModel.laterRadius ∧ ¬physicalExpansionWithoutHistoryModel.selectedRadiusHistoryRealizesNonstationarySolutionAt .closedFriedmann .uniformDust .friedmannWorld .friedmannSmall .friedmannEarly .friedmannLarge .friedmannLate .friedmannRelativistic ∧ ¬physicalExpansionWithoutInterpretationModel.homogeneousCosmicExpansionAt .friedmannWorld .friedmannSmall .friedmannEarly .friedmannLarge .friedmannLate .friedmannRelativistic ∧ lemaitreExpansionControlContext.EstablishedHomogeneousCosmicExpansionAt lemaitreExpansionControlModel := by obtain ⟨distinctTimes, distinctRadii, _distinctCosmos, _distinctGeometry, _evolution, _increase, noHistory, _interpretedIncrease, noExpansion, _friedmann, lemaitre⟩ := cosmic_expansion_distinguishes_times_radii_histories_geometries_and_controls exact ⟨distinctTimes, distinctRadii, noHistory, noExpansion, lemaitre⟩ end HomogeneousExpansionPhysicalReuseAudit namespace DensityFunctionalPhysicalReuseAudit open PhysicsLogic.Framework.DensityFunctionalContext.Countermodels example : physicalGroundDensityModel.potential ≠ harmonicGroundDensityControlModel.potential ∧ wrongParticleNumberDensityModel.particleNumberOf wrongParticleNumberDensityModel.density = 1 ∧ wrongParticleNumberDensityModel.particleNumber = 2 ∧ ¬wrongParticleNumberDensityModel.SelectedGroundDensitySetup ∧ minimizingDensityWithoutPhysicalEnergyModel.MinimizesOverAdmissibleDensities ∧ ¬minimizingDensityWithoutPhysicalEnergyModel.AttainsPhysicalGroundEnergy ∧ harmonicGroundDensityControlContext.EstablishedGroundDensityVariationalAt harmonicGroundDensityControlModel := by obtain ⟨distinctPotentials, _distinctStates, _distinctDensity, actualNumber, selectedNumber, invalidSetup, minimizing, wrongEnergy, _coulomb, harmonic⟩ := density_functional_distinguishes_potentials_states_densities_particle_sectors_and_controls exact ⟨distinctPotentials, actualNumber, selectedNumber, invalidSetup, minimizing, wrongEnergy, harmonic⟩ end DensityFunctionalPhysicalReuseAudit namespace BogoliubovProductionPhysicalReuseAudit open PhysicsLogic.Framework.BogoliubovParticleProductionContext.Countermodels example : physicalCosmologicalProductionModel.incomingMode ≠ physicalCosmologicalProductionModel.outgoingMode ∧ physicalCosmologicalProductionModel.incomingState ≠ hawkingProductionControlModel.incomingState ∧ ¬physicalZeroMixingProductionModel.negativeFrequencyMixingNonzero .vanishingBeta ∧ positiveOccupationWithoutProductionModel.outgoingOccupationPositive .cosmologicalParticles ∧ ¬positiveOccupationWithoutProductionModel.particlesProducedInOutgoingMode .cosmologicalVacuum .expandingOutgoing .cosmologicalParticles ∧ hawkingProductionControlContext.EstablishedBogoliubovParticleProductionAt hawkingProductionControlModel := by obtain ⟨distinctModes, distinctVacua, _distinctMixings, _vacuum, _expansion, zeroMixing, _zeroOccupation, positiveOccupation, noInterpretation, _cosmological, hawking⟩ := particle_production_distinguishes_vacua_modes_mixing_occupation_and_controls exact ⟨distinctModes, distinctVacua, zeroMixing, positiveOccupation, noInterpretation, hawking⟩ end BogoliubovProductionPhysicalReuseAudit namespace DefectUnbindingPhysicalReuseAudit open PhysicsLogic.Framework.DefectUnbindingTransitionContext.Countermodels example : physicalVortexUnbindingModel.orderedPhase ≠ physicalVortexUnbindingModel.unboundPhase ∧ physicalVortexUnbindingModel.fugacity ≠ physicalVortexUnbindingModel.stiffness ∧ ¬physicalFugacityWithoutStiffnessFlowContext.UnbindingFlowCalculationAt physicalFugacityWithoutStiffnessFlowModel ∧ ¬physicalFlowWithoutUnbindingModel.defectPairsUnbind .vortexAntivortexPair .vortexHighTemperature ∧ crystalDislocationUnbindingControlContext.EstablishedDefectUnbindingTransitionAt crystalDislocationUnbindingControlModel := by obtain ⟨distinctPhases, _distinctRegimes, _distinctDefects, distinctCouplings, _fugacityGrows, _stiffnessPersists, noFullFlow, _flowGrows, noUnbinding, _vortex, dislocation⟩ := defect_unbinding_distinguishes_phases_defects_couplings_regimes_and_controls exact ⟨distinctPhases, distinctCouplings, noFullFlow, noUnbinding, dislocation⟩ end DefectUnbindingPhysicalReuseAudit namespace EquilibriumSamplingPhysicalReuseAudit open PhysicsLogic.Framework.EquilibriumSamplingContext.Countermodels example : physicalMetropolisSamplingModel.update ≠ gaugeHeatBathSamplingControlModel.update ∧ physicalBalancedButTrappedSamplingModel.detailedBalance .trappedProposal .molecularBoltzmann ∧ ¬physicalBalancedButTrappedSamplingModel.targetSpaceAccessibleBy .trappedProposal .molecularBoltzmann ∧ ¬physicalAccessibleWithoutConvergenceModel.repeatedUpdatesConvergeTo .metropolisProposal .molecularBoltzmann ∧ ¬physicalMetropolisSamplingContext.selectedFiniteRunConverged .metropolisProposal .molecularBoltzmann ∧ gaugeHeatBathSamplingControlContext.EstablishedEquilibriumSamplingAt gaugeHeatBathSamplingControlModel := by obtain ⟨distinctUpdates, _distinctTargets, balance, inaccessible, _noTrappedSetup, _accessible, noConvergence, noFiniteRun, _metropolis, gauge⟩ := equilibrium_sampling_distinguishes_updates_targets_accessibility_convergence_and_controls exact ⟨distinctUpdates, balance, inaccessible, noConvergence, noFiniteRun, gauge⟩ end EquilibriumSamplingPhysicalReuseAudit namespace QuantumNoCloningPhysicalReuseAudit open PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels example : physicalPolarizationReplicationModel.firstBasisState ≠ physicalPolarizationReplicationModel.secondBasisState ∧ physicalPolarizationReplicationModel.linearOutput ≠ physicalPolarizationReplicationModel.cloneOutput .polarizedSuperposition ∧ physicalBasisReplicationOnlyModel.SelectedReplicationSetup ∧ ¬physicalBasisReplicationOnlyModel.evolvesToAt .polarizationDuplicator .polarizedSuperposition .polarizationReady .linearPolarizationOutput .polarizationUnitary ∧ ¬physicalNondeterministicReplicationModel.SelectedEvolutionOutputUnique ∧ nonorthogonalReplicationControlContext.EstablishedLinearStateReplicationObstructionAt nonorthogonalReplicationControlModel := by obtain ⟨distinctBasis, _distinctSuperposition, outputMismatch, _distinctDevice, basisOnly, noSuperposition, _linearOutput, _requiredClone, nondeterministic, _polarization, nonorthogonal⟩ := quantum_no_cloning_distinguishes_basis_superposition_outputs_devices_and_controls exact ⟨distinctBasis, outputMismatch, basisOnly, noSuperposition, nondeterministic, nonorthogonal⟩ end QuantumNoCloningPhysicalReuseAudit namespace GravitationalRayPhysicalReuseAudit open PhysicsLogic.Framework.GravitationalRayDeflectionContext.Countermodels example : physicalFirstOrderRayModel.propagationSetup ≠ staticMetricRayControlModel.propagationSetup ∧ physicalSolarAngleHundredths physicalFirstOrderRayModel.deflection = 83 ∧ physicalSolarAngleHundredths staticMetricRayControlModel.deflection = 170 ∧ ¬physicalRayPropagationWithoutCalculationModel.deflectionCalculatedAt .acceleratedCoordinateLight .huygensGrazingRay .firstOrderSolarAngle .firstOrderSolar ∧ ¬physicalCalculatedZeroRayModel.deflectionMagnitudeNonzero .zeroAngle ∧ staticMetricRayControlContext.EstablishedGravitationalRayDeflectionAt staticMetricRayControlModel := by obtain ⟨_distinctDescriptions, distinctRays, _distinctAngles, firstAngle, metricAngle, _propagation, noCalculation, _zeroCalculated, zeroNotNonzero, _firstOrder, metric⟩ := gravitational_ray_deflection_distinguishes_descriptions_rays_angles_and_metric_controls exact ⟨distinctRays, firstAngle, metricAngle, noCalculation, zeroNotNonzero, metric⟩ example : PhysicalEinstein1911PotentialGradientHuygensDeflectionAt physicalEinstein1911RayReadout := physical_einstein_1911_coordinate_speed_huygens_solar_deflection_is_nonvacuous example : PhysicalEinstein1916StaticMetricNullHuygensDeflectionAt physicalEinstein1916RayReadout := physical_einstein_1916_static_metric_null_huygens_deflection_is_nonvacuous example : PhysicalEinsteinSolarDeflectionOpenEvidenceAt physicalEinstein1911RayReadout := physical_original_solar_deflection_predictions_remain_unconfirmed.1 example : PhysicalEinsteinSolarDeflectionOpenEvidenceAt physicalEinstein1916RayReadout := physical_original_solar_deflection_predictions_remain_unconfirmed.2 example : ¬PhysicalEinstein1911PotentialGradientHuygensDeflectionAt physical1911MissingHuygensWavefrontControl := einstein_1911_huygens_gradient_half_deflection_and_eclipse_scope_are_independent.2.1 example : ¬PhysicalEinstein1911PotentialGradientHuygensDeflectionAt physical1911MissingPotentialGradientControl := einstein_1911_huygens_gradient_half_deflection_and_eclipse_scope_are_independent.2.2.1 example : ¬PhysicalEinstein1911PotentialGradientHuygensDeflectionAt physical1911WrongReportedSolarAngleControl := einstein_1911_huygens_gradient_half_deflection_and_eclipse_scope_are_independent.2.2.2.1 example : ¬PhysicalEinstein1911PotentialGradientHuygensDeflectionAt physical1911RetrofittedSpatialMetricControl := einstein_1911_huygens_gradient_half_deflection_and_eclipse_scope_are_independent.2.2.2.2.1 example : ¬PhysicalEinstein1911PotentialGradientHuygensDeflectionAt physical1911ClaimedConfirmedEclipseControl := einstein_1911_huygens_gradient_half_deflection_and_eclipse_scope_are_independent.2.2.2.2.2 example : ¬PhysicalEinstein1916StaticMetricNullHuygensDeflectionAt physical1916MissingHuygensPrincipleControl := einstein_1916_metric_null_huygens_spatial_bending_and_reports_are_independent.2.1 example : ¬PhysicalEinstein1916StaticMetricNullHuygensDeflectionAt physical1916MissingSpatialMetricContributionControl := einstein_1916_metric_null_huygens_spatial_bending_and_reports_are_independent.2.2.1 example : ¬PhysicalEinstein1916StaticMetricNullHuygensDeflectionAt physical1916MissingNullMetricConditionControl := einstein_1916_metric_null_huygens_spatial_bending_and_reports_are_independent.2.2.2.1 example : ¬PhysicalEinstein1916StaticMetricNullHuygensDeflectionAt physical1916MissingPrintedMetricSpeedControl := einstein_1916_printed_speed_and_metric_null_signs_are_independently_retained.2.1 example : ¬PhysicalEinstein1916StaticMetricNullHuygensDeflectionAt physical1916SignCorrectedPrintedMetricSpeedControl := einstein_1916_printed_speed_and_metric_null_signs_are_independently_retained.2.2.1 example : physicalEinstein1916RayReadout.printedMetricSpeedExpansionValue ≠ physicalEinstein1916RayReadout.nullMetricConsistentFirstOrderSpeedValue := printed_1916_coordinate_speed_sign_differs_from_metric_null_expansion example : ¬PhysicalEinstein1916StaticMetricNullHuygensDeflectionAt physical1916WrongFullDeflectionCoefficientControl := einstein_1916_metric_null_huygens_spatial_bending_and_reports_are_independent.2.2.2.2.1 example : ¬PhysicalEinstein1916StaticMetricNullHuygensDeflectionAt physical1916WrongReportedSolarAngleControl := einstein_1916_metric_null_huygens_spatial_bending_and_reports_are_independent.2.2.2.2.2.1 example : ¬PhysicalEinstein1916StaticMetricNullHuygensDeflectionAt physical1916ClaimedConfirmedEclipseControl := einstein_1916_metric_null_huygens_spatial_bending_and_reports_are_independent.2.2.2.2.2.2 example : ¬PhysicalEinsteinSolarDeflectionOpenEvidenceAt physical1916ClaimedConfirmedEclipseControl := einstein_1916_eclipse_and_strong_field_overclaims_are_independent.2.1 example : ¬PhysicalEinsteinSolarDeflectionOpenEvidenceAt physical1916ClaimedExactStrongFieldControl := einstein_1916_eclipse_and_strong_field_overclaims_are_independent.2.2 example : physicalEinstein1916RayReadout.totalDeflectionAngleValue = 2 * physicalEinstein1911RayReadout.totalDeflectionAngleValue := temporal_and_spatial_metric_bending_double_first_order_unrounded_prediction.2.2.1 example : physicalEinstein1916RayReadout.reportedSolarDeflectionHundredthsValue ≠ 2 * physicalEinstein1911RayReadout.reportedSolarDeflectionHundredthsValue := temporal_and_spatial_metric_bending_double_first_order_unrounded_prediction.2.2.2.2.2 example : physicalEinstein1911RayReadout.reportedSolarDeflectionHundredthsValue = PhysicsLogic.Papers.Einstein1911.solarDeflection1911.hundredths := by rfl example : physicalEinstein1916RayReadout.reportedSolarDeflectionHundredthsValue = 10 * PhysicsLogic.Papers.Einstein1916.solarLimbDeflectionPaperValue.tenthsOfArcsecond := by rfl example : PhysicalEinstein1911PotentialGradientHuygensDeflectionAt physicalEinstein1911RayReadout ∧ PhysicsLogic.Papers.Einstein1911.FirstOrderDeflectionFormulaReadout.Countermodels.HasNamedDeflectionCalculation PhysicsLogic.Papers.Einstein1911.FirstOrderDeflectionFormulaReadout.Countermodels.namedDeflectionCalculationWithoutPrintedFormulas := ⟨physical_einstein_1911_coordinate_speed_huygens_solar_deflection_is_nonvacuous, PhysicsLogic.Papers.Einstein1911.FirstOrderDeflectionFormulaReadout.Countermodels.named_deflection_calculation_does_not_supply_printed_formulas.1⟩ end GravitationalRayPhysicalReuseAudit namespace NonequilibriumWorkPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.NonequilibriumWorkIdentityContext.Countermodels example (model : NonequilibriumWorkCalculationModel Unit Empty) : ¬model.NonvacuousNonequilibriumWorkIdentityAndBoundAt := model.empty_realization_space_cannot_exhibit_work_identity_and_bound example : unnormalizedWorkIdentityModel.ExponentialWorkIdentity ∧ unnormalizedWorkIdentityModel.MeanWorkDefinition ∧ unnormalizedWorkIdentityModel.MeanWorkBound ∧ ¬unnormalizedWorkIdentityModel.SelectedSetup := by obtain ⟨identity, meanDefinition, bound, _notNormalized, noSetup, _⟩ := scalar_identity_and_mean_bound_do_not_supply_normalized_work_ensemble exact ⟨identity, meanDefinition, bound, noSetup⟩ example {Configuration Protocol Realization : Type} (framework : PhysicsLogic.Papers.Jarzynski1997.SwitchingFramework Configuration Protocol Realization) (protocol : Protocol) (domain : framework.NormalizedExponentialWorkEnsembleAt protocol) (meanDefinition : framework.MeanWorkDefinition protocol) : framework.ExponentialJensenBound protocol := framework.normalized_work_ensemble_derives_exponential_jensen protocol domain meanDefinition example {State Path Protocol Ensemble : Type} (framework : PhysicsLogic.Papers.Crooks1999.FluctuationFramework State Path Protocol Ensemble) (protocol : Protocol) (domain : framework.NormalizedExponentialWorkEnsembleAt protocol) : framework.WorkJensenCalculation protocol := framework.normalized_path_ensemble_derives_work_jensen protocol domain example : physicalHamiltonianWorkModel.process ≠ reversePathWorkControlModel.process ∧ physicalHamiltonianWorkModel.work .hamiltonianForward .hamiltonianTrajectory = 0 ∧ physicalHamiltonianWorkModel.work .hamiltonianForward .reversePath = 1 ∧ physicalForwardWithoutCanonicalPreparationModel.ExponentialWorkIdentity ∧ ¬physicalForwardWithoutCanonicalPreparationModel.SelectedSetup ∧ ¬physicalIdentityWithoutMeanDefinitionModel.MeanWorkDefinition ∧ reversePathWorkControlContext.EstablishedExponentialWorkIdentityAndBoundAt reversePathWorkControlModel := by obtain ⟨distinctProcesses, _distinctPaths, forwardWork, reverseWork, unpreparedIdentity, _notPrepared, noSetup, _wrongMeanIdentity, noMean, _hamiltonian, reverse⟩ := nonequilibrium_work_distinguishes_protocols_paths_preparation_means_and_controls exact ⟨distinctProcesses, forwardWork, reverseWork, unpreparedIdentity, noSetup, noMean, reverse⟩ end NonequilibriumWorkPhysicalReuseAudit namespace ResonantSpectroscopyPhysicalReuseAudit open PhysicsLogic.Framework.ResonantTransitionSpectroscopyContext.Countermodels example : physicalRabiSpectroscopyModel.initialState ≠ physicalRabiSpectroscopyModel.finalState ∧ physicalRabiSpectroscopyModel.drive ≠ separatedFieldSpectroscopyControlModel.drive ∧ physicalRabiSpectroscopyModel.resonanceReadout ≠ separatedFieldSpectroscopyControlModel.resonanceReadout ∧ ¬physicalOffResonantSpectroscopyModel.driveResonantWithCharacteristicAt .offResonance .molecularResonance .molecularBeam .molecularBeamExperiment ∧ ¬physicalSpectroscopyWithoutReadoutModel.transitionReadoutEstablishedAt .molecularTransition .gradientRefocusingSignal .molecularBeam .molecularBeamExperiment ∧ separatedFieldSpectroscopyControlContext.EstablishedResonantTransitionSpectroscopyAt separatedFieldSpectroscopyControlModel := by obtain ⟨distinctStates, distinctDrives, _distinctFrequencies, distinctReadouts, _distinctQuantities, _offResonantResponse, notResonant, noReadout, _rabi, ramsey⟩ := resonant_spectroscopy_distinguishes_states_drives_frequencies_readouts_and_controls exact ⟨distinctStates, distinctDrives, distinctReadouts, notResonant, noReadout, ramsey⟩ example : PhysicalRabiWeakOscillatingTransitionAt physicalRabiWeakTransitionReadout ∧ PhysicalRabiGradientRefocusingAt physicalRabiGradientReadout ∧ PhysicalRabiCrossMoleculeMomentMagnitudeAt physicalRabiMomentReadout ∧ physicalRabiMomentReadout.displayedLithiumSevenGFactorValue * physicalRabiMomentReadout.lithiumSevenSpinValue ≠ physicalRabiMomentReadout.displayedLithiumSevenMomentValue ∧ physicalRabiWeakTransitionReadout.momentSignDetermined = false := by exact ⟨physical_rabi_larmor_weak_field_transition_and_sign_scope_are_nonvacuous.1, physical_rabi_gradient_refocusing_produces_a_detector_minimum.1, physical_rabi_li7_moment_distinguishes_exact_value_and_printed_digits.1, physical_rabi_li7_moment_distinguishes_exact_value_and_printed_digits.2.2.2, physical_rabi_larmor_weak_field_transition_and_sign_scope_are_nonvacuous.2.2.2.2⟩ example : PhysicalRamseySeparatedPulseInterferenceAt physicalRamseyPulseReadout ∧ PhysicalRamseyVelocityAveragedFringeAt physicalRamseyVelocityAverageReadout ∧ PhysicalRamseyPulseAreasAndWidthScalesAt physicalRamseyWidthReadout ∧ physicalRamseyWidthReadout.centralHalfWidthValue = 8 / 25 ∧ PhysicalRamseyDriveFreeIntermediateAccessAt physicalRamseyDriveFreeAccessReadout ∧ physicalRamseyDriveFreeAccessReadout.shieldedRegionMeasurementPerformed = false := by exact ⟨physical_ramsey_pulse_free_pulse_interference_is_nonvacuous.1, physical_ramsey_velocity_distribution_and_interpolated_interval_are_distinct.1, physical_ramsey_pulse_areas_and_three_printed_width_scales_are_nonvacuous.1, physical_ramsey_pulse_areas_and_three_printed_width_scales_are_nonvacuous.2.1, physical_ramsey_drive_free_mean_access_is_not_a_performed_application.1, physical_ramsey_drive_free_mean_access_is_not_a_performed_application.2.2.2⟩ example : ¬physicalSpectroscopyEstablishment.establishedAt .oscillatingFieldDeterminesMomentSign .rabiMillmanKuschZacharias1939 ∧ ¬physicalSpectroscopyEstablishment.establishedAt .comparativePointSixEqualsPointSixFourCoefficient .ramsey1950 ∧ physicalSpectroscopyPossibility.TentativeAt physicalSpectroscopyEstablishment .ramseyProposedShieldedRegionMeasurement .ramsey1950 := by exact ⟨spectroscopy_source_overclaims_remain_epistemically_unestablished.1, spectroscopy_source_overclaims_remain_epistemically_unestablished.2.2.2.2.2.2.2.1, spectroscopy_source_overclaims_remain_epistemically_unestablished.2.2.2.2.2.2.2.2.2.2.2⟩ end ResonantSpectroscopyPhysicalReuseAudit namespace RamseyNominalWidthClient open PhysicsLogic.Papers.Ramsey1950 set_option backward.isDefEq.respectTransparency false noncomputable section def beam : ConventionalBeamParameters := ⟨40 / 101, 303 / 100, 2⟩ def gapOnlyBeam : ConventionalBeamParameters := ⟨2 / 5, 3, 2⟩ def centralNominalWidth : Real := (16 : Real) / 25 * 2 / 3 theorem the_independent_beam_is_positive_and_nominally_tuned : beam.PhysicalAt ∧ beam.beta = (6 : Real) / 5 * Real.pi := by refine ⟨by norm_num [beam, ConventionalBeamParameters.PhysicalAt], ?_⟩ dsimp [beam, ConventionalBeamParameters.beta] ring theorem the_driven_length_includes_both_pulse_regions : beam.drivenLength = (3 : Real) + 2 * (3 / 200) := by norm_num [beam] theorem the_library_computes_the_length_dependent_nominal_ratio : centralNominalWidth / beam.nominalWidth = 202 / 335 := by calc centralNominalWidth / beam.nominalWidth = (40 : Real) / 67 * (beam.drivenLength / 3) := beam.nominal_width_ratio_retains_the_two_distinct_lengths the_independent_beam_is_positive_and_nominally_tuned.1 3 (by norm_num) _ = 202 / 335 := by norm_num [beam] theorem the_independent_nominal_comparison_has_nonzero_error_and_narrowing : |centralNominalWidth / beam.nominalWidth - 3 / 5| = 1 / 335 ∧ ¬centralNominalWidth / beam.nominalWidth = 3 / 5 ∧ centralNominalWidth < beam.nominalWidth := by rw [the_library_computes_the_length_dependent_nominal_ratio] norm_num [centralNominalWidth, beam, ConventionalBeamParameters.nominalWidth] theorem the_library_scales_an_actual_conventional_fwhm_conditionally (width : Real) (calculation : ResonanceLineShape.CentralFWHMAt (ResonanceLineShape.conventionalResponse beam.beta) width) : ResonanceLineShape.CentralFWHMAt beam.responseAt (width * (40 / 101)) := beam.an_actual_conventional_full_width_has_its_cyclic_frequency_scale the_independent_beam_is_positive_and_nominally_tuned.1 width calculation theorem retuning_at_a_different_length_keeps_beta_but_not_nominal_width : gapOnlyBeam.PhysicalAt ∧ gapOnlyBeam.beta = beam.beta ∧ gapOnlyBeam.nominalWidth ≠ beam.nominalWidth := by refine ⟨by norm_num [gapOnlyBeam, ConventionalBeamParameters.PhysicalAt], ?_, ?_⟩ · dsimp [gapOnlyBeam, beam, ConventionalBeamParameters.beta] ring · norm_num [gapOnlyBeam, beam, ConventionalBeamParameters.nominalWidth] end end RamseyNominalWidthClient run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.RamseyNominalWidthClient let mut traversal : Lean.CollectAxioms.State := {} let mut theorems := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theorems := theorems + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theorems >= 6 do throwError "Missing independent Ramsey nominal-width client theorems" for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational Ramsey nominal-width client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Ramsey nominal-width client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``RamseyNominalWidthClient.the_library_computes_the_length_dependent_nominal_ratio, ``PhysicsLogic.Papers.Ramsey1950.ConventionalBeamParameters.nominal_width_ratio_retains_the_two_distinct_lengths), (``RamseyNominalWidthClient.the_library_scales_an_actual_conventional_fwhm_conditionally, ``PhysicsLogic.Papers.Ramsey1950.ConventionalBeamParameters.an_actual_conventional_full_width_has_its_cyclic_frequency_scale)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Ramsey nominal-width client audit passed: {theorems} elaborated theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, independent positive geometry and retuned beam, no production diagnostic dependencies." namespace RamseyWidthValidationClient open PhysicsLogic.Papers.Ramsey1950 set_option backward.isDefEq.respectTransparency false noncomputable section def profile (width frequency : Real) : Real := max 0 (width - |frequency|) / (2 * width) def comparison : WidthComparisonReadout := ⟨3 / 5, 1, 61 / 100, 103 / 100, 1 / 50, 1 / 25⟩ theorem profile_is_a_continuous_probability (width : Real) (positive : 0 < width) : Continuous (profile width) ∧ ∀ frequency, 0 ≤ profile width frequency ∧ profile width frequency ≤ 1 / 2 := by refine ⟨by unfold profile; fun_prop, ?_⟩ intro frequency have denominator : 0 < 2 * width := by positivity refine ⟨div_nonneg (le_max_left _ _) (le_of_lt denominator), ?_⟩ apply (div_le_iff₀ denominator).2 have upper : max 0 (width - |frequency|) ≤ width := max_le (le_of_lt positive) (sub_le_self _ (abs_nonneg _)) nlinarith theorem profile_has_its_actual_full_width (width : Real) (positive : 0 < width) : ResonanceLineShape.CentralFWHMAt (profile width) width := by have nonzero := ne_of_gt positive have center : profile width 0 = 1 / 2 := by simp only [profile, abs_zero, sub_zero, max_eq_right (le_of_lt positive)] field_simp refine ⟨positive, by rw [center]; norm_num, ?_, ?_, ?_, ?_⟩ · intro frequency rw [center] exact (profile_is_a_continuous_probability width positive).2 frequency |>.2 · rw [center] have half : 0 < width / 2 := by positivity have value : width - |width / 2| = width / 2 := by rw [abs_of_pos half]; ring rw [profile, value, max_eq_right (le_of_lt half)] field_simp · intro frequency inside rw [center] apply (lt_div_iff₀ (by positivity : 0 < 2 * width)).2 have comparison := le_max_right (0 : Real) (width - |frequency|) nlinarith · intro frequency simp [profile] theorem the_comparison_keeps_two_actual_profile_widths : ResonanceLineShape.CentralFWHMAt (profile comparison.actualNarrow) comparison.actualNarrow ∧ ResonanceLineShape.CentralFWHMAt (profile comparison.actualReference) comparison.actualReference := ⟨profile_has_its_actual_full_width _ (by norm_num [comparison]), profile_has_its_actual_full_width _ (by norm_num [comparison])⟩ theorem both_numerical_errors_are_controlled : comparison.ErrorsControlledAt := by norm_num [comparison, WidthComparisonReadout.ErrorsControlledAt] theorem the_nominal_gap_and_reference_lower_bound_are_resolved : comparison.NominalGapResolvedAt ∧ comparison.ReferenceResolvedAt := by norm_num [comparison, WidthComparisonReadout.NominalGapResolvedAt, WidthComparisonReadout.ReferenceResolvedAt] theorem the_library_derives_actual_profile_narrowing : ResonanceLineShape.CentralFWHMAt (profile comparison.actualNarrow) comparison.actualNarrow ∧ ResonanceLineShape.CentralFWHMAt (profile comparison.actualReference) comparison.actualReference ∧ comparison.actualNarrow < comparison.actualReference := ⟨the_comparison_keeps_two_actual_profile_widths.1, the_comparison_keeps_two_actual_profile_widths.2, comparison.errors_and_resolved_gap_derive_actual_narrowing both_numerical_errors_are_controlled the_nominal_gap_and_reference_lower_bound_are_resolved.1⟩ theorem the_library_bounds_the_calculated_ratio_with_both_errors : |comparison.actualNarrow / comparison.actualReference - comparison.nominalNarrow / comparison.nominalReference| ≤ 11 / 240 := by have bound := (comparison.errors_and_resolved_reference_control_the_ratio both_numerical_errors_are_controlled the_nominal_gap_and_reference_lower_bound_are_resolved.2).2 have allowance : comparison.ratioErrorAllowance = 11 / 240 := by norm_num [comparison, WidthComparisonReadout.ratioErrorAllowance] rw [allowance] at bound exact bound theorem the_library_transfers_the_nominal_point_six_comparison : |comparison.actualNarrow / comparison.actualReference - 3 / 5| ≤ 11 / 240 ∧ |comparison.actualNarrow / comparison.actualReference - 3 / 5| = 4 / 515 := by have bound := comparison.a_nominal_ratio_bound_transfers_with_both_width_errors both_numerical_errors_are_controlled the_nominal_gap_and_reference_lower_bound_are_resolved.2 (3 / 5) 0 (by norm_num [comparison]) have allowance : comparison.ratioErrorAllowance = 11 / 240 := by norm_num [comparison, WidthComparisonReadout.ratioErrorAllowance] rw [zero_add, allowance] at bound exact ⟨bound, by norm_num [comparison]⟩ end end RamseyWidthValidationClient run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.RamseyWidthValidationClient let expectedSourceTheorems : Array Lean.Name := #[ ``RamseyWidthValidationClient.profile_is_a_continuous_probability, ``RamseyWidthValidationClient.profile_has_its_actual_full_width, ``RamseyWidthValidationClient.the_comparison_keeps_two_actual_profile_widths, ``RamseyWidthValidationClient.both_numerical_errors_are_controlled, ``RamseyWidthValidationClient.the_nominal_gap_and_reference_lower_bound_are_resolved, ``RamseyWidthValidationClient.the_library_derives_actual_profile_narrowing, ``RamseyWidthValidationClient.the_library_bounds_the_calculated_ratio_with_both_errors, ``RamseyWidthValidationClient.the_library_transfers_the_nominal_point_six_comparison] for declaration in expectedSourceTheorems do match environment.find? declaration with | some (.thmInfo _) => pure () | _ => throwError "Missing width-validation source theorem {declaration}" let mut traversal : Lean.CollectAxioms.State := {} for (name, _) in environment.constants do unless clientNamespace.isPrefixOf name do continue let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational width-validation client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Width-validation client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``RamseyWidthValidationClient.the_library_derives_actual_profile_narrowing, ``PhysicsLogic.Papers.Ramsey1950.WidthComparisonReadout.errors_and_resolved_gap_derive_actual_narrowing), (``RamseyWidthValidationClient.the_library_bounds_the_calculated_ratio_with_both_errors, ``PhysicsLogic.Papers.Ramsey1950.WidthComparisonReadout.errors_and_resolved_reference_control_the_ratio), (``RamseyWidthValidationClient.the_library_transfers_the_nominal_point_six_comparison, ``PhysicsLogic.Papers.Ramsey1950.WidthComparisonReadout.a_nominal_ratio_bound_transfers_with_both_width_errors)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Ramsey width-validation client audit passed: {expectedSourceTheorems.size} named source theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, independent bounded profiles with nonzero width errors, no production diagnostic dependencies." namespace RamseyVelocitySelectionClient open PhysicsLogic.Papers.Ramsey1950 open WeakPerturbationSelection MolecularBeamVelocityAverageReadout Filter open scoped Topology set_option backward.isDefEq.respectTransparency false noncomputable section def beam : SourceBeamParameters := ⟨Real.pi / 12, 0, 0, 2, 7, 1⟩ theorem independently_selected_beam_is_physical : beam.PhysicalParametersAt ∧ beam.pulseArea = Real.pi / 3 := by constructor · norm_num [beam, SourceBeamParameters.PhysicalParametersAt, Real.pi_ne_zero] · dsimp [beam, SourceBeamParameters.pulseArea]; ring theorem area_retuning_really_halves_the_coupling : (beam.withPulseArea (Real.pi / 6)).PhysicalParametersAt ∧ (beam.withPulseArea (Real.pi / 6)).pulseArea = Real.pi / 6 ∧ (beam.withPulseArea (Real.pi / 6)).oscillatingCoupling = beam.oscillatingCoupling / 2 := by refine ⟨beam.a_positive_area_replacement_has_physical_parameters independently_selected_beam_is_physical.1 _ (by positivity), beam.replacing_the_pulse_area_retains_its_physical_definition independently_selected_beam_is_physical.1 _, ?_⟩ dsimp [beam, SourceBeamParameters.withPulseArea] ring theorem the_conditioned_density_uses_actual_coherent_transitions : molecularBeamWeightAt 2 * beam.exactProbabilityAt 2 / beam.exactAverage = selectedSpeedDensity (Real.pi / 3) 2 := by have identification := physical_double_resonance_realizes_selected_speed_density beam independently_selected_beam_is_physical.1 rfl rfl 2 (by norm_num) simpa [independently_selected_beam_is_physical.2] using identification theorem a_half_field_retains_more_of_the_slow_population : transitionProbability ((Real.pi / 3) / 2) 5 / transitionProbability (Real.pi / 3) 5 < transitionProbability ((Real.pi / 3) / 2) 2 / transitionProbability (Real.pi / 3) 2 := by apply halving_the_field_retains_a_larger_fraction_of_slow_transitions (Real.pi / 3) 2 5 (by positivity) (by norm_num) (by norm_num) linarith [Real.pi_pos] theorem weak_limit_has_quadratic_signal_and_reweighted_density : Tendsto (fun area => SourceBeamParameters.equationNineteenAverage area / area ^ 2) (𝓝[>] (0 : Real)) (𝓝 1) ∧ Tendsto (fun area => selectedSpeedDensity area 2) (𝓝[>] (0 : Real)) (𝓝 (4 * Real.exp (-4))) := by have result := weak_field_limit_retains_quadratic_signal_and_selected_density 2 (by norm_num) norm_num [weakSelectedWeight] at result exact result theorem speed_selection_is_relative_to_the_incident_density : weakSelectedWeight 2 / molecularBeamWeightAt 2 = 1 / 4 ∧ weakSelectedWeight 5 / molecularBeamWeightAt 5 = 1 / 25 ∧ weakSelectedWeight 5 / molecularBeamWeightAt 5 < weakSelectedWeight 2 / molecularBeamWeightAt 2 := by have result := weak_selection_enhances_slow_speeds_relative_to_incident_beam 2 5 (by norm_num) (by norm_num) norm_num at result exact result end end RamseyVelocitySelectionClient run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.RamseyVelocitySelectionClient let expectedSourceTheorems : Array Lean.Name := #[ ``RamseyVelocitySelectionClient.independently_selected_beam_is_physical, ``RamseyVelocitySelectionClient.area_retuning_really_halves_the_coupling, ``RamseyVelocitySelectionClient.the_conditioned_density_uses_actual_coherent_transitions, ``RamseyVelocitySelectionClient.a_half_field_retains_more_of_the_slow_population, ``RamseyVelocitySelectionClient.weak_limit_has_quadratic_signal_and_reweighted_density, ``RamseyVelocitySelectionClient.speed_selection_is_relative_to_the_incident_density] for declaration in expectedSourceTheorems do match environment.find? declaration with | some (.thmInfo _) => pure () | _ => throwError "Missing velocity-selection source theorem {declaration}" let mut traversal : Lean.CollectAxioms.State := {} for (name, _) in environment.constants do unless clientNamespace.isPrefixOf name do continue let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational velocity-selection client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Velocity-selection client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``RamseyVelocitySelectionClient.the_conditioned_density_uses_actual_coherent_transitions, ``PhysicsLogic.Papers.Ramsey1950.WeakPerturbationSelection.physical_double_resonance_realizes_selected_speed_density), (``RamseyVelocitySelectionClient.a_half_field_retains_more_of_the_slow_population, ``PhysicsLogic.Papers.Ramsey1950.WeakPerturbationSelection.halving_the_field_retains_a_larger_fraction_of_slow_transitions), (``RamseyVelocitySelectionClient.weak_limit_has_quadratic_signal_and_reweighted_density, ``PhysicsLogic.Papers.Ramsey1950.WeakPerturbationSelection.weak_field_limit_retains_quadratic_signal_and_selected_density), (``RamseyVelocitySelectionClient.speed_selection_is_relative_to_the_incident_density, ``PhysicsLogic.Papers.Ramsey1950.WeakPerturbationSelection.weak_selection_enhances_slow_speeds_relative_to_incident_beam)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Ramsey velocity-selection client audit passed: {expectedSourceTheorems.size} named source theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, independently calibrated beam and velocities, no production diagnostic dependencies." namespace RamseyNormalizedFringeClient open PhysicsLogic.Papers.Ramsey1950 ResonanceLineShape MolecularBeamVelocityAverageReadout open Filter open scoped Topology set_option backward.isDefEq.respectTransparency false noncomputable section def attenuated (coordinate : Real) : Real := (1 / 50) * RamseyWidthValidationClient.profile (3 / 7) coordinate theorem independent_attenuated_profile_has_its_actual_width : CentralFWHMAt attenuated (3 / 7) := positive_amplitude_scaling_preserves_the_full_width _ _ (1 / 50) (RamseyWidthValidationClient.profile_has_its_actual_full_width (3 / 7) (by norm_num)) (by norm_num) theorem peak_normalization_keeps_the_actual_width : attenuated 0 = 1 / 100 ∧ CentralFWHMAt (fun coordinate => attenuated coordinate / attenuated 0) (3 / 7) := by have peak : attenuated 0 = 1 / 100 := by norm_num [attenuated, RamseyWidthValidationClient.profile] exact ⟨peak, (a_positive_peak_normalization_preserves_and_reflects_the_full_width _ _ (by rw [peak]; norm_num)).2 independent_attenuated_profile_has_its_actual_width⟩ theorem the_profile_is_a_selected_velocity_interference_average : normalizedNearResponse (Real.pi / 7) (2 / 5) = sourceIntegral (fun speed => WeakPerturbationSelection.selectedSpeedDensity (Real.pi / 7) speed * fringeKernel (2 / 5) speed) := normalized_fringe_is_the_transition_selected_interference_average _ _ theorem sufficiently_weak_fields_have_actual_unit_peak_profiles : ∀ᶠ area in 𝓝[>] (0 : Real), normalizedNearResponse area 0 = 1 ∧ ∀ coordinate : Real, 0 ≤ normalizedNearResponse area coordinate ∧ normalizedNearResponse area coordinate ≤ 1 := WeakPerturbationSelection.sufficiently_weak_positive_fields_admit_conditioning.mono (fun area admissible => positive_transition_mass_makes_a_unit_peak_probability_profile area admissible) theorem the_weak_normalized_profile_uses_the_reweighted_beam : Tendsto (fun area => normalizedNearResponse area (2 / 5)) (𝓝[>] (0 : Real)) (𝓝 (sourceIntegral (fun speed => WeakPerturbationSelection.weakSelectedWeight speed * fringeKernel (2 / 5) speed))) := the_positive_field_normalized_fringe_converges_to_the_selected_weak_profile (2 / 5) theorem a_small_absolute_error_is_amplified_by_a_weak_reference_peak : |(1 / 250 : Real) / (1 / 100) - (1 / 200) / (1 / 100)| ≤ 1 / 10 := by have transferred := common_reference_peak_normalization_transfers_error (1 / 250) (1 / 200) (1 / 100) (1 / 1000) (by norm_num) (by norm_num) convert transferred using 1 norm_num theorem finite_detuning_cannot_be_kept_in_the_near_regime_by_weakening_the_field : ∀ tolerance : Real, ¬(SourceBeamParameters.mk (1 / 2) (3 / 2) 1 2 3 1).NearPulseRegimeAt tolerance := by intro tolerance near have bound := SourceBeamParameters.near_regime_at_nonzero_detuning_requires_larger_coupling (SourceBeamParameters.mk (1 / 2) (3 / 2) 1 2 3 1) tolerance near (by norm_num) norm_num at bound end end RamseyNormalizedFringeClient run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.RamseyNormalizedFringeClient let expectedSourceTheorems : Array Lean.Name := #[ ``RamseyNormalizedFringeClient.independent_attenuated_profile_has_its_actual_width, ``RamseyNormalizedFringeClient.peak_normalization_keeps_the_actual_width, ``RamseyNormalizedFringeClient.the_profile_is_a_selected_velocity_interference_average, ``RamseyNormalizedFringeClient.sufficiently_weak_fields_have_actual_unit_peak_profiles, ``RamseyNormalizedFringeClient.the_weak_normalized_profile_uses_the_reweighted_beam, ``RamseyNormalizedFringeClient.a_small_absolute_error_is_amplified_by_a_weak_reference_peak, ``RamseyNormalizedFringeClient.finite_detuning_cannot_be_kept_in_the_near_regime_by_weakening_the_field] for declaration in expectedSourceTheorems do match environment.find? declaration with | some (.thmInfo _) => pure () | _ => throwError "Missing normalized-fringe source theorem {declaration}" let mut traversal : Lean.CollectAxioms.State := {} for (name, _) in environment.constants do unless clientNamespace.isPrefixOf name do continue let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational normalized-fringe client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Normalized-fringe client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``RamseyNormalizedFringeClient.independent_attenuated_profile_has_its_actual_width, ``PhysicsLogic.Papers.Ramsey1950.ResonanceLineShape.positive_amplitude_scaling_preserves_the_full_width), (``RamseyNormalizedFringeClient.peak_normalization_keeps_the_actual_width, ``PhysicsLogic.Papers.Ramsey1950.ResonanceLineShape.a_positive_peak_normalization_preserves_and_reflects_the_full_width), (``RamseyNormalizedFringeClient.the_profile_is_a_selected_velocity_interference_average, ``PhysicsLogic.Papers.Ramsey1950.ResonanceLineShape.normalized_fringe_is_the_transition_selected_interference_average), (``RamseyNormalizedFringeClient.sufficiently_weak_fields_have_actual_unit_peak_profiles, ``PhysicsLogic.Papers.Ramsey1950.ResonanceLineShape.positive_transition_mass_makes_a_unit_peak_probability_profile), (``RamseyNormalizedFringeClient.the_weak_normalized_profile_uses_the_reweighted_beam, ``PhysicsLogic.Papers.Ramsey1950.ResonanceLineShape.the_positive_field_normalized_fringe_converges_to_the_selected_weak_profile), (``RamseyNormalizedFringeClient.a_small_absolute_error_is_amplified_by_a_weak_reference_peak, ``PhysicsLogic.Papers.Ramsey1950.ResonanceLineShape.common_reference_peak_normalization_transfers_error), (``RamseyNormalizedFringeClient.finite_detuning_cannot_be_kept_in_the_near_regime_by_weakening_the_field, ``PhysicsLogic.Papers.Ramsey1950.SourceBeamParameters.near_regime_at_nonzero_detuning_requires_larger_coupling)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Ramsey normalized-fringe client audit passed: {expectedSourceTheorems.size} named source theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, independent attenuated profile and peak calibration, no production diagnostic dependencies." namespace RamseyPrintedWidthClient open PhysicsLogic.Papers.Ramsey1950 SourceWidthCoefficients set_option backward.isDefEq.respectTransparency false noncomputable section def beam : SourceBeamParameters := ⟨(21 / 20) * Real.pi, 0, 0, 2, 3, 7⟩ theorem independent_beam_has_physical_geometry_and_selected_area : beam.PhysicalParametersAt ∧ beam.pulseArea = (3 / 5) * Real.pi := by norm_num [beam, SourceBeamParameters.PhysicalParametersAt, SourceBeamParameters.pulseArea, Real.pi_ne_zero] ring theorem the_printed_near_gaps_have_the_client_frequency_scale : figureTwo.centralWidth beam - equationTwentyTwo.centralWidth beam = 7 / 300 ∧ equationTwentyTwo.subsidiarySpacing beam - figureTwo.subsidiarySpacing beam = 7 / 300 ∧ figureTwo.broadWidth beam = equationTwentyTwo.broadWidth beam := by obtain ⟨central, spacing, broad, _positive⟩ := printed_near_labels_differ_with_the_same_positive_frequency_scale beam independent_beam_has_physical_geometry_and_selected_area.1 norm_num [beam] at central spacing exact ⟨central, spacing, broad⟩ theorem an_independent_actual_profile_fits_both_displays : ResonanceLineShape.CentralFWHMAt (RamseyWidthValidationClient.profile (301 / 200)) (301 / 200) ∧ (centralComparison beam (301 / 200) (7 / 600) (7 / 600)).ErrorsControlledAt := by refine ⟨RamseyWidthValidationClient.profile_has_its_actual_full_width _ (by norm_num), ?_⟩ norm_num [centralComparison, beam, figureTwo, equationTwentyTwo, centralWidth, WidthComparisonReadout.ErrorsControlledAt] theorem the_library_enforces_the_dimensional_gap_budget (actualWidth equationError figureError : Real) (errors : (centralComparison beam actualWidth equationError figureError).ErrorsControlledAt) : (7 : Real) / 300 ≤ equationError + figureError := by have bound := fitting_one_width_to_both_displays_requires_a_sufficient_error_budget beam independent_beam_has_physical_geometry_and_selected_area.1 actualWidth equationError figureError errors convert bound using 1 norm_num [beam] theorem no_actual_width_fits_both_prints_with_zero_error : ∀ actualWidth : Real, ¬(centralComparison beam actualWidth 0 0).ErrorsControlledAt := by intro actualWidth errors have impossible := the_library_enforces_the_dimensional_gap_budget actualWidth 0 0 errors norm_num at impossible end end RamseyPrintedWidthClient run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.RamseyPrintedWidthClient let expectedSourceTheorems : Array Lean.Name := #[ ``RamseyPrintedWidthClient.independent_beam_has_physical_geometry_and_selected_area, ``RamseyPrintedWidthClient.the_printed_near_gaps_have_the_client_frequency_scale, ``RamseyPrintedWidthClient.an_independent_actual_profile_fits_both_displays, ``RamseyPrintedWidthClient.the_library_enforces_the_dimensional_gap_budget, ``RamseyPrintedWidthClient.no_actual_width_fits_both_prints_with_zero_error] for declaration in expectedSourceTheorems do match environment.find? declaration with | some (.thmInfo _) => pure () | _ => throwError "Missing printed-width source theorem {declaration}" let mut traversal : Lean.CollectAxioms.State := {} for (name, _) in environment.constants do unless clientNamespace.isPrefixOf name do continue let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational printed-width client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Printed-width client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``RamseyPrintedWidthClient.the_printed_near_gaps_have_the_client_frequency_scale, ``PhysicsLogic.Papers.Ramsey1950.SourceWidthCoefficients.printed_near_labels_differ_with_the_same_positive_frequency_scale), (``RamseyPrintedWidthClient.the_library_enforces_the_dimensional_gap_budget, ``PhysicsLogic.Papers.Ramsey1950.SourceWidthCoefficients.fitting_one_width_to_both_displays_requires_a_sufficient_error_budget), (``RamseyPrintedWidthClient.no_actual_width_fits_both_prints_with_zero_error, ``PhysicsLogic.Papers.Ramsey1950.SourceWidthCoefficients.fitting_one_width_to_both_displays_requires_a_sufficient_error_budget)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" let isolatedApis : Array Lean.Name := #[ ``PhysicsLogic.Papers.Ramsey1950.SourceWidthCoefficients.printed_near_labels_differ_with_the_same_positive_frequency_scale, ``PhysicsLogic.Papers.Ramsey1950.SourceWidthCoefficients.fitting_one_width_to_both_displays_requires_a_sufficient_error_budget, ``PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.equation_twenty_two_and_apparatus_derive_exact_table_residuals, ``PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.figure_two_and_apparatus_have_distinct_table_residuals, ``PhysicsLogic.Papers.Ramsey1950.SeparatedFieldFramework.established_empirical_comparison_retains_the_nonrounding_residuals] for declaration in isolatedApis do let (_, dependencies) := ((Lean.CollectAxioms.collect declaration).run environment).run {} for dependency in dependencies.visited.toList do unless (dependency.toString.splitOn ".").contains "PhysicsLogic" do continue let mut ancestor := dependency while !ancestor.isAnonymous do let part := match ancestor with | .str _ value => value | _ => "" if part == "deprecated" || part.endsWith "Countermodels" || (part.endsWith "Controls" && !Lean.isStructure environment ancestor && !environment.isProjectionFn ancestor) then throwError "Public printed-width API {declaration} depends on a diagnostic {dependency}" ancestor := ancestor.getPrefix Lean.logInfo m!"Ramsey printed-width client audit passed: {expectedSourceTheorems.size} named source theorems, {uses.size} public API uses, {isolatedApis.size} public APIs independently isolated, {traversal.axioms.size} foundational axioms, independent geometry and actual profile, no production diagnostic dependencies." namespace KSWPionlessFlowClient open PhysicsLogic.Papers.KaplanSavageWise1998 noncomputable section set_option backward.isDefEq.respectTransparency false def flow : PionlessRGFlow := ⟨8 * Real.pi, -1 / 2, 3, fun mu => (1 / 2) * (-mu - 1 / 2)⁻¹, fun mu => (3 / 4) * (-mu - 1 / 2)⁻¹ ^ 2⟩ theorem the_independent_flow_realizes_the_full_nonpole_solution : flow.EquationElevenAt := by refine ⟨by dsimp [PionlessRGFlow.PhysicalAt, flow]; positivity, ?_⟩ intro mu nonpole constructor <;> dsimp [flow, PionlessRGFlow.c0Solution, PionlessRGFlow.c2Solution] <;> norm_num <;> field_simp <;> ring theorem the_library_derives_both_actual_contact_derivatives : flow.ExactBetaDerivativesAt 1 := flow.equation_eleven_derives_actual_beta_derivatives the_independent_flow_realizes_the_full_nonpole_solution 1 (by norm_num [PionlessRGFlow.NonpoleAt, flow]) theorem the_actual_flow_has_computed_nonzero_beta_values : deriv flow.c0 1 = 2 / 9 ∧ deriv flow.c2 1 = -(4 / 9) := by have derivatives := the_library_derives_both_actual_contact_derivatives rw [derivatives.1.deriv, derivatives.2.deriv] norm_num [flow, Real.pi_ne_zero] constructor <;> field_simp <;> ring theorem the_library_retains_the_printed_scale_weight : 1 * deriv flow.c0 1 = flow.beta0 1 ∧ 1 * deriv flow.c2 1 = flow.beta2 1 := flow.actual_beta_derivatives_retain_the_printed_scale_weight 1 the_library_derives_both_actual_contact_derivatives theorem the_library_recovers_the_matched_zero_scale_contacts : flow.c0 0 = -1 ∧ flow.c2 0 = 3 := by have boundary := flow.equation_eleven_retains_zero_scale_boundary_data the_independent_flow_realizes_the_full_nonpole_solution (by norm_num [flow]) convert boundary using 1 <;> norm_num [flow, Real.pi_ne_zero] <;> field_simp <;> ring theorem the_independent_couplings_really_run : flow.c0 0 ≠ flow.c0 1 ∧ flow.c2 0 ≠ flow.c2 1 := by norm_num [flow] end end KSWPionlessFlowClient run_cmd do let environment <- Lean.getEnv let expected : Array Lean.Name := #[ ``KSWPionlessFlowClient.the_independent_flow_realizes_the_full_nonpole_solution, ``KSWPionlessFlowClient.the_library_derives_both_actual_contact_derivatives, ``KSWPionlessFlowClient.the_actual_flow_has_computed_nonzero_beta_values, ``KSWPionlessFlowClient.the_library_retains_the_printed_scale_weight, ``KSWPionlessFlowClient.the_library_recovers_the_matched_zero_scale_contacts, ``KSWPionlessFlowClient.the_independent_couplings_really_run] let mut traversal : Lean.CollectAxioms.State := {} for declaration in expected do match environment.find? declaration with | some (.thmInfo _) => pure () | _ => throwError "Missing independent KSW flow theorem {declaration}" let (_, updated) := ((Lean.CollectAxioms.collect declaration).run environment).run traversal traversal := updated for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational KSW flow client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "KSW flow client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``KSWPionlessFlowClient.the_library_derives_both_actual_contact_derivatives, ``PhysicsLogic.Papers.KaplanSavageWise1998.PionlessRGFlow.equation_eleven_derives_actual_beta_derivatives), (``KSWPionlessFlowClient.the_library_retains_the_printed_scale_weight, ``PhysicsLogic.Papers.KaplanSavageWise1998.PionlessRGFlow.actual_beta_derivatives_retain_the_printed_scale_weight), (``KSWPionlessFlowClient.the_library_recovers_the_matched_zero_scale_contacts, ``PhysicsLogic.Papers.KaplanSavageWise1998.PionlessRGFlow.equation_eleven_retains_zero_scale_boundary_data)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"KSW pionless-flow client audit passed: {expected.size} named theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, independent mass and effective-range data, actual nonzero beta values, no production diagnostic dependencies." namespace KSWPionInclusiveClient open PhysicsLogic.Papers.KaplanSavageWise1998 noncomputable section set_option backward.isDefEq.respectTransparency false def flow : PionInclusiveRGFlow := ⟨8 * Real.pi, 2, 2, fun mu => 1 / 2 + (3 / 2) * (mu - 2), fun mu => 3 + 6 * (mu - 2), fun mu => (mu - 2) / 4⟩ theorem the_independent_jet_realizes_the_retained_beta_equations : flow.PhysicalAt ∧ flow.RetainedBetaEquationsAt 2 := by refine ⟨⟨by dsimp [flow]; positivity, by norm_num [flow]⟩, by norm_num, ?_⟩ intro kind have base := (hasDerivAt_id (2 : Real)).sub_const 2 cases kind <;> dsimp [PionInclusiveRGFlow.coupling, PionInclusiveRGFlow.beta, flow] · convert (base.const_mul (3 / 2)).const_add (1 / 2) using 1 norm_num field_simp ring · convert (base.const_mul 6).const_add 3 using 1 norm_num field_simp ring · convert base.div_const 4 using 1 norm_num field_simp ring theorem the_library_recovers_each_actual_weighted_equation : ∀ kind, 2 * deriv (flow.coupling kind) 2 = flow.beta kind 2 := flow.retained_derivatives_give_the_printed_weighted_equations 2 the_independent_jet_realizes_the_retained_beta_equations.2 theorem the_library_detects_generation_from_zero_d2 : flow.d2 2 = 0 ∧ 0 < flow.beta .d2 2 := by refine ⟨by norm_num [flow], ?_⟩ exact flow.a_vanishing_d2_is_generated_by_the_inhomogeneous_term 2 the_independent_jet_realizes_the_retained_beta_equations.1 (by norm_num [flow]) (by norm_num [flow]) (by norm_num [flow]) def expansion : PionAmplitudeScaleExpansion Real where retained := fun scale n => if n = 2 then 7 * (scale : Complex) else 3 omitted := fun scale n => if n = 2 then -(7 * (scale : Complex)) else 0 full := fun _ n => if n = 2 then 0 else 3 theorem the_independent_completion_has_only_higher_order_omissions : expansion.CompletesAt ∧ expansion.OmittedBeyondSubleadingAt ∧ expansion.FullScaleIndependentAt := by refine ⟨?_, ?_, ?_⟩ · intro scale n by_cases h : n = 2 <;> simp [expansion, h] · intro scale n low have h : n ≠ 2 := by omega simp [expansion, h] · intro left right n rfl theorem the_library_derives_the_retained_coefficient_agreement : expansion.ThroughSubleadingAt := expansion.completion_and_full_invariance_derive_computed_order_invariance the_independent_completion_has_only_higher_order_omissions.1 the_independent_completion_has_only_higher_order_omissions.2.1 the_independent_completion_has_only_higher_order_omissions.2.2 theorem the_library_identifies_the_nonzero_residual_compensation : expansion.retained 0 2 - expansion.retained 1 2 = -(expansion.omitted 0 2 - expansion.omitted 1 2) ∧ expansion.retained 0 2 - expansion.retained 1 2 = -7 := by refine ⟨expansion.omitted_terms_cancel_the_actual_retained_residual the_independent_completion_has_only_higher_order_omissions.1 the_independent_completion_has_only_higher_order_omissions.2.2 0 1 2, ?_⟩ norm_num [expansion] theorem the_verified_retained_orders_do_not_give_all_order_invariance : expansion.ThroughSubleadingAt ∧ ¬expansion.ExactRetainedScaleIndependentAt := by refine ⟨the_library_derives_the_retained_coefficient_agreement, ?_⟩ intro invariant have impossible := invariant 0 1 2 norm_num [expansion] at impossible end end KSWPionInclusiveClient run_cmd do let environment <- Lean.getEnv let expected : Array Lean.Name := #[ ``KSWPionInclusiveClient.the_independent_jet_realizes_the_retained_beta_equations, ``KSWPionInclusiveClient.the_library_recovers_each_actual_weighted_equation, ``KSWPionInclusiveClient.the_library_detects_generation_from_zero_d2, ``KSWPionInclusiveClient.the_independent_completion_has_only_higher_order_omissions, ``KSWPionInclusiveClient.the_library_derives_the_retained_coefficient_agreement, ``KSWPionInclusiveClient.the_library_identifies_the_nonzero_residual_compensation, ``KSWPionInclusiveClient.the_verified_retained_orders_do_not_give_all_order_invariance] let mut traversal : Lean.CollectAxioms.State := {} for declaration in expected do match environment.find? declaration with | some (.thmInfo _) => pure () | _ => throwError "Missing independent pion-inclusive source theorem {declaration}" let (_, updated) := ((Lean.CollectAxioms.collect declaration).run environment).run traversal traversal := updated for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational pion-inclusive client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Pion-inclusive client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``KSWPionInclusiveClient.the_library_recovers_each_actual_weighted_equation, ``PhysicsLogic.Papers.KaplanSavageWise1998.PionInclusiveRGFlow.retained_derivatives_give_the_printed_weighted_equations), (``KSWPionInclusiveClient.the_library_detects_generation_from_zero_d2, ``PhysicsLogic.Papers.KaplanSavageWise1998.PionInclusiveRGFlow.a_vanishing_d2_is_generated_by_the_inhomogeneous_term), (``KSWPionInclusiveClient.the_library_derives_the_retained_coefficient_agreement, ``PhysicsLogic.Papers.KaplanSavageWise1998.PionAmplitudeScaleExpansion.completion_and_full_invariance_derive_computed_order_invariance), (``KSWPionInclusiveClient.the_library_identifies_the_nonzero_residual_compensation, ``PhysicsLogic.Papers.KaplanSavageWise1998.PionAmplitudeScaleExpansion.omitted_terms_cancel_the_actual_retained_residual)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"KSW pion-inclusive client audit passed: {expected.size} named theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, independent positive-scale jet and nonzero higher-order compensation, no production diagnostic dependencies." namespace HohenbergKohnResponseClient open PhysicsLogic.Papers.HohenbergKohn1964.ResponseKernel open Filter open scoped Topology noncomputable section set_option backward.isDefEq.respectTransparency false def evenKernel (momentum : Real) : Real := 2 - 3 * momentum ^ 2 + momentum ^ 6 theorem the_independent_gradient_kernel_has_its_actual_polynomial : FiniteIsotropicGradientKernelAt evenKernel := by refine ⟨Polynomial.C 2 - Polynomial.C 3 * Polynomial.X + Polynomial.X ^ 3, ?_⟩ intro momentum simp [evenKernel] ring theorem the_library_derives_analyticity_at_the_selected_fermi_scale : AnalyticAt Real evenKernel 3 := a_finite_isotropic_gradient_kernel_is_analytic evenKernel the_independent_gradient_kernel_has_its_actual_polynomial 3 theorem the_library_excludes_the_source_derivative_divergence_for_this_polynomial : ¬Tendsto (deriv evenKernel) (𝓝[≠] (3 : Real)) atTop := by intro singular exact positive_derivative_divergence_excludes_analyticity evenKernel 3 singular the_library_derives_analyticity_at_the_selected_fermi_scale theorem the_independent_polynomial_cannot_agree_near_a_source_singularity (sourceKernel : Real -> Real) (singular : Tendsto (deriv sourceKernel) (𝓝[≠] (3 : Real)) atTop) : ¬evenKernel =ᶠ[𝓝 (3 : Real)] sourceKernel := a_finite_gradient_kernel_cannot_agree_near_a_nonanalytic_kernel evenKernel sourceKernel 3 the_independent_gradient_kernel_has_its_actual_polynomial (positive_derivative_divergence_excludes_analyticity sourceKernel 3 singular) def correctedKernel (radius : Real) : Real := friedelProfile 3 2 (Real.pi / 3) radius + 7 / radius ^ 4 theorem the_library_retains_the_independent_nonexact_friedel_tail : FriedelTailAt correctedKernel 3 2 (Real.pi / 3) := by have small : Tendsto (fun radius : Real => radius ^ 3 * (7 / radius ^ 4)) atTop (𝓝 0) := by have inverse : Tendsto (fun radius : Real => 7 * radius⁻¹) atTop (𝓝 0) := by simpa using (tendsto_inv_atTop_zero : Tendsto (fun radius : Real => radius⁻¹) atTop (𝓝 0)).const_mul 7 apply inverse.congr' filter_upwards [eventually_gt_atTop (0 : Real)] with radius positive field_simp exact inverse_cube_small_remainders_preserve_the_friedel_tail (friedelProfile 3 2 (Real.pi / 3)) (fun radius => 7 / radius ^ 4) 3 2 (Real.pi / 3) (the_leading_profile_has_its_own_friedel_tail 3 2 (Real.pi / 3) (by norm_num) (by norm_num)) small theorem the_library_recovers_the_actual_scaled_envelope : Tendsto (fun radius => radius ^ 3 * correctedKernel radius - 2 * Real.cos (3 * radius + Real.pi / 3)) atTop (𝓝 0) := a_friedel_tail_is_an_actual_scaled_envelope_limit correctedKernel 3 2 (Real.pi / 3) the_library_retains_the_independent_nonexact_friedel_tail theorem the_independent_asymptotic_profile_is_not_an_exact_identity : ¬correctedKernel = friedelProfile 3 2 (Real.pi / 3) := by intro same have impossible := congrFun same 1 norm_num [correctedKernel] at impossible theorem the_corrected_kernel_is_nonzero_at_a_leading_profile_node : friedelProfile 3 2 (Real.pi / 3) (Real.pi / 18) = 0 ∧ 0 < correctedKernel (Real.pi / 18) := by have angle : (3 : Real) * (Real.pi / 18) + Real.pi / 3 = Real.pi / 2 := by ring have node : friedelProfile 3 2 (Real.pi / 3) (Real.pi / 18) = 0 := by simp [friedelProfile, angle, Real.cos_pi_div_two] refine ⟨node, ?_⟩ dsimp [correctedKernel] rw [node] positivity end end HohenbergKohnResponseClient run_cmd do let environment <- Lean.getEnv let expected : Array Lean.Name := #[ ``HohenbergKohnResponseClient.the_independent_gradient_kernel_has_its_actual_polynomial, ``HohenbergKohnResponseClient.the_library_derives_analyticity_at_the_selected_fermi_scale, ``HohenbergKohnResponseClient.the_library_excludes_the_source_derivative_divergence_for_this_polynomial, ``HohenbergKohnResponseClient.the_independent_polynomial_cannot_agree_near_a_source_singularity, ``HohenbergKohnResponseClient.the_library_retains_the_independent_nonexact_friedel_tail, ``HohenbergKohnResponseClient.the_library_recovers_the_actual_scaled_envelope, ``HohenbergKohnResponseClient.the_independent_asymptotic_profile_is_not_an_exact_identity, ``HohenbergKohnResponseClient.the_corrected_kernel_is_nonzero_at_a_leading_profile_node] let mut traversal : Lean.CollectAxioms.State := {} for declaration in expected do match environment.find? declaration with | some (.thmInfo _) => pure () | _ => throwError "Missing Hohenberg-Kohn response client theorem {declaration}" let (_, updated) := ((Lean.CollectAxioms.collect declaration).run environment).run traversal traversal := updated for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational response client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Response client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``HohenbergKohnResponseClient.the_library_derives_analyticity_at_the_selected_fermi_scale, ``PhysicsLogic.Papers.HohenbergKohn1964.ResponseKernel.a_finite_isotropic_gradient_kernel_is_analytic), (``HohenbergKohnResponseClient.the_library_excludes_the_source_derivative_divergence_for_this_polynomial, ``PhysicsLogic.Papers.HohenbergKohn1964.ResponseKernel.positive_derivative_divergence_excludes_analyticity), (``HohenbergKohnResponseClient.the_independent_polynomial_cannot_agree_near_a_source_singularity, ``PhysicsLogic.Papers.HohenbergKohn1964.ResponseKernel.a_finite_gradient_kernel_cannot_agree_near_a_nonanalytic_kernel), (``HohenbergKohnResponseClient.the_library_retains_the_independent_nonexact_friedel_tail, ``PhysicsLogic.Papers.HohenbergKohn1964.ResponseKernel.the_leading_profile_has_its_own_friedel_tail), (``HohenbergKohnResponseClient.the_library_retains_the_independent_nonexact_friedel_tail, ``PhysicsLogic.Papers.HohenbergKohn1964.ResponseKernel.inverse_cube_small_remainders_preserve_the_friedel_tail), (``HohenbergKohnResponseClient.the_library_recovers_the_actual_scaled_envelope, ``PhysicsLogic.Papers.HohenbergKohn1964.ResponseKernel.a_friedel_tail_is_an_actual_scaled_envelope_limit)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Hohenberg-Kohn response client audit passed: {expected.size} named theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, independent finite polynomial and nonexact oscillatory tail, no production diagnostic dependencies." namespace HohenbergKohnGradientSeriesClient open PhysicsLogic.Papers.HohenbergKohn1964 open Filter open scoped Topology noncomputable section set_option backward.isDefEq.respectTransparency false def baseline : GradientSeries where coefficient := fun order => if order = 0 then 3 else 0 valueAtScale := fun _ => 3 def corrected : GradientSeries := baseline.withRemainder (fun scale => 5 * Real.exp (-scale)) theorem the_independent_partial_sums_are_three (order : Nat) (scale : Real) : baseline.partialSum order scale = 3 := by classical rw [GradientSeries.partialSum, Finset.sum_eq_single 0] · simp [baseline] · intro index _ nonzero simp [baseline, nonzero] · simp theorem the_independent_baseline_has_actual_asymptotics : baseline.AsymptoticAtLargeScale := by intro order change Tendsto (fun scale => scale ^ order * (3 - baseline.partialSum order scale)) atTop (𝓝 0) simp only [the_independent_partial_sums_are_three, sub_self, mul_zero] exact tendsto_const_nhds theorem the_library_preserves_every_order_under_the_independent_correction : corrected.AsymptoticAtLargeScale := by apply GradientSeries.beyond_all_orders_remainders_preserve_the_expansion baseline (fun scale => 5 * Real.exp (-scale)) the_independent_baseline_has_actual_asymptotics intro order have limit := (Real.tendsto_pow_mul_exp_neg_atTop_nhds_zero order).const_mul 5 simpa only [mul_zero, mul_left_comm] using limit theorem the_library_supplies_a_fixed_order_eventual_error_budget (order : Nat) : ∀ᶠ scale in atTop, 0 < scale ∧ |corrected.valueAtScale scale - corrected.partialSum order scale| < (1 / 100 : Real) / scale ^ order := GradientSeries.asymptotic_series_has_an_eventual_orderwise_error_budget corrected the_library_preserves_every_order_under_the_independent_correction order (1 / 100) (by norm_num) theorem the_library_bounds_the_difference_between_equal_coefficient_readouts (order : Nat) : Tendsto (fun scale => scale ^ order * (corrected.valueAtScale scale - baseline.valueAtScale scale)) atTop (𝓝 0) := GradientSeries.same_asymptotic_coefficients_allow_only_beyond_all_orders_differences corrected baseline rfl the_library_preserves_every_order_under_the_independent_correction the_independent_baseline_has_actual_asymptotics order theorem the_independent_correction_is_not_exact_equality : corrected.valueAtScale 2 ≠ baseline.valueAtScale 2 := by intro same have positive := Real.exp_pos (-2) dsimp [corrected, baseline, GradientSeries.withRemainder] at same linarith theorem the_independent_order_limit_exists_but_is_not_the_physical_target : corrected.ConvergesAt 2 ∧ ¬Tendsto (fun order => corrected.partialSum order 2) atTop (𝓝 (corrected.valueAtScale 2)) := by have limit : Tendsto (fun order => corrected.partialSum order 2) atTop (𝓝 3) := by change Tendsto (fun order => baseline.partialSum order 2) atTop (𝓝 3) simp only [the_independent_partial_sums_are_three] exact tendsto_const_nhds refine ⟨⟨3, limit⟩, ?_⟩ intro targetLimit exact the_independent_correction_is_not_exact_equality (tendsto_nhds_unique limit targetLimit).symm def assessment (empirical : Nat -> Real -> Prop) : PhysicsLogic.Paradigm.EffectiveTheory.TruncationAssessmentContext Nat Real where calculationCompleteAt := fun order scale => order = 4 ∧ 0 < scale remainderControlledAt := fun order scale => |corrected.valueAtScale scale - corrected.partialSum order scale| < (1 / 100 : Real) / scale ^ order empiricallyAdequateAt := empirical theorem the_independent_error_budget_establishes_shared_truncation_control (empirical : Nat -> Real -> Prop) : ∀ᶠ scale in atTop, (assessment empirical).ControlledAt 4 scale := by filter_upwards [the_library_supplies_a_fixed_order_eventual_error_budget 4] with scale bound exact ⟨⟨rfl, bound.1⟩, bound.2⟩ theorem the_shared_truncation_api_preserves_the_independent_energy_error (empirical : Nat -> Real -> Prop) (scale : Real) (controlled : (assessment empirical).ControlledAt 4 scale) : |corrected.valueAtScale scale - corrected.partialSum 4 scale| < (1 / 100 : Real) / scale ^ 4 := (PhysicsLogic.Paradigm.EffectiveTheory.controlled_truncation_exposes_calculation_and_remainder_control (assessment empirical) 4 scale controlled).2 end end HohenbergKohnGradientSeriesClient run_cmd do let environment <- Lean.getEnv let expected : Array Lean.Name := #[ ``HohenbergKohnGradientSeriesClient.the_independent_partial_sums_are_three, ``HohenbergKohnGradientSeriesClient.the_independent_baseline_has_actual_asymptotics, ``HohenbergKohnGradientSeriesClient.the_library_preserves_every_order_under_the_independent_correction, ``HohenbergKohnGradientSeriesClient.the_library_supplies_a_fixed_order_eventual_error_budget, ``HohenbergKohnGradientSeriesClient.the_library_bounds_the_difference_between_equal_coefficient_readouts, ``HohenbergKohnGradientSeriesClient.the_independent_correction_is_not_exact_equality, ``HohenbergKohnGradientSeriesClient.the_independent_order_limit_exists_but_is_not_the_physical_target, ``HohenbergKohnGradientSeriesClient.the_independent_error_budget_establishes_shared_truncation_control, ``HohenbergKohnGradientSeriesClient.the_shared_truncation_api_preserves_the_independent_energy_error] let mut traversal : Lean.CollectAxioms.State := {} for declaration in expected do match environment.find? declaration with | some (.thmInfo _) => pure () | _ => throwError "Missing independent gradient-series theorem {declaration}" let (_, updated) := ((Lean.CollectAxioms.collect declaration).run environment).run traversal traversal := updated for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational gradient-series axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Gradient-series client uses production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``HohenbergKohnGradientSeriesClient.the_library_preserves_every_order_under_the_independent_correction, ``PhysicsLogic.Papers.HohenbergKohn1964.GradientSeries.beyond_all_orders_remainders_preserve_the_expansion), (``HohenbergKohnGradientSeriesClient.the_library_supplies_a_fixed_order_eventual_error_budget, ``PhysicsLogic.Papers.HohenbergKohn1964.GradientSeries.asymptotic_series_has_an_eventual_orderwise_error_budget), (``HohenbergKohnGradientSeriesClient.the_library_bounds_the_difference_between_equal_coefficient_readouts, ``PhysicsLogic.Papers.HohenbergKohn1964.GradientSeries.same_asymptotic_coefficients_allow_only_beyond_all_orders_differences), (``HohenbergKohnGradientSeriesClient.the_shared_truncation_api_preserves_the_independent_energy_error, ``PhysicsLogic.Paradigm.EffectiveTheory.controlled_truncation_exposes_calculation_and_remainder_control)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Hohenberg-Kohn gradient-series client audit passed: {expected.size} named theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, independent coefficients and nonzero flat correction, no production diagnostic dependencies." namespace KTScreeningApproximationClient open PhysicsLogic.Papers.KosterlitzThouless1973 Filter Topology noncomputable section def dielectricCurve : MeanFieldScreeningTrajectory where value := fun scale => physicalDielectricScreeningValue 2 3 (1 + scale) domain := Set.Ici 0 def descendingCurve : MeanFieldScreeningTrajectory where value := fun scale => 1 - 2 * scale domain := Set.Ici 0 theorem the_independent_dielectric_curve_has_the_physical_floor (scale : Real) (nonnegative : 0 <= scale) : -4 < dielectricCurve.value scale := positive_dielectric_readout_stays_above_minus_four 2 3 (1 + scale) (by norm_num) (by norm_num) (by linarith) theorem a_negative_screening_value_can_still_be_physical : dielectricCurve.value 3 = -1 ∧ -4 < dielectricCurve.value 3 := by exact ⟨by norm_num [dielectricCurve, physicalDielectricScreeningValue], the_independent_dielectric_curve_has_the_physical_floor 3 (by norm_num)⟩ theorem the_positive_dielectric_curve_cannot_run_away : ¬MeanFieldRunawayBranch dielectricCurve := by intro runaway obtain ⟨scale, nonnegative, _within, below⟩ := runaway (-4) linarith [the_independent_dielectric_curve_has_the_physical_floor scale nonnegative] theorem the_independent_core_has_nonzero_physical_radius : physicalRadiusAtLogScale 7 0 = 7 := zero_logarithmic_scale_is_the_core_radius 7 theorem the_reciprocal_rescalings_preserve_the_exponent : ((8 / (3 / 2) : Real) * ((3 / 2) * (5 / 2))) = 8 * (5 / 2) := the_mean_field_rescaling_preserves_the_exponential_argument (3 / 2) 8 (5 / 2) (by norm_num) theorem the_independent_descending_curve_has_a_runaway_branch : MeanFieldRunawayBranch descendingCurve := by intro level refine ⟨|level| + 2, by positivity, by change 0 <= |level| + 2; positivity, ?_⟩ change 1 - 2 * (|level| + 2) < level nlinarith [abs_nonneg level, neg_abs_le level] theorem the_library_derives_a_negative_scale_from_the_independent_runaway : ∃ scale, 0 <= scale ∧ scale ∈ descendingCurve.domain ∧ descendingCurve.value scale < 0 := mean_field_runaway_reaches_a_negative_scale descendingCurve the_independent_descending_curve_has_a_runaway_branch theorem a_qualitative_runaway_does_not_supply_equation_twenty_one : ¬EquationTwentyOneOn descendingCurve := by intro equation have derivative : HasDerivAt descendingCurve.value (-2) 0 := by convert ((hasDerivAt_id (0 : Real)).const_mul 2).const_sub 1 using 1 norm_num [descendingCurve] have equal := derivative.unique (equation 0 (by change (0 : Real) <= 0; rfl)) simp only [descendingCurve, mul_zero, sub_zero, neg_zero, zero_mul, Real.exp_zero] at equal norm_num at equal theorem the_library_retains_the_frozen_unscaled_equation (curve : MeanFieldScreeningTrajectory) (scale : Real) (equation : EquationTwentyOneOn curve) (withinDomain : 3 * scale ∈ curve.domain) : HasDerivAt (unscaledMeanFieldValue 3 curve) (-9 * Real.exp (-scale * unscaledMeanFieldValue 3 curve scale)) scale := by convert rescaled_equation_realizes_the_frozen_unscaled_equation 3 curve scale equation withinDomain using 1 norm_num end end KTScreeningApproximationClient run_cmd do let environment <- Lean.getEnv let expected : Array Lean.Name := #[ ``KTScreeningApproximationClient.the_independent_dielectric_curve_has_the_physical_floor, ``KTScreeningApproximationClient.a_negative_screening_value_can_still_be_physical, ``KTScreeningApproximationClient.the_positive_dielectric_curve_cannot_run_away, ``KTScreeningApproximationClient.the_independent_core_has_nonzero_physical_radius, ``KTScreeningApproximationClient.the_reciprocal_rescalings_preserve_the_exponent, ``KTScreeningApproximationClient.the_independent_descending_curve_has_a_runaway_branch, ``KTScreeningApproximationClient.the_library_derives_a_negative_scale_from_the_independent_runaway, ``KTScreeningApproximationClient.a_qualitative_runaway_does_not_supply_equation_twenty_one, ``KTScreeningApproximationClient.the_library_retains_the_frozen_unscaled_equation] let mut traversal : Lean.CollectAxioms.State := {} for declaration in expected do match environment.find? declaration with | some (.thmInfo _) => pure () | _ => throwError "Missing independent KT approximation theorem {declaration}" let (_, updated) := ((Lean.CollectAxioms.collect declaration).run environment).run traversal traversal := updated for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational KT approximation axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "KT approximation client uses production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``KTScreeningApproximationClient.the_independent_dielectric_curve_has_the_physical_floor, ``PhysicsLogic.Papers.KosterlitzThouless1973.positive_dielectric_readout_stays_above_minus_four), (``KTScreeningApproximationClient.the_independent_core_has_nonzero_physical_radius, ``PhysicsLogic.Papers.KosterlitzThouless1973.zero_logarithmic_scale_is_the_core_radius), (``KTScreeningApproximationClient.the_reciprocal_rescalings_preserve_the_exponent, ``PhysicsLogic.Papers.KosterlitzThouless1973.the_mean_field_rescaling_preserves_the_exponential_argument), (``KTScreeningApproximationClient.the_library_derives_a_negative_scale_from_the_independent_runaway, ``PhysicsLogic.Papers.KosterlitzThouless1973.mean_field_runaway_reaches_a_negative_scale), (``KTScreeningApproximationClient.the_library_retains_the_frozen_unscaled_equation, ``PhysicsLogic.Papers.KosterlitzThouless1973.rescaled_equation_realizes_the_frozen_unscaled_equation)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"KT screening-approximation client audit passed: {expected.size} named theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, independent dielectric and continuation data, no production diagnostic dependencies." namespace ChandrasekharPressureReductionClient open PhysicsLogic.Papers.Chandrasekhar1931 noncomputable section def model : CompositionPressureModel where pressureScale := 17 / 3 particleMass := 2 massDensity := 2 ionElectronRatio := 1 / 8 set_option backward.isDefEq.respectTransparency false in theorem the_independent_pressure_data_are_physical : model.PhysicalDomain := by norm_num [CompositionPressureModel.PhysicalDomain, model] theorem the_independent_reduced_pressure_is_seventeen_thirds : model.neglectedIonPressure = 17 / 3 := by norm_num [CompositionPressureModel.neglectedIonPressure, model] theorem the_library_keeps_positive_nonidentical_pressures : 0 < model.retainedPressure ∧ model.retainedPressure < model.neglectedIonPressure := ⟨(model.both_pressure_readouts_are_strictly_positive the_independent_pressure_data_are_physical).1, model.a_positive_ion_fraction_strictly_lowers_the_retained_pressure the_independent_pressure_data_are_physical (by norm_num [model])⟩ theorem the_library_computes_the_actual_relative_pressure_error : (model.neglectedIonPressure - model.retainedPressure) / model.neglectedIonPressure = 1 - 1 / Real.rpow (9 / 8 : Real) (4 / 3 : Real) := by have error := model.the_relative_pressure_error_is_an_actual_composition_quantity the_independent_pressure_data_are_physical convert error using 1 norm_num [model] theorem the_library_derives_the_unrounded_coefficient_law : model.neglectedIonPressure = model.neglectedIonCoefficient * Real.rpow model.massDensity (4 / 3 : Real) := model.the_reduced_pressure_has_the_derived_four_thirds_coefficient the_independent_pressure_data_are_physical theorem the_library_rejects_exact_pressure_agreement : model.retainedPressure ≠ model.neglectedIonPressure := by intro equal have zero := (model.exact_pressure_agreement_iff_zero_ion_fraction the_independent_pressure_data_are_physical).mp equal norm_num [model] at zero theorem the_exact_zero_fraction_does_restore_agreement : ({ model with ionElectronRatio := 0 } : CompositionPressureModel).retainedPressure = ({ model with ionElectronRatio := 0 } : CompositionPressureModel).neglectedIonPressure := by apply (CompositionPressureModel.exact_pressure_agreement_iff_zero_ion_fraction _ ?_).mpr rfl norm_num [CompositionPressureModel.PhysicalDomain, model] theorem the_source_adapter_keeps_nonzero_composition_error {T : ChandrasekharTypes} (framework : Framework T) (source : framework.EstablishedRelativisticEquationOfState) (positive : 0 < framework.ionElectronRatioValueAt framework.fullyIonizedMatter) : framework.pressureValueAt framework.relativisticPressureFormula framework.fullyIonizedMatter framework.selectedStar .extremeRelativisticDegeneracy < framework.equationOfStateCoefficientValueAt framework.relativisticEquationOfState * Real.rpow (framework.massDensityValueAt framework.fullyIonizedMatter framework.selectedStar .extremeRelativisticDegeneracy) (4 / 3 : Real) := framework.equation_four_positive_ion_fraction_keeps_nonzero_pressure_error source positive end end ChandrasekharPressureReductionClient run_cmd do let environment <- Lean.getEnv let expected : Array Lean.Name := #[ ``ChandrasekharPressureReductionClient.the_independent_pressure_data_are_physical, ``ChandrasekharPressureReductionClient.the_independent_reduced_pressure_is_seventeen_thirds, ``ChandrasekharPressureReductionClient.the_library_keeps_positive_nonidentical_pressures, ``ChandrasekharPressureReductionClient.the_library_computes_the_actual_relative_pressure_error, ``ChandrasekharPressureReductionClient.the_library_derives_the_unrounded_coefficient_law, ``ChandrasekharPressureReductionClient.the_library_rejects_exact_pressure_agreement, ``ChandrasekharPressureReductionClient.the_exact_zero_fraction_does_restore_agreement, ``ChandrasekharPressureReductionClient.the_source_adapter_keeps_nonzero_composition_error] let mut traversal : Lean.CollectAxioms.State := {} for declaration in expected do match environment.find? declaration with | some (.thmInfo _) => pure () | _ => throwError "Missing independent pressure-reduction theorem {declaration}" let (_, updated) := ((Lean.CollectAxioms.collect declaration).run environment).run traversal traversal := updated for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational pressure-reduction axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Pressure-reduction client uses production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``ChandrasekharPressureReductionClient.the_library_keeps_positive_nonidentical_pressures, ``PhysicsLogic.Papers.Chandrasekhar1931.CompositionPressureModel.a_positive_ion_fraction_strictly_lowers_the_retained_pressure), (``ChandrasekharPressureReductionClient.the_library_computes_the_actual_relative_pressure_error, ``PhysicsLogic.Papers.Chandrasekhar1931.CompositionPressureModel.the_relative_pressure_error_is_an_actual_composition_quantity), (``ChandrasekharPressureReductionClient.the_library_derives_the_unrounded_coefficient_law, ``PhysicsLogic.Papers.Chandrasekhar1931.CompositionPressureModel.the_reduced_pressure_has_the_derived_four_thirds_coefficient), (``ChandrasekharPressureReductionClient.the_library_rejects_exact_pressure_agreement, ``PhysicsLogic.Papers.Chandrasekhar1931.CompositionPressureModel.exact_pressure_agreement_iff_zero_ion_fraction), (``ChandrasekharPressureReductionClient.the_source_adapter_keeps_nonzero_composition_error, ``PhysicsLogic.Papers.Chandrasekhar1931.Framework.equation_four_positive_ion_fraction_keeps_nonzero_pressure_error)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Chandrasekhar pressure-reduction client audit passed: {expected.size} named theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, independent pressure and composition data, no production diagnostic dependencies." namespace StanleyJointRemainderClient open PhysicsLogic.Papers.Stanley1968 open Filter Topology noncomputable section def family : FluctuationRemainderFamily Nat where siteSize := fun site => 2 * site + 1 admissible := fun site => 0 < 2 * site + 1 factor := fun dimension site => Real.exp (3 * (dimension : Real) + 2 * (2 * (site : Real) + 1)) theorem odd_site_sizes_are_cofinal : family.CofinalSiteSizes := by intro bound exact ⟨⟨bound, by dsimp [family]; omega⟩, by dsimp [family]; omega⟩ theorem the_independent_factor_has_an_actual_joint_limit : family.JointLimitAt 0 := by refine ⟨odd_site_sizes_are_cofinal, Filter.Eventually.of_forall (fun _ => Real.exp_pos _), ?_⟩ have dimensions : Tendsto (fun pair : Nat × family.AdmissibleSite => (pair.1 : Real)) family.jointApproach atTop := tendsto_natCast_atTop_atTop.comp Filter.tendsto_fst have sites : Tendsto (fun pair : Nat × family.AdmissibleSite => (family.siteSize pair.2.val : Real)) family.jointApproach atTop := by have sizes : Tendsto (fun site : family.AdmissibleSite => family.siteSize site.val) family.siteApproach atTop := Filter.tendsto_comap exact tendsto_natCast_atTop_atTop.comp (sizes.comp Filter.tendsto_snd) have limit := ((tendsto_inv_atTop_zero.comp dimensions).const_mul 2).add ((tendsto_inv_atTop_zero.comp sites).const_mul 3) simp only [mul_zero, add_zero] at limit apply limit.congr' filter_upwards [family.dimensions_and_admissible_site_sizes_are_eventually_positive] with pair positive have dimensionNonzero : (pair.1 : Real) ≠ 0 := by exact_mod_cast Nat.ne_of_gt positive.1 have sitePositive : (0 : Real) < 2 * (pair.2.val : Real) + 1 := by positivity have siteNonzero := ne_of_gt sitePositive simp only [Function.comp_apply, FluctuationRemainderFamily.normalizedLogAt, family, Real.log_exp, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat] field_simp ring theorem the_independent_finite_remainder_is_nonzero : family.normalizedLogAt 2 3 = 10 / 7 := by norm_num [FluctuationRemainderFamily.normalizedLogAt, family, Real.log_exp] theorem the_library_checks_nontriviality_of_the_odd_site_domain : family.siteApproach.NeBot := family.cofinal_site_sizes_give_a_nontrivial_approach odd_site_sizes_are_cofinal theorem the_library_supplies_simultaneous_size_thresholds : ∃ dimensionBound siteBound : Nat, ∀ dimension : Nat, ∀ site : family.AdmissibleSite, dimensionBound ≤ dimension -> siteBound ≤ family.siteSize site.val -> |family.normalizedLogAt dimension site.val| < (1 / 100 : Real) := by simpa only [sub_zero] using family.a_joint_limit_has_rectangular_error_control 0 the_independent_factor_has_an_actual_joint_limit (1 / 100) (by norm_num) theorem the_library_transports_the_limit_along_the_independent_path : Tendsto (fun index : Nat => family.normalizedLogAt (index + 1) index) atTop (nhds 0) := by apply family.a_joint_limit_retains_every_cofinal_path 0 the_independent_factor_has_an_actual_joint_limit (fun index => index + 1) (fun index => ⟨index, by dsimp [family]; omega⟩) (tendsto_add_atTop_nat 1) exact tendsto_atTop_mono (fun index : Nat => by change index ≤ 2 * index + 1; omega) tendsto_id theorem the_library_rejects_a_different_recorded_limit (value : Real) (recorded : family.JointLimitAt value) : value = 0 := family.actual_joint_limits_have_a_unique_value value 0 recorded the_independent_factor_has_an_actual_joint_limit end end StanleyJointRemainderClient run_cmd do let environment <- Lean.getEnv let expected : Array Lean.Name := #[ ``StanleyJointRemainderClient.odd_site_sizes_are_cofinal, ``StanleyJointRemainderClient.the_independent_factor_has_an_actual_joint_limit, ``StanleyJointRemainderClient.the_independent_finite_remainder_is_nonzero, ``StanleyJointRemainderClient.the_library_checks_nontriviality_of_the_odd_site_domain, ``StanleyJointRemainderClient.the_library_supplies_simultaneous_size_thresholds, ``StanleyJointRemainderClient.the_library_transports_the_limit_along_the_independent_path, ``StanleyJointRemainderClient.the_library_rejects_a_different_recorded_limit] let mut traversal : Lean.CollectAxioms.State := {} for declaration in expected do match environment.find? declaration with | some (.thmInfo _) => pure () | _ => throwError "Missing independent joint-remainder theorem {declaration}" let (_, updated) := ((Lean.CollectAxioms.collect declaration).run environment).run traversal traversal := updated for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational joint-remainder axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Joint-remainder client uses production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``StanleyJointRemainderClient.the_library_checks_nontriviality_of_the_odd_site_domain, ``PhysicsLogic.Papers.Stanley1968.FluctuationRemainderFamily.cofinal_site_sizes_give_a_nontrivial_approach), (``StanleyJointRemainderClient.the_library_supplies_simultaneous_size_thresholds, ``PhysicsLogic.Papers.Stanley1968.FluctuationRemainderFamily.a_joint_limit_has_rectangular_error_control), (``StanleyJointRemainderClient.the_library_transports_the_limit_along_the_independent_path, ``PhysicsLogic.Papers.Stanley1968.FluctuationRemainderFamily.a_joint_limit_retains_every_cofinal_path), (``StanleyJointRemainderClient.the_library_rejects_a_different_recorded_limit, ``PhysicsLogic.Papers.Stanley1968.FluctuationRemainderFamily.actual_joint_limits_have_a_unique_value)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Stanley joint-remainder client audit passed: {expected.size} named theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, independent factor and odd site sizes, no production diagnostic dependencies." namespace BoundaryMajoranaPhysicalReuseAudit open PhysicsLogic.Framework.BoundaryMajoranaModeContext.Countermodels example : physicalKitaevBoundaryModel.majoranaPhase ≠ physicalKitaevBoundaryModel.trivialPhase ∧ physicalKitaevBoundaryModel.mode ≠ physicalOrdinaryZeroModeModel.mode ∧ ¬physicalTopologicalBulkWithoutBoundaryModel.modeSolvesBoundaryEquation .chainEndMajorana .chainEnd .kitaevChain .chainGapped ∧ physicalOrdinaryZeroModeModel.modeAtZeroEnergyFor .ordinaryZeroMode .kitaevChain .chainGapped ∧ ¬physicalOrdinaryZeroModeModel.modeSelfConjugate .ordinaryZeroMode ∧ helicalWireMajoranaControlContext.EstablishedBoundaryMajoranaModeAt helicalWireMajoranaControlModel ∧ fuKaneMajoranaControlContext.EstablishedBoundaryMajoranaModeAt fuKaneMajoranaControlModel := by obtain ⟨_distinctSystems, distinctPhases, _distinctHamiltonians, _distinctInterfaces, distinctModes, _classifiedBulk, noBoundary, _ordinaryLocalized, ordinaryZero, notMajorana, _kitaev, helical, fuKane⟩ := boundary_majorana_distinguishes_systems_phases_interfaces_modes_and_controls exact ⟨distinctPhases, distinctModes, noBoundary, ordinaryZero, notMajorana, helical, fuKane⟩ end BoundaryMajoranaPhysicalReuseAudit namespace SolitaryWavePhysicalReuseAudit open PhysicsLogic.Framework.SolitaryTravelingWaveContext.Countermodels example : physicalShallowWaterSolitaryModel.equation ≠ physicalShallowWaterSolitaryModel.reducedEquation ∧ physicalShallowWaterSolitaryModel.pulse ≠ numericalKdVSolitaryWaveControlModel.pulse ∧ ¬physicalWaveWithoutStationaryReductionModel.stationaryEquationReducedAt .shallowWaterStationaryODE .waterWaveEquation .canalComovingFrame .shallowCanal .weaklyNonlinearShallowWater ∧ physicalFlatBackgroundWaveModel.localizedPulseProfileObtainedAt .flatBackground .zeroContrast .canalPulseWidth .waterWaveEquation .shallowWaterStationaryODE .canalComovingFrame .stillWater .shallowCanal .weaklyNonlinearShallowWater ∧ ¬physicalFlatBackgroundWaveModel.pulseHasNonzeroBackgroundContrastAt .flatBackground .zeroContrast .stillWater .canalComovingFrame .shallowCanal .weaklyNonlinearShallowWater ∧ numericalKdVSolitaryWaveControlContext.EstablishedSolitaryTravelingWaveAt numericalKdVSolitaryWaveControlModel := by obtain ⟨distinctEquations, _distinctSystems, distinctPulses, _distinctBackgrounds, noReduction, flatProfile, noContrast, _shallowWater, numerical⟩ := solitary_waves_distinguish_equations_pulses_backgrounds_reductions_and_controls exact ⟨distinctEquations, distinctPulses, noReduction, flatProfile, noContrast, numerical⟩ end SolitaryWavePhysicalReuseAudit namespace CausalDefectPhysicalReuseAudit open PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels example : physicalKibbleDefectFormationModel.transition ≠ guthBubbleMonopoleControlModel.transition ∧ physicalKibbleDefectFormationModel.localOrderParameter .kibbleLeft ≠ physicalKibbleDefectFormationModel.localOrderParameter .kibbleRight ∧ ¬physicalKibbleDefectFormationModel.domainFamily .exterior ∧ ¬physicalAssemblyWithoutObstructionModel.topologicalObstructionAt .noncontractibleVacuumSphere physicalKibbleDomainFamily physicalCausalOrderAssignment .fourDomainVertex .symmetryBreaking .postTransition ∧ ¬physicalObstructionWithoutDefectModel.assemblyProducesDefectAt physicalKibbleDomainFamily physicalCausalOrderAssignment .fourDomainVertex .noncontractibleVacuumSphere .protodomainMonopole .symmetryBreaking .postTransition ∧ guthBubbleMonopoleControlContext.EstablishedCausalDomainDefectFormationAt guthBubbleMonopoleControlModel := by obtain ⟨distinctTransitions, _distinctDefects, distinctChoices, excludedExterior, _extension, noObstruction, _obstruction, noDefect, _kibble, guth⟩ := causal_defects_distinguish_domains_choices_obstructions_monopoles_and_controls exact ⟨distinctTransitions, distinctChoices, excludedExterior, noObstruction, noDefect, guth⟩ end CausalDefectPhysicalReuseAudit namespace SpectralGapPhysicalReuseAudit open PhysicsLogic.Framework.SelfConsistentSpectralGapContext.Countermodels example : physicalBcsSpectralGapModel.gap ≠ nambuSpectralGapControlModel.gap ∧ ¬physicalGapWithoutSelfConsistentSolutionModel.nontrivialSolutionAt .bcsVariationalPairing .bcsPairingGap .bcsCondensed ∧ physicalFiniteGapWithoutSpectrumModel.finiteGap .bcsPairingGap ∧ ¬physicalFiniteGapWithoutSpectrumModel.spectrumGappedBy .bcsQuasiparticleBand .bcsPairingGap ∧ nambuSpectralGapControlModel.excitationInSpectrum .nambuCollectiveMode .nambuCollectiveBranch ∧ ¬nambuSpectralGapControlModel.spectrumGappedBy .nambuCollectiveBranch .nambuAnomalousGap ∧ nambuSpectralGapControlContext.EstablishedSelfConsistentSpectralGapAt nambuSpectralGapControlModel := by obtain ⟨_distinctDescriptions, distinctGaps, _distinctSpectra, _problem, noSolution, finiteGap, noSpectrumGap, collectiveExcitation, collectiveUngapped, _bcs, nambu⟩ := spectral_gap_distinguishes_pairing_gaps_excitations_spectra_and_controls exact ⟨distinctGaps, noSolution, finiteGap, noSpectrumGap, collectiveExcitation, collectiveUngapped, nambu⟩ end SpectralGapPhysicalReuseAudit namespace NuclearBarrierPhysicalReuseAudit open PhysicsLogic.Framework.FiniteBarrierMetastableDecayContext.Countermodels example : physicalGamowBarrierModel.lowerEnergySelection.energy ≠ physicalGamowBarrierModel.higherEnergySelection.energy ∧ physicalGamowBarrierModel.lowerEnergySelection.decayRate ≠ physicalGamowBarrierModel.higherEnergySelection.decayRate ∧ ¬physicalBarrierAttenuationWithoutCouplingModel.nonzeroInteriorExteriorCouplingAt .gamowLowerTunnelWave .gamowLowerNucleus .gamowInterior .gamowAlphaChannel .gamowCoulombBarrier .gamowLowerEnergy .gamowNuclearDecay ∧ ¬physicalHigherEnergyWithoutPenetrationModel.greaterBarrierPenetrationAt .gamowHigherTunnelWave .gamowHigherNucleus .gamowCoulombBarrier .gamowHigherEnergy .gamowLowerTunnelWave .gamowLowerNucleus .gamowCoulombBarrier .gamowLowerEnergy .gamowNuclearDecay ∧ gurneyCondonBarrierControlContext.EstablishedFiniteBarrierMetastableDecayComparisonAt gurneyCondonBarrierControlModel := by obtain ⟨distinctEnergies, distinctRates, _distinctChannels, _attenuation, noCoupling, _higherEnergy, noPenetration, _gamow, gurney⟩ := nuclear_barrier_decay_distinguishes_energies_channels_couplings_rates_and_controls exact ⟨distinctEnergies, distinctRates, noCoupling, noPenetration, gurney⟩ example : PhysicalGamowRadialPreparationAndDecayAt physicalGamow1928RadialDecayReadout ∧ PhysicalGurneyCondonFiniteOverlapChanceAt physicalGurneyCondon1928LowerOverlapReadout ∧ physicalGamow1928RadialDecayReadout.probabilityBarrierExponentValue = 2 * physicalGamow1928RadialDecayReadout.amplitudeBarrierExponentValue ∧ physicalGamow1928RadialDecayReadout.preparationTransitionGammaMechanismEstablished = false ∧ physicalGurneyCondon1928LowerOverlapReadout.lowerWaveOverlapValue = 1 / 10 ∧ physicalGurneyCondon1928LowerOverlapReadout.printedQuantitativeDecayFormulaRetained = false := by refine ⟨ physical_gamow_radial_integral_s_wave_preparation_and_open_gamma_cause_are_nonvacuous, physical_gurney_condon_small_overlap_and_history_independent_chance_are_nonvacuous, ?_, ?_, ?_, ?_⟩ all_goals norm_num [physicalGamow1928RadialDecayReadout, physicalGurneyCondon1928LowerOverlapReadout] example : PhysicalGamowApproximateGeigerNuttallAt physicalGamow1928ApproximateCoulombComparisonReadout ∧ PhysicalGurneyCondonScatteringAndAnalogyAt physicalGurneyCondon1928ConditionalComparisonReadout ∧ physicalGamow1928ApproximateCoulombComparisonReadout.derivation ≠ physicalGamow1928RadialDecayReadout.derivation ∧ physicalGamow1928ApproximateCoulombComparisonReadout.branch ≠ physicalGamow1928RadialDecayReadout.branch ∧ physicalGamow1928ApproximateCoulombComparisonReadout.exactLinearGeigerNuttallLawEstablished = false ∧ physicalGurneyCondon1928ConditionalComparisonReadout.incomingAlphaReachesNuclearInterior = false ∧ ¬PhysicalGurneyCondonConditionalOverlapOrderingAt physicalGurneyMissingComparablePotentialControl ∧ ¬PhysicalGurneyCondonScatteringAndAnalogyAt physicalGurneyAnalogyClaimedAsNuclearDerivationControl := by obtain ⟨gamow, gurney, _distinctSources, _distinctDerivations, _wrongExponent, _gamma, _exactLaw, _printedFormula, noUnqualifiedComparison, noAnalogyDerivation⟩ := nuclear_decay_sources_distinguish_rectangular_damping_radial_gamma_overlap_chance_and_scope refine ⟨gamow, gurney, ?_, ?_, ?_, ?_, noUnqualifiedComparison, noAnalogyDerivation⟩ all_goals norm_num [physicalGamow1928ApproximateCoulombComparisonReadout, physicalGamow1928RadialDecayReadout, physicalGurneyCondon1928ConditionalComparisonReadout, physicalGurneyCondon1928LowerOverlapReadout] <;> simp example : physicalNuclearDecayScopePossibility.TentativeAt physicalNuclearDecayScopeEstablishment .gamowTransitionGammaCausationIsEstablished .gamow1928 ∧ physicalNuclearDecayScopePossibility.TentativeAt physicalNuclearDecayScopeEstablishment .gamowGeigerNuttallLawIsExactlyLinear .gamow1928 ∧ physicalNuclearDecayScopePossibility.TentativeAt physicalNuclearDecayScopeEstablishment .gurneyCondonLetterDerivesQuantitativeTunnelingExponent .gurneyCondon1928 ∧ physicalNuclearDecayScopePossibility.TentativeAt physicalNuclearDecayScopeEstablishment .gurneyCondonFieldEmissionAnalogyDerivesNuclearDecay .gurneyCondon1928 := by obtain ⟨_potential, gammaCausation, exactLaw, _angularMomentum, printedExponent, analogyDerivation, _universalOrdering, _nuclearPenetration, _classicalConfinement⟩ := nuclear_decay_source_overclaims_remain_epistemically_open exact ⟨gammaCausation, exactLaw, printedExponent, analogyDerivation⟩ end NuclearBarrierPhysicalReuseAudit namespace AmplitudeProbabilityPhysicalReuseAudit open PhysicsLogic.Framework.AmplitudeOutcomeProbabilityCountermodels example : physicalBornAmplitudeProbabilityModel.state ≠ kennardAmplitudeProbabilityControlModel.state ∧ physicalBornAmplitudeProbabilityModel.representation ≠ kennardAmplitudeProbabilityControlModel.representation ∧ ¬physicalAmplitudeWithoutCalculatedWeightModel.weightCalculatedFromAmplitudeAt .bornChannelWeight .bornScatteringAmplitude .outgoingScatteringChannel .incidentChannelState .scatteringChannelBasis .elasticCollision ∧ ¬physicalCalculatedWeightWithoutInterpretationModel.weightRepresentsOutcomeProbabilityAt .bornChannelWeight .outgoingScatteringChannel .incidentChannelState .bornScattering .collisionAtom .elasticCollision ∧ kennardAmplitudeProbabilityControlContext.EstablishedAmplitudeOutcomeProbabilityAt kennardAmplitudeProbabilityControlModel := by obtain ⟨distinctStates, distinctRepresentations, _distinctAmplitudes, _distinctOutcomes, _distinctWeights, _admissible, noWeight, _calculated, noInterpretation, _born, kennard⟩ := amplitude_probability_distinguishes_states_representations_outcomes_weights_and_controls exact ⟨distinctStates, distinctRepresentations, noWeight, noInterpretation, kennard⟩ end AmplitudeProbabilityPhysicalReuseAudit namespace ControlledParityPhysicalReuseAudit open PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels example : physicalCobaltParityModel.experiment ≠ muonParityFalsificationControlModel.experiment ∧ physicalCobaltParityModel.readout.forwardCountRate ≠ muonParityFalsificationControlModel.readout.forwardCountRate ∧ ¬physicalParityWithoutCalibrationModel.experimental.calibrationPerformedAt .cobaltGammaCalibration .orientedCobaltBetaDecay .polarizedCobalt ∧ ¬physicalParityWithoutArtifactExclusionModel.experimental.SelectedAlternativesExcludedAt physicalCobaltAlternative .orientedCobaltBetaDecay selectedReadout .polarizedCobalt ∧ ¬physicalInvariantParityModel.NonvacuousControlledSymmetryFalsificationAt ∧ muonParityFalsificationControlContext.EstablishedControlledSymmetryFalsificationAt muonParityFalsificationControlModel := by obtain ⟨distinctExperiments, _distinctCalibrations, _distinctObservables, distinctReadouts, _distinctStates, noCalibration, noArtifactExclusion, _invariant, noFalsification, _cobalt, muon⟩ := controlled_parity_tests_distinguish_experiments_readouts_controls_laws_and_artifacts exact ⟨distinctExperiments, distinctReadouts, noCalibration, noArtifactExclusion, noFalsification, muon⟩ end ControlledParityPhysicalReuseAudit namespace ResolventLocalizationPhysicalReuseAudit open PhysicsLogic.Framework.ResolventDecayLocalizationContext.Countermodels example : physicalAndersonLocalizationModel.kernel ≠ finiteBarLocalizationControlModel.kernel ∧ ¬physicalDescriptorWithoutSpatialDecayModel.spatialDecayEstablishedAt .disorderAveragedResolvent .probabilisticLocatorDecay .randomTightBindingLattice .strongDisorderFiniteEnergy ∧ ¬physicalDecayWithoutLocalizationInterpretationModel.localizationIdentifiedFromDecayAt .randomTightBindingLattice .probabilisticLocatorDecay .strongDisorderFiniteEnergy ∧ physicalUnqualifiedLocalizationClaimModel.unqualifiedAllStateLocalizationEstablishedAt .randomTightBindingLattice .strongDisorderFiniteEnergy ∧ ¬physicalUnqualifiedLocalizationClaimModel.NonvacuousResolventDecayLocalizationAt ∧ finiteBarLocalizationControlContext.EstablishedResolventDecayLocalizationAt finiteBarLocalizationControlModel := by obtain ⟨_distinctSystems, distinctKernels, _distinctDescriptors, _calculatedDescriptor, noDecay, _decay, noInterpretation, unqualified, noScopedResult, _anderson, finiteBar⟩ := resolvent_localization_distinguishes_disorder_kernels_descriptors_scopes_and_controls exact ⟨distinctKernels, noDecay, noInterpretation, unqualified, noScopedResult, finiteBar⟩ end ResolventLocalizationPhysicalReuseAudit namespace CosmologicalSamplingPhysicalReuseAudit open PhysicsLogic.Framework.PowerSpectrumSamplingVarianceContext.Countermodels example : physicalSmallAreaVarianceModel.model ≠ fullSkyMultipoleVarianceControlModel.model ∧ physicalSmallAreaVarianceModel.realization ≠ fullSkyMultipoleVarianceControlModel.realization ∧ ¬physicalNoiseFreeWithoutSamplingVarianceModel.positiveSamplingVarianceIn .smallAreaSurvey .patchTemperatureVariance ∧ ¬physicalVarianceWithoutDistinctSkiesModel.DistinctPossibleEstimates ∧ ¬physicalMultipleObservedSkiesModel.SelectedRealizationIsOnlyObservedAt ∧ fullSkyMultipoleVarianceControlContext.EstablishedSamplingVarianceAt fullSkyMultipoleVarianceControlModel := by obtain ⟨distinctEnsembles, distinctSkies, _distinctObservables, _distinctEstimates, _noiseExcluded, noVariance, _varianceLabel, noDistinctSkies, _alternateObserved, notSingle, _smallArea, fullSky⟩ := cosmological_sampling_distinguishes_ensembles_skies_estimates_noise_and_controls exact ⟨distinctEnsembles, distinctSkies, noVariance, noDistinctSkies, notSingle, fullSky⟩ example : PhysicalScottFinitePatchMomentsAt physicalScottFinitePatchReadout ∧ PhysicalScottSmallAreaLimitAt physicalScottSmallAreaLimitReadout ∧ PhysicalScottLargeAreaScalingAt physicalScottLargeAreaReadout ∧ PhysicalScottIndependentPatchHeuristicAt physicalScottIndependentPatchReadout ∧ physicalScottLinearScanReadout.effectiveSolidAngleValue ≠ physicalScottAlternateLinearScanReadout.effectiveSolidAngleValue := by exact ⟨physical_scott_finite_patch_gaussian_moments_are_nonvacuous.1, physical_scott_small_and_large_patch_limits_are_distinct.1, physical_scott_small_and_large_patch_limits_are_distinct.2.1, physical_scott_independent_patch_heuristic_is_qualified.1, physical_scott_linear_scan_depends_on_cosmological_correlation_angle.2.2.2.2.1⟩ example : PhysicalKnoxNoiseFreeCosmicVarianceAt physicalKnoxNoiseFreeReadout ∧ 0 < physicalKnoxNoiseFreeReadout.noiseFreeVarianceValue ∧ PhysicalKnoxBeamDetectorNoiseAt physicalKnoxBeamNoiseReadout ∧ physicalKnoxBeamNoiseReadout.estimatorVarianceValue < physicalKnoxBeamNoiseReadout.comparisonEstimatorVarianceValue ∧ PhysicalKnoxConditionalForecastAt physicalKnoxConditionalForecastReadout ∧ physicalKnoxConditionalForecastReadout.scalarTiltForecastUncertaintyValue < physicalKnoxConditionalForecastReadout.nuisanceTiltShiftValue ∧ PhysicalKnoxCorrectedTensorConsistencyAt physicalKnoxCorrectedTensorReadout := by exact ⟨physical_knox_noise_free_full_sky_cosmic_variance_is_positive.1, physical_knox_noise_free_full_sky_cosmic_variance_is_positive.2.2.2, physical_knox_beam_deconvolution_retains_finite_detector_noise.1, physical_knox_smaller_beam_at_fixed_detector_weight_is_better.2.2, physical_knox_forecast_precision_is_conditional_and_nuisance_limited.1, physical_knox_forecast_precision_is_conditional_and_nuisance_limited.2.2, physical_knox_tensor_consistency_retains_cosmological_correction.1⟩ example : ¬physicalCosmologicalSamplingEstablishment.establishedAt .exactInverseAreaLawForEveryPatch .scottSrednickiWhite1993 ∧ ¬physicalCosmologicalSamplingEstablishment.establishedAt .unconditionalInflationaryParameterMeasurement .knox1995 ∧ physicalCosmologicalSamplingPossibility.TentativeAt physicalCosmologicalSamplingEstablishment .higherMultipoleDegeneracyResolution .knox1995 := by exact ⟨cosmological_sampling_source_overclaims_remain_epistemically_unestablished.1, cosmological_sampling_source_overclaims_remain_epistemically_unestablished.2.2.2.1, cosmological_sampling_source_overclaims_remain_epistemically_unestablished.2.2.2.2.2.2.2⟩ end CosmologicalSamplingPhysicalReuseAudit namespace StateDispersionPhysicalReuseAudit open PhysicsLogic.Framework.StateDispersionUncertaintyContext.Countermodels example : physicalKennardDispersionModel.state ≠ robertsonZeroBoundControlModel.state ∧ physicalKennardDispersionModel.lowerBound ≠ robertsonZeroBoundControlModel.lowerBound ∧ ¬physicalPreparedWithoutSpreadModel.dispersionRepresentsStateSpreadAt .kennardCoordinatePrecision .position .gaussianWavePacket .kennardTransformationMechanics .canonicalParticle .canonicalTransformationPair ∧ ¬physicalSpreadWithoutProductModel.dispersionProductBoundedBelowAt .kennardCoordinatePrecision .kennardMomentumPrecision .positiveCanonicalBound .gaussianWavePacket .kennardTransformationMechanics .canonicalParticle .canonicalTransformationPair ∧ robertsonZeroBoundControlContext.EstablishedStateDispersionUncertaintyAt robertsonZeroBoundControlModel ∧ ¬robertsonZeroBoundControlContext.strictlyPositiveBoundAt .zeroCommutatorExpectationBound .angularMomentumZeroMeanCommutator := by obtain ⟨distinctStates, _distinctLeftObservables, _distinctRightObservables, _distinctDispersions, distinctBounds, _prepared, noSpread, _boundCalculated, noProduct, _kennard, robertson, notPositive⟩ := state_dispersion_distinguishes_preparations_observables_spreads_bounds_and_zero_controls exact ⟨distinctStates, distinctBounds, noSpread, noProduct, robertson, notPositive⟩ example : PhysicalKennardGeneralFactorizedBoundAt physicalKennard1927GeneralFactorizationReadout ∧ PhysicalKennardAlternativeExactObservationsAt physicalKennard1927AlternativeObservationReadout ∧ PhysicalRobertsonCanonicalCommutatorAt physicalRobertson1929CanonicalCommutatorReadout ∧ PhysicalRobertsonZeroCommutatorBoundAt physicalRobertson1929ZeroCommutatorReadout ∧ PhysicalRobertsonExtremalAngularSaturationAt physicalRobertson1929ExtremalAngularReadout ∧ physicalKennard1927GaussianMinimumReadout.canonicalPrecisionLowerBoundValue = 2 * physicalRobertson1929CanonicalCommutatorReadout.operatorLowerBoundValue ∧ physicalRobertson1929ZeroCommutatorReadout.operatorLowerBoundValue = 0 ∧ ¬PhysicalKennardAlternativeExactObservationsAt physicalKennardJointExactObservationControl ∧ ¬PhysicalRobertsonOperatorProofAt physicalRobertsonClaimedPolynomialProofControl ∧ ¬PhysicalRobertsonZeroCommutatorBoundAt physicalRobertsonClaimedUniversalPositiveBoundControl := by obtain ⟨general, alternatives, canonical, zero, extremal, _distinctSources, _distinctDerivations, normalization, _gaussian, _factorization, noJoint, noPolynomial, noUniversal, _saturation⟩ := uncertainty_sources_distinguish_gaussian_factorization_rms_zero_bound_angular_saturation_and_measurement_scope refine ⟨general, alternatives, canonical, zero, extremal, normalization, ?_, noJoint, noPolynomial, noUniversal⟩ rfl example : physicalUncertaintyScopePossibility.TentativeAt physicalUncertaintyScopeEstablishment .kennardGaussianEqualityProvesGeneralInequality .kennard1927 ∧ physicalUncertaintyScopePossibility.TentativeAt physicalUncertaintyScopeEstablishment .kennardPreparedStateBoundAloneProvesMeasurementDisturbance .kennard1927 ∧ physicalUncertaintyScopePossibility.TentativeAt physicalUncertaintyScopeEstablishment .robertsonEveryStateHasStrictlyPositiveBound .robertson1929 ∧ physicalUncertaintyScopePossibility.TentativeAt physicalUncertaintyScopeEstablishment .robertsonPolynomialMomentumProofAppearsInLetter .robertson1929 := by obtain ⟨gaussian, _joint, disturbance, _classical, positive, polynomial, _apparatus, _saturation⟩ := uncertainty_source_overclaims_remain_epistemically_open exact ⟨gaussian, disturbance, positive, polynomial⟩ end StateDispersionPhysicalReuseAudit namespace GaugeMassPhysicalReuseAudit open PhysicsLogic.Framework.GaugeFieldMassGenerationContext.Countermodels example : physicalEnglertBroutGaugeMassModel.system ≠ higgsSemisimpleGaugeMassControlModel.system ∧ physicalEnglertBroutGaugeMassModel.brokenGenerator ≠ physicalEnglertBroutGaugeMassModel.residualGenerator ∧ ¬physicalBrokenVacuumWithoutGaugeMassModel.massGenerationCalculatedAt .compactPolarizationMass .compactMassiveVector .matterVacuumOrder .englertBroutCompactGaugeMatter .lowestOrderCompactVacuum ∧ ¬physicalGaugeMassWithoutLongitudinalModel.brokenVacuumModeSuppliesLongitudinalResponseAt .brokenVacuumPolarizationPole .compactMassiveVector .compactBrokenDirection .englertBroutCompactGaugeMatter .lowestOrderCompactVacuum ∧ ¬physicalGaugeMassWithoutQualifiedScopeContext.NonvacuousGaugeFieldMassGenerationAt physicalGaugeMassWithoutQualifiedScopeModel ∧ higgsSemisimpleGaugeMassControlContext.EstablishedGaugeFieldMassGenerationAt higgsSemisimpleGaugeMassControlModel := by obtain ⟨distinctSystems, _distinctSymmetries, distinctGenerators, _distinctFields, _distinctModes, noMass, _generatedMass, noLongitudinal, _noScope, noQualifiedResult, _englertBrout, higgs⟩ := gauge_mass_distinguishes_symmetries_generators_modes_residual_fields_and_scoped_controls exact ⟨distinctSystems, distinctGenerators, noMass, noLongitudinal, noQualifiedResult, higgs⟩ end GaugeMassPhysicalReuseAudit namespace ThreeBodyPhysicalReuseAudit open PhysicsLogic.Framework.ThreeBodyPairwiseInputContext.Countermodels example : physicalThomasPairwiseModel.system ≠ bhvkRunningCountertermControlModel.system ∧ ¬physicalCalibratedPairWithoutCollapseModel.shortDistanceInstabilityAt .deuteronTritonSystem .fixedDeuteronInteraction .tritonGroundStateBinding .singularNuclearRangeLimit ∧ physicalCalibratedPairWithoutCollapseModel.stableThreeBodyPredictionAt .deuteronTritonSystem .fixedDeuteronInteraction .tritonGroundStateBinding .singularNuclearRangeLimit ∧ ¬physicalThomasUnselectedRemedyReadout.HasUniqueSelectedRemedyAt ∧ physicalRunningCountertermControlReadout.HasRunningCountertermControlAt ∧ bhvkRunningCountertermControlContext.EstablishedThreeBodyPairwiseInputInsufficiencyAt bhvkRunningCountertermControlModel := by obtain ⟨distinctSystems, _distinctInputs, _distinctObservables, _distinctRegimes, _calibrated, noCollapse, stable, _insufficientLabel, _noObstruction, _matches, _candidates, noSelectedRemedy, runningControl, _thomas, bhvk⟩ := three_body_obstructions_distinguish_systems_inputs_collapse_remedies_and_running_controls exact ⟨distinctSystems, noCollapse, stable, noSelectedRemedy, runningControl, bhvk⟩ example : PhysicalThomasFixedDeuteronVariationalCollapseAt physicalThomasOriginalSourceReadout := physical_thomas_fixed_deuteron_variational_collapse_is_nonvacuous example (lowerBound : Real) : ∃ index, physicalThomasOriginalSourceReadout.threeBodyTotalEnergyAt index < lowerBound := by obtain ⟨_, _, _, _, _, _, _, _, _, _, _, unbounded, _⟩ := physical_thomas_fixed_deuteron_variational_collapse_is_nonvacuous exact unbounded lowerBound example : PhysicalThomasUnselectedAlternativesAt physicalThomasOriginalSourceReadout := physical_thomas_original_alternatives_are_nonvacuous example : ¬PhysicalThomasFixedDeuteronVariationalCollapseAt physicalThomasFixedPotentialDepthControl := thomas_zero_range_calibration_depth_energy_and_remedies_are_independent.2.1 example : ¬PhysicalThomasFixedDeuteronVariationalCollapseAt physicalThomasUnpreservedDeuteronDatumControl := thomas_zero_range_calibration_depth_energy_and_remedies_are_independent.2.2.1 example : ¬PhysicalThomasFixedDeuteronVariationalCollapseAt physicalThomasPositiveEnergySignControl := thomas_zero_range_calibration_depth_energy_and_remedies_are_independent.2.2.2.1 example : ¬PhysicalThomasFixedDeuteronVariationalCollapseAt physicalThomasFiniteGroundEnergyControl := thomas_zero_range_calibration_depth_energy_and_remedies_are_independent.2.2.2.2.1 example : ¬PhysicalThomasUnselectedAlternativesAt physicalThomasSelectedFiniteRangeControl := thomas_zero_range_calibration_depth_energy_and_remedies_are_independent.2.2.2.2.2.2.1 example : ¬PhysicalThomasUnselectedAlternativesAt physicalThomasInventedRunningForceControl := thomas_zero_range_calibration_depth_energy_and_remedies_are_independent.2.2.2.2.2.2.2 example : PhysicalBHvKBosonDimerExchangeKernelAt physicalBHvKOriginalSourceReadout := physical_bhvk_boson_dimer_exchange_kernel_is_nonvacuous example : ¬PhysicalBHvKBosonDimerExchangeKernelAt physicalBHvKNuclearChannelSubstitutionControl := bhvk_bosonic_kernel_imaginary_roots_and_quartet_scope_are_independent.2.1 example : ¬PhysicalBHvKBosonDimerExchangeKernelAt physicalBHvKMissingResummationControl := bhvk_bosonic_kernel_imaginary_roots_and_quartet_scope_are_independent.2.2.1 example : ¬PhysicalBHvKBosonDimerExchangeKernelAt physicalBHvKWrongExchangeKernelSignControl := bhvk_bosonic_kernel_imaginary_roots_and_quartet_scope_are_independent.2.2.2.1 example : PhysicalBHvKImaginaryRootsAndCutoffAmbiguityAt physicalBHvKOriginalSourceReadout := physical_bhvk_imaginary_roots_and_cutoff_ambiguity_are_nonvacuous example : ¬PhysicalBHvKImaginaryRootsAndCutoffAmbiguityAt physicalBHvKArtificialInverseMomentumControl := bhvk_bosonic_kernel_imaginary_roots_and_quartet_scope_are_independent.2.2.2.2.2.1 example : ¬PhysicalBHvKImaginaryRootsAndCutoffAmbiguityAt physicalBHvKTwoBodyDeterminedPhaseControl := bhvk_bosonic_kernel_imaginary_roots_and_quartet_scope_are_independent.2.2.2.2.2.2.1 example : ¬PhysicalBHvKImaginaryRootsAndCutoffAmbiguityAt physicalBHvKQuartetForceOverclaimControl := bhvk_bosonic_kernel_imaginary_roots_and_quartet_scope_are_independent.2.2.2.2.2.2.2 example : PhysicalBHvKIndependentRunningCountertermAt physicalBHvKOriginalSourceReadout := physical_bhvk_independently_fitted_running_force_is_nonvacuous example (phase : Real) : physicalThreeBodyRunningForceAtPhase physicalBHvKOriginalSourceReadout.positiveImaginaryExponentValue (phase + Real.pi) = physicalThreeBodyRunningForceAtPhase physicalBHvKOriginalSourceReadout.positiveImaginaryExponentValue phase := physical_running_force_is_pi_periodic physicalBHvKOriginalSourceReadout.positiveImaginaryExponentValue phase example : physicalBHvKOriginalSourceReadout.firstRunningCountertermValue = physicalThreeBodyRunningForceAtCutoff physicalBHvKOriginalSourceReadout.positiveImaginaryExponentValue physicalBHvKOriginalSourceReadout.firstCutoffValue physicalBHvKOriginalSourceReadout.referenceThreeBodyScaleValue := physical_bhvk_first_running_counterterm_matches_original_formula example : physicalBHvKOriginalSourceReadout.secondRunningCountertermValue = physicalThreeBodyRunningForceAtCutoff physicalBHvKOriginalSourceReadout.positiveImaginaryExponentValue physicalBHvKOriginalSourceReadout.secondCutoffValue physicalBHvKOriginalSourceReadout.referenceThreeBodyScaleValue := physical_bhvk_second_running_counterterm_matches_original_formula example : ¬PhysicalBHvKIndependentRunningCountertermAt physicalBHvKFixedCountertermControl := bhvk_running_datum_finite_cutoff_and_shallow_state_scopes_are_independent.2.1 example : ¬PhysicalBHvKIndependentRunningCountertermAt physicalBHvKMissingIndependentDatumControl := bhvk_running_datum_finite_cutoff_and_shallow_state_scopes_are_independent.2.2.1 example : PhysicalBHvKScopedFiniteCutoffPredictionAt physicalBHvKOriginalSourceReadout := physical_bhvk_finite_cutoff_and_shallow_state_scopes_are_nonvacuous example : ¬PhysicalBHvKScopedFiniteCutoffPredictionAt physicalBHvKExactFiniteCutoffOverclaimControl := bhvk_running_datum_finite_cutoff_and_shallow_state_scopes_are_independent.2.2.2.2.1 example : ¬PhysicalBHvKScopedFiniteCutoffPredictionAt physicalBHvKAllBoundStatesValidOverclaimControl := bhvk_running_datum_finite_cutoff_and_shallow_state_scopes_are_independent.2.2.2.2.2.1 example : ¬PhysicalBHvKScopedFiniteCutoffPredictionAt physicalBHvKMicroscopicPotentialOverclaimControl := bhvk_running_datum_finite_cutoff_and_shallow_state_scopes_are_independent.2.2.2.2.2.2.1 example : ¬PhysicalBHvKScopedFiniteCutoffPredictionAt physicalBHvKCoulombCorrectionOverclaimControl := bhvk_running_datum_finite_cutoff_and_shallow_state_scopes_are_independent.2.2.2.2.2.2.2 example : physicalThreeBodyRemedySelection.OpenAlternativePairAt .finiteNeutronProtonInteractionRange .nonnegligibleNeutronNeutronRepulsion .thomas1935 := three_body_source_overclaims_remain_epistemically_open.2.2.2.2.2.2.2.2.1 example : physicalThreeBodyRemedySelection.uniquelySelectedAt .leadingRunningThreeBodyForce .bedaqueHammerVanKolck1998 := three_body_source_overclaims_remain_epistemically_open.2.2.2.2.2.2.2.2.2 example : PhysicalThomasUnselectedAlternativesAt physicalThomasOriginalSourceReadout ∧ PhysicsLogic.Papers.Thomas1935.ThomasFramework.Countermodels.thomasAlternativesWithoutSelectionReadout.HasCandidateRemediesAt := ⟨physical_thomas_original_alternatives_are_nonvacuous, PhysicsLogic.Papers.Thomas1935.ThomasFramework.Countermodels.thomas_realizes_shared_three_body_remedy_boundary.2.1⟩ example : PhysicalBHvKIndependentRunningCountertermAt physicalBHvKOriginalSourceReadout ∧ (PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework.Countermodels.hasNamedRunningFormula PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework.Countermodels.wrongEquationHModel ∧ ¬PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework.Countermodels.retainsEquationHSineRatio PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework.Countermodels.wrongEquationHModel) := ⟨physical_bhvk_independently_fitted_running_force_is_nonvacuous, PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyRenormalizationFramework.Countermodels.named_running_formula_does_not_supply_equation_h_sine_ratio⟩ end ThreeBodyPhysicalReuseAudit namespace CalibratedBackgroundPhysicalReuseAudit open PhysicsLogic.Framework.CalibratedResidualBackgroundContext.Countermodels example : physicalHornBackgroundModel.instrument ≠ firasFullMissionBackgroundControlModel.instrument ∧ physicalHornBackgroundModel.backgroundSignal ≠ firasFullMissionBackgroundControlModel.backgroundSignal ∧ ¬physicalCalibrationWithoutSeparationModel.backgroundSeparatedAt .hornAntennaTemperature .atmosphericAndAntennaLosses .excessAntennaTemperature .singleFrequencyRadioResidual ∧ ¬physicalSeparatedBackgroundWithoutUncertaintyModel.uncertaintyAssignedAt .excessAntennaTemperature .hornTemperatureError .singleFrequencyRadioResidual ∧ ¬physicalBackgroundWithoutForegroundExclusionModel.NonvacuousCalibratedResidualBackgroundDetectionAt ∧ firasFullMissionBackgroundControlContext.EstablishedCalibratedResidualBackgroundDetectionAt firasFullMissionBackgroundControlModel := by obtain ⟨distinctInstruments, _distinctObservations, _distinctReferences, _distinctRawSignals, distinctBackgrounds, _distinctContaminants, _calibrated, noSeparation, _separated, noUncertainty, _detected, _noForegroundExclusion, noDetection, _horn, firas⟩ := calibrated_background_distinguishes_instruments_residuals_foregrounds_uncertainty_and_controls exact ⟨distinctInstruments, distinctBackgrounds, noSeparation, noUncertainty, noDetection, firas⟩ end CalibratedBackgroundPhysicalReuseAudit namespace InertialKinematicsPhysicalReuseAudit open PhysicsLogic.Framework.InertialFrameKinematicsCountermodels example : physicalEinsteinInertialModel.firstFrame ≠ physicalEinsteinInertialModel.secondFrame ∧ physicalEinsteinInertialModel.lightStructure ≠ minkowskiNullConeControlModel.lightStructure ∧ ¬physicalCoincidentInertialFrameModel.NonvacuousInertialFrameKinematicsAt ∧ ¬physicalTransformationWithoutLightPreservationModel.preservesLightStructureAt .synchronizedLorentzBoost .invariantLightSphere .synchronizedMovingCoordinates ∧ ¬physicalCovarianceWithPreferredFrameModel.NonvacuousInertialFrameKinematicsAt ∧ minkowskiNullConeControlContext.EstablishedInertialFrameKinematicsAt minkowskiNullConeControlModel := by obtain ⟨distinctFrames, _distinctBoosts, _distinctLaws, distinctLightStructures, noDistinctFrameRoute, _admissible, noLightPreservation, _preferred, noPreferenceRoute, _einstein, minkowski⟩ := inertial_kinematics_distinguishes_frames_boosts_light_cones_preference_and_controls exact ⟨distinctFrames, distinctLightStructures, noDistinctFrameRoute, noLightPreservation, noPreferenceRoute, minkowski⟩ example : PhysicalEinsteinRoundTripSynchronizationAt physicalEinsteinSynchronizationReadout ∧ physicalEinsteinSynchronizationReadout.reflectionTimeValue = (physicalEinsteinSynchronizationReadout.emissionTimeValue + physicalEinsteinSynchronizationReadout.returnTimeValue) / 2 := physical_einstein_synchronization_uses_the_actual_round_trip_midpoint example : PhysicalEinsteinLorentzCoordinateTransformationAt physicalEinsteinLorentzReadout ∧ physicalEinsteinLorentzReadout.movingTimeValue ≠ physicalEinsteinLorentzReadout.stationaryTimeValue := physical_einstein_lorentz_boost_retains_both_postulates_and_time_mixing example : PhysicalEinsteinRodAndClockReadoutAt physicalEinsteinRodAndClockReadout ∧ physicalEinsteinRodAndClockReadout.movingIntrinsicLengthValue = physicalEinsteinRodAndClockReadout.restLengthValue := physical_einstein_rod_contraction_is_relative_and_clock_dilation_is_oriented example : PhysicalMinkowskiUpperHyperboloidAt physicalMinkowskiHyperboloidReadout ∧ 0 < physicalMinkowskiHyperboloidReadout.hyperboloidMovingTimeValue := physical_minkowski_transformation_retains_the_upper_unit_hyperboloid example : ¬PhysicalEinsteinLorentzCoordinateTransformationAt physicalEinsteinGalileanTimeControl := einstein_synchronization_postulates_scale_and_galilean_time_are_independent.2.2.2.1 example : ¬PhysicalMinkowskiMichelsonWorldPostulateBoundaryAt physicalMinkowskiWorldPostulateInferredFromNullResultControl := by obtain ⟨_, _, _, _, _, _, _, missingPostulate, _, _, _⟩ := minkowski_hyperboloid_branch_null_cone_timelike_interval_and_postulates_are_independent exact missingPostulate example : ¬physicalInertialEstablishment.establishedAt .june1905ArticleDerivesMassEnergyEquivalence .einstein1905 := inertial_source_overclaims_remain_epistemically_unestablished.2.2.2.2.1 end InertialKinematicsPhysicalReuseAudit namespace PartitionZeroPhysicalReuseAudit open PhysicsLogic.Framework.PartitionZeroPhaseBoundaryContext.Countermodels example : physicalLeeYangBoundaryModel.selectedControl ≠ contourCoexistenceBoundaryControlModel.selectedControl ∧ physicalLeeYangBoundaryModel.limitingDescription ≠ contourCoexistenceBoundaryControlModel.limitingDescription ∧ ¬physicalLeeYangBoundaryModel.finitePartitionZeroAt .latticeGasFiniteVolume .positiveActivityTransition ∧ physicalLeeYangBoundaryModel.finitePartitionZeroAt .latticeGasFiniteVolume .upperComplexFiniteZero ∧ ¬physicalPartitionZerosWithoutSingularityModel.NonvacuousPartitionZeroPhaseBoundaryAt ∧ contourCoexistenceBoundaryControlContext.EstablishedPartitionZeroPhaseBoundaryAt contourCoexistenceBoundaryControlModel := by obtain ⟨distinctControls, _distinctRegions, distinctLimits, _boundary, noFinitePositiveZero, finiteComplexZero, _conjugateComplexZero, _boundaryWithoutZeros, _noComplexZero, _zero, _noSingularity, noBoundaryResult, _leeYang, contour⟩ := partition_zeros_distinguish_finite_volumes_controls_accumulation_singularity_and_coexistence exact ⟨distinctControls, distinctLimits, noFinitePositiveZero, finiteComplexZero, noBoundaryResult, contour⟩ end PartitionZeroPhysicalReuseAudit namespace FractionalHallPhysicalReuseAudit open PhysicsLogic.Framework.FractionalHallQuasiholeChargeContext.Countermodels example : physicalLaughlinFractionalChargeModel.system ≠ arovasAdiabaticFractionalChargeControlModel.system ∧ physicalLaughlinFractionalChargeModel.filling ≠ arovasAdiabaticFractionalChargeControlModel.filling ∧ ¬physicalFractionalValueWithoutAssignmentContext.NonvacuousFractionalHallQuasiholeChargeAt physicalFractionalValueWithoutAssignmentModel ∧ ¬physicalIntegerFillingChargeContext.EstablishedFractionalHallQuasiholeChargeAt physicalIntegerFillingChargeModel ∧ arovasAdiabaticFractionalChargeControlContext.EstablishedFractionalHallQuasiholeChargeAt arovasAdiabaticFractionalChargeControlModel := by obtain ⟨distinctSystems, _distinctQuasiholes, distinctFillings, _unassignedValue, noAssignment, _integerValue, noFractionality, _notLocalized, _laughlin, arovas⟩ := fractional_hall_charge_distinguishes_fluids_holes_fillings_assignment_and_localized_controls exact ⟨distinctSystems, distinctFillings, noAssignment, noFractionality, arovas⟩ end FractionalHallPhysicalReuseAudit namespace CollapseAbundancePhysicalReuseAudit open PhysicsLogic.Framework.ThresholdedCollapseAbundanceContext.Countermodels example : physicalPressSchechterAbundanceModel.field ≠ bondSharpKFirstCrossingControlModel.field ∧ physicalPressSchechterAbundanceModel.rawFraction ≠ physicalPressSchechterAbundanceModel.completedFraction ∧ ¬physicalRawCollapseWithoutMassAccountingModel.missingMassAccountedFor .gaussianDensityField .selfSimilarCondensates .gaussianUpperTailFraction .missingMassCompletedFraction .maximumVarianceGaussianCounting .selfSimilarCondensationSpectrum ∧ ¬physicalAccountedCollapseWithoutSpectrumModel.NonvacuousThresholdedCollapseAbundanceAt ∧ bondSharpKFirstCrossingControlContext.EstablishedThresholdedCollapseAbundanceAt bondSharpKFirstCrossingControlModel := by obtain ⟨distinctFields, _distinctPopulations, distinctFractions, _distinctSpectra, _rawFraction, noAccounting, _accounted, _noSpectrum, noAbundance, _pressSchechter, bond⟩ := collapse_abundance_distinguishes_fields_fractions_accounting_spectra_and_first_crossing_controls exact ⟨distinctFields, distinctFractions, noAccounting, noAbundance, bond⟩ example : PhysicalPressSchechterFiniteBootstrapEvidenceAt physicalPressSchechter1974AbundanceReadout ∧ PhysicalBondEquationThreeNineMassFunctionAt physicalBond1991SharpKAbundanceReadout ∧ physicalPressSchechter1974AbundanceReadout.rawTailFractionValue = PhysicsLogic.Framework.CosmologicalPerturbationCollapseContext.Countermodels.physicalPressSchechter1974CondensationReadout.rawGaussianTailFractionValue ∧ physicalPressSchechter1974AbundanceReadout.completedFractionValue = PhysicsLogic.Framework.CosmologicalPerturbationCollapseContext.Countermodels.physicalPressSchechter1974CondensationReadout.completedCollapsedMassFractionValue ∧ physicalPressSchechter1974AbundanceReadout.factorTwoDerivedByFirstPassage = false ∧ physicalBond1991SharpKAbundanceReadout.factorTwoDerivedByFirstPassage = true := by refine ⟨ physical_press_schechter_two_thousand_body_experiments_remain_finite, physical_bond_sharp_k_mass_function_retains_equation_three_nine, ?_, ?_, press_schechter_and_bond_factor_two_mechanisms_are_physically_distinct.2.2.2.2.1, press_schechter_and_bond_factor_two_mechanisms_are_physically_distinct.2.2.2.2.2.1⟩ all_goals norm_num [physicalPressSchechter1974AbundanceReadout, PhysicsLogic.Framework.CosmologicalPerturbationCollapseContext.Countermodels.physicalPressSchechter1974CondensationReadout] example : PhysicalBondCorrelatedGeneralFilterAssessmentAt physicalBond1991SharpKAbundanceReadout ∧ ¬PhysicalBondCorrelatedGeneralFilterAssessmentAt physicalBondExactGeneralFilterControl ∧ ¬PhysicalBondQualifiedSimulationAndCalibrationAt physicalBondPerfectParticleIdentificationControl ∧ ¬PhysicalBondTwoBarrierConditionalHistoryAt physicalBondNoSecondBarrierControl ∧ PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetFramework.Countermodels.HasSharpKMarkovIncrementsAt PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetFramework.Countermodels.sharpKMarkovWithCorrelatedGeneralFilter ∧ ¬PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetFramework.Countermodels.HasGeneralFilterMarkovIncrementsAt PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetFramework.Countermodels.sharpKMarkovWithCorrelatedGeneralFilter := ⟨physical_bond_general_filters_retain_correlations_bounds_and_numerics, bond_correlated_general_filters_do_not_inherit_sharp_k_exactness.2.2.2.1, bond_mean_abundance_calibration_and_low_mass_identity_are_independent.2.1, bond_one_barrier_abundance_does_not_supply_a_two_barrier_history.2.2.1, PhysicsLogic.Papers.BondColeEfstathiouKaiser1991.ExcursionSetFramework.Countermodels.sharp_k_markov_exactness_does_not_transfer_to_general_filters⟩ example : physicalCollapseAbundanceScopePossibility.TentativeAt physicalCollapseAbundanceScopeEstablishment .rigorousPressSchechterBoundaryOverlapAccounting .pressSchechter1974 ∧ physicalCollapseAbundanceScopePossibility.TentativeAt physicalCollapseAbundanceScopeEstablishment .exactGeneralFilterFirstCrossingFormula .bondColeEfstathiouKaiser1991 ∧ physicalCollapseAbundanceScopePossibility.TentativeAt physicalCollapseAbundanceScopeEstablishment .meanMassFunctionFitIdentifiesIndividualHaloParticles .bondColeEfstathiouKaiser1991 := ⟨collapse_abundance_source_overclaims_remain_epistemically_open.1, collapse_abundance_source_overclaims_remain_epistemically_open.2.2.2.1, collapse_abundance_source_overclaims_remain_epistemically_open.2.2.2.2.1⟩ end CollapseAbundancePhysicalReuseAudit namespace GoldstoneSpectrumPhysicalReuseAudit open PhysicsLogic.Framework.BrokenSymmetrySpectrumContext.Countermodels example : physicalRelativisticGoldstoneModel.symmetry ≠ andersonUncoupledMatterSpectrumControlModel.symmetry ∧ ¬physicalBrokenPhaseWithoutGaplessModeModel.gaplessMode .relativisticInternalSymmetry .relativisticBrokenVacuum .masslessGoldstoneParticle ∧ ¬physicalGoldstoneWithoutSpectralMembershipModel.modeInSpectrum .masslessGoldstoneParticle .relativisticParticleSpectrum ∧ ¬physicalElectromagneticallyScreenedSpectrumModel.NonvacuousBrokenSymmetrySpectrumAt ∧ andersonUncoupledMatterSpectrumControlContext.EstablishedGoldstoneSpectrumAt andersonUncoupledMatterSpectrumControlModel := by obtain ⟨distinctSymmetries, _distinctPhases, _distinctSpectra, noGaplessMode, _gapless, noMembership, _plasma, noGoldstoneSpectrum, _relativistic, matter⟩ := goldstone_spectra_distinguish_symmetries_phases_modes_screening_and_matter_controls exact ⟨distinctSymmetries, noGaplessMode, noMembership, noGoldstoneSpectrum, matter⟩ end GoldstoneSpectrumPhysicalReuseAudit namespace CosmologicalCollapsePhysicalReuseAudit open PhysicsLogic.Framework.CosmologicalPerturbationCollapseContext.Countermodels example : physicalWeinbergCollapseModel.background ≠ pressSchechterSphericalCollapseControlModel.background ∧ physicalSubcriticalExactCriterionContext.EstablishedCollapseCriterionAt physicalSubcriticalExactCriterionModel ∧ ¬physicalSubcriticalExactCriterionModel.gravitationallyCollapsesAt .subcriticalOverdensity .weinbergVacuumEnergyBackground .vacuumEnergyStructureFormation ∧ ¬physicalCriterionWithoutDynamicalCollapseModel.NonvacuousExactCollapseCriterionAt ∧ pressSchechterSphericalCollapseControlContext.EstablishedCollapseCriterionAt pressSchechterSphericalCollapseControlModel := by obtain ⟨distinctBackgrounds, _distinctPerturbations, _distinctRegimes, exactSubcritical, noSubcriticalCollapse, _satisfied, _noCollapse, noExactCriterion, _weinberg, pressSchechter⟩ := cosmological_collapse_distinguishes_backgrounds_perturbations_thresholds_and_spherical_controls exact ⟨distinctBackgrounds, exactSubcritical, noSubcriticalCollapse, noExactCriterion, pressSchechter⟩ end CosmologicalCollapsePhysicalReuseAudit namespace RemoteCorrelationPhysicalReuseAudit open PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels example : physicalEprRemoteCorrelationModel.experiment ≠ bellSingletRemoteCorrelationControlModel.experiment ∧ physicalEprRemoteCorrelationModel.localSubsystem ≠ physicalEprRemoteCorrelationModel.remoteSubsystem ∧ ¬physicalPredictionWithoutCalculatedCorrelationModel.remoteCorrelationCalculatedAt .leftPositionMeasurement .eprLeftParticle .remotePosition .eprRightParticle .eprContinuousVariablePair .spatiallySeparatedCanonicalPair ∧ ¬physicalPredictionWithoutNonDisturbanceModel.observation.leavesUndisturbed .leftPositionMeasurement .remotePosition .spatiallySeparatedCanonicalPair ∧ ¬physicalEmptyRemoteQuantityModel.NonvacuousRemoteCorrelationPredictionAt ∧ (physicalEprRemoteCorrelationModel.observation |>.MatchesRealityCriterion physicalEprRemoteRealityInterpretation) ∧ ¬physicalEprRemoteRealityInterpretation.physicallyAdmissible .remoteSpinAlongA .separatedSingletCorrelations ∧ bellSingletRemoteCorrelationControlContext.EstablishedRemoteCorrelationPredictionAt bellSingletRemoteCorrelationControlModel := by obtain ⟨distinctExperiments, distinctSubsystems, _distinctProcedures, _position, _momentum, _certain, noCorrelation, noNonDisturbance, noSelectedResult, realityInterpretation, notAdmissible, _epr, bell⟩ := remote_correlations_distinguish_experiments_subsystems_observables_disturbance_and_singlet_controls exact ⟨distinctExperiments, distinctSubsystems, noCorrelation, noNonDisturbance, noSelectedResult, realityInterpretation, notAdmissible, bell⟩ end RemoteCorrelationPhysicalReuseAudit namespace CommutatorConePhysicalReuseAudit open PhysicsLogic.Framework.FiniteVelocityCommutatorConeContext.Countermodels example : physicalLiebRobinsonConeModel.system ≠ exponentialLatticeCommutatorConeControlModel.system ∧ physicalLiebRobinsonConeModel.sourceRegion ≠ physicalLiebRobinsonConeModel.targetRegion ∧ ¬physicalSeparatedSupportsWithoutConeCalculationModel.commutatorConeCalculatedFor .finiteRangeSpinDynamics .finiteRangeSpinCoupling .spinSourceObservable .spinTargetObservable .spinSourceRegion .spinTargetRegion .finiteRangeExponentialCone .boundedFiniteRangeDynamics ∧ ¬physicalCalculatedConeWithoutFiniteVelocityModel.finiteInformationVelocityFor .finiteRangeQuantumSpinChain .finiteRangeSpinCoupling .finiteRangeExponentialCone .boundedFiniteRangeDynamics ∧ ¬physicalEffectiveConeAllowsExactSignals.NoSuperluminalSignalling ∧ exponentialLatticeCommutatorConeControlContext.EstablishedFiniteVelocityCommutatorConeAt exponentialLatticeCommutatorConeControlModel := by obtain ⟨distinctSystems, _distinctInteractions, _distinctObservables, distinctRegions, _distinctBounds, _separated, noConeCalculation, _calculated, noVelocity, noExactCausality, _liebRobinson, exponentialControl⟩ := commutator_cones_distinguish_systems_interactions_supports_bounds_and_exact_causality exact ⟨distinctSystems, distinctRegions, noConeCalculation, noVelocity, noExactCausality, exponentialControl⟩ end CommutatorConePhysicalReuseAudit namespace InfraredOrderPhysicalReuseAudit open PhysicsLogic.Framework.InfraredOrderSuppressionContext.Countermodels example : let context := physicalMerminWagnerInfraredContext let model := context.selectedModel context.EstablishedInfraredOrderSuppressionAt model /\ context.DirectStructuralInfraredOrderSuppressionExplanationAt model /\ ¬context.SourceStructuralInfraredOrderSuppressionExplanationAt model .absent missingInfraredOrderSuppressionDerivationObtains := completed_infrared_route_and_ordinary_explanation_without_source_derivation_have_no_source_explanation example : physicalMerminWagnerInfraredModel.system ≠ boundVortexPlanarInfraredControlModel.system ∧ boundVortexPlanarInfraredControlModel.system ≠ higherFormWilsonLoopInfraredControlModel.system ∧ ¬physicalUnchargedMagnetizationModel.SelectedInfraredSetupAt ∧ ¬physicalSuppressedButBrokenSpinModel.SuppressedOrderImpliesUnbrokenAt ∧ ¬chargedLoopWithoutInfraredSuppressionModel.ExpectationSuppressedAt ∧ boundVortexPlanarInfraredControlModel.otherOrderValue = 1 ∧ higherFormWilsonLoopInfraredControlContext.EstablishedInfraredOrderSuppressionAt higherFormWilsonLoopInfraredControlModel := by obtain ⟨distinctMagnets, distinctGaugeSystem, _distinctSymmetries, _distinctMagneticChannels, _distinctWilsonLoop, _unchargedSuppressed, missingCharge, _brokenSuppressed, _broken, missingBridge, unsuppressedLoop, survivingRigidity, _ferromagnetic, _antiferromagnetic, _boundVortex, higherForm⟩ := infrared_order_distinguishes_magnets_vortices_wilson_loops_charge_and_surviving_order exact ⟨distinctMagnets, distinctGaugeSystem, missingCharge, missingBridge, unsuppressedLoop, survivingRigidity, higherForm⟩ example : PhysicalMerminWagnerFiniteBogoliubovModeAt physicalMerminWagnerBogoliubovReadout ∧ physicalMerminWagnerBogoliubovReadout.modeDenominatorValue = 17 / 8 := physical_mermin_wagner_finite_mode_retains_exchange_and_field_denominator example : PhysicalMerminWagnerOrderedLimitAndOtherOrderAt physicalMerminWagnerOrderedLimitReadout ∧ physicalMerminWagnerOrderedLimitReadout.unrelatedOrderValue = 1 := physical_mermin_wagner_zero_magnetization_permits_other_order_and_susceptibility example : PhysicalKosterlitzThoulessBoundVortexFreeEnergyAt physicalKosterlitzThoulessBoundFreeEnergyReadout ∧ PhysicalKosterlitzThoulessFreeVortexFreeEnergyAt physicalKosterlitzThoulessFreeVortexReadout := physical_kosterlitz_thouless_vortex_free_energy_changes_sign_across_bare_bound example : PhysicalKosterlitzThoulessNeutralTopologicalOrderAt physicalKosterlitzThoulessNeutralOrderReadout ∧ physicalKosterlitzThoulessNeutralOrderReadout.thermodynamicZeroFieldMagnetizationValue = 0 ∧ physicalKosterlitzThoulessNeutralOrderReadout.topologicalOrderValue = 1 := physical_kosterlitz_thouless_neutral_bound_vortices_retain_nonmagnetic_order example : PhysicalHigherFormContinuousDimensionThresholdAt physicalHigherFormContinuousThresholdReadout ∧ PhysicalHigherFormDiscreteDimensionThresholdAt physicalHigherFormDiscreteThresholdReadout ∧ physicalHigherFormContinuousThresholdReadout.spacetimeDimensionValue ≠ physicalHigherFormDiscreteThresholdReadout.spacetimeDimensionValue := physical_higher_form_continuous_and_discrete_dimension_thresholds_differ example : PhysicalHigherFormCoulombWilsonExpectationAt physicalHigherFormCoulombExpectationReadout ∧ 0 < physicalHigherFormCoulombExpectationReadout.finiteWilsonExpectationValue := physical_higher_form_finite_wilson_expectation_is_positive_before_infrared_limit example : ¬PhysicalMerminWagnerOrderedLimitAndOtherOrderAt physicalMerminWagnerLimitsReversedControl := by obtain ⟨_, _, _, _, _, _, _, _, _, _, _, reversedLimits, _, _, _⟩ := mermin_wagner_temperature_isotropy_channels_bogoliubov_limits_and_exception_are_independent exact reversedLimits example : ¬PhysicalKosterlitzThoulessNeutralTopologicalOrderAt physicalKosterlitzThoulessTopologicalOrderErasedControl := by obtain ⟨_, _, _, _, _, _, erasedOrder, _, _, _, _, _, _⟩ := vortex_entropy_neutrality_surviving_order_screening_and_source_exclusions_are_independent exact erasedOrder example : ¬PhysicalHigherFormContinuousDimensionThresholdAt physicalHigherFormDegreeAboveDimensionControl := higher_form_thresholds_degree_domain_wilson_exponential_other_order_and_ward_bridge_are_independent.2.2.1 example : ¬physicalInfraredEstablishment.establishedAt .absenceOfMagneticOrderExcludesEveryPhaseTransition .merminWagner1966 := infrared_source_overclaims_remain_epistemically_unestablished.1 end InfraredOrderPhysicalReuseAudit namespace RelativeMeasurementPhysicalReuseAudit open PhysicsLogic.Framework.LinearRelativeStateMeasurementContext.Countermodels example : physicalEverettRelativeMeasurementModel.finalState ≠ physicalZehPointerMeasurementControlModel.finalState ∧ physicalInterbranchCoherenceSuppressed .zehLeftPointerBranch .zehRightPointerBranch .environmentallyDecoupledPointer ∧ physicalZehPointerMeasurementControlModel.containsBranch .zehEnvironmentallyDecoupledState .zehLeftPointerBranch .environmentallyDecoupledPointer ∧ physicalZehPointerMeasurementControlModel.containsBranch .zehEnvironmentallyDecoupledState .zehRightPointerBranch .environmentallyDecoupledPointer ∧ ¬physicalZehPointerMeasurementControlModel.globallySelectsSingleOutcome .zehEnvironmentallyDecoupledState .environmentallyDecoupledPointer ∧ ¬physicalEverettBranchesWithoutRelativeRecordsModel.NonvacuousLinearRelativeStateMeasurementAt ∧ ¬physicalObserverRecordsComparedAt .rovelliFirstObserver .rovelliSecondObserver .rovelliSystemObserverCoupling .observerRelativeCorrelation ∧ physicalObserverRecordsComparedAt .rovelliFirstObserver .rovelliSecondObserver .rovelliObserverRecordComparison .observerRecordComparison ∧ ¬physicalObserverWithoutRecordAccessContext.OperationalObserverOfAt .everettInternalObserver .everettSpinSystem .everettUnitarySpinCoupling .everettUpMemory .closedEverettUnitaryEvolution ∧ physicalRovelliRelativeMeasurementControlContext.EstablishedLinearRelativeStateMeasurementAt physicalRovelliRelativeMeasurementControlModel := by obtain ⟨distinctEverettZeh, _distinctZehRovelli, _distinctSystemAlternatives, _distinctObservers, _distinctRecords, _distinctBranches, _distinctEverettInteraction, _distinctZehInteraction, decoherence, firstRetained, secondRetained, noGlobalSelection, _noEverettDecoherence, _noCalibration, _noUncalibratedResult, _noRelativeRecord, noRecordResult, _globalSelection, _noGloballySelectedResult, _everettObserver, _rovelliObserver, noPrematureComparison, explicitComparison, _secondObserver, noRecordAccess, _everett, _zeh, rovelli⟩ := relative_measurement_distinguishes_observers_records_branches_decoherence_and_comparison exact ⟨distinctEverettZeh, decoherence, firstRetained, secondRetained, noGlobalSelection, noRecordResult, noPrematureComparison, explicitComparison, noRecordAccess, rovelli⟩ end RelativeMeasurementPhysicalReuseAudit namespace CoarseGrainedScalingPhysicalReuseAudit open PhysicsLogic.Framework.CoarseGrainedScalingContext.Countermodels example : physicalKadanoffBlockScalingModel.microscopicDescription ≠ dynamicHydrodynamicShellScalingControlModel.microscopicDescription ∧ physicalKadanoffBlockScalingModel.sector .singularFreeEnergy ∧ physicalKadanoffBlockScalingModel.sector .longDistanceSpinCorrelation ∧ dynamicHydrodynamicShellScalingControlModel.sector .dynamicVelocityCorrelation ∧ ¬physicalKadanoffBlockScalingModel.effectivePredicts .restoredCriticalIsingModel .singularFreeEnergy .singularFreeEnergyValue ∧ ¬physicalBlockWithoutCoarseGrainingControlModel.NonvacuousControlledSectorScalingAt ∧ ¬physicalBlockWithoutPredictionMatchingModel.NonvacuousControlledSectorScalingAt ∧ ¬physicalEmptyCriticalObservableSectorModel.NonvacuousControlledSectorScalingAt ∧ dynamicHydrodynamicShellScalingControlContext.EstablishedSectorScalingAt .randomlyStirredIncompressibleFluid .selectedHydrodynamicMomentumShell .stochasticInfraredHydrodynamics dynamicHydrodynamicShellScalingControlContext.selectedSector ∧ physicalKadanoffBlockScalingContext.toReexpressedScaleMatchingContext.MatchedAt .criticalIsingLattice .selectedKadanoffCellScale .equilibriumCriticalScaling physicalKadanoffBlockScalingContext.selectedSector := by obtain ⟨distinctDescriptions, _distinctReduced, _distinctEffective, _distinctScales, _distinctRegimes, freeEnergy, spinCorrelation, _excludedVelocity, velocityCorrelation, _excludedFreeEnergy, _reexpressedObservable, _reexpressedValue, _microscopicPrediction, _transformedPrediction, noRawPrediction, _noCoarseControl, noUncontrolledScaling, _noRescalingControl, _noUncontrolledRescaling, _controlledWithoutMatching, noPredictionMatching, noEmptySector, _noAlternateScale, _blockScaling, dynamicScaling, blockMatching, _dynamicMatching⟩ := coarse_graining_distinguishes_block_spins_shells_scales_observables_and_reexpressed_predictions exact ⟨distinctDescriptions, freeEnergy, spinCorrelation, velocityCorrelation, noRawPrediction, noUncontrolledScaling, noPredictionMatching, noEmptySector, dynamicScaling, blockMatching⟩ end CoarseGrainedScalingPhysicalReuseAudit namespace RenormalizedNuclearEftPhysicalReuseAudit open PhysicsLogic.Framework.RenormalizedPredictionContext.Countermodels open PhysicsLogic.Framework.ResummedSectorRenormalizedPredictionHandoff.Countermodels example : physicalPdsNuclearRenormalizationModel.system ≠ physicalThreeBosonRenormalizationControlModel.system ∧ physicalRenormalizationScaleFamilyFor physicalPdsNuclearRenormalizationModel.system ≠ physicalRenormalizationScaleFamilyFor physicalThreeBosonRenormalizationControlModel.system ∧ ¬physicalPdsWithFrozenContactCouplingModel.foundation.ParametersRun ∧ ¬physicalThreeBosonWithoutCutoffCancellationModel.NonvacuousRenormalizedPredictionAt ∧ physicalThreeBosonSubsystemRenormalizationControlHandoff.resummationModel.system = .shortRangeTwoBosonPair ∧ physicalThreeBosonSubsystemRenormalizationControlHandoff.predictionModel.system = .threeBosonParticleDimerScattering ∧ ¬physicalThreeBosonWithoutSubsystemHandoff.SelectedResummedRenormalizationHandoffAt ∧ physicalPdsSubsystemRenormalizationHandoff.NonvacuousResummationGroundedRenormalizedPredictionAt ∧ physicalThreeBosonSubsystemRenormalizationControlHandoff.NonvacuousResummationGroundedRenormalizedPredictionAt := by obtain ⟨distinctPredictionSystems, _distinctTargets, _distinctRegimes, distinctScaleFamilies, _distinctEffectiveDescriptions, _runningPdsCoupling, _runningThreeBodyCounterterm, _singlet, _triplet, _pdsExcludesDimer, _dimer, _threeBodyExcludesSinglet, _missingRunningCalculation, _noMissingRunningPrediction, _frozenCalculation, noFrozenRunning, _noFrozenPrediction, _noTargetPrediction, _noTargetResult, _noCutoffMatching, noCutoffResult, _pdsEstablished, _threeBodyEstablished⟩ := renormalized_predictions_distinguish_pds_channels_three_body_cutoffs_running_and_matching obtain ⟨_distinctSubsystems, _distinctMomenta, _distinctSubsystemRegimes, _distinctDressedDescriptions, _distinctRecoveredTargets, _distinctFullPredictionSystems, twoBosonSubsystem, threeBosonPrediction, _pdsHandoff, _threeBodyHandoff, _alignedMissingHandoff, missingHandoff, _missingComposition, pdsComposed, threeBodyComposed⟩ := renormalized_eft_distinguishes_two_body_subsystems_three_body_targets_and_missing_handoffs exact ⟨distinctPredictionSystems, distinctScaleFamilies, noFrozenRunning, noCutoffResult, twoBosonSubsystem, threeBosonPrediction, missingHandoff, pdsComposed, threeBodyComposed⟩ end RenormalizedNuclearEftPhysicalReuseAudit namespace ExactExteriorGravityPhysicalReuseAudit open PhysicsLogic.Framework.ExactExteriorVacuumMetricContext.Countermodels example : physicalSchwarzschildExteriorModel.metric ≠ physicalKerrRotatingExteriorControlModel.metric ∧ physicalSchwarzschildExteriorModel.boundary ≠ physicalKerrRotatingExteriorControlModel.boundary ∧ physicalExteriorParameterRetainedAt .stationaryAxisymmetricRotatingMetric .rotationAngularMomentum ∧ ¬physicalExteriorParameterRetainedAt .staticSphericalPointMassMetric .rotationAngularMomentum ∧ ¬physicalPointMetricWithRotatingRingBoundaryModel.ExteriorVacuumSetup ∧ ¬physicalKerrWithoutRingRegularityModel.NonvacuousExactExteriorVacuumMetricSolutionAt ∧ physicalSchwarzschildExteriorContext.EstablishedExactExteriorVacuumMetricSolutionAt physicalSchwarzschildExteriorModel ∧ physicalKerrRotatingExteriorControlContext.EstablishedExactExteriorVacuumMetricSolutionAt physicalKerrRotatingExteriorControlModel := by obtain ⟨distinctMetrics, _distinctSituations, distinctBoundaries, _distinctConstructions, _spherical, _axisymmetric, _noStaticAxisymmetry, _staticMass, _rotatingMass, kerrAngularMomentum, noStaticAngularMomentum, _polarCoordinates, _asymptoticCoordinates, _noVacuumCalculation, _expressionWithoutVacuum, _noVacuumResult, _vacuumWithoutExpression, _noExpressionResult, _vacuumWithoutAsymptotics, _noAsymptoticResult, invalidBoundary, _invalidBoundaryResult, _kerrVacuum, _missingRingScope, noRingScopeResult, schwarzschild, kerr⟩ := exterior_vacuum_metrics_distinguish_point_masses_rotating_rings_symmetries_and_scoped_outputs exact ⟨distinctMetrics, distinctBoundaries, kerrAngularMomentum, noStaticAngularMomentum, invalidBoundary, noRingScopeResult, schwarzschild, kerr⟩ end ExactExteriorGravityPhysicalReuseAudit namespace TopologicalHallPhysicalReuseAudit open PhysicsLogic.Framework.TopologicalTransverseResponseContext.Countermodels example : physicalMagneticBlochHallModel.system ≠ physicalTwistedManyBodyHallControlModel.system ∧ physicalMagneticBlochHallModel.parameterSpace ≠ physicalTwistedManyBodyHallControlModel.parameterSpace ∧ physicalHallMicroscopicScopeAt .cleanPeriodicMagneticElectronBandSystem .periodicNoninteractingCleanBands ∧ ¬physicalHallMicroscopicScopeAt .cleanPeriodicMagneticElectronBandSystem .interactingDisorderedManyBodyBulk ∧ physicalHallMicroscopicScopeAt .interactingDisorderedManyBodyHallSystem .interactingDisorderedManyBodyBulk ∧ ¬physicalManyBodyFixedTwistWithoutThermodynamicAveragingModel.NonvacuousTopologicalTransverseResponseAt ∧ physicalHallGroundStateDegeneracyValueAt .degenerateTwistedManyBodyGroundSpace = 3 ∧ physicalFractionalGroundSpaceNormalizationAt .degenerateFractionalHallConductance .degenerateTwistedManyBodyGroundSpace .degenerateGroundSpaceTotalChernNumber .electronSquaredOverPlanckConductanceQuantum ∧ ¬physicalDegenerateFractionalManyBodyHallModel.NonvacuousTopologicalTransverseResponseAt ∧ physicalMagneticBlochHallContext.EstablishedTopologicalTransverseResponseAt .magneticBlochPeriodicOneParticleDescription .cleanPeriodicMagneticElectronBandSystem .magneticBrillouinMomentumTwoTorus .filledMagneticBlochSubband .magneticBlochIntegerHallConductance .magneticBlochBerryCurvatureIntegral .magneticBlochBandChernNumber .electronSquaredOverPlanckConductanceQuantum .periodicBandWeakFieldZeroTemperature ∧ physicalTwistedManyBodyHallControlContext.EstablishedTopologicalTransverseResponseAt .interactingDisorderedTwistedManyBodyDescription .interactingDisorderedManyBodyHallSystem .gaugeCovariantBoundaryTwistTwoTorus .nondegenerateTwistedManyBodyGroundState .thermodynamicTwistAveragedIntegerHallConductance .nondegenerateTwistFirstChernIntegral .nondegenerateTwistFirstChernNumber .electronSquaredOverPlanckConductanceQuantum .interactingManyBodyThermodynamicWeakField := by obtain ⟨_distinctDescriptions, distinctSystems, distinctParameterSpaces, _distinctSectors, _distinctResponses, _distinctCurvatures, _distinctInvariants, _sameUnit, _distinctRegimes, cleanScope, cleanExcludesInteractions, interactingScope, _interactingExcludesClean, _noMomentumFamily, _noGapResult, _noCurvatureResult, _noIntegralityResult, _noNormalizationResult, _wrongBlochTorus, _wrongManyBodyTorus, _bulkTwistIndependence, _noFiniteTwistIndependence, _finiteGap, _noFixedTwistCurvature, noFiniteQuantization, _nondegeneracy, degeneracy, fractionalNormalization, _noNondegenerateFraction, _noFractionalIntegerNormalization, noFractionalIntegerRoute, blochEstablished, manyBodyEstablished⟩ := topological_hall_response_distinguishes_magnetic_bands_many_body_twists_and_degenerate_fractions exact ⟨distinctSystems, distinctParameterSpaces, cleanScope, cleanExcludesInteractions, interactingScope, noFiniteQuantization, degeneracy, fractionalNormalization, noFractionalIntegerRoute, blochEstablished, manyBodyEstablished⟩ end TopologicalHallPhysicalReuseAudit namespace InflationaryCosmologyPhysicalReuseAudit open PhysicsLogic.Framework.InflationaryPatchContext.Countermodels open PhysicsLogic.Framework.BubbleTransitionCompletionContext.Countermodels open PhysicsLogic.Framework.InflationaryPatchBubbleObstructionHandoff.Countermodels example : physicalGuthFalseVacuumPatchModel.mechanism ≠ physicalAlbrechtSteinhardtFluctuationPatchModel.mechanism ∧ physicalInflationObligationStatusAt .radiativelyInducedThermalFluctuationCosmology .fluctuationCurvatureCorrections = .deferred ∧ physicalInflationObligationStatusAt .radiativelyInducedThermalFluctuationCosmology .fluctuationThermalEquilibration = .deferred ∧ physicalInflationObligationStatusAt .radiativelyInducedThermalFluctuationCosmology .fluctuationBaryogenesis = .deferred ∧ ¬physicalInflationPostTransitionAccountingCompleteAt .radiativelyInducedThermalFluctuationCosmology ∧ ¬physicalGuthPatchAndBubbleWithoutSourceHandoff.SelectedPatchBubbleObstructionHandoffAt ∧ ¬physicalReplacementMisclassifiedAsSameMechanism.SelectedPatchBubbleAlignmentAt ∧ physicalGuthSameMechanismInflationBubbleHandoff.NonvacuousInflationaryPatchBubbleObstructionAt ∧ physicalAlbrechtSteinhardtReplacementInflationBubbleHandoff.NonvacuousInflationaryPatchBubbleObstructionAt := by obtain ⟨_distinctModels, _distinctInitials, _distinctRegions, distinctMechanisms, _sameHomogeneity, _sameFlatness, _initialRadius, _expandedRadius, _coverage, _noSmallCoverage, _noExpansion, _noHomogeneity, _noFlatness, _noFlatPotential, _noCoverageResult, _wrongBubbleMechanism, _coincidentTargets, _oldFailedExit, _noOldReplacement, _newReplacement, _oldTrigger, _oldParticles, deferredCurvature, deferredThermalization, deferredBaryogenesis, _oldIncomplete, newIncomplete, _oldEstablished, _newEstablished⟩ := inflationary_patches_distinguish_false_vacua_fluctuation_regions_exit_failure_and_deferred_physics obtain ⟨_distinctBubbleModels, _distinctBubbleInitials, _distinctBubbleRegions, _distinctBubbleMechanisms, _distinctNucleationRegimes, _oldSameMechanism, _newReplacementMechanism, _oldSourceHandoff, _newSourceHandoff, _missingPatch, _missingBubble, noSourceHandoff, _noMissingComposition, noSameClassification, _noMismatchedModel, _noPercolatingObstruction, oldComposition, newComposition⟩ := inflationary_bubble_handoffs_distinguish_failed_false_vacuum_exit_and_fluctuation_replacement exact ⟨distinctMechanisms, deferredCurvature, deferredThermalization, deferredBaryogenesis, newIncomplete, noSourceHandoff, noSameClassification, oldComposition, newComposition⟩ end InflationaryCosmologyPhysicalReuseAudit namespace NonequilibriumFluctuationPhysicalReuseAudit open PhysicsLogic.Framework.EntropyProductionFluctuationContext.Countermodels example : physicalFluctuationAssumptionStatusAt .conjecturedFiniteSegmentLyapunovWeight .conjecturedFiniteSegmentNaturalMeasure = .conjectured ∧ physicalFluctuationAssumptionStatusAt .conjecturedFiniteSegmentLyapunovWeight .assumedWcaFluidHyperbolicity = .assumed ∧ physicalFluctuationAssumptionStatusAt .conjecturedFiniteSegmentLyapunovWeight .finiteSegmentConditionalRatioDerivation = .conditionallyDerived ∧ physicalFluctuationAssumptionStatusAt .assumedHyperbolicSrbMarkovWeight .unprovedManyBodyChaoticHypothesis = .assumed ∧ physicalFluctuationAssumptionStatusAt .assumedHyperbolicSrbMarkovWeight .importedHyperbolicSrbConstruction = .importedConditional ∧ physicalFluctuationAssumptionStatusAt .assumedHyperbolicSrbMarkovWeight .boundedSrbConditionalRatioDerivation = .conditionallyDerived ∧ physicalFluctuationSourceBridgeAt .conjecturedFiniteSegmentLyapunovWeight .gaussConstraintIdentifiesShearStressWithPhaseContraction ∧ physicalFluctuationSourceBridgeAt .conjecturedFiniteSegmentLyapunovWeight .fixedShearSpatialTimeReversalNegatesLocalLyapunovSpectrum ∧ physicalFluctuationSourceBridgeAt .assumedHyperbolicSrbMarkovWeight .markovTilesRepresentUnstableManifoldSrbWeight ∧ physicalFluctuationSourceBridgeAt .assumedHyperbolicSrbMarkovWeight .stableUnstableTransversalityBoundsJacobianAngleFactor ∧ ¬physicalFluctuationExactFiniteTimeRatioAt .conjecturedFiniteSegmentLyapunovWeight ∧ ¬physicalFluctuationExactFiniteTimeRatioAt .assumedHyperbolicSrbMarkovWeight ∧ ¬physicalEvansWithoutGaussThermostatContractionBridgeModel.NonvacuousEntropyProductionFluctuationSymmetryAt ∧ ¬physicalEvansWithoutReversedLyapunovSpectrumBridgeModel.NonvacuousEntropyProductionFluctuationSymmetryAt ∧ ¬physicalGallavottiWithoutUnstableManifoldMarkovBridgeModel.NonvacuousEntropyProductionFluctuationSymmetryAt ∧ ¬physicalGallavottiWithoutTransverseAngleBoundBridgeModel.NonvacuousEntropyProductionFluctuationSymmetryAt ∧ ¬physicalGallavottiWithEvansSegmentConstructionModel.NonvacuousEntropyProductionFluctuationSymmetryAt ∧ ¬physicalEvansWithSrbConstructionModel.NonvacuousEntropyProductionFluctuationSymmetryAt ∧ physicalEvansFiniteSegmentFluctuationModel.selectedTemporalAsymmetryContext.DirectedTemporalAsymmetryAt physicalEvansFiniteSegmentFluctuationModel.system physicalEvansFiniteSegmentFluctuationModel.dynamics physicalEvansFiniteSegmentFluctuationModel.probabilityConstruction physicalEvansFiniteSegmentFluctuationModel.positiveWindow physicalEvansFiniteSegmentFluctuationModel.negativeWindow physicalEvansFiniteSegmentFluctuationModel.positiveProbability physicalEvansFiniteSegmentFluctuationModel.negativeProbability physicalEvansFiniteSegmentFluctuationModel.regime ∧ physicalGallavottiChaoticSrbFluctuationControlModel.selectedTemporalAsymmetryContext.DirectedTemporalAsymmetryAt physicalGallavottiChaoticSrbFluctuationControlModel.system physicalGallavottiChaoticSrbFluctuationControlModel.dynamics physicalGallavottiChaoticSrbFluctuationControlModel.probabilityConstruction physicalGallavottiChaoticSrbFluctuationControlModel.positiveWindow physicalGallavottiChaoticSrbFluctuationControlModel.negativeWindow physicalGallavottiChaoticSrbFluctuationControlModel.positiveProbability physicalGallavottiChaoticSrbFluctuationControlModel.negativeProbability physicalGallavottiChaoticSrbFluctuationControlModel.regime ∧ physicalFluctuationProductionContext.EnsembleSecondLaw .isoenergeticSllodShearedWcaDiskFluid ∧ ¬physicalFluctuationProductionContext.PathwiseSecondLaw .isoenergeticSllodShearedWcaDiskFluid ∧ physicalFluctuationProductionContext.EnsembleSecondLaw .chaoticHypothesisShearedManyParticleFluid ∧ ¬physicalFluctuationProductionContext.PathwiseSecondLaw .chaoticHypothesisShearedManyParticleFluid := by obtain ⟨_distinctSystems, _distinctDynamics, _distinctPositiveWindows, _distinctNegativeWindows, _distinctPositiveProductions, _distinctNegativeProductions, _distinctMeanProductions, _distinctPositiveProbabilities, _distinctNegativeProbabilities, _distinctConstructions, _distinctObservationTimes, _distinctLongTimes, _distinctRegimes, _fixedShearReversal, _bareReversalRejected, _chaoticReversal, _wrongChaoticReversal, gaussContractionBridge, fixedShearLyapunovBridge, _noEvansMarkovWeight, unstableMarkovBridge, transverseAngleBridge, _noGallavottiFixedShearBridge, conjecturedMeasure, assumedWcaHyperbolicity, conditionalFiniteSegmentDerivation, _finiteNumericalAgreement, assumedChaoticHypothesis, conditionalSrbConstruction, conditionalSrbRatioDerivation, _positiveMeanNumericalAgreement, _boundedCorrection, _noFiniteSegmentBoundedCorrection, noEvansExactFiniteRatio, noGallavottiExactFiniteRatio, _noConjecturedMeasure, _noAssumedWcaHyperbolicity, noGaussContractionBridge, _noShearPreservingReversal, noReversedLyapunovBridge, _noOppositeProduction, _noRatioCalculation, _noFiniteNegativeWindow, _noLongTimeSuppression, _noChaoticHypothesis, _noImportedConstruction, noUnstableMarkovBridge, _noBoundedCorrectionScope, noTransverseAngleBridge, noWrongGallavottiConstruction, noWrongEvansConstruction, _evansEstablished, _gallavottiEstablished, evansArrow, gallavottiArrow, evansEnsembleLaw, noEvansPathwiseLaw, gallavottiEnsembleLaw, noGallavottiPathwiseLaw⟩ := entropy_fluctuations_distinguish_sheared_segments_srb_measures_reversal_assumptions_and_temporal_arrows exact ⟨conjecturedMeasure, assumedWcaHyperbolicity, conditionalFiniteSegmentDerivation, assumedChaoticHypothesis, conditionalSrbConstruction, conditionalSrbRatioDerivation, gaussContractionBridge, fixedShearLyapunovBridge, unstableMarkovBridge, transverseAngleBridge, noEvansExactFiniteRatio, noGallavottiExactFiniteRatio, noGaussContractionBridge, noReversedLyapunovBridge, noUnstableMarkovBridge, noTransverseAngleBridge, noWrongGallavottiConstruction, noWrongEvansConstruction, evansArrow, gallavottiArrow, evansEnsembleLaw, noEvansPathwiseLaw, gallavottiEnsembleLaw, noGallavottiPathwiseLaw⟩ end NonequilibriumFluctuationPhysicalReuseAudit namespace MacroscopicCoherencePhysicalReuseAudit open PhysicsLogic.Framework.MacroscopicCoherenceSuppressionContext.Countermodels example : physicalZehEnvironmentalCoherenceModel.state ≠ physicalGrwObjectiveLocalizationCoherenceModel.state ∧ physicalZehEnvironmentalCoherenceModel.mechanism ≠ physicalGrwObjectiveLocalizationCoherenceModel.mechanism ∧ physicalCoherenceDynamicsAt .universalUnitaryEnvironmentalEntanglement .universalLinearSchrodingerEvolution ∧ physicalCoherenceDynamicsAt .constituentPoissonianSpontaneousLocalization .stochasticNonHamiltonianLocalizationEvolution ∧ physicalZehPureEnvironmentalBoundaryReadout.HasGloballyPureSelectedStateAt ∧ ¬physicalZehPureEnvironmentalBoundaryReadout.HasDynamicallyProducedStatisticalMixtureAt ∧ physicalGrwNonselectiveLocalizedMixtureReadout.HasDynamicallyProducedStatisticalMixtureAt ∧ ¬physicalGrwNonselectiveLocalizedMixtureReadout.HasOneActualizedBranchAt ∧ physicalCoherencePredictionContext.PredictionEquivalentIn .universallyPureEnvironmentEntangledPointerState .environmentalReducedPointerMixtureComparator .denseSpectrumUniversalEnvironmentalBranchDecoupling physicalCoherenceLocalPointerSector ∧ physicalCoherencePredictionContext.InterferenceWitnessAgainst .universallyPureEnvironmentEntangledPointerState .environmentalReducedPointerMixtureComparator .environmentCorrelatedFirstPointerBranch .environmentCorrelatedSecondPointerBranch .denseSpectrumUniversalEnvironmentalBranchDecoupling physicalCoherenceGlobalRecoherenceSector ∧ ¬physicalCoherenceMeasurementAccessibleAt .hypotheticalGlobalBranchRecoherenceMeasurement .denseSpectrumUniversalEnvironmentalBranchDecoupling ∧ ¬physicalCoherenceOutcomeActualizationContext.SelectsUniquePossibleOutcomeIn .universallyPureEnvironmentEntangledPointerState .universalUnitaryEnvironmentalEntanglement .universalUncollapsedEnvironmentalHistory .denseSpectrumUniversalEnvironmentalBranchDecoupling ∧ ¬physicalCoherenceOutcomeActualizationContext.SelectsUniquePossibleOutcomeIn .objectivelyLocalizedNonselectivePointerEnsemble .constituentPoissonianSpontaneousLocalization .nonselectiveStochasticPointerEnsembleHistory .amplifiedObjectiveMacroscopicLocalization ∧ physicalCoherenceOutcomeActualizationContext.SelectsUniquePossibleOutcomeIn .objectivelyLocalizedNonselectivePointerEnsemble .constituentPoissonianSpontaneousLocalization .firstIndividualLocalizedPointerHistory .amplifiedObjectiveMacroscopicLocalization ∧ physicalCoherenceOutcomeActualizationContext.SelectsUniquePossibleOutcomeIn .objectivelyLocalizedNonselectivePointerEnsemble .constituentPoissonianSpontaneousLocalization .secondIndividualLocalizedPointerHistory .amplifiedObjectiveMacroscopicLocalization ∧ physicalCoherenceEnergyStatusAt .constituentPoissonianSpontaneousLocalization = .positiveLocalizationHeating ∧ physicalCoherenceEnergyStatusAt .constituentPoissonianSpontaneousLocalization ≠ .exactEnergyConservation := by obtain ⟨distinctStates, _distinctFirstBranches, _distinctSecondBranches, distinctMechanisms, _distinctRegimes, unitaryEnvironmentalDynamics, _noEnvironmentalStochasticDynamics, objectiveStochasticDynamics, _noObjectiveUnitaryDynamics, _denseEnvironmentalSpectrum, _distinctEnvironmentRecords, _decayingMatrixElement, _noEnvironmentalPoissonProcess, _rareConstituentLocalization, _constituentAmplificationBridge, _nonselectiveMasterEquation, _individualStochasticHistory, _amplifiedLocalizationRate, _noUnitaryConstituentAmplification, _noZehEnergyAssessment, positiveLocalizationHeating, noExactLocalizationEnergyConservation, _noDenseSpectrum, _noDistinctRecords, _noHamiltonianDecay, _noZehInterferenceBridge, _wrongZehMechanism, _coincidentZehBranches, _noConstituentLocalization, _noConstituentAmplification, _noMacroscopicSeparation, _noNonselectiveEquation, _noGrwInterferenceBridge, _wrongGrwMechanism, _wrongMicroscopicScope, _zehEstablished, _grwEstablished, _zehSuppressed, _zehReducedPointerDiagonal, zehGloballyPure, noZehProperMixture, _noZehSelectedBranch, _grwSuppressed, grwNonselectiveMixture, _noGrwGlobalPurity, noGrwEnsembleOutcome, localPointerEquivalence, globalCoherenceWitness, noOperationalRecoherence, noZehOutcome, noNonselectiveOutcome, firstIndividualOutcome, secondIndividualOutcome⟩ := macroscopic_coherence_distinguishes_unitary_environmental_branches_stochastic_localization_and_outcomes exact ⟨distinctStates, distinctMechanisms, unitaryEnvironmentalDynamics, objectiveStochasticDynamics, zehGloballyPure, noZehProperMixture, grwNonselectiveMixture, noGrwEnsembleOutcome, localPointerEquivalence, globalCoherenceWitness, noOperationalRecoherence, noZehOutcome, noNonselectiveOutcome, firstIndividualOutcome, secondIndividualOutcome, positiveLocalizationHeating, noExactLocalizationEnergyConservation⟩ end MacroscopicCoherencePhysicalReuseAudit namespace ReciprocalTransportPhysicalReuseAudit open PhysicsLogic.Framework.ReciprocalTransport open PhysicsLogic.Framework.ReciprocalTransport.Countermodels example : physicalReciprocalScopeStatusAt .zeroFieldAnisotropicCrystalHeatRegression .anisotropicZeroFieldHeatReciprocity = .provedInSelectedOriginalPaper ∧ physicalReciprocalScopeStatusAt .zeroFieldAnisotropicCrystalHeatRegression .generalTransportTheoremBeyondOnsagerPartI = .deferredBeyondSelectedOriginalPaper ∧ physicalReciprocalScopeStatusAt .magneticMixedParityFourPoleRegression .independentFluctuationRegressionAssumption = .separatelyAssumed ∧ physicalReciprocalRegressionValidity.RegimeRestricted .zeroFieldAnisotropicCrystalHeatRegression ∧ physicalReciprocalRegressionValidity.RegimeRestricted .magneticMixedParityFourPoleRegression ∧ ¬physicalReciprocalOrdinarySameFieldReversalAt .zeroFieldAnisotropicCrystalHeatRegression .positiveMagneticField ∧ ¬physicalReciprocalOrdinarySameFieldReversalAt .zeroFieldAnisotropicCrystalHeatRegression .positiveCoriolisRotation ∧ physicalReciprocalVariableParity .capacitiveTerminalCharge = .even ∧ physicalReciprocalVariableParity .inductiveCircuitCurrent = .odd ∧ physicalReciprocalJointFieldReversalAt .magneticMixedParityFourPoleRegression .positiveMagneticField .reversedMagneticField ∧ ¬physicalOnsagerWithoutIndependentRegressionModel.NonvacuousReciprocalTransportAt ∧ ¬physicalCasimirWithoutJointMagneticFieldReversalModel.NonvacuousReciprocalTransportAt ∧ ¬physicalCasimirWithoutIndependentRegressionModel.NonvacuousReciprocalTransportAt ∧ physicalHallSymmetricCrossCoefficientAt .positiveMagneticField = 3 ∧ physicalHallSymmetricCrossCoefficientAt .reversedMagneticField = 3 ∧ physicalHallAntisymmetricCrossCoefficientAt .positiveMagneticField = 1 ∧ physicalHallAntisymmetricCrossCoefficientAt .reversedMagneticField = -1 ∧ (∀ firstForce secondForce, physicalAntisymmetricHallEntropyContributionAt .positiveMagneticField firstForce secondForce = 0) ∧ physicalOnsagerZeroFieldHeatTransportContext.EstablishedReciprocalTransportAt physicalOnsagerZeroFieldHeatTransportModel ∧ physicalCasimirMagneticParityTransportContext.EstablishedReciprocalTransportAt physicalCasimirMagneticParityTransportModel := by obtain ⟨_distinctProcesses, _distinctFields, _distinctLeftVariables, _distinctRightVariables, _zeroFieldReversal, _magneticFieldReversal, _coriolisRotationReversal, _heatXAxisParity, _heatYAxisParity, capacitiveParity, inductiveParity, _zeroFieldOrdinaryReversal, noSameMagneticReversal, noSameCoriolisReversal, jointFieldReversal, _noUnreversedMagneticRoute, provedHeatResult, deferredGeneralResult, _excludedMagneticScope, _excludedCoriolisScope, independentlyAssumedRegression, _restoredMagneticScope, _nondissipativeHallScope, onsagerRegressionWindow, casimirRegressionWindow, noOnsagerRegression, _noOnsagerWindow, _noOnsagerCorrelation, _noOnsagerMagnetic, _noOnsagerCoriolis, _noCoincidentHeatVariables, noCasimirJointReversal, noCasimirRegression, _noCasimirWindow, _noCasimirCorrelation, _noIncorrectCasimirParity, _noCoincidentCasimirVariables, _matchedFieldReversalBoundary, _fieldReversedParityBoundary, _noSameFieldMicroscopicBalance, _noSameFieldParityReciprocity, symmetricPositiveField, symmetricReversedField, antisymmetricPositiveField, antisymmetricReversedField, noAntisymmetricEntropy, onsagerEstablished, casimirEstablished⟩ := reciprocal_transport_distinguishes_zero_field_heat_magnetic_parity_hall_entropy_and_regression exact ⟨provedHeatResult, deferredGeneralResult, independentlyAssumedRegression, onsagerRegressionWindow, casimirRegressionWindow, noSameMagneticReversal, noSameCoriolisReversal, capacitiveParity, inductiveParity, jointFieldReversal, noOnsagerRegression, noCasimirJointReversal, noCasimirRegression, symmetricPositiveField, symmetricReversedField, antisymmetricPositiveField, antisymmetricReversedField, noAntisymmetricEntropy, onsagerEstablished, casimirEstablished⟩ end ReciprocalTransportPhysicalReuseAudit namespace NuclearBoundStateExtractionPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.EffectiveRangeBoundStateExtractionContext.Countermodels example : physicalNuclearExtractionSourceBridgeAt .betheComparisonGroundStateSpecialization .exactRadialComparisonWronskianIdentity ∧ physicalNuclearExtractionSourceBridgeAt .betheComparisonGroundStateSpecialization .deuteronGroundStateComparisonSpecialization ∧ ¬physicalNuclearExtractionSourceBridgeAt .betheComparisonGroundStateSpecialization .scatteringMatrixZeroAtDeuteronPole ∧ physicalNuclearExtractionSourceBridgeAt .blattJacksonVariationalScatteringMatrixContinuation .stationarySchwingerVariationalFunctional ∧ physicalNuclearExtractionSourceBridgeAt .blattJacksonVariationalScatteringMatrixContinuation .analyticContinuationToImaginaryBoundMomentum ∧ physicalNuclearExtractionSourceBridgeAt .blattJacksonVariationalScatteringMatrixContinuation .scatteringMatrixZeroAtDeuteronPole ∧ physicalNuclearExtractionScopeStatusAt .betheComparisonGroundStateSpecialization .neutronProtonSingletEffectiveRangeEstimate = .borrowedUsingApproximateChargeEquality ∧ physicalNuclearExtractionScopeStatusAt .blattJacksonVariationalScatteringMatrixContinuation .neutronProtonSingletEffectiveRangeEstimate = .underdeterminedByAvailableData ∧ physicalNuclearExtractionScopeStatusAt .betheComparisonGroundStateSpecialization .protonProtonVersusNeutronProtonSingletComparison = .qualifiedChargeDifferenceReported ∧ physicalNuclearExtractionScopeStatusAt .blattJacksonVariationalScatteringMatrixContinuation .protonProtonVersusNeutronProtonSingletComparison = .outsideSelectedSourceScope ∧ ¬physicalNuclearSingletRangeDeterminedAt .blattJacksonVariationalScatteringMatrixContinuation ∧ ¬physicalNuclearTensorContributionIncludedAt .betheComparisonGroundStateSpecialization ∧ ¬physicalNuclearPotentialShapeUniquelyResolvedAt .blattJacksonVariationalScatteringMatrixContinuation ∧ ¬physicalNuclearZeroRemainderBoundStateRelationAt .betheComparisonGroundStateSpecialization ∧ ¬physicalNuclearZeroRemainderBoundStateRelationAt .blattJacksonVariationalScatteringMatrixContinuation ∧ ¬physicalBetheWithoutParahydrogenDatumModel.NonvacuousEffectiveRangeBoundStateExtractionAt ∧ ¬physicalBetheWithSingletDeuteronSubstitutionModel.NonvacuousEffectiveRangeBoundStateExtractionAt ∧ ¬physicalBlattJacksonWithoutCoherentAmplitudeModel.NonvacuousEffectiveRangeBoundStateExtractionAt ∧ ¬physicalBlattJacksonWithBetheGroundStateSubstitutionModel.NonvacuousEffectiveRangeBoundStateExtractionAt ∧ ¬physicalBlattJacksonWithTensorExtensionModel.NonvacuousEffectiveRangeBoundStateExtractionAt ∧ ¬physicalBlattJacksonWithSingletRangeClaimModel.NonvacuousEffectiveRangeBoundStateExtractionAt ∧ physicalBetheGroundStateExtractionModel.QualifiedParameterEvidenceSupportAt ∧ physicalBlattJacksonPoleExtractionModel.QualifiedParameterEvidenceSupportAt ∧ physicalBetheGroundStateExtractionContext.EstablishedEffectiveRangeBoundStateExtractionAt physicalBetheGroundStateExtractionModel ∧ physicalBlattJacksonPoleExtractionContext.EstablishedEffectiveRangeBoundStateExtractionAt physicalBlattJacksonPoleExtractionModel := by obtain ⟨_distinctSources, _distinctEvidence, _betheTriplet, _blattJacksonTriplet, _distinctBoundAndScatteringEnergies, betheComparisonIdentity, betheGroundStateSpecialization, noBetheScatteringMatrixRoute, blattJacksonStationaryFunctional, _blattJacksonZeroEnergyTrial, blattJacksonAnalyticContinuation, blattJacksonScatteringMatrixPole, _noBlattJacksonComparisonRoute, _betheFreeProtonDatum, _betheParahydrogenDatum, _noBetheCoherentDatum, _blattJacksonEpithermalDatum, _blattJacksonCoherentDatum, _noBlattJacksonParahydrogenDatum, _qualifiedBetheTriplet, borrowedBetheSinglet, underdeterminedBlattJacksonSinglet, betheQualifiedChargeComparison, noBlattJacksonChargeComparison, _betheDeferredTensorScope, _blattJacksonDeferredTensorScope, _betheControlledTruncation, _blattJacksonControlledTruncation, _noBetheSingletMeasurement, noBlattJacksonSingletMeasurement, noBetheTensorContribution, _noBlattJacksonTensorContribution, _consistentSquareWell, _consistentYukawaWell, _noBetheShapeResolution, noBlattJacksonShapeResolution, _betheControlledBoundStateRelation, noExactBetheBoundStateRelation, _blattJacksonControlledBoundStateRelation, noExactBlattJacksonBoundStateRelation, _noBetheFreeProtonDatum, noBetheParahydrogenDatum, _noBetheScatteringMatrixSubstitution, _noBetheZeroEnergyReference, noBetheSingletDeuteron, _noBlattJacksonEpithermalDatum, noBlattJacksonCoherentDatum, noBlattJacksonBetheSubstitution, _noBlattJacksonControlledRelation, noBlattJacksonTensorExtension, noBlattJacksonSingletClaim, betheEvidence, blattJacksonEvidence, betheEstablished, blattJacksonEstablished⟩ := nuclear_bound_state_extraction_distinguishes_comparison_poles_triplet_data_and_open_scopes exact ⟨betheComparisonIdentity, betheGroundStateSpecialization, noBetheScatteringMatrixRoute, blattJacksonStationaryFunctional, blattJacksonAnalyticContinuation, blattJacksonScatteringMatrixPole, borrowedBetheSinglet, underdeterminedBlattJacksonSinglet, betheQualifiedChargeComparison, noBlattJacksonChargeComparison, noBlattJacksonSingletMeasurement, noBetheTensorContribution, noBlattJacksonShapeResolution, noExactBetheBoundStateRelation, noExactBlattJacksonBoundStateRelation, noBetheParahydrogenDatum, noBetheSingletDeuteron, noBlattJacksonCoherentDatum, noBlattJacksonBetheSubstitution, noBlattJacksonTensorExtension, noBlattJacksonSingletClaim, betheEvidence, blattJacksonEvidence, betheEstablished, blattJacksonEstablished⟩ end NuclearBoundStateExtractionPhysicalReuseAudit namespace NebularCalibrationPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.NebularDistanceVelocityCalibrationContext.Countermodels example : physicalLemaitreWeightedNebularCalibrationModel.distanceData ≠ physicalHubbleAdoptedNebularCalibrationModel.distanceData ∧ physicalLemaitreWeightedNebularCalibrationModel.velocityData ≠ physicalHubbleAdoptedNebularCalibrationModel.velocityData ∧ physicalNebularSampleCountAt .lemaitreFortyTwoWeightedNebulae = 42 ∧ physicalNebularSampleCountAt .hubbleTwentyFourIndividualNebulae = 24 ∧ physicalNebularSampleCountAt .hubbleNineDirectionalDistanceGroups = 9 ∧ physicalNebularSampleCountAt .hubbleFourVirgoClusterNebulae = 4 ∧ physicalNebularVelocityCatalogCountAt .lemaitreStrombergFortyThreeMeasuredVelocities = 43 ∧ physicalNebularVelocityCatalogCountAt .hubbleFortySixMeasuredVelocities = 46 ∧ physicalNebularCoefficientValueAt .lemaitreWeighted625 = 625 ∧ physicalNebularCoefficientValueAt .lemaitreUnweighted575 = 575 ∧ physicalNebularCoefficientValueAt .hubbleIndividual465 = 465 ∧ physicalNebularCoefficientValueAt .hubbleGrouped513 = 513 ∧ physicalNebularCoefficientValueAt .hubbleAdopted500 = 500 ∧ physicalNebularCoefficientUncertaintyAt .hubbleIndividual465 = some 50 ∧ physicalNebularCoefficientUncertaintyAt .hubbleGrouped513 = some 60 ∧ physicalNebularCoefficientValueAt .hubbleIndividual465 < physicalNebularCoefficientValueAt .hubbleAdopted500 ∧ physicalNebularCoefficientValueAt .hubbleAdopted500 < physicalNebularCoefficientValueAt .hubbleGrouped513 ∧ physicalNebularCalibrationSourceBridgeAt .lemaitre1927MetricRedshiftWeightedCalibration .constantMassIncreasingRadiusFieldEquation ∧ physicalNebularCalibrationSourceBridgeAt .lemaitre1927MetricRedshiftWeightedCalibration .properPeriodRatioFromExpandingMetric ∧ physicalNebularCalibrationSourceBridgeAt .lemaitre1927MetricRedshiftWeightedCalibration .nearbyFirstOrderVelocityDistanceApproximation ∧ physicalNebularCalibrationSourceBridgeAt .hubble1929SolarMotionIndividualGroupedCalibration .cepheidPeriodLuminosityZeroPoint ∧ physicalNebularCalibrationSourceBridgeAt .hubble1929SolarMotionIndividualGroupedCalibration .solarMotionXyzDirectionalCorrection ∧ physicalNebularCalibrationSourceBridgeAt .hubble1929SolarMotionIndividualGroupedCalibration .individualTwentyFourObjectFit ∧ physicalNebularCalibrationSourceBridgeAt .hubble1929SolarMotionIndividualGroupedCalibration .groupedNineDirectionDistanceBinFit ∧ physicalNebularCalibrationSourceBridgeAt .hubble1929SolarMotionIndividualGroupedCalibration .fourVirgoClusterMembersSharePeculiarMotion ∧ ¬physicalNebularCalibrationSourceBridgeAt .hubble1929SolarMotionIndividualGroupedCalibration .constantMassIncreasingRadiusFieldEquation ∧ ¬physicalNebularCalibrationSourceBridgeAt .lemaitre1927MetricRedshiftWeightedCalibration .individualTwentyFourObjectFit ∧ physicalNebularCalibrationScopeStatusAt .lemaitre1927MetricRedshiftWeightedCalibration .metricExpansionExplanation = .derivedFromSourceMetric ∧ physicalNebularCalibrationScopeStatusAt .hubble1929SolarMotionIndividualGroupedCalibration .metricExpansionExplanation = .outsideSelectedSourceScope ∧ physicalNebularCalibrationScopeStatusAt .hubble1929SolarMotionIndividualGroupedCalibration .deSitterInterpretation = .suggestedButUnestablished ∧ physicalNebularCalibrationScopeStatusAt .lemaitre1927MetricRedshiftWeightedCalibration .radiationPressureCausalMechanism = .suggestedButUnestablished ∧ physicalNebularCalibrationScopeStatusAt .hubble1929SolarMotionIndividualGroupedCalibration .virgoPeculiarMotionBias = .acknowledgedSystematicBias ∧ physicalNebularReportedReductionContrastAt physicalLemaitreWeightedNebularCalibrationModel physicalLemaitreUnweightedNebularCalibrationModel ∧ physicalNebularReportedReductionContrastAt physicalHubbleIndividualNebularCalibrationModel physicalHubbleGroupedNebularCalibrationModel ∧ physicalNebularInterpretationPossibility.TentativeAt physicalNebularInterpretationEstablishment .deSitterRedshiftExplanation .hubble1929SolarMotionIndividualGroupedCalibration ∧ physicalNebularInterpretationPossibility.TentativeAt physicalNebularInterpretationEstablishment .radiationPressureDrivesExpansion .lemaitre1927MetricRedshiftWeightedCalibration ∧ ¬physicalLemaitreWithoutMetricRedshiftBridgeModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ ¬physicalLemaitreWithoutWeightedFitModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ ¬physicalLemaitreWithHubbleDistanceSubstitutionModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ ¬physicalHubbleWithoutCepheidZeroPointModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ ¬physicalHubbleWithoutSolarMotionCorrectionModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ ¬physicalHubbleWithoutIndividualFitModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ ¬physicalHubbleWithoutGroupedFitModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ ¬physicalHubbleWithoutVirgoPeculiarMotionQualificationModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ ¬physicalHubbleWithVelocityOnlySampleModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ ¬physicalHubbleBeyondObservedDistanceRangeModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ ¬physicalHubbleWithExactUniversalRecessionClaimModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ ¬physicalHubbleWithLemaitreCoefficientSubstitutionModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ physicalLemaitreUnweightedNebularCalibrationModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ physicalHubbleIndividualNebularCalibrationModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ physicalHubbleGroupedNebularCalibrationModel.NonvacuousQualifiedDistanceVelocityCalibrationAt ∧ physicalLemaitreWeightedNebularCalibrationContext.EstablishedQualifiedDistanceVelocityCalibrationAt physicalLemaitreWeightedNebularCalibrationModel ∧ physicalHubbleAdoptedNebularCalibrationContext.EstablishedQualifiedDistanceVelocityCalibrationAt physicalHubbleAdoptedNebularCalibrationModel := by obtain ⟨_distinctDescriptions, _distinctSamples, distinctDistanceCatalogs, distinctVelocityCatalogs, _distinctRelations, _distinctCoefficients, _distinctRegimes, lemaitreSampleCount, hubbleIndividualSampleCount, hubbleGroupedSampleCount, virgoSampleCount, lemaitreCatalogCount, hubbleCatalogCount, lemaitreWeightedValue, lemaitreUnweightedValue, hubbleIndividualValue, hubbleGroupedValue, hubbleAdoptedValue, hubbleIndividualUncertainty, hubbleGroupedUncertainty, individualBelowAdopted, adoptedBelowGrouped, lemaitreFieldEquation, lemaitrePeriodRatio, lemaitreNearbyApproximation, hubbleCepheidZeroPoint, hubbleSolarMotionDipole, hubbleIndividualFit, hubbleGroupedFit, hubbleVirgoQualification, noHubbleMetricEquation, noLemaitreIndividualFit, lemaitreExpansionDerived, hubbleExpansionOutsideScope, hubbleDeSitterSuggested, lemaitreRadiationSuggested, virgoSystematicAcknowledged, _lemaitreNonuniversal, _hubbleHistoricallyUncalibrated, lemaitreReductionContrast, hubbleReductionContrast, hubbleDeSitterTentative, lemaitreRadiationTentative, noLemaitreMetricBridge, noLemaitreWeightedFit, noLemaitreHubbleDistanceSubstitution, noHubbleCepheidZeroPoint, noHubbleSolarMotionCorrection, noHubbleIndividualFit, noHubbleGroupedFit, noHubbleVirgoQualification, noHubbleVelocityOnlySample, noHubbleBeyondObservedRange, noHubbleExactUniversalClaim, noHubbleLemaitreCoefficient, lemaitreUnweightedNonvacuous, hubbleIndividualNonvacuous, hubbleGroupedNonvacuous, lemaitreEstablished, hubbleEstablished⟩ := nebular_calibration_distinguishes_expanding_metrics_solar_dipoles_samples_fits_and_interpretations exact ⟨distinctDistanceCatalogs, distinctVelocityCatalogs, lemaitreSampleCount, hubbleIndividualSampleCount, hubbleGroupedSampleCount, virgoSampleCount, lemaitreCatalogCount, hubbleCatalogCount, lemaitreWeightedValue, lemaitreUnweightedValue, hubbleIndividualValue, hubbleGroupedValue, hubbleAdoptedValue, hubbleIndividualUncertainty, hubbleGroupedUncertainty, individualBelowAdopted, adoptedBelowGrouped, lemaitreFieldEquation, lemaitrePeriodRatio, lemaitreNearbyApproximation, hubbleCepheidZeroPoint, hubbleSolarMotionDipole, hubbleIndividualFit, hubbleGroupedFit, hubbleVirgoQualification, noHubbleMetricEquation, noLemaitreIndividualFit, lemaitreExpansionDerived, hubbleExpansionOutsideScope, hubbleDeSitterSuggested, lemaitreRadiationSuggested, virgoSystematicAcknowledged, lemaitreReductionContrast, hubbleReductionContrast, hubbleDeSitterTentative, lemaitreRadiationTentative, noLemaitreMetricBridge, noLemaitreWeightedFit, noLemaitreHubbleDistanceSubstitution, noHubbleCepheidZeroPoint, noHubbleSolarMotionCorrection, noHubbleIndividualFit, noHubbleGroupedFit, noHubbleVirgoQualification, noHubbleVelocityOnlySample, noHubbleBeyondObservedRange, noHubbleExactUniversalClaim, noHubbleLemaitreCoefficient, lemaitreUnweightedNonvacuous, hubbleIndividualNonvacuous, hubbleGroupedNonvacuous, lemaitreEstablished, hubbleEstablished⟩ end NebularCalibrationPhysicalReuseAudit namespace ActionPhaseCompositionPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.ActionPhaseCompositionContext.Countermodels example : physicalDiracActionPhaseCompositionModel.system ≠ physicalFeynmanActionPhaseCompositionModel.system ∧ physicalDiracActionPhaseCompositionModel.history ≠ physicalFeynmanActionPhaseCompositionModel.history ∧ physicalDiracActionPhaseCompositionModel.partition ≠ physicalFeynmanActionPhaseCompositionModel.partition ∧ physicalDiracActionPhaseCompositionModel.action ≠ physicalFeynmanActionPhaseCompositionModel.action ∧ physicalDiracActionPhaseCompositionModel.localAmplitude ≠ physicalFeynmanActionPhaseCompositionModel.localAmplitude ∧ physicalDiracActionPhaseCompositionModel.composedAmplitude ≠ physicalFeynmanActionPhaseCompositionModel.composedAmplitude ∧ physicalActionPhaseSourceBridgeAt .dirac1933OrderedTransformationFunction .diracWellOrderedMixedRepresentation ∧ physicalActionPhaseSourceBridgeAt .dirac1933OrderedTransformationFunction .diracCanonicalContactGeneratingFunction ∧ physicalActionPhaseSourceBridgeAt .dirac1933OrderedTransformationFunction .diracIntermediateCanonicalCoordinateIntegration ∧ physicalActionPhaseSourceBridgeAt .feynman1948NormalizedPositionPathPostulates .feynmanUndistinguishedAlternativesAmplitudeSum ∧ physicalActionPhaseSourceBridgeAt .feynman1948NormalizedPositionPathPostulates .feynmanDistinguishedAlternativesProbabilitySum ∧ physicalActionPhaseSourceBridgeAt .feynman1948NormalizedPositionPathPostulates .feynmanEqualMagnitudeActionPhasePostulate ∧ physicalActionPhaseSourceBridgeAt .feynman1948NormalizedPositionPathPostulates .feynmanNonzeroSliceNormalization ∧ physicalActionPhaseSourceBridgeAt .feynman1948NormalizedPositionPathPostulates .feynmanCoherentAllPathConfigurationIntegral ∧ ¬physicalActionPhaseSourceBridgeAt .dirac1933OrderedTransformationFunction .feynmanEqualMagnitudeActionPhasePostulate ∧ ¬physicalActionPhaseSourceBridgeAt .feynman1948NormalizedPositionPathPostulates .diracWellOrderedMixedRepresentation ∧ physicalActionPhaseSliceNormalizationAt .feynmanEqualTimeCoordinateSubdivision = 2 ∧ physicalActionPhaseSliceNormalizationAt .feynmanUnnormalizedCoordinateSubdivision = 0 ∧ physicalActionPhaseCoherentAlternativeWeightAt 1 1 = 4 ∧ physicalActionPhaseResolvedAlternativeWeightAt 1 1 = 2 ∧ physicalActionPhaseCoherentAlternativeWeightAt 1 (-1) = 0 ∧ physicalActionPhaseResolvedAlternativeWeightAt 1 (-1) = 2 ∧ physicalActionPhaseAlternativeMeasurement.DisturbsAt .resolvedIntermediateAlternativeProcedure .feynmanNonrelativisticPositionPath .feynmanVanishingPlanckQuadraticVelocityLimit ∧ physicalActionPhaseScopeStatusAt .dirac1933OrderedTransformationFunction .intermediateAlternativeComposition = .transformationFunctionIntegration ∧ physicalActionPhaseScopeStatusAt .feynman1948NormalizedPositionPathPostulates .intermediateAlternativeComposition = .coherentAmplitudeBeforeSquaring ∧ physicalActionPhaseScopeStatusAt .feynman1948NormalizedPositionPathPostulates .measuredAlternativeComposition = .incoherentProbabilityAfterDistinguishing ∧ physicalActionPhaseScopeStatusAt .feynman1948NormalizedPositionPathPostulates .continuumComplexPathMeasure = .acknowledgedMathematicallyUnresolved ∧ physicalActionPhaseScopeStatusAt .feynman1948NormalizedPositionPathPostulates .preciseIdealMeasurementDefinition = .acknowledgedOperationallyUndefined ∧ physicalActionPhaseScopeStatusAt .feynman1948NormalizedPositionPathPostulates .schrodingerEquivalence = .establishedOnlyForQuadraticVelocity ∧ physicalActionPhaseSchrodingerEquivalenceAt .feynman1948NormalizedPositionPathPostulates .feynmanVanishingPlanckQuadraticVelocityLimit ∧ ¬physicalActionPhaseSchrodingerEquivalenceAt .feynman1948NormalizedPositionPathPostulates .feynmanNonquadraticVelocityExtension ∧ ¬physicalActionPhaseSchrodingerEquivalenceAt .dirac1933OrderedTransformationFunction .diracVanishingPlanckCanonicalLimit ∧ physicalActionPhaseInterpretationPossibility.TentativeAt physicalActionPhaseInterpretationEstablishment .rigorousContinuumComplexPathMeasure .feynman1948NormalizedPositionPathPostulates ∧ physicalActionPhaseInterpretationPossibility.TentativeAt physicalActionPhaseInterpretationEstablishment .fullyDerivedRelativisticPathDynamics .feynman1948NormalizedPositionPathPostulates ∧ physicalActionPhaseInterpretationPossibility.TentativeAt physicalActionPhaseInterpretationEstablishment .fourDimensionalGeneralizedBoundaryTransformation .dirac1933OrderedTransformationFunction ∧ ¬physicalDiracWithoutOperatorOrderingModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalDiracWithoutIntermediateIntegrationModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalDiracWithFeynmanPathAmplitudeModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalDiracWithoutVanishingPlanckLimitModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalFeynmanWithoutIdealMeasurementModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalFeynmanWithoutActionPhasePostulateModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalFeynmanWithoutTimeSliceSubdivisionModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalFeynmanWithoutSliceNormalizationModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalFeynmanWithoutConfigurationIntegralModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalFeynmanWithMeasuredAlternativeCompositionModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalFeynmanWithDiracTransformationSubstitutionModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalFeynmanWithoutVanishingPlanckLimitModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalFeynmanWithNonquadraticWaveEquationClaimModel.NonvacuousActionPhaseCompositionAt ∧ ¬physicalFeynmanWithHigherDerivativePathModel.NonvacuousActionPhaseCompositionAt ∧ physicalDiracActionPhaseCompositionContext.EstablishedActionPhaseCompositionAt physicalDiracActionPhaseCompositionModel ∧ physicalFeynmanActionPhaseCompositionContext.EstablishedActionPhaseCompositionAt physicalFeynmanActionPhaseCompositionModel := by obtain ⟨distinctSystems, _distinctInitialConfigurations, _distinctIntermediateConfigurations, _distinctFinalConfigurations, distinctHistories, distinctPartitions, distinctActions, distinctLocalAmplitudes, distinctComposedAmplitudes, _distinctRegimes, diracOrdering, diracGeneratingFunction, diracIntermediateIntegration, feynmanCoherentAlternatives, feynmanMeasuredAlternatives, feynmanActionPhase, feynmanNormalization, feynmanAllPathIntegral, _feynmanQuadraticRestriction, noDiracPathPostulate, noFeynmanOperatorOrdering, normalizationValue, unnormalizedValue, constructiveCoherentWeight, constructiveMeasuredWeight, destructiveCoherentWeight, destructiveMeasuredWeight, measuredAlternativeDisturbance, diracIntegrationScope, feynmanCoherentScope, feynmanMeasuredScope, _feynmanNormalizationScope, unresolvedContinuumMeasure, undefinedIdealMeasurement, restrictedSchrodingerScope, _higherDerivativeObstruction, _formalRelativisticExtension, quadraticSchrodingerEquivalence, noNonquadraticSchrodingerEquivalence, noDiracSchrodingerEquivalence, tentativeContinuumMeasure, _tentativeIdealMeasurement, tentativeRelativisticExtension, tentativeBoundaryGeneralization, noDiracOrdering, noDiracIntegration, noDiracFeynmanAmplitude, noDiracZeroLimit, noFeynmanIdealMeasurement, noFeynmanPhasePostulate, noFeynmanSubdivision, noFeynmanNormalization, noFeynmanIntegral, noFeynmanMeasuredComposition, noFeynmanDiracSubstitution, noFeynmanZeroLimit, noFeynmanNonquadraticClaim, noFeynmanHigherDerivativePath, diracEstablished, feynmanEstablished⟩ := action_phase_composition_distinguishes_ordered_transformations_normalized_paths_measurements_and_classical_limits exact ⟨distinctSystems, distinctHistories, distinctPartitions, distinctActions, distinctLocalAmplitudes, distinctComposedAmplitudes, diracOrdering, diracGeneratingFunction, diracIntermediateIntegration, feynmanCoherentAlternatives, feynmanMeasuredAlternatives, feynmanActionPhase, feynmanNormalization, feynmanAllPathIntegral, noDiracPathPostulate, noFeynmanOperatorOrdering, normalizationValue, unnormalizedValue, constructiveCoherentWeight, constructiveMeasuredWeight, destructiveCoherentWeight, destructiveMeasuredWeight, measuredAlternativeDisturbance, diracIntegrationScope, feynmanCoherentScope, feynmanMeasuredScope, unresolvedContinuumMeasure, undefinedIdealMeasurement, restrictedSchrodingerScope, quadraticSchrodingerEquivalence, noNonquadraticSchrodingerEquivalence, noDiracSchrodingerEquivalence, tentativeContinuumMeasure, tentativeRelativisticExtension, tentativeBoundaryGeneralization, noDiracOrdering, noDiracIntegration, noDiracFeynmanAmplitude, noDiracZeroLimit, noFeynmanIdealMeasurement, noFeynmanPhasePostulate, noFeynmanSubdivision, noFeynmanNormalization, noFeynmanIntegral, noFeynmanMeasuredComposition, noFeynmanDiracSubstitution, noFeynmanZeroLimit, noFeynmanNonquadraticClaim, noFeynmanHigherDerivativePath, diracEstablished, feynmanEstablished⟩ end ActionPhaseCompositionPhysicalReuseAudit namespace SchwarzschildMasterPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.SchwarzschildMasterEquationContext.Countermodels example : physicalReggeWheelerOddMasterModel.parity ≠ physicalZerilliEvenMasterModel.parity ∧ physicalReggeWheelerOddMasterModel.masterVariable ≠ physicalZerilliEvenMasterModel.masterVariable ∧ physicalReggeWheelerOddMasterModel.effectivePotential ≠ physicalZerilliEvenMasterModel.effectivePotential ∧ physicalReggeWheelerOddMasterModel.backgroundMetric ≠ physicalVishveshwaraOddMasterModel.backgroundMetric ∧ physicalSchwarzschildAngularMomentumValueAt .radiativeQuadrupole = 2 ∧ physicalSchwarzschildAngularMomentumValueAt .rotationalDipole = 1 ∧ physicalSchwarzschildSeparationParameterAt .radiativeQuadrupole = 2 ∧ physicalSchwarzschildPotentialValueAt .reggeWheelerOddAxialPotential 1 4 .radiativeQuadrupole = 9 / 64 ∧ physicalSchwarzschildPotentialValueAt .vishveshwaraOddAxialPotential 1 4 .radiativeQuadrupole = 9 / 64 ∧ physicalSchwarzschildPotentialValueAt .zerilliEvenPolarPotential 1 4 .radiativeQuadrupole = 1041 / 7744 ∧ physicalSchwarzschildPotentialValueAt .reggeWheelerOddAxialPotential 1 4 .radiativeQuadrupole ≠ physicalSchwarzschildPotentialValueAt .zerilliEvenPolarPotential 1 4 .radiativeQuadrupole ∧ physicalSchwarzschildSourceBridgeAt .reggeWheeler1957OddMasterAndWormholeBoundary .reggeWheelerOddMasterQDefinition ∧ physicalSchwarzschildSourceBridgeAt .reggeWheeler1957OddMasterAndWormholeBoundary .reggeWheelerEvenConstraintWithoutEvenPotential ∧ ¬physicalSchwarzschildSourceBridgeAt .reggeWheeler1957OddMasterAndWormholeBoundary .zerilliEvenPotentialFormula ∧ physicalSchwarzschildSourceBridgeAt .zerilli1970CorrectedEvenMasterPotential .zerilliErrorsInOldEvenEquationsIdentified ∧ physicalSchwarzschildSourceBridgeAt .zerilli1970CorrectedEvenMasterPotential .zerilliCorrectedEvenRadialEquations ∧ physicalSchwarzschildSourceBridgeAt .zerilli1970CorrectedEvenMasterPotential .zerilliEvenAlgebraicConstraint ∧ physicalSchwarzschildSourceBridgeAt .zerilli1970CorrectedEvenMasterPotential .zerilliSolvedTransformationCoefficients ∧ ¬physicalSchwarzschildSourceBridgeAt .zerilli1970CorrectedEvenMasterPotential .vishveshwaraKruskalHorizonRegularity ∧ physicalSchwarzschildSourceBridgeAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .vishveshwaraKruskalHorizonRegularity ∧ physicalSchwarzschildSourceBridgeAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .vishveshwaraOddGrowingModeHorizonDivergence ∧ physicalSchwarzschildSourceBridgeAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .vishveshwaraEvenGrowingModeHorizonDivergence ∧ physicalSchwarzschildGrowingModeExcludedAt .reggeWheeler1957OddMasterAndWormholeBoundary .wormholeThroatMatching .oddAxial .imaginaryExponentiallyGrowingMode ∧ physicalSchwarzschildGrowingModeExcludedAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .kruskalHorizonRegularity .oddAxial .imaginaryExponentiallyGrowingMode ∧ physicalSchwarzschildGrowingModeExcludedAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .kruskalHorizonRegularity .evenPolar .imaginaryExponentiallyGrowingMode ∧ ¬physicalSchwarzschildGrowingModeExcludedAt .zerilli1970CorrectedEvenMasterPotential .kruskalHorizonRegularity .evenPolar .imaginaryExponentiallyGrowingMode ∧ ¬physicalSchwarzschildGrowingModeExcludedWithWithheldBridgeAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .vishveshwaraKruskalHorizonRegularity .kruskalHorizonRegularity .oddAxial .imaginaryExponentiallyGrowingMode ∧ physicalSchwarzschildModeCompletenessAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .oddAxial ∧ ¬physicalSchwarzschildModeCompletenessAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .evenPolar ∧ physicalSchwarzschildDipoleStatusAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .oddAxial .rotationalDipole .imaginaryExponentiallyGrowingMode = .removableNonstationaryGaugeMode ∧ physicalSchwarzschildDipoleStatusAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .oddAxial .rotationalDipole .zeroStationaryMode = .physicalStationaryRotation ∧ physicalSchwarzschildScopeStatusAt .reggeWheeler1957OddMasterAndWormholeBoundary .evenParityMasterPotential = .explicitlyUnavailableInSource ∧ physicalSchwarzschildScopeStatusAt .zerilli1970CorrectedEvenMasterPotential .evenParityMasterPotential = .derivedCorrectedEvenMasterPotential ∧ physicalSchwarzschildScopeStatusAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .evenSectorCompleteness = .evenCompletenessExplicitlyUnproved ∧ physicalSchwarzschildScopeStatusAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .arbitraryInitialDataStability = .modeByModeArgumentNotArbitraryDataProof ∧ physicalSchwarzschildScopeStatusAt .zerilli1970CorrectedEvenMasterPotential .sourcedRadiationSolution = .radiationApplicationAnticipatedNotCalculated ∧ physicalSchwarzschildInterpretationPossibility.TentativeAt physicalSchwarzschildInterpretationEstablishment .completeEvenParityArbitraryInitialDataExpansion .vishveshwara1970KruskalBoundaryAndCompletenessGap ∧ physicalSchwarzschildInterpretationPossibility.TentativeAt physicalSchwarzschildInterpretationEstablishment .fullyNonlinearSchwarzschildStability .vishveshwara1970KruskalBoundaryAndCompletenessGap ∧ physicalSchwarzschildBackgroundParadigm.FixedPhysicalBackground .fixedVacuumSchwarzschildExterior ∧ physicalSchwarzschildCoordinateCovariance.NonvacuousLawCovariant .firstOrderVacuumRicciEquation ∧ ¬physicalReggeWheelerWithoutOddGaugeModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalReggeWheelerWithoutMasterTransformationModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalReggeWheelerWithZerilliEvenPotentialModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalReggeWheelerWithUnconstructedEvenMasterModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalZerilliWithoutCorrectedEvenEquationsModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalZerilliWithoutAlgebraicConstraintModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalZerilliWithoutSolvedTransformationModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalZerilliWithReggeWheelerOddPotentialModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalZerilliWithoutTortoiseCoordinateModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalZerilliWithSourcedRadiationModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalZerilliWithUnprovedStabilityClaimModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalVishveshwaraWithoutCorrectedOddEquationsModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalVishveshwaraWithEvenMasterSubstitutionModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ ¬physicalVishveshwaraWithRotationalDipoleSubstitutionModel.NonvacuousSchwarzschildMasterEquationReductionAt ∧ physicalReggeWheelerOddMasterContext.EstablishedSchwarzschildMasterEquationReductionAt physicalReggeWheelerOddMasterModel ∧ physicalZerilliEvenMasterContext.EstablishedSchwarzschildMasterEquationReductionAt physicalZerilliEvenMasterModel ∧ physicalVishveshwaraOddMasterContext.EstablishedSchwarzschildMasterEquationReductionAt physicalVishveshwaraOddMasterModel := by obtain ⟨distinctParities, _distinctPerturbations, _distinctGauges, _distinctCoupledEquations, distinctMasterVariables, distinctPotentials, _distinctMasterEquations, _distinctRegimes, distinctBackgrounds, _distinctFrequencies, radiativeAngularMomentum, dipoleAngularMomentum, separationParameter, reggeWheelerPotential, vishveshwaraPotential, zerilliPotential, unequalPotentials, reggeWheelerOddVariable, reggeWheelerEvenConstraint, noReggeWheelerEvenPotential, zerilliPriorEquationErrors, zerilliCorrectedEquations, zerilliAlgebraicConstraint, _zerilliNonuniqueTransformation, zerilliSolvedCoefficients, _zerilliPotentialBridge, noZerilliKruskalBoundary, vishveshwaraKruskalBoundary, vishveshwaraOddDivergence, vishveshwaraEvenDivergence, _vishveshwaraDipoleGauge, _vishveshwaraStationaryRotation, _vishveshwaraCompletenessAcknowledgment, reggeWheelerOddExclusion, vishveshwaraOddExclusion, vishveshwaraEvenExclusion, noZerilliBoundaryExclusion, noVishveshwaraExclusionWithoutKruskal, vishveshwaraOddCompleteness, noVishveshwaraEvenCompleteness, nonstationaryDipoleGauge, stationaryDipoleRotation, reggeWheelerEvenUnavailable, zerilliEvenDerived, _vishveshwaraKruskalRequired, vishveshwaraEvenCompletenessGap, vishveshwaraNoArbitraryDataProof, zerilliRadiationOnlyAnticipated, tentativeEvenCompleteness, tentativeNonlinearStability, fixedSchwarzschildBackground, coordinateCovariance, noReggeWheelerOddGauge, noReggeWheelerTransformation, noReggeWheelerZerilliPotential, noReggeWheelerEvenMaster, noZerilliCorrectedEquations, noZerilliConstraint, noZerilliSolvedTransformation, noZerilliOddPotential, noZerilliTortoise, noZerilliRadiationSource, noZerilliStabilityClaim, noVishveshwaraCorrectedEquations, noVishveshwaraEvenSubstitution, noVishveshwaraRotationalDipoleSubstitution, reggeWheelerEstablished, zerilliEstablished, vishveshwaraEstablished⟩ := schwarzschild_master_reduction_distinguishes_axial_polar_kruskal_gauge_potentials_and_scope exact ⟨distinctParities, distinctMasterVariables, distinctPotentials, distinctBackgrounds, radiativeAngularMomentum, dipoleAngularMomentum, separationParameter, reggeWheelerPotential, vishveshwaraPotential, zerilliPotential, unequalPotentials, reggeWheelerOddVariable, reggeWheelerEvenConstraint, noReggeWheelerEvenPotential, zerilliPriorEquationErrors, zerilliCorrectedEquations, zerilliAlgebraicConstraint, zerilliSolvedCoefficients, noZerilliKruskalBoundary, vishveshwaraKruskalBoundary, vishveshwaraOddDivergence, vishveshwaraEvenDivergence, reggeWheelerOddExclusion, vishveshwaraOddExclusion, vishveshwaraEvenExclusion, noZerilliBoundaryExclusion, noVishveshwaraExclusionWithoutKruskal, vishveshwaraOddCompleteness, noVishveshwaraEvenCompleteness, nonstationaryDipoleGauge, stationaryDipoleRotation, reggeWheelerEvenUnavailable, zerilliEvenDerived, vishveshwaraEvenCompletenessGap, vishveshwaraNoArbitraryDataProof, zerilliRadiationOnlyAnticipated, tentativeEvenCompleteness, tentativeNonlinearStability, fixedSchwarzschildBackground, coordinateCovariance, noReggeWheelerOddGauge, noReggeWheelerTransformation, noReggeWheelerZerilliPotential, noReggeWheelerEvenMaster, noZerilliCorrectedEquations, noZerilliConstraint, noZerilliSolvedTransformation, noZerilliOddPotential, noZerilliTortoise, noZerilliRadiationSource, noZerilliStabilityClaim, noVishveshwaraCorrectedEquations, noVishveshwaraEvenSubstitution, noVishveshwaraRotationalDipoleSubstitution, reggeWheelerEstablished, zerilliEstablished, vishveshwaraEstablished⟩ end SchwarzschildMasterPhysicalReuseAudit namespace MagneticInterferencePhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.FieldFreeMagneticInterferenceContext.Countermodels example : physicalAharonovBohmSplitBeamModel.system ≠ physicalBerryTransportedBoxModel.system ∧ physicalAharonovBohmSplitBeamModel.closedCircuit ≠ physicalBerryTransportedBoxModel.closedCircuit ∧ physicalAharonovBohmSplitBeamModel.relativePhase ≠ physicalBerryTransportedBoxModel.relativePhase ∧ physicalMagneticPhaseValueAt .aharonovBohmGaussianFluxPhase = 3 ∧ physicalMagneticPhaseValueAt .berryTransportedBoxFluxPhase = 6 ∧ physicalMagneticWindingNumberAt .splitBeamWindingCircuit = 1 ∧ physicalMagneticWindingNumberAt .transportedBoxOnceWindingCircuit = 1 ∧ physicalMagneticWindingNumberAt .nonwindingCircuit = 0 ∧ physicalMagneticExactScatteringReadoutAt .integerFluxQuantum = 0 ∧ physicalMagneticExactScatteringReadoutAt .halfIntegerFluxQuantum = 1 ∧ physicalMagneticSourceBridgeAt .aharonovBohm1959GaussianSplitBeamShieldingAndScattering .aharonovBohmGaussianChargeFluxOverPlanckAndLightSpeed ∧ ¬physicalMagneticSourceBridgeAt .aharonovBohm1959GaussianSplitBeamShieldingAndScattering .berryBoxTransportDoesNotRequireAdiabaticApproximation ∧ physicalMagneticSourceBridgeAt .berry1984ExactTranslatedBoxSingleValuedStates .berryBoxTransportDoesNotRequireAdiabaticApproximation ∧ ¬physicalMagneticSourceBridgeAt .berry1984ExactTranslatedBoxSingleValuedStates .aharonovBohmExactFixedFluxFractionalBesselScattering ∧ physicalMagneticLocalGaugeRemovalAt .aharonovBohm1959GaussianSplitBeamShieldingAndScattering .gaussianSolenoidGauge .upperElectronBeam ∧ ¬physicalMagneticGlobalSingleValuedGaugeRemovalAt .aharonovBohm1959GaussianSplitBeamShieldingAndScattering .gaussianSolenoidGauge .splitBeamWindingCircuit .aharonovBohmShieldedSolenoidFlux ∧ physicalMagneticGlobalSingleValuedGaugeRemovalAt .aharonovBohm1959GaussianSplitBeamShieldingAndScattering .gaussianSolenoidGauge .splitBeamWindingCircuit .integerFluxQuantum ∧ physicalMagneticWavefunctionSingleValuedAt .berry1984ExactTranslatedBoxSingleValuedStates .berryTranslatedBoxEigenstate ∧ ¬physicalMagneticAdiabaticRequirementAt .berry1984ExactTranslatedBoxSingleValuedStates .berryTranslatedChargedBox ∧ physicalMagneticAdiabaticRequirementAt .berry1984ExactTranslatedBoxSingleValuedStates .berryGenericAdiabaticEigenstateCycle ∧ physicalMagneticPhaseAtLevel .berry1984ExactTranslatedBoxSingleValuedStates .groundBoxEigenstate = physicalMagneticPhaseAtLevel .berry1984ExactTranslatedBoxSingleValuedStates .excitedBoxEigenstate ∧ physicalMagneticScopeStatusAt .aharonovBohm1959GaussianSplitBeamShieldingAndScattering .directContemporaryFluxMeasurement = .direct1959ExperimentProposedButNotReported ∧ physicalMagneticScopeStatusAt .berry1984ExactTranslatedBoxSingleValuedStates .adiabaticTransportRequirement = .transportedBoxApproximationExplicitlyNotRequired ∧ physicalMagneticInterpretationPossibility.TentativeAt physicalMagneticInterpretationEstablishment .direct1959MagneticExperimentalConfirmation .aharonovBohm1959GaussianSplitBeamShieldingAndScattering ∧ physicalMagneticInterpretiveAlternativeSelection.OpenAlternativePairAt .localElectromagneticPotentialOntology .nonlocalExcludedFieldOntology .aharonovBohm1959GaussianSplitBeamShieldingAndScattering ∧ ¬physicalAharonovBohmSplitBeamContext.NonvacuousFieldFreeMagneticInterferenceAt physicalAharonovBohmWithLeakedExteriorFieldModel ∧ ¬physicalAharonovBohmSplitBeamContext.NonvacuousFieldFreeMagneticInterferenceAt physicalAharonovBohmWithoutWindingCircuitModel ∧ ¬physicalAharonovBohmSplitBeamContext.NonvacuousFieldFreeMagneticInterferenceAt physicalAharonovBohmWithGloballyRemovedGaugeModel ∧ ¬physicalAharonovBohmSplitBeamContext.NonvacuousFieldFreeMagneticInterferenceAt physicalAharonovBohmWithoutLightSpeedFactorModel ∧ ¬physicalAharonovBohmWithoutShieldingBridgeContext.NonvacuousFieldFreeMagneticInterferenceAt physicalAharonovBohmWithoutShieldingBridgeModel ∧ ¬physicalAharonovBohmSplitBeamContext.NonvacuousFieldFreeMagneticInterferenceAt physicalAharonovBohmWithoutRecombinationModel ∧ ¬physicalAharonovBohmSplitBeamContext.NonvacuousFieldFreeMagneticInterferenceAt physicalAharonovBohmWithoutFluxModel ∧ ¬physicalBerryTransportedBoxContext.NonvacuousFieldFreeMagneticInterferenceAt physicalBerryWithSplitBeamSubstitutionModel ∧ ¬physicalBerryTransportedBoxContext.NonvacuousFieldFreeMagneticInterferenceAt physicalBerryWithoutOnceThreadedCircuitModel ∧ ¬physicalBerryTransportedBoxContext.NonvacuousFieldFreeMagneticInterferenceAt physicalBerryWithMultivaluedStateModel ∧ ¬physicalBerryTransportedBoxContext.NonvacuousFieldFreeMagneticInterferenceAt physicalBerryWithGaussianConventionSubstitutionModel ∧ ¬physicalBerryWithSpuriousAdiabaticRequirementContext.NonvacuousFieldFreeMagneticInterferenceAt physicalBerryWithSpuriousAdiabaticRequirementModel ∧ ¬physicalBerryWithoutNormalizedStateBridgeContext.NonvacuousFieldFreeMagneticInterferenceAt physicalBerryWithoutNormalizedStateBridgeModel ∧ ¬physicalBerryWithLevelDependentPhaseContext.NonvacuousFieldFreeMagneticInterferenceAt physicalBerryWithLevelDependentPhaseModel ∧ physicalAharonovBohmSplitBeamContext.EstablishedFieldFreeMagneticInterferenceAt physicalAharonovBohmSplitBeamModel ∧ physicalBerryTransportedBoxContext.EstablishedFieldFreeMagneticInterferenceAt physicalBerryTransportedBoxModel := by obtain ⟨distinctSystems, _distinctStates, _distinctRegions, _distinctSupports, _distinctFirstPaths, _distinctSecondPaths, distinctCircuits, _distinctPotentials, _distinctFields, _distinctFluxes, distinctPhases, _distinctExperiments, _distinctUnitConventions, gaussianPhase, transportedPhase, splitBeamWinding, transportedBoxWinding, zeroWinding, integerFluxScattering, halfIntegerScattering, gaussianSource, noAdiabaticExceptionInAharonovBohm, berryAdiabaticException, noExactScatteringDerivationInBerry, _aharonovBohmTopology, _berryTopology, localGaugeRemoval, noGlobalGaugeRemoval, quantizedGlobalGaugeRemoval, singleValuedBoxState, _notSingleValuedAnsatz, boxNotAdiabatic, genericCycleAdiabatic, levelIndependentPhase, directMeasurementOnlyProposed, _boundedElectricTransient, transportedBoxNoApproximation, unconfirmed1959Experiment, openInterpretiveAlternatives, noLeakedExteriorField, noMissingWinding, noGloballyRemovedGauge, noMissingLightSpeed, noMissingShielding, noMissingRecombination, noMissingFlux, noSplitBeamSubstitution, noMissingBoxWinding, noMultivaluedBoxState, noGaussianConventionSubstitution, noSpuriousAdiabaticRequirement, noMissingNormalization, noLevelDependentPhase, aharonovBohmEstablished, berryEstablished⟩ := magnetic_interference_distinguishes_split_beams_transported_boxes_gauge_topology_and_source_scope exact ⟨distinctSystems, distinctCircuits, distinctPhases, gaussianPhase, transportedPhase, splitBeamWinding, transportedBoxWinding, zeroWinding, integerFluxScattering, halfIntegerScattering, gaussianSource, noAdiabaticExceptionInAharonovBohm, berryAdiabaticException, noExactScatteringDerivationInBerry, localGaugeRemoval, noGlobalGaugeRemoval, quantizedGlobalGaugeRemoval, singleValuedBoxState, boxNotAdiabatic, genericCycleAdiabatic, levelIndependentPhase, directMeasurementOnlyProposed, transportedBoxNoApproximation, unconfirmed1959Experiment, openInterpretiveAlternatives, noLeakedExteriorField, noMissingWinding, noGloballyRemovedGauge, noMissingLightSpeed, noMissingShielding, noMissingRecombination, noMissingFlux, noSplitBeamSubstitution, noMissingBoxWinding, noMultivaluedBoxState, noGaussianConventionSubstitution, noSpuriousAdiabaticRequirement, noMissingNormalization, noLevelDependentPhase, aharonovBohmEstablished, berryEstablished⟩ end MagneticInterferencePhysicalReuseAudit namespace PhaseWindingPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.PhaseWindingCirculationContext.Countermodels example : physicalFeynmanToroidalRingContext.selectedDescription ≠ physicalKosterlitzPlanarVortexContext.selectedDescription ∧ physicalFeynmanToroidalRingContext.selectedContour ≠ physicalKosterlitzPlanarVortexContext.selectedContour ∧ physicalKosterlitzPlanarVortexContext.selectedDescription ≠ physicalKosterlitzNeutralFilmContext.selectedDescription ∧ physicalWindingDescriptionDimensionAt .feynmanNeutralHeliumToroidalRing = 3 ∧ physicalWindingDescriptionDimensionAt .kosterlitzThoulessPlanarXYVortexGas = 2 ∧ physicalWindingDescriptionDimensionAt .kosterlitzThoulessNeutralSuperfluidFilm = 2 ∧ physicalWindingDefectChargeAt .kosterlitzThoulessPositiveUnitVortex = 1 ∧ physicalWindingDefectChargeAt .kosterlitzThoulessNegativeUnitAntivortex = -1 ∧ physicalWindingDefectChargeAt .kosterlitzThoulessPositiveUnitVortex + physicalWindingDefectChargeAt .kosterlitzThoulessNegativeUnitAntivortex = 0 ∧ physicalWindingNeutralPairAt .kosterlitzThouless1973PlanarVortexAndNeutralFilm .kosterlitzThoulessPositiveUnitVortex .kosterlitzThoulessNegativeUnitAntivortex ∧ physicalFeynmanToroidalRingContext.circulationQuantum = Real.pi ∧ physicalKosterlitzPlanarVortexContext.circulationQuantum = 2 * Real.pi ∧ physicalFeynmanToroidalRingContext.circulationQuantum ≠ physicalKosterlitzPlanarVortexContext.circulationQuantum ∧ physicalWindingRigidityAt .kosterlitzThoulessPlanarXYVortexGas .kosterlitzThoulessColdBoundVortexPhase ∧ ¬physicalWindingRigidityAt .kosterlitzThoulessPlanarXYVortexGas .kosterlitzThoulessHotFreeVortexPhase ∧ ¬physicalWindingConventionalLongRangeOrderAt .kosterlitzThoulessPlanarXYVortexGas .kosterlitzThoulessColdBoundVortexPhase ∧ physicalWindingScopeStatusAt .feynman1954ManyBodyNeutralHeliumRing .feynmanLiteralTwoFluidSpecies = .twoFluidSpeciesPartitionIsArtificial ∧ physicalWindingScopeStatusAt .kosterlitzThouless1973PlanarVortexAndNeutralFilm .absoluteFiniteVolumeMetastability = .finiteSystemBarrierNotAbsolutelyInfinite ∧ physicalWindingScopeStatusAt .kosterlitzThouless1973PlanarVortexAndNeutralFilm .chargedSuperconductorApplicability = .chargedFilmScreeningInvalidatesSameMechanism ∧ physicalWindingScopeStatusAt .kosterlitzThouless1973PlanarVortexAndNeutralFilm .isotropicHeisenbergApplicability = .heisenbergSecondAngleContinuouslyUnwinds ∧ physicalWindingInterpretationPossibility.TentativeAt physicalWindingInterpretationEstablishment .feynmanDerivedLambdaCriticalTheory .feynman1954ManyBodyNeutralHeliumRing ∧ physicalWindingInterpretationPossibility.TentativeAt physicalWindingInterpretationEstablishment .kosterlitzThoulessModernUniversalStiffnessJump .kosterlitzThouless1973PlanarVortexAndNeutralFilm ∧ ¬PhysicalSourcePhaseWindingAt .feynman1954ManyBodyNeutralHeliumRing physicalFeynmanWithoutNoncontractibleContour .multiplyConnectedThreeDimensionalHeliumTorus .feynmanManyBodyNeutralHeliumPhase .feynmanPersistentToroidalFlowSector .multiplyConnectedHeliumTorusHole ∧ ¬PhysicalSourcePhaseWindingAt .feynman1954ManyBodyNeutralHeliumRing physicalFeynmanWithPlanarAngularQuantum .multiplyConnectedThreeDimensionalHeliumTorus .feynmanManyBodyNeutralHeliumPhase .feynmanPersistentToroidalFlowSector .multiplyConnectedHeliumTorusHole ∧ ¬PhysicalSourcePhaseWindingAt .feynman1954ManyBodyNeutralHeliumRing physicalFeynmanWithoutPhaseFlow .multiplyConnectedThreeDimensionalHeliumTorus .feynmanManyBodyNeutralHeliumPhase .feynmanPersistentToroidalFlowSector .multiplyConnectedHeliumTorusHole ∧ ¬PhysicalSourcePhaseWindingAt .feynman1954ManyBodyNeutralHeliumRing physicalFeynmanZeroWindingSector .multiplyConnectedThreeDimensionalHeliumTorus .feynmanManyBodyNeutralHeliumPhase .feynmanPersistentToroidalFlowSector .multiplyConnectedHeliumTorusHole ∧ ¬PhysicalSourcePhaseWindingAt .kosterlitzThouless1973PlanarVortexAndNeutralFilm physicalKosterlitzCoreCrossingContour .puncturedTwoDimensionalPlanarSpinDomain .kosterlitzThoulessCompactPlanarSpinAngle .kosterlitzThoulessPositiveUnitVortex .excludedFinitePlanarVortexCore ∧ ¬PhysicalSourcePhaseWindingAt .kosterlitzThouless1973PlanarVortexAndNeutralFilm physicalKosterlitzChargedFilmSubstitution .puncturedTwoDimensionalPlanarSpinDomain .chargedGaugeCoupledOrderParameter .screenedChargedFiniteEnergyFluxLine .finitePenetrationDepthFluxCore ∧ ¬PhysicalSourcePhaseWindingAt .kosterlitzThouless1973PlanarVortexAndNeutralFilm physicalKosterlitzHeisenbergSubstitution .continuouslyUnwoundTwoAngleSphere .heisenbergTwoAngleMagnetizationTexture .continuouslyUnwoundHeisenbergTexture .excludedFinitePlanarVortexCore ∧ ¬PhysicalSourcePhaseWindingAt .kosterlitzThouless1973PlanarVortexAndNeutralFilm physicalKosterlitzHeliumQuantumSubstitution .puncturedTwoDimensionalPlanarSpinDomain .kosterlitzThoulessCompactPlanarSpinAngle .kosterlitzThoulessPositiveUnitVortex .excludedFinitePlanarVortexCore ∧ ¬PhysicalSourcePhaseWindingAt .kosterlitzThouless1973PlanarVortexAndNeutralFilm physicalKosterlitzZeroWindingSector .puncturedTwoDimensionalPlanarSpinDomain .kosterlitzThoulessCompactPlanarSpinAngle .kosterlitzThoulessPositiveUnitVortex .excludedFinitePlanarVortexCore ∧ PhysicalSourcePhaseWindingAt .feynman1954ManyBodyNeutralHeliumRing physicalFeynmanToroidalRingContext .multiplyConnectedThreeDimensionalHeliumTorus .feynmanManyBodyNeutralHeliumPhase .feynmanPersistentToroidalFlowSector .multiplyConnectedHeliumTorusHole ∧ PhysicalSourcePhaseWindingAt .kosterlitzThouless1973PlanarVortexAndNeutralFilm physicalKosterlitzPlanarVortexContext .puncturedTwoDimensionalPlanarSpinDomain .kosterlitzThoulessCompactPlanarSpinAngle .kosterlitzThoulessPositiveUnitVortex .excludedFinitePlanarVortexCore ∧ PhysicalSourcePhaseWindingAt .kosterlitzThouless1973PlanarVortexAndNeutralFilm physicalKosterlitzNeutralFilmContext .periodicTwoDimensionalNeutralFilm .kosterlitzThoulessLocalNeutralCondensatePhase .neutralFilmPersistentFlowSector .excludedFiniteNeutralFilmVortexCore := by obtain ⟨distinctDescriptions, _distinctRegimes, distinctContours, distinctPlanarFilmDescriptions, _distinctPlanarFilmContours, heliumDimension, planarDimension, filmDimension, _heliumWinding, _planarWinding, positiveCharge, negativeCharge, neutralCharge, neutralPair, heliumQuantum, planarQuantum, distinctQuanta, _feynmanPhaseBridge, _artificialCurrentSplit, _noKTCoreBridgeInFeynman, _neutralityBridge, _logarithmicBarrier, _chargedScreening, _heisenbergUnwinding, _noFeynmanPhaseInKT, coldRigidity, noHotRigidity, noConventionalOrder, artificialTwoFluids, finiteBarrierOnly, screenedFilmExcluded, heisenbergExcluded, lambdaUnproved, universalJumpUnproved, noContractibleFeynmanRing, _noPlanarFeynmanSubstitution, noWrongFeynmanQuantum, noFeynmanPhaseFlow, noZeroFeynmanWinding, noCoreCrossing, _noFreeIsolatedDefect, noChargedFilm, noHeisenbergTwist, noWrongKTQuantum, noZeroKTWinding, actualFeynman, actualPlanarVortex, actualNeutralFilm⟩ := phase_winding_distinguishes_helium_tori_planar_vortices_film_topology_quanta_and_thermodynamic_scope exact ⟨distinctDescriptions, distinctContours, distinctPlanarFilmDescriptions, heliumDimension, planarDimension, filmDimension, positiveCharge, negativeCharge, neutralCharge, neutralPair, heliumQuantum, planarQuantum, distinctQuanta, coldRigidity, noHotRigidity, noConventionalOrder, artificialTwoFluids, finiteBarrierOnly, screenedFilmExcluded, heisenbergExcluded, lambdaUnproved, universalJumpUnproved, noContractibleFeynmanRing, noWrongFeynmanQuantum, noFeynmanPhaseFlow, noZeroFeynmanWinding, noCoreCrossing, noChargedFilm, noHeisenbergTwist, noWrongKTQuantum, noZeroKTWinding, actualFeynman, actualPlanarVortex, actualNeutralFilm⟩ example : physicalWindingThermodynamicContext.EstablishedThermodynamicPropertyAt .boundVortexTopologicalRigidity ∧ physicalWindingThermodynamicContext.limitHasPropertyAt physicalWindingThermodynamicContext.limitingDescription .absolutelyInfiniteWindingBarrier ∧ ¬physicalWindingThermodynamicContext.finiteHasPropertyAt (physicalWindingThermodynamicContext.finiteDescription .finiteLarge) .absolutelyInfiniteWindingBarrier ∧ ¬physicalWindingThermodynamicContext.EstablishedThermodynamicPropertyAt .absolutelyInfiniteWindingBarrier ∧ ¬physicalWindingThermodynamicContext.limitHasPropertyAt physicalWindingThermodynamicContext.limitingDescription .conventionalSpontaneousMagnetization := finite_vortex_rigidity_transfers_but_absolute_metastability_is_only_limiting end PhaseWindingPhysicalReuseAudit namespace LogarithmicDefectPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.LogarithmicDefectThermodynamicsContext.Countermodels example : physicalLogSourceChargeRankAt .kosterlitzThouless1973ScalarVortexEnergyEntropy = 1 ∧ physicalLogSourceChargeRankAt .halperinNelson1978TriangularElasticDislocations = 2 ∧ physicalLogSourceChargeRankAt .young1979TriangularVectorCoulombGas = 2 ∧ physicalLogNeutralPairAt .kosterlitzThouless1973ScalarVortexEnergyEntropy .kosterlitzThoulessPositiveScalarVortex .kosterlitzThoulessNegativeScalarAntivortex ∧ physicalLogNeutralPairAt .halperinNelson1978TriangularElasticDislocations .halperinNelsonPositivePrimitiveBurgersDislocation .halperinNelsonNegativePrimitiveBurgersDislocation ∧ physicalLogNeutralPairAt .young1979TriangularVectorCoulombGas .youngPositivePrimitiveBurgersDislocation .youngNegativePrimitiveBurgersDislocation ∧ (physicalLogDefectChargeAt .youngPositivePrimitiveBurgersDislocation).1 + (physicalLogDefectChargeAt .youngSecondPrimitiveBurgersDislocation).1 = (physicalLogDefectChargeAt .youngThirdPrimitiveBurgersDislocation).1 ∧ (physicalLogDefectChargeAt .youngPositivePrimitiveBurgersDislocation).2 + (physicalLogDefectChargeAt .youngSecondPrimitiveBurgersDislocation).2 = (physicalLogDefectChargeAt .youngThirdPrimitiveBurgersDislocation).2 ∧ ¬physicalLogNeutralPairAt .young1979TriangularVectorCoulombGas .youngPositivePrimitiveBurgersDislocation .youngSecondPrimitiveBurgersDislocation := scalar_vortex_and_vector_burgers_neutrality_are_physically_distinct example : physicalLogSourceCriticalCouplingAt .halperinNelson1978TriangularElasticDislocations = 16 * Real.pi ∧ physicalLogSourceCriticalCouplingAt .young1979TriangularVectorCoulombGas = 2 / Real.pi ∧ physicalLogSourceCriticalCouplingAt .halperinNelson1978TriangularElasticDislocations ≠ physicalLogSourceCriticalCouplingAt .young1979TriangularVectorCoulombGas ∧ physicalLogSourceApproximateExponentAt .halperinNelson1978TriangularElasticDislocations = (44817 : Real) / 100000 ∧ physicalLogSourceApproximateExponentAt .young1979TriangularVectorCoulombGas = (3696 : Real) / 10000 ∧ physicalLogSourceApproximateExponentAt .halperinNelson1978TriangularElasticDislocations ≠ physicalLogSourceApproximateExponentAt .young1979TriangularVectorCoulombGas := by obtain ⟨_scalarCritical, elasticCritical, vectorCritical, distinctCritical, elasticExponent, vectorExponent, distinctExponents⟩ := triangular_crystal_critical_normalizations_and_exponents_do_not_collapse exact ⟨elasticCritical, vectorCritical, distinctCritical, elasticExponent, vectorExponent, distinctExponents⟩ example : physicalHotScalarVortexWithFiniteCoreCost.LogarithmicDefectDomainAt ∧ physicalHotScalarVortexWithFiniteCoreCost.bareUnbindingTemperature < physicalHotScalarVortexWithFiniteCoreCost.temperature ∧ 0 < physicalHotScalarVortexWithFiniteCoreCost.isolatedDefectFreeEnergy := positive_core_cost_prevents_automatic_finite_size_hot_unbinding example : physicalLogInterpretationPossibility.TentativeAt physicalLogInterpretationEstablishment .halperinNelsonFirstOrderMeltingAlternative .halperinNelson1978TriangularElasticDislocations ∧ physicalLogInterpretationPossibility.TentativeAt physicalLogInterpretationEstablishment .halperinNelsonPrematureDisclinationAlternative .halperinNelson1978TriangularElasticDislocations ∧ physicalLogInterpretationPossibility.TentativeAt physicalLogInterpretationEstablishment .youngSquareLatticeCriticalExponentFullyDerived .young1979TriangularVectorCoulombGas ∧ physicalLogInterpretationPossibility.TentativeAt physicalLogInterpretationEstablishment .youngPhysicalBulkModulusFullyCalculated .young1979TriangularVectorCoulombGas ∧ physicalLogInterpretationPossibility.TentativeAt physicalLogInterpretationEstablishment .youngHighDensityAlternativeFlow .young1979TriangularVectorCoulombGas := by exact melting_mechanisms_and_calculation_limits_remain_epistemically_open.2 example : ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .kosterlitzThouless1973ScalarVortexEnergyEntropy physicalKosterlitzVectorDefectSubstitution .kosterlitzThoulessNegativeScalarAntivortex ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .kosterlitzThouless1973ScalarVortexEnergyEntropy physicalKosterlitzWithoutPositionalEntropy .kosterlitzThoulessNegativeScalarAntivortex ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .kosterlitzThouless1973ScalarVortexEnergyEntropy physicalKosterlitzWithoutPairScreening .kosterlitzThoulessNegativeScalarAntivortex ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .kosterlitzThouless1973ScalarVortexEnergyEntropy physicalKosterlitzChargedFilmSubstitution .kosterlitzThoulessNegativeScalarAntivortex ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .halperinNelson1978TriangularElasticDislocations physicalHalperinNelsonScalarVortexSubstitution .halperinNelsonNegativePrimitiveBurgersDislocation ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .halperinNelson1978TriangularElasticDislocations physicalHalperinNelsonWithoutCoreEnergy .halperinNelsonNegativePrimitiveBurgersDislocation ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .halperinNelson1978TriangularElasticDislocations physicalHalperinNelsonDisclinationSubstitution .halperinNelsonNegativePrimitiveBurgersDislocation ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .halperinNelson1978TriangularElasticDislocations physicalHalperinNelsonPinnedSubstrateSubstitution .halperinNelsonNegativePrimitiveBurgersDislocation ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .young1979TriangularVectorCoulombGas physicalYoungScalarVortexSubstitution .youngNegativePrimitiveBurgersDislocation ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .young1979TriangularVectorCoulombGas physicalYoungWithoutAngularInteraction .youngNegativePrimitiveBurgersDislocation ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .young1979TriangularVectorCoulombGas physicalYoungWithoutPairScreening .youngNegativePrimitiveBurgersDislocation ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .young1979TriangularVectorCoulombGas physicalYoungSquareLatticeExtrapolation .youngNegativePrimitiveBurgersDislocation ∧ ¬PhysicalSourceLogarithmicDefectThermodynamicsAt .young1979TriangularVectorCoulombGas physicalYoungHighDensityFlowSubstitution .youngNegativePrimitiveBurgersDislocation ∧ PhysicalSourceLogarithmicDefectThermodynamicsAt .kosterlitzThouless1973ScalarVortexEnergyEntropy physicalKosterlitzScalarVortexContext .kosterlitzThoulessNegativeScalarAntivortex ∧ PhysicalSourceLogarithmicDefectThermodynamicsAt .halperinNelson1978TriangularElasticDislocations physicalHalperinNelsonTriangularDislocationContext .halperinNelsonNegativePrimitiveBurgersDislocation ∧ PhysicalSourceLogarithmicDefectThermodynamicsAt .young1979TriangularVectorCoulombGas physicalYoungTriangularVectorDislocationContext .youngNegativePrimitiveBurgersDislocation := by obtain ⟨_distinctDescriptions, _distinctCrystalDescriptions, _distinctDefects, _distinctCrystalDefects, _scalarRank, _elasticRank, _vectorRank, _scalarNeutral, _elasticNeutral, _vectorNeutral, _scalarCore, _elasticCore, _vectorCore, _scalarEntropy, _elasticEntropy, _vectorEntropy, _scalarBridge, _noElasticBridgeInScalar, _higherDisclinations, _noYoungFusionInLetter, _youngAngular, _youngFusion, _noScalarAreaBridge, _bareOnly, _higherTransition, _firstOrderOpen, _squareIncomplete, _bulkIncomplete, _firstOrderTentative, _highDensityTentative, noVectorInScalar, noScalarEntropy, noScalarScreening, noChargedFilm, noScalarInElastic, noMissingCrystalCore, noDisclinationSubstitution, noPinnedSubstrate, noScalarInYoung, noMissingAngularForce, noMissingYoungScreening, noSquareExtrapolation, noUncontrolledHighDensity, actualScalar, actualElastic, actualVector⟩ := logarithmic_defects_distinguish_scalar_vortices_vector_dislocations_screening_core_cost_and_source_scope exact ⟨noVectorInScalar, noScalarEntropy, noScalarScreening, noChargedFilm, noScalarInElastic, noMissingCrystalCore, noDisclinationSubstitution, noPinnedSubstrate, noScalarInYoung, noMissingAngularForce, noMissingYoungScreening, noSquareExtrapolation, noUncontrolledHighDensity, actualScalar, actualElastic, actualVector⟩ end LogarithmicDefectPhysicalReuseAudit namespace FinitePrecisionPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.FinitePrecisionTrajectoryContext.Countermodels example : physicalPredictabilitySourceParticleCountAt .miller1964ControlledParallelStellarExperiment = 32 ∧ physicalPredictabilitySourceParticleCountAt .portegiesZwartBoekholt2014ConvergedThreeBodyEnsemble = 3 ∧ physicalPredictabilitySourceEnsembleCountAt .miller1964ControlledParallelStellarExperiment = 0 ∧ physicalPredictabilitySourceEnsembleCountAt .portegiesZwartBoekholt2014ConvergedThreeBodyEnsemble = 10000 ∧ ¬PhysicalSourcePredictabilityBoundaryAt .lorenz1963DissipativeConvectionInstability physicalLorenzWithoutNearbyInitialPair ∧ ¬PhysicalSourcePredictabilityBoundaryAt .miller1964ControlledParallelStellarExperiment physicalMillerWithoutSeparatedTrajectories ∧ ¬PhysicalSourcePredictabilityBoundaryAt .miller1964ControlledParallelStellarExperiment physicalMillerOnticBranchingSubstitution ∧ ¬PhysicalSourcePredictabilityBoundaryAt .portegiesZwartBoekholt2014ConvergedThreeBodyEnsemble physicalPZBWithoutConvergedReference ∧ ¬PhysicalSourcePredictabilityBoundaryAt .portegiesZwartBoekholt2014ConvergedThreeBodyEnsemble physicalPZBWithoutTestedEnsemble ∧ ¬PhysicalSourcePredictabilityBoundaryAt .portegiesZwartBoekholt2014ConvergedThreeBodyEnsemble physicalPZBOutsideStatisticalTolerance ∧ PhysicalSourcePredictabilityBoundaryAt .lorenz1963DissipativeConvectionInstability physicalLorenzConvectionBoundary ∧ PhysicalSourcePredictabilityBoundaryAt .miller1964ControlledParallelStellarExperiment physicalMillerParallelStellarBoundary ∧ PhysicalSourcePredictabilityBoundaryAt .portegiesZwartBoekholt2014ConvergedThreeBodyEnsemble physicalPortegiesZwartBoekholtThreeBodyBoundary := by obtain ⟨_distinctLorenzMiller, _distinctMillerPZB, millerParticles, pzbParticles, millerEnsemble, pzbEnsemble, _lorenzTruncation, _noBrutusInLorenz, _millerEncounters, _noPZBEnsembleInMiller, _brutusConvergence, _testedEnsemble, _noMillerUncertaintyInPZB, noLorenzInitialPair, noMillerSeparation, noMillerBranching, noPZBReference, noPZBEnsemble, noPZBTolerance, actualLorenz, actualMiller, actualPZB⟩ := deterministic_predictability_distinguishes_convection_stellar_encounters_three_body_convergence_and_ensemble_scope exact ⟨millerParticles, pzbParticles, millerEnsemble, pzbEnsemble, noLorenzInitialPair, noMillerSeparation, noMillerBranching, noPZBReference, noPZBEnsemble, noPZBTolerance, actualLorenz, actualMiller, actualPZB⟩ example : physicalLorenzConvectionBoundary.HasUniqueReferenceEvolutionAt ∧ ¬physicalLorenzConvectionBoundary.HasOnticIndeterminismAt ∧ ¬physicalLorenzConvectionBoundary.HasTestedEnsembleAgreementAt ∧ physicalMillerParallelStellarBoundary.HasUniqueReferenceEvolutionAt ∧ ¬physicalMillerParallelStellarBoundary.HasTestedEnsembleAgreementAt ∧ physicalPortegiesZwartBoekholtThreeBodyBoundary.HasTestedEnsembleAgreementAt := by obtain ⟨lorenz, noLorenzBranching, noLorenzEnsemble⟩ := physical_lorenz_convection_boundary_is_deterministic_and_nonvacuous obtain ⟨miller, _noMillerBranching, noMillerEnsemble⟩ := physical_miller_stellar_boundary_is_deterministic_without_ensemble_claim obtain ⟨_pzb, pzbEnsemble, _noPZBBranching⟩ := physical_converged_three_body_boundary_retains_tested_ensemble_agreement exact ⟨lorenz.2.1, noLorenzBranching, noLorenzEnsemble, miller.2.1, noMillerEnsemble, pzbEnsemble⟩ end FinitePrecisionPhysicalReuseAudit namespace NeutralDefectPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.NeutralDefectConfigurationContext.Countermodels example : physicalKTScalarVortexConfiguration.chargeRank = 1 ∧ physicalHNTriangularDislocationConfiguration.chargeRank = 2 ∧ physicalYoungTriangularFusionConfiguration.chargeRank = 2 ∧ physicalKTScalarVortexConfiguration.charges.length = 2 ∧ physicalHNTriangularDislocationConfiguration.charges.length = 4 ∧ physicalYoungTriangularFusionConfiguration.charges.length = 6 ∧ ¬PhysicalSourceNeutralConfigurationAt .kosterlitzThouless1973CompactScalarWinding physicalKTIsolatedNonneutralVortex ∧ ¬PhysicalSourceNeutralConfigurationAt .kosterlitzThouless1973CompactScalarWinding physicalKTTransverseRankOneSubstitution ∧ ¬PhysicalSourceNeutralConfigurationAt .kosterlitzThouless1973CompactScalarWinding physicalKTVectorDislocationSubstitution ∧ ¬PhysicalSourceNeutralConfigurationAt .halperinNelson1978TriangularElasticBurgersGas physicalHNScalarRankSubstitution ∧ ¬PhysicalSourceNeutralConfigurationAt .halperinNelson1978TriangularElasticBurgersGas physicalHNNonneutralBurgersConfiguration ∧ ¬PhysicalSourceNeutralConfigurationAt .halperinNelson1978TriangularElasticBurgersGas physicalHNDisclinationSubstitution ∧ ¬PhysicalSourceNeutralConfigurationAt .young1979TriangularVectorCoulombGas physicalYoungEmptyVectorGas ∧ ¬PhysicalSourceNeutralConfigurationAt .young1979TriangularVectorCoulombGas physicalYoungMissingTriangularFusionDirections ∧ ¬PhysicalSourceNeutralConfigurationAt .young1979TriangularVectorCoulombGas physicalYoungNonneutralFusionConfiguration ∧ ¬PhysicalSourceNeutralConfigurationAt .young1979TriangularVectorCoulombGas physicalYoungSquareLatticeExtrapolation ∧ ¬PhysicalSourceNeutralConfigurationAt .young1979TriangularVectorCoulombGas physicalYoungHotUnboundRegime ∧ PhysicalSourceNeutralConfigurationAt .kosterlitzThouless1973CompactScalarWinding physicalKTScalarVortexConfiguration ∧ PhysicalSourceNeutralConfigurationAt .halperinNelson1978TriangularElasticBurgersGas physicalHNTriangularDislocationConfiguration ∧ PhysicalSourceNeutralConfigurationAt .young1979TriangularVectorCoulombGas physicalYoungTriangularFusionConfiguration := by obtain ⟨_differentKTAndHNDescriptions, _differentHNAndYoungDescriptions, _differentKTAndHNDefects, _differentHNAndYoungDefects, scalarRank, elasticRank, vectorRank, scalarCount, elasticCount, vectorCount, _scalarNeutrality, _noVectorBridgeInScalar, _disclinationIsSeparate, _noYoungFusionInHN, _sixPrimitiveDirections, _youngFusion, _noScalarBridgeInYoung, noIsolatedScalar, noTransverseScalar, noWrongScalarDefect, noWrongElasticRank, noNonneutralElastic, noDisclinationSubstitution, noEmptyVectorGas, noMissingFusionDirections, noNonneutralFusion, noSquareExtrapolation, noHotUnboundPhase, actualScalar, actualElastic, actualVector⟩ := neutral_defects_distinguish_scalar_vortices_triangular_burgers_vectors_fusion_and_source_scope exact ⟨scalarRank, elasticRank, vectorRank, scalarCount, elasticCount, vectorCount, noIsolatedScalar, noTransverseScalar, noWrongScalarDefect, noWrongElasticRank, noNonneutralElastic, noDisclinationSubstitution, noEmptyVectorGas, noMissingFusionDirections, noNonneutralFusion, noSquareExtrapolation, noHotUnboundPhase, actualScalar, actualElastic, actualVector⟩ end NeutralDefectPhysicalReuseAudit namespace AcceleratedGravityPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.AcceleratedFrameGravityContext.Countermodels example : physicalEinstein1911GravityModel.referenceFrame.gravitationalAcceleration = physicalEinstein1911GravityModel.acceleratedFrame.coordinateAcceleration ∧ physicalEinstein1911GravityModel.regime.firstOrderOnly = true ∧ physicalEinstein1916GravityModel.regime.spacetimeDimension = 4 ∧ ¬PhysicalSourceAcceleratedFrameGravityAt .einstein1911HomogeneousFieldMechanicalComparison physical1911MismatchedAccelerationModel ∧ ¬PhysicalSourceAcceleratedFrameGravityAt .einstein1911HomogeneousFieldMechanicalComparison physical1911NonmaterialLightSubstitutionModel ∧ ¬PhysicalSourceAcceleratedFrameGravityAt .einstein1911HomogeneousFieldMechanicalComparison physical1911BeyondFirstOrderModel ∧ ¬PhysicalSourceAcceleratedFrameGravityAt .einstein1916CompositionIndependentAcceleratedCoordinates physical1916NoninertialReferenceModel ∧ ¬PhysicalSourceAcceleratedFrameGravityAt .einstein1916CompositionIndependentAcceleratedCoordinates physical1916CompositionDependentFallModel ∧ ¬PhysicalSourceAcceleratedFrameGravityAt .einstein1916CompositionIndependentAcceleratedCoordinates physical1916ExtendedTidalSubstitutionModel ∧ PhysicalTidalDistinguishabilityAt physicalExtendedTidalBodyPair physicalExtendedTidalRegime ∧ PhysicalSourceAcceleratedFrameGravityAt .einstein1911HomogeneousFieldMechanicalComparison physicalEinstein1911GravityModel ∧ PhysicalSourceAcceleratedFrameGravityAt .einstein1916CompositionIndependentAcceleratedCoordinates physicalEinstein1916GravityModel := by obtain ⟨_differentReferenceFrames, _differentAcceleratedFrames, _distinct1911Bodies, _distinct1916Bodies, firstOrder1911, fourDimensional1916, matched1911Acceleration, _matched1911Bridge, _arbitraryFieldLimitation, _noEotvosBridgeIn1911, _eotvosBridge1916, _fieldEquationRequiresIndependentSelection, _noFirstOrderBridgeIn1916, noMismatched1911Acceleration, _noSame1911Composition, no1911LightSubstitution, no1911BeyondFirstOrder, no1916NoninertialReference, _no1916NonuniformFrame, no1916CompositionDependentFall, _no1916SameBody, no1916ExtendedTidalRegion, _no1916ModelIn1911Source, _no1911ModelIn1916Source, actualTidalDistinguishability, actual1911, actual1916⟩ := accelerated_gravity_distinguishes_homogeneous_fields_inertial_frames_compositions_tidal_scope_and_source_limits exact ⟨matched1911Acceleration, firstOrder1911, fourDimensional1916, noMismatched1911Acceleration, no1911LightSubstitution, no1911BeyondFirstOrder, no1916NoninertialReference, no1916CompositionDependentFall, no1916ExtendedTidalRegion, actualTidalDistinguishability, actual1911, actual1916⟩ example : physicalGravityInterpretationPossibility.TentativeAt physicalGravityInterpretationEstablishment .allPhysicalProcessesEquivalentWithoutAdditionalHypothesis1911 .einstein1911HomogeneousFieldMechanicalComparison ∧ physicalGravityInterpretationPossibility.TentativeAt physicalGravityInterpretationEstablishment .metricFieldEquationDerivedFromFrameComparison1916 .einstein1916CompositionIndependentAcceleratedCoordinates ∧ physicalGravityInterpretationPossibility.TentativeAt physicalGravityInterpretationEstablishment .strongEquivalenceForSelfGravitatingBodies1916 .einstein1916CompositionIndependentAcceleratedCoordinates := by obtain ⟨allProcessOpen, _firstOrderOpen, _globalRemovalOpen, fieldEquationOpen, strongEquivalenceOpen⟩ := accelerated_gravity_source_overclaims_remain_epistemically_open exact ⟨allProcessOpen, fieldEquationOpen, strongEquivalenceOpen⟩ end AcceleratedGravityPhysicalReuseAudit namespace AdiabaticCyclePhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.AdiabaticCyclePhaseContext.Countermodels example : physicalBerryGeometricCycleModel.parameterSpace.dimension = 3 ∧ physicalArovasQuasiholeCycleModel.parameterSpace.dimension = 2 ∧ physicalBerryGeometricCycleModel.sector.variationalApproximation = false ∧ physicalArovasQuasiholeCycleModel.sector.variationalApproximation = true ∧ physicalArovasQuasiholeCycleModel.family.fillingNumerator = 1 ∧ physicalArovasQuasiholeCycleModel.family.fillingDenominator = 3 ∧ physicalArovasQuasiholeCycleModel.phase.piNumerator = -2 ∧ physicalArovasQuasiholeCycleModel.phase.piDenominator = 3 ∧ ¬PhysicalSourceAdiabaticCyclePhaseAt .berry1984NondegenerateGeometricPhase physicalBerryDegeneracyCrossingModel ∧ ¬PhysicalSourceAdiabaticCyclePhaseAt .berry1984NondegenerateGeometricPhase physicalBerryFastTraversalModel ∧ ¬PhysicalSourceAdiabaticCyclePhaseAt .arovasSchriefferWilczek1984LaughlinQuasiholeTransport physicalArovasSmallLoopModel ∧ ¬PhysicalSourceAdiabaticCyclePhaseAt .arovasSchriefferWilczek1984LaughlinQuasiholeTransport physicalArovasWrongFluxPhaseModel ∧ ¬PhysicalFractionalQuasiholeExchangeAt physicalArovasSingleHoleWithoutBraiding ∧ ¬PhysicalFractionalQuasiholeExchangeAt physicalArovasWrongExchangeHalfPhase ∧ ¬PhysicalFractionalQuasiholeExchangeAt physicalArovasThirdHolePathDependence ∧ PhysicalSourceAdiabaticCyclePhaseAt .berry1984NondegenerateGeometricPhase physicalBerryLongerTraversalModel ∧ PhysicalSourceAdiabaticCyclePhaseAt .arovasSchriefferWilczek1984LaughlinQuasiholeTransport physicalArovasLongerTraversalModel ∧ PhysicalFractionalQuasiholeExchangeAt physicalArovasOneThirdQuasiholeBraiding := by obtain ⟨_differentFamilies, berryDimension, arovasDimension, berryNonvariational, arovasVariational, _berryPhaseNumerator, _berryPhaseDenominator, arovasFillingNumerator, arovasFillingDenominator, arovasPhaseNumerator, arovasPhaseDenominator, _berryHalfSolidAngleBridge, _noArovasBraidingBridgeInBerry, _arovasInsideOutsideBridge, _arovasVariationalQualification, _noBerryEigenstateBridgeInArovas, noDegeneracyCrossing, _noOpenPath, noFastBerryPath, _noDegenerateBerrySector, _noPurelyDynamicalBerryPhase, noSmallArovasLoop, _noIntegralFilling, _noUnnormalizedState, noWrongFluxPhase, noSingleHoleBraiding, noWrongExchangeHalfPhase, noThirdHoleIndependentExchange, longerBerry, longerArovas, actualFractionalBraiding, _actualBerry, _actualArovas⟩ := adiabatic_cycles_distinguish_berry_eigenstate_holonomy_laughlin_quasiholes_fractional_braiding_and_source_scope exact ⟨berryDimension, arovasDimension, berryNonvariational, arovasVariational, arovasFillingNumerator, arovasFillingDenominator, arovasPhaseNumerator, arovasPhaseDenominator, noDegeneracyCrossing, noFastBerryPath, noSmallArovasLoop, noWrongFluxPhase, noSingleHoleBraiding, noWrongExchangeHalfPhase, noThirdHoleIndependentExchange, longerBerry, longerArovas, actualFractionalBraiding⟩ example : physicalBerryReversedPhase.piNumerator = -physicalBerryHalfSolidAnglePhase.piNumerator ∧ physicalArovasReversedPhase.piNumerator = -physicalArovasOneThirdFluxPhase.piNumerator ∧ physicalArovasOneThirdQuasiholeBraiding.fullWindingPhasePiNumerator = 2 * physicalArovasOneThirdQuasiholeBraiding.exchangePhasePiNumerator := by obtain ⟨_berryFormula, _arovasFormula, berryReversal, arovasReversal⟩ := reversing_berry_and_quasihole_loops_reverses_their_geometric_phases obtain ⟨braiding, _numerator, _denominator, _fullWinding, _exchange⟩ := two_quasiholes_realize_fractional_full_winding_and_exchange_phases exact ⟨berryReversal, arovasReversal, braiding.2.2.2.2.2.2.2.2.2.1⟩ end AdiabaticCyclePhysicalReuseAudit namespace CorrelationDecayPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.CorrelationDecayChannelContext.Countermodels example : physicalKTAlgebraicSpinWaveChannel.algebraicExponent = 1 / 4 ∧ physicalHKPowerLawCorrectedBosonicChannel.algebraicExponent = 2 ∧ physicalHKExponentialCorrectedBosonicChannel.exponentialRate = 1 ∧ physicalHKExponentialReadout.groundSectorDegeneracy = 2 ∧ ¬PhysicalSourceCorrelationDecayAt .kosterlitzThouless1973SpinWavePowerLaw physicalKTSpinWaveReadout physicalKTFullCorrelationSubstitution ∧ ¬PhysicalSourceCorrelationDecayAt .kosterlitzThouless1973SpinWavePowerLaw physicalKTZeroTemperatureReadout physicalKTAlgebraicSpinWaveChannel ∧ ¬PhysicalSourceCorrelationDecayAt .hastingsKoma2006ExponentialCorrectedBosonicClustering physicalHKGaplessExponentialReadout physicalHKExponentialCorrectedBosonicChannel ∧ ¬PhysicalSourceCorrelationDecayAt .hastingsKoma2006ExponentialCorrectedBosonicClustering physicalHKNonuniformGapReadout physicalHKExponentialCorrectedBosonicChannel ∧ ¬PhysicalSourceCorrelationDecayAt .hastingsKoma2006ExponentialCorrectedBosonicClustering physicalHKNoWeakStarLimitReadout physicalHKExponentialCorrectedBosonicChannel ∧ ¬PhysicalSourceCorrelationDecayAt .hastingsKoma2006ExponentialCorrectedBosonicClustering physicalHKWrongDressedRateReadout physicalHKExponentialCorrectedBosonicChannel ∧ ¬PhysicalSourceCorrelationDecayAt .hastingsKoma2006ExponentialCorrectedBosonicClustering physicalHKPowerLawAsExponentialReadout physicalHKExponentialCorrectedBosonicChannel ∧ ¬PhysicalSourceCorrelationDecayAt .hastingsKoma2006ExponentialCorrectedBosonicClustering physicalHKExponentialReadout physicalHKRawBosonicSubstitution ∧ ¬PhysicalRawBosonicDecayUpgradeAt physicalHKExponentialReadout ∧ PhysicalRawBosonicDecayUpgradeAt physicalHKSymmetryUpgradedReadout ∧ PhysicalSourceCorrelationDecayAt .kosterlitzThouless1973SpinWavePowerLaw physicalKTSpinWaveReadout physicalKTAlgebraicSpinWaveChannel ∧ PhysicalSourceCorrelationDecayAt .hastingsKoma2006PowerLawCorrectedBosonicClustering physicalHKPowerLawReadout physicalHKPowerLawCorrectedBosonicChannel ∧ PhysicalSourceCorrelationDecayAt .hastingsKoma2006ExponentialCorrectedBosonicClustering physicalHKExponentialReadout physicalHKExponentialCorrectedBosonicChannel := by obtain ⟨_differentDescriptions, _differentChannels, ktExponent, hkPowerExponent, hkExponentialRate, groundDegeneracy, _ktFactorizationBridge, _noGroundProjectorInKT, _hkPowerLawBridge, _noExponentialBridgeInPowerLaw, _hkExponentialBridge, _hkRawNeedsCancellation, noKTFullCorrelation, noZeroTemperature, _noZeroStiffness, noGaplessHK, noNonuniformGap, noWeakStarLimit, noWrongDressedRate, noWrongInteractionBranch, noRawBosonicSubstitution, _noZeroCorrectedChannel, noUpgradeWithoutSymmetry, upgradedRaw, actualKT, actualHKPower, actualHKExponential⟩ := correlation_decay_distinguishes_spin_waves_corrected_ground_sectors_interaction_tails_and_thermodynamic_scope exact ⟨ktExponent, hkPowerExponent, hkExponentialRate, groundDegeneracy, noKTFullCorrelation, noZeroTemperature, noGaplessHK, noNonuniformGap, noWeakStarLimit, noWrongDressedRate, noWrongInteractionBranch, noRawBosonicSubstitution, noUpgradeWithoutSymmetry, upgradedRaw, actualKT, actualHKPower, actualHKExponential⟩ example : physicalHKFiniteVolumeCorrelationLimit.EstablishedThermodynamicPropertyAt .uniformGroundSectorCorrectedDecay ∧ ¬physicalHKFiniteVolumeCorrelationLimit.limitHasPropertyAt .infiniteWeakStarGroundSector .unsubtractedBosonicDecay := uniform_finite_volume_corrected_bounds_transfer_but_raw_decay_does_not example : PhysicalSourceCorrelationDecayAt .hastingsKoma2006ExponentialCorrectedBosonicClustering physicalHKExponentialReadout physicalHKExponentialCorrectedBosonicChannel ∧ ¬Filter.Tendsto (fun separation : Real => 1 + Real.exp (-separation)) Filter.atTop (nhds 0) := by obtain ⟨corrected, _forwardGroundTerm, _reverseGroundTerm, notRaw⟩ := corrected_bosonic_decay_does_not_remove_a_nonzero_ground_sector_background exact ⟨corrected, notRaw⟩ end CorrelationDecayPhysicalReuseAudit namespace NuclearEffectiveInteractionPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.EffectiveInteractionDynamicsContext.Countermodels example : PhysicalSourceNuclearEffectiveDynamicsAt .ordonezRayVanKolck1995 physicalOrdonez1995Model ∧ PhysicalSourceNuclearEffectiveDynamicsAt .epelbaumGloeckleMeissner1998 physicalEpelbaum1998Model ∧ PhysicalSourceNuclearEffectiveDynamicsAt .epelbaumGloeckleMeissner1999 physicalEpelbaum1999Model := by obtain ⟨ordonez, formalUnitaryProjection, regulatedScattering, _⟩ := nuclear_effective_dynamics_distinguishes_chiral_potentials_unitary_projection_regulated_iteration_and_source_scope exact ⟨ordonez, formalUnitaryProjection, regulatedScattering⟩ example : physicalEpelbaum1999NLOInteraction.regulatorCutoffMeV = some 500 ∧ physicalEpelbaum1999Interaction.regulatorCutoffMeV = some 875 ∧ physicalEpelbaum1999NLOInteraction.maximumChiralPower < physicalEpelbaum1999Interaction.maximumChiralPower := by obtain ⟨_nloPotential, _nloRegulator, _nnloPotential, _nnloRegulator, nloCutoff, nnloCutoff, orderIncrease⟩ := physical_epelbaum_1999_nlo_and_nnlo_regulators_are_order_distinct exact ⟨nloCutoff, nnloCutoff, orderIncrease⟩ example : ¬PhysicalNuclearRegulatorConsistencyAt physicalEpelbaum1999WrongRegulatorExponent physicalEpelbaum1999Dynamics physicalEpelbaum1999Regime ∧ PhysicalNuclearRegulatorConsistencyAt physicalEpelbaum1999ExponentialRegulator physicalEpelbaum1999Dynamics physicalEpelbaum1999Regime := by obtain ⟨_notBlochHorowitz, _noUnfittedContacts, wrongExponent, allowedExponent, _⟩ := unitary_projection_contact_fits_and_regulator_powers_do_not_collapse exact ⟨wrongExponent, allowedExponent⟩ example : physicalEpelbaum1998Dynamics.deuteronBoundStateEstablished = false ∧ physicalEpelbaum1999Dynamics.deuteronBoundStateEstablished = true := by obtain ⟨_first, _second, _third, formalHasNoNumericalDeuteron, numericalDeuteron, _⟩ := nuclear_sources_cannot_exchange_derivations_fit_windows_or_dynamical_claims exact ⟨formalHasNoNumericalDeuteron, numericalDeuteron⟩ end NuclearEffectiveInteractionPhysicalReuseAudit namespace StatisticalMemoryKernelPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.MemoryKernelReductionContext.Countermodels example : PhysicalSourceMemoryReductionAt .mori1965ProjectedCollectiveTransport physicalMoriReadout physicalMoriModel ∧ PhysicalSourceMemoryReductionAt .zwanzig1973NonlinearShiftedHarmonicBath physicalZwanzigReadout physicalZwanzigModel := by obtain ⟨mori, zwanzig, _⟩ := memory_kernel_reduction_distinguishes_projected_forces_shifted_baths_thermal_noise_and_markov_scope exact ⟨mori, zwanzig⟩ example : PhysicalZwanzigShiftedCanonicalBathAt physicalZwanzigNoSlowMotionReadout ∧ ¬PhysicalMemoryShortMemoryLicenseAt .zwanzig1973NonlinearShiftedHarmonicBath physicalZwanzigNoSlowMotionReadout := by obtain ⟨_exactWithoutContinuum, _noContinuumLocalization, exactWithoutSlowMotion, noSlowMotionLocalization, _⟩ := finite_bath_continuum_and_slow_motion_are_independent_memory_stages exact ⟨exactWithoutSlowMotion, noSlowMotionLocalization⟩ example : PhysicalMoriProjectedForceAt physicalMoriReadout ∧ ¬PhysicalMoriProjectedForceAt physicalMoriBareForceSubstitution := by obtain ⟨projected, _notUnprojected, notBare, _⟩ := orthogonal_projection_cannot_be_replaced_by_bare_force_statistics exact ⟨projected, notBare⟩ end StatisticalMemoryKernelPhysicalReuseAudit namespace SelfGravityEquivalencePhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.BodyDependentSelfGravityContext.Countermodels example : PhysicalSourceSelfGravityResponseAt .nordtvedt1968StationaryMassiveBody physicalNordtvedtReadout physicalNordtvedtModel ∧ PhysicalSourceSelfGravityResponseAt .eardley1975CompactScalarBinary physicalEardleyReadout physicalEardleyModel := by obtain ⟨nordtvedt, eardley, _⟩ := self_gravity_distinguishes_test_bodies_metric_binding_compact_sensitivities_dipoles_and_strong_equivalence exact ⟨nordtvedt, eardley⟩ example : physicalNordtvedtReadout.testBodyMassRatio = 1 ∧ physicalNordtvedtReadout.massiveBodyMassRatio ≠ 1 := by obtain ⟨_response, testBody, massiveBody, _⟩ := test_body_equivalence_does_not_force_massive_body_equivalence exact ⟨testBody, massiveBody⟩ example : physicalEardleyEqualSensitivityBinary.dipoleDecayStrength = 0 ∧ physicalEardleyBlackHoleCompanion.clockVariationAmplitude = 0 := by obtain ⟨_equalSensitivities, noDipole, _blackHole, _noCharge, noClockVariation, _⟩ := equal_sensitivities_remove_dipoles_but_black_hole_companions_remove_clock_variation exact ⟨noDipole, noClockVariation⟩ end SelfGravityEquivalencePhysicalReuseAudit namespace CondensedMatterLocalizationPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.OneParameterLocalizationScalingContext.Countermodels example : PhysicalSourceLocalizationScalingAt .abrahamsAndersonLicciardelloRamakrishnan1979 physicalAALRReadout physicalAALRModel ∧ PhysicalSourceLocalizationScalingAt .macKinnonKramer1981 physicalMacKinnonKramerReadout physicalMacKinnonKramerModel := by obtain ⟨aalr, macKinnon, _⟩ := localization_scaling_distinguishes_thouless_blocks_anderson_bars_dimensions_mobility_edges_and_source_scope exact ⟨aalr, macKinnon⟩ example : physicalMacKinnonKramerReadout.criticalDisorderEstimate = 16 ∧ physicalMacKinnonKramerReadout.criticalDisorderUncertainty = 1 / 2 ∧ physicalMacKinnonKramerReadout.criticalExponentEstimate = 6 / 5 ∧ physicalMacKinnonKramerReadout.criticalExponentUncertainty = 3 / 10 := by norm_num [physicalMacKinnonKramerReadout] example : PhysicalMacKinnonFiniteSizeScalingAt physicalMacKinnonKramerReadout ∧ ¬PhysicalMacKinnonFiniteSizeScalingAt physicalMacKinnonUnsampledWeakDisorder := by obtain ⟨sampled, notWeak, _⟩ := finite_size_localization_retains_sample_windows_recursions_and_fitted_uncertainty exact ⟨sampled, notWeak⟩ example : PhysicalAALRBlockScalingAt physicalAALRReadout ∧ ¬PhysicalAALRBlockScalingAt physicalAALRNoBlockClosure := by obtain ⟨aalr, _notCoherent, noClosure, _⟩ := thouless_scaling_requires_coherence_block_assumptions_and_correct_normalization exact ⟨aalr, noClosure⟩ end CondensedMatterLocalizationPhysicalReuseAudit namespace QuantumInformationBlackHoleSharingPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.TripartiteSharingContext.Countermodels example : PhysicalSourceTripartiteSharingAt .coffmanKunduWootters2000 physicalCKWSharingReadout physicalCKWSharingModel ∧ PhysicalSourceTripartiteSharingAt .almheiriMarolfPolchinskiSully2012 physicalAMPSInformationLossBoundaryReadout physicalAMPSSharingModel := by obtain ⟨ckw, amps, _⟩ := tripartite_sharing_distinguishes_qubit_tangle_hawking_entropy_factorization_and_source_scope exact ⟨ckw, amps⟩ example : physicalCKWFormationCounterexample.pairABFormation = 601 / 1000 ∧ physicalCKWFormationCounterexample.pairACFormation = 601 / 1000 ∧ physicalCKWFormationCounterexample.compositeFormation < physicalCKWFormationCounterexample.pairABFormation + physicalCKWFormationCounterexample.pairACFormation := by norm_num [physicalCKWFormationCounterexample] example : PhysicalAMPSOldHolePremisesAt physicalAMPSOldHoleConflictReadout ∧ ¬PhysicalAMPSStrongSubadditivityAt physicalAMPSOldHoleConflictReadout := by obtain ⟨premises, conflict, _⟩ := amps_old_hole_premises_require_factorization_and_violate_strong_subadditivity exact ⟨premises, conflict⟩ example : PhysicalAMPSPremisesWithoutFactorizationAt physicalAMPSMissingPairFactorization ∧ ¬PhysicalAMPSOldHolePremisesAt physicalAMPSMissingPairFactorization := by obtain ⟨_premises, _conflict, retained, notComplete, _⟩ := amps_old_hole_premises_require_factorization_and_violate_strong_subadditivity exact ⟨retained, notComplete⟩ example : ¬PhysicalAMPSHorizonApplicabilityAt physicalAMPSYoungBlackHole ∧ ¬PhysicalAMPSHorizonApplicabilityAt physicalAMPSRindlerHorizon ∧ ¬PhysicalAMPSHorizonApplicabilityAt physicalAMPSDeSitterHorizon := by obtain ⟨_old, young, rindler, deSitter⟩ := amps_horizon_scope_excludes_young_rindler_and_nonevaporating_desitter exact ⟨young, rindler, deSitter⟩ end QuantumInformationBlackHoleSharingPhysicalReuseAudit namespace ClassicalCompleteIntegralPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.CompleteIntegralDynamicsCountermodels example : PhysicalSourceCompleteIntegralDynamicsAt .hamiltonGeneralMethod1834 physicalHamiltonCompleteIntegralReadout physicalHamiltonCompleteIntegralModel ∧ PhysicalSourceCompleteIntegralDynamicsAt .jacobiPDEReduction1837 physicalJacobiCompleteIntegralReadout physicalJacobiCompleteIntegralModel := by obtain ⟨hamilton, jacobi, _⟩ := complete_integral_dynamics_distinguishes_hamilton_endpoint_actions_jacobi_arbitrary_integrals_and_time_dependence exact ⟨hamilton, jacobi⟩ example : physicalHamiltonCompleteIntegralReadout.finalEndpointDerivative = 2 ∧ physicalHamiltonCompleteIntegralReadout.initialEndpointDerivative = -3 ∧ physicalHamiltonCompleteIntegralReadout.actionVariationValue = 5 := by norm_num [physicalHamiltonCompleteIntegralReadout] example : PhysicalHamiltonConservativeMethodScopeAt physicalHamiltonCompleteIntegralReadout ∧ ¬PhysicalHamiltonConservativeMethodScopeAt physicalHamiltonNonconservativeForces := by obtain ⟨_source, _sign, _action, _stationary, _parameters, _initial, scopeWitness, notConservative⟩ := hamilton_endpoint_signs_stationary_comparisons_completeness_and_scope_are_independent exact ⟨scopeWitness, notConservative⟩ example : PhysicalJacobiTimeDependentRecoveryAt physicalJacobiCompleteIntegralReadout ∧ physicalJacobiCompleteIntegralReadout.energyDriftValue ≠ 0 ∧ physicalJacobiCompleteIntegralReadout.secondEndpointEquationSatisfied = false := by obtain ⟨recovery, _arbitrary, _family, _balance, _constant, _notConserved, energyDrift, noSecondEquation, _⟩ := jacobi_arbitrary_integrals_parameter_constants_and_time_dependent_energy_are_independent exact ⟨recovery, energyDrift, noSecondEquation⟩ example : PhysicalJacobiAutonomousReductionAt physicalJacobiAutonomousEnergyControl ∧ ¬PhysicalJacobiTimeDependentRecoveryAt physicalJacobiAutonomousEnergyControl := by obtain ⟨_timeDependent, _notAutonomous, autonomous, notTimeDependent⟩ := jacobi_autonomous_energy_reduction_and_explicit_time_dependence_are_distinct exact ⟨autonomous, notTimeDependent⟩ end ClassicalCompleteIntegralPhysicalReuseAudit namespace NuclearLargeScatteringLengthPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.LargeScatteringLengthResummationContext.Countermodels example : PhysicalSourceLargeLengthResummationAt .kaplanSavageWise1998 physicalKSWPDSLargeLengthReadout physicalKSWLargeLengthModel ∧ PhysicalSourceLargeLengthResummationAt .bedaqueHammerVanKolck1998 physicalBHvKDressedDibaryonReadout physicalBHvKLargeLengthModel := by obtain ⟨ksw, bedaque, _⟩ := nuclear_large_length_resummation_distinguishes_pds_bubbles_dressed_dibaryons_running_contacts_and_three_body_scope exact ⟨ksw, bedaque⟩ example : physicalKSWPDSLargeLengthReadout.leadingContactValue ≠ physicalKSWAlternativeSubtractionScale.leadingContactValue ∧ physicalKSWPDSLargeLengthReadout.recoveredAmplitudeRealValue = physicalKSWAlternativeSubtractionScale.recoveredAmplitudeRealValue ∧ physicalKSWPDSLargeLengthReadout.recoveredAmplitudeImaginaryValue = physicalKSWAlternativeSubtractionScale.recoveredAmplitudeImaginaryValue := by obtain ⟨_original, _changed, running, sameReal, sameImaginary, _⟩ := ksw_pds_poles_loop_bubble_running_and_ms_window_are_independent exact ⟨running, sameReal, sameImaginary⟩ example : physicalBHvKDressedDibaryonReadout.auxiliaryDibaryonScaleValue ≠ physicalBHvKAlternativeAuxiliaryNormalization.auxiliaryDibaryonScaleValue ∧ physicalBHvKDressedDibaryonReadout.dressedDibaryonRealValue ≠ physicalBHvKAlternativeAuxiliaryNormalization.dressedDibaryonRealValue ∧ physicalBHvKDressedDibaryonReadout.externalLineAmplitudeRealValue = physicalBHvKAlternativeAuxiliaryNormalization.externalLineAmplitudeRealValue := by obtain ⟨_original, _changed, differentScale, differentPropagator, sameAmplitude, _⟩ := bhvk_dibaryon_normalization_dressing_external_lines_and_cutoff_accuracy_are_independent exact ⟨differentScale, differentPropagator, sameAmplitude⟩ example : PhysicalDressedDibaryonAt physicalBHvKDressedDibaryonReadout ∧ ¬PhysicalThreeBodyCountertermEstablishedAt physicalBHvKDressedDibaryonReadout ∧ PhysicalThreeBodyCountertermEstablishedAt physicalBHvKWithIndependentThreeBodyForce := by obtain ⟨twoBody, noThreeBody, _extended, independentlyEstablished, _⟩ := two_body_resummation_does_not_determine_three_body_counterterm exact ⟨twoBody, noThreeBody, independentlyEstablished⟩ example : ¬PhysicalOrdinaryMinimalSubtractionWindowAt physicalKSWOrdinaryMinimalSubtractionAtLargeMomentum ∧ PhysicalOrdinaryMinimalSubtractionWindowAt physicalKSWOrdinaryMinimalSubtractionWindow := by obtain ⟨_original, _changed, _running, _real, _imaginary, _pole, _loop, _bubble, _truncation, excluded, allowed⟩ := ksw_pds_poles_loop_bubble_running_and_ms_window_are_independent exact ⟨excluded, allowed⟩ end NuclearLargeScatteringLengthPhysicalReuseAudit namespace QuantumProjectorValuationPhysicalReuseAudit open PhysicsLogic.Framework.ProjectorValuationObstruction open PhysicsLogic.Framework.ProjectorValuationObstruction.Countermodels example : PhysicalSourceProjectorValuationReadoutAt .bellHiddenVariables1966 physicalBellProjectorValuationReadout ∧ PhysicalSourceProjectorValuationReadoutAt .kochenSpeckerHiddenVariables1967 physicalKochenSpeckerSpinOneValuationReadout := by obtain ⟨bell, kochen, _⟩ := projector_valuations_distinguish_gleason_contextual_apparatus_finite_117_spin_one_polarity_and_experimental_scope exact ⟨bell, kochen⟩ example : physicalBellProjectorValuationReadout.explicitContextualModelConstructed = true ∧ physicalBellProjectorValuationReadout.contextualOutcomesDeterministic = true ∧ PhysicalPositiveTwoDimensionalHiddenModelAt physicalBellProjectorValuationReadout := by obtain ⟨_obstruction, twoDimensional, constructed, deterministic, _⟩ := contextual_no_go_preserves_deterministic_bell_and_positive_dimension_two_models exact ⟨constructed, deterministic, twoDimensional⟩ example : physicalKochenSpeckerSpinOneValuationReadout.finiteNoColoringDirectionCount = 117 ∧ physicalKochenSpeckerSpinOneValuationReadout.smallerNonembeddingDirectionCount = 17 ∧ physicalKochenSpeckerSpinOneValuationReadout.zeroSpinProjectorRank = 1 ∧ physicalKochenSpeckerSpinOneValuationReadout.nonzeroSquaredSpinProjectorRank = 2 := by norm_num [physicalKochenSpeckerSpinOneValuationReadout] example : PhysicalSpinOneOrthogonalFrameAt physicalKochenCommutationWithoutMeasurementBridge ∧ ¬PhysicalKochenSpeckerRhombicMeasurementAt physicalKochenCommutationWithoutMeasurementBridge := by obtain ⟨commuting, notMeasured, _⟩ := rhombic_measurement_and_proposed_experiment_do_not_follow_from_commutation exact ⟨commuting, notMeasured⟩ example : PhysicalKochenSpeckerRhombicMeasurementAt physicalKochenClaimedDirectExperiment ∧ ¬PhysicalKochenSpeckerExperimentalQualificationAt physicalKochenClaimedDirectExperiment := by obtain ⟨_commuting, _missingBridge, _degenerate, proposedMeasurement, notPerformed, _⟩ := rhombic_measurement_and_proposed_experiment_do_not_follow_from_commutation exact ⟨proposedMeasurement, notPerformed⟩ end QuantumProjectorValuationPhysicalReuseAudit namespace CondensedMatterDipoleScreeningPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.DipoleScreeningResponseContext open PhysicsLogic.Framework.DipoleScreeningResponseContext.Countermodels example : PhysicalSourceDipoleScreeningAt .kosterlitzThouless1973ScalarVortexScreening physicalKosterlitzScalarDipoleReadout ∧ PhysicalSourceDipoleScreeningAt .young1979TriangularVectorDislocationScreening physicalYoungTriangularDipoleReadout := by obtain ⟨scalar, triangular, _⟩ := dipole_screening_distinguishes_scalar_annuli_vector_burgers_angular_response_composite_fugacity_and_source_scope exact ⟨scalar, triangular⟩ example : physicalKosterlitzScalarDipoleReadout.chargeRank = 1 ∧ physicalYoungTriangularDipoleReadout.chargeRank = 2 ∧ physicalKosterlitzScalarDipoleReadout.oppositePairSpeciesCount = 1 ∧ physicalYoungTriangularDipoleReadout.oppositePairSpeciesCount = 3 := by norm_num [physicalKosterlitzScalarDipoleReadout, physicalYoungTriangularDipoleReadout] example : physicalKosterlitzScalarDipoleContext.outerBindingCoefficient < physicalKosterlitzScalarDipoleContext.innerBindingCoefficient ∧ physicalYoungTriangularDipoleContext.outerBindingCoefficient < physicalYoungTriangularDipoleContext.innerBindingCoefficient := by obtain ⟨_scalar, _triangular, scalarNonvacuous, triangularNonvacuous, _⟩ := dipole_screening_distinguishes_scalar_annuli_vector_burgers_angular_response_composite_fugacity_and_source_scope exact ⟨physicalKosterlitzScalarDipoleContext |>.nonvacuous_dipole_screening_strictly_weakens_binding scalarNonvacuous, physicalYoungTriangularDipoleContext |>.nonvacuous_dipole_screening_strictly_weakens_binding triangularNonvacuous⟩ example : PhysicalFourPiDielectricScreeningAt physicalYoungRadialOnlyScreeningControl ∧ ¬PhysicalTriangularVectorAngularPolarizabilityAt physicalYoungRadialOnlyScreeningControl := by obtain ⟨_original, _species, commonScreening, notAngular, _⟩ := triangular_species_angular_bessel_and_composite_fugacity_are_independent exact ⟨commonScreening, notAngular⟩ example : PhysicalTriangularCompositeFugacityAt physicalYoungTriangularDipoleReadout ∧ ¬PhysicalTriangularCompositeFugacityAt physicalYoungMissingQuadraticFugacityTerm := by obtain ⟨_angular, _species, _common, _radial, _bessel, composite, _fusion, noQuadratic, _⟩ := triangular_species_angular_bessel_and_composite_fugacity_are_independent exact ⟨composite, noQuadratic⟩ example : PhysicalTriangularSourceScopeAt physicalYoungTriangularDipoleReadout ∧ ¬PhysicalTriangularSourceScopeAt physicalYoungClaimedCompleteSquareLattice ∧ ¬PhysicalTriangularSourceScopeAt physicalYoungClaimedLaterDisclinationAnalysis := by obtain ⟨retained, incompleteSquare, _bulk, laterTransition⟩ := triangular_dilute_screening_does_not_complete_square_bulk_or_disclinations exact ⟨retained, incompleteSquare, laterTransition⟩ end CondensedMatterDipoleScreeningPhysicalReuseAudit namespace GeneralRelativisticSchwarzschildModeStabilityPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.SchwarzschildMasterEquationContext.Countermodels open PhysicsLogic.Framework.SchwarzschildLinearModeStabilityContext.Countermodels example : PhysicalSourceSchwarzschildLinearModeStabilityAt .reggeWheeler1957HistoricalWormholeArgument physicalReggeWheelerHistoricalWormholeReadout ∧ PhysicalSourceSchwarzschildLinearModeStabilityAt .vishveshwara1970CorrectedKruskalArgument physicalVishveshwaraCorrectedKruskalReadout := by obtain ⟨historical, corrected, _⟩ := schwarzschild_stability_distinguishes_wormhole_kruskal_parity_dipole_packets_completeness_and_source_scope exact ⟨historical, corrected⟩ example : physicalReggeWheelerHistoricalWormholeReadout.selectedNearHorizonBranchExponent = 1 ∧ physicalVishveshwaraCorrectedKruskalReadout.selectedNearHorizonBranchExponent = -1 ∧ physicalReggeWheelerHistoricalWormholeReadout.innerBoundary ≠ physicalVishveshwaraCorrectedKruskalReadout.innerBoundary := by obtain ⟨_, _, historical, corrected, distinct, _⟩ := schwarzschild_stability_distinguishes_wormhole_kruskal_parity_dipole_packets_completeness_and_source_scope exact ⟨historical, corrected, distinct⟩ example : PhysicalOddGrowingModeExclusionAt physicalVishveshwaraOddOnlyBoundaryControl ∧ ¬PhysicalEvenGrowingModeExclusionAt physicalVishveshwaraOddOnlyBoundaryControl := by obtain ⟨_, _, _, _, _, odd, even, _⟩ := schwarzschild_stability_distinguishes_wormhole_kruskal_parity_dipole_packets_completeness_and_source_scope exact ⟨odd, even⟩ example : physicalSchwarzschildDipoleStatusAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .oddAxial .rotationalDipole .imaginaryExponentiallyGrowingMode = .removableNonstationaryGaugeMode ∧ physicalSchwarzschildDipoleStatusAt .vishveshwara1970KruskalBoundaryAndCompletenessGap .oddAxial .rotationalDipole .zeroStationaryMode = .physicalStationaryRotation := by obtain ⟨_, _, _, growing, stationary, _⟩ := nonstationary_odd_dipole_is_gauge_but_stationary_dipole_is_rotation exact ⟨growing, stationary⟩ example : PhysicalRegularRealFrequencyPacketAt physicalVishveshwaraCorrectedKruskalReadout ∧ PhysicalOddSturmLiouvilleCompletenessAt physicalVishveshwaraCorrectedKruskalReadout ∧ ¬PhysicalEvenArbitraryDataCompletenessAt physicalVishveshwaraCorrectedKruskalReadout := by obtain ⟨packet, odd, even, _⟩ := regular_real_frequency_packets_do_not_supply_even_mode_completeness exact ⟨packet, odd, even⟩ example : PhysicalComplexRadiationBoundaryAt physicalVishveshwaraCorrectedKruskalReadout ∧ physicalVishveshwaraCorrectedKruskalReadout.nonlinearStabilityEstablished = false ∧ physicalVishveshwaraCorrectedKruskalReadout.globalDecayEstablished = false := by obtain ⟨_, complex, nonlinear, decay, _⟩ := source_qualified_linear_mode_exclusion_does_not_supply_nonlinear_stability_or_global_decay exact ⟨complex, nonlinear, decay⟩ end GeneralRelativisticSchwarzschildModeStabilityPhysicalReuseAudit namespace QFTAndNuclearUnitarySimilarityPhysicalReuseAudit open PhysicsLogic.Framework open PhysicsLogic.Framework.UnitarySimilarityDecouplingContext.Countermodels example : PhysicalSourceUnitarySimilarityDecouplingAt .glazekWilson1993FiniteCountertermSimilarity physicalGlazekWilsonFiniteSimilarityReadout ∧ PhysicalSourceUnitarySimilarityDecouplingAt .bognerFurnstahlPerry2007ContinuousNuclearSRG physicalBognerContinuousNuclearSRGReadout := by obtain ⟨finite, continuous, _⟩ := unitary_similarity_distinguishes_finite_counterterms_continuous_nuclear_flow_radial_kernels_and_omitted_three_body_operators exact ⟨finite, continuous⟩ example : PhysicalFiniteUnitaryBandSimilarityAt physicalGlazekWilsonFiniteSimilarityReadout ∧ PhysicalContinuousKineticSRGFlowAt physicalBognerContinuousNuclearSRGReadout ∧ physicalGlazekWilsonFiniteSimilarityReadout.construction ≠ physicalBognerContinuousNuclearSRGReadout.construction := by obtain ⟨finite, continuous, distinct, _⟩ := finite_unitary_similarity_and_continuous_double_commutator_flow_are_distinct exact ⟨finite, continuous, distinct⟩ example : PhysicalGlazekWilsonCountertermAt physicalGlazekWilsonFiniteSimilarityReadout ∧ ¬PhysicalGlazekWilsonCountertermAt physicalGlazekWilsonWrongRunningCountertermControl := by obtain ⟨counterterm, _, _, changed, _⟩ := finite_counterterm_recursion_does_not_imply_nonperturbative_control exact ⟨counterterm, changed⟩ example : PhysicalBognerPartialWaveFlowAt physicalBognerContinuousNuclearSRGReadout ∧ physicalBognerContinuousNuclearSRGReadout.nonlinearIntegralTerm = -(1 / 4 : Real) ∧ ¬PhysicalBognerPartialWaveFlowAt physicalBognerMissingNonlinearIntegralControl := by obtain ⟨flow, nonlinear, _, missing, _⟩ := bogner_double_commutator_retains_nonlinear_partial_wave_kernel exact ⟨flow, nonlinear, missing⟩ example : PhysicalBognerTwoBodyObservablesAt physicalBognerContinuousNuclearSRGReadout ∧ PhysicalBognerThreeBodyOmissionAt physicalBognerContinuousNuclearSRGReadout ∧ physicalBognerContinuousNuclearSRGReadout.initialTritonEnergy ≠ physicalBognerContinuousNuclearSRGReadout.twoBodyOnlyEvolvedTritonEnergy := by obtain ⟨twoBody, threeBody, running, _⟩ := two_body_unitarity_does_not_preserve_omitted_induced_three_body_physics exact ⟨twoBody, threeBody, running⟩ example : PhysicalBognerConvergenceEvidenceAt physicalBognerContinuousNuclearSRGReadout ∧ ¬PhysicalBognerConvergenceEvidenceAt physicalBognerClaimedUniversalConvergenceControl := by obtain ⟨evidence, _, _, universal⟩ := selected_weinberg_and_basis_improvement_is_not_universal_convergence exact ⟨evidence, universal⟩ end QFTAndNuclearUnitarySimilarityPhysicalReuseAudit namespace QuantumBellCHSHLocalResponsePhysicalReuseAudit open PhysicsLogic.Framework.DeterministicLocalResponse open PhysicsLogic.Framework.DeterministicLocalResponse.Countermodels example : PhysicalSourceLocalFactorizationAt .bell1964PerfectSinglet physicalBellReadout ∧ PhysicalSourceLocalFactorizationAt .chsh1969ImperfectOpticalPair physicalCHSHReadout := by obtain ⟨bell, chsh, _⟩ := deterministic_local_responses_distinguish_bell_anticorrelation_chsh_imperfection_hidden_independence_and_detector_loopholes exact ⟨bell, chsh⟩ example : physicalBellReadout.referenceCorrelation = -1 ∧ physicalCHSHReadout.referenceCorrelation = 1 / 2 := by obtain ⟨_, _, bell, chsh, _⟩ := bell_perfect_anticorrelation_is_distinct_from_chsh_imperfect_reference exact ⟨bell, chsh⟩ example : PhysicalCHSHQuantumCountingViolationAt physicalCHSHSettingDependentDetectionControl ∧ ¬PhysicalCHSHDetectorAndFluxAt physicalCHSHSettingDependentDetectionControl := by obtain ⟨_, detected, missing, _⟩ := local_emergence_does_not_supply_detector_or_constant_flux_bridge exact ⟨detected, missing⟩ example : ¬PhysicalCHSHDetectorAndFluxAt physicalCHSHClaimedCalibrationControl ∧ ¬PhysicalCHSHQuantumCountingViolationAt physicalCHSHWideAcceptanceControl := by obtain ⟨_, _, calibration, acceptance⟩ := calibration_and_prior_quantum_agreement_do_not_close_detection_loophole exact ⟨calibration, acceptance⟩ example : ¬Factorizes physicalBellRemoteSettingDependent (physicalLocalResponses .bell1964PerfectSinglet) := remote_setting_dependence_does_not_admit_bell_local_factorization.2 end QuantumBellCHSHLocalResponsePhysicalReuseAudit namespace CondensedMatterMomentumVariationalPhysicalReuseAudit open PhysicsLogic.Framework.MomentumResolvedVariationalBoundContext.Countermodels example : PhysicalSourceMomentumVariationalAt .feynman1954HeliumDensityMode physicalFeynman1954VariationalReadout ∧ PhysicalSourceMomentumVariationalAt .feynmanCohen1956CurrentConservingBackflow physicalFeynmanCohen1956BackflowReadout ∧ PhysicalSourceMomentumVariationalAt .girvinMacDonaldPlatzman1986ProjectedHallMode physicalGMP1986ProjectedHallReadout := by obtain ⟨helium, backflow, hall, _⟩ := momentum_variational_bounds_distinguish_helium_density_backflow_projected_hall_algebra_quartic_scaling_and_gap_scope exact ⟨helium, backflow, hall⟩ example : physicalFeynman1954VariationalReadout.lowestSectorEnergy < physicalFeynman1954VariationalReadout.trialEnergy ∧ ¬PhysicalFeynman1954DensityModeAt physicalFeynmanClaimedExactEnergyControl := by obtain ⟨_, strict, _, exact⟩ := helium_density_structure_factor_bound_is_not_exact_sector_energy exact ⟨strict, exact⟩ example : physicalFeynmanCohen1956BackflowReadout.currentDivergenceAfter = 0 ∧ physicalFeynmanCohen1956BackflowReadout.thermodynamicGap < physicalFeynmanCohen1956BackflowReadout.reportedImprovedGap ∧ ¬PhysicalFeynmanCohen1956BackflowAt physicalFeynmanCohenMissingReturnFlowControl := by obtain ⟨_, conserved, _, remaining, missing, _⟩ := backflow_conserves_current_and_lowers_but_does_not_exactly_fix_gap exact ⟨conserved, remaining, missing⟩ example : ¬PhysicalGMP1986ProjectedHallModeAt physicalGMPUnprojectedCyclotronControl ∧ ¬PhysicalGMP1986ProjectedHallModeAt physicalGMPCommutingProjectedDensityControl := by obtain ⟨_, _, unprojected, commuting⟩ := hall_intra_landau_trial_requires_projection_and_noncommuting_density exact ⟨unprojected, commuting⟩ example : physicalGMP1986ProjectedHallReadout.projectedQuadraticStructure = 0 ∧ 0 < physicalGMP1986ProjectedHallReadout.projectedQuarticStructure ∧ physicalGMP1986ProjectedHallReadout.exactGapEstablished = false ∧ ¬PhysicalGMP1986ProjectedHallModeAt physicalGMPSaturatedNumericalModeControl := by obtain ⟨_, quadratic, quartic, gap, _⟩ := quartic_projected_structure_and_first_moment_do_not_prove_exact_gap obtain ⟨_, _, _, saturation⟩ := ninety_eight_percent_overlap_is_not_single_mode_saturation exact ⟨quadratic, quartic, gap, saturation⟩ end CondensedMatterMomentumVariationalPhysicalReuseAudit namespace ClassicalNumericalNearReturnPhysicalReuseAudit open PhysicsLogic.Framework.ValidatedNumericalNearReturnContext.Countermodels example : PhysicalSourceNumericalNearReturnAt .fermiPastaUlam1955FixedEndpointChain physicalFPU1955FixedEndpointReadout ∧ PhysicalSourceNumericalNearReturnAt .zabuskyKruskal1965PeriodicKdVSolitons physicalZK1965PeriodicKdVReadout := by obtain ⟨fpu, kdv, _⟩ := numerical_near_returns_distinguish_fpu_chains_kdv_solitons_observables_validation_recurrence_and_thermalization_scope exact ⟨fpu, kdv⟩ example : physicalFPU1955FixedEndpointReadout.firstModeReturnEnergy = 99 / 100 * physicalFPU1955FixedEndpointReadout.firstModeInitialEnergy ∧ physicalFPU1955FixedEndpointReadout.fullStateResidual ≠ 0 ∧ ¬PhysicalFPUValidatedModalNearReturnAt physicalFPUMissingLinearControl := by obtain ⟨_, modal, residual, control, _⟩ := first_mode_energy_return_and_linear_control_do_not_fix_full_state exact ⟨modal, residual, control⟩ example : physicalZK1965PeriodicKdVReadout.momentumDrift = 0 ∧ physicalZK1965PeriodicKdVReadout.numericalEnergyDrift = 1 / 100000 ∧ ¬PhysicalKdVValidatedSolitonTrainAt physicalKdVClaimedExactEnergyControl := by obtain ⟨_, momentum, energy, exact⟩ := exact_kdv_momentum_does_not_imply_exact_numerical_energy exact ⟨momentum, energy, exact⟩ example : physicalZK1965PeriodicKdVReadout.pulseCount = 8 ∧ physicalZK1965PeriodicKdVReadout.closeInteractionResolved = false ∧ ¬PhysicalKdVValidatedSolitonTrainAt physicalKdVClaimedResolvedInteractionControl := by obtain ⟨_, count, unresolved, _, resolved⟩ := isolated_soliton_width_speed_and_eight_pulses_do_not_resolve_overlap exact ⟨count, unresolved, resolved⟩ example : physicalZK1965PeriodicKdVReadout.firstProfileResidual < physicalZK1965PeriodicKdVReadout.secondProfileResidual ∧ ¬PhysicalKdVWaveformNearRecurrenceAt physicalKdVClaimedExactRecurrenceControl ∧ ¬PhysicalKdVWaveformNearRecurrenceAt physicalKdVClaimedNoThermalizationControl := by obtain ⟨_, _, worsening, exact, thermalization⟩ := successive_waveform_near_returns_do_not_establish_exact_recurrence exact ⟨worsening, exact, thermalization⟩ end ClassicalNumericalNearReturnPhysicalReuseAudit namespace BohmianQuantumEquilibriumPhysicalReuseAudit open PhysicsLogic.Framework.GuidedDensityEquivarianceContext.Countermodels example : PhysicalSourceGuidanceAt .bohm1952QuantumPotentialGuidance physicalBohm1952GuidanceReadout ∧ PhysicalSourceGuidanceAt .durrGoldsteinZanghi1992QuantumEquilibrium physicalDGZ1992EquilibriumReadout := by obtain ⟨bohm, dgz, _⟩ := guided_equivariance_distinguishes_quantum_potential_configurations_conditional_typicality_and_absolute_uncertainty exact ⟨bohm, dgz⟩ example : physicalBohm1952GuidanceReadout.quantumPotential = -1 / 2 ∧ ¬PhysicalGuidanceAndEquivarianceAt physicalBohmMissingActualConfigurationControl := by obtain ⟨_, potential, configuration, _⟩ := polar_continuity_does_not_supply_configuration_or_nodal_guidance exact ⟨potential, configuration⟩ example : physicalBohm1952GuidanceReadout.laterNonequilibriumDensity ≠ physicalBohm1952GuidanceReadout.laterWaveDensity ∧ ¬PhysicalBohmQuantumPotentialGuidanceAt physicalBohmClaimedRelaxationControl := by obtain ⟨_, _, _, nonequilibrium, relaxation⟩ := equivariance_preserves_equilibrium_without_proving_nonequilibrium_relaxation exact ⟨nonequilibrium, relaxation⟩ example : physicalDGZ1992EquilibriumReadout.conditionalBornProbability = 1 / 2 ∧ ¬PhysicalDGZConditionalTypicalityAt physicalDGZMissingTypicalityControl ∧ ¬PhysicalDGZConditionalTypicalityAt physicalDGZWrongSingleWorldFrequencyControl := by obtain ⟨_, probability, typicality, _, frequency, _⟩ := conditional_born_statistics_require_typicality_and_a_selected_world exact ⟨probability, typicality, frequency⟩ example : physicalDGZ1992EquilibriumReadout.conditionalWaveAutonomous = false ∧ 0 < physicalDGZ1992EquilibriumReadout.irreducibleUncertaintyBound ∧ physicalDGZ1992EquilibriumReadout.actualConfigurationPostulated = true := by obtain ⟨_, autonomous, uncertainty, configuration, _⟩ := conditional_wave_need_not_evolve_autonomously_and_uncertainty_preserves_configuration exact ⟨autonomous, uncertainty, configuration⟩ end BohmianQuantumEquilibriumPhysicalReuseAudit namespace GeneralRelativityVacuumMetricPhysicalReuseAudit open PhysicsLogic.Framework.VacuumMetricDynamicsContext.Countermodels example : PhysicalSourceVacuumMetricAt .einstein1916VacuumFieldSelection physicalEinstein1916VacuumReadout ∧ PhysicalSourceVacuumMetricAt .schwarzschild1916ExactPointMassExterior physicalSchwarzschild1916ExteriorReadout := by obtain ⟨einstein, schwarzschild, _⟩ := vacuum_metric_dynamics_distinguishes_ricci_curvature_unit_jacobian_point_source_continuity_and_exterior_scope exact ⟨einstein, schwarzschild⟩ example : physicalEinstein1916VacuumReadout.ricciContraction = 0 ∧ physicalEinstein1916VacuumReadout.riemannCurvature ≠ 0 ∧ ¬PhysicalRicciFlatMetricFreeMotionAt physicalEinsteinMissingGeodesicExtensionControl := by obtain ⟨_, ricci, _, curvature, _, _, _, _, _, motion⟩ := ricci_flat_vacuum_retains_curvature_without_deriving_free_motion exact ⟨ricci, curvature, motion⟩ example : (∀ first second : Fin 4, physicalEinstein1916VacuumReadout.ricciTensorComponent first second = 0) ∧ physicalEinstein1916VacuumReadout.riemannTensorComponent 1 0 1 0 ≠ 0 ∧ ¬PhysicalRicciFlatMetricFreeMotionAt physicalEinsteinNonzeroRicciTensorControl ∧ ¬PhysicalRicciFlatMetricFreeMotionAt physicalEinsteinUnbalancedCurvatureContractionControl := by obtain ⟨_, _, ricci, _, curvature, _, _, nonzeroRicci, contraction, _⟩ := ricci_flat_vacuum_retains_curvature_without_deriving_free_motion exact ⟨ricci, curvature, nonzeroRicci, contraction⟩ example : physicalEinstein1916VacuumReadout.traceAdjustedMatterSourceRetained = true ∧ ¬PhysicalEinsteinVacuumFieldSelectionAt physicalEinsteinRelativityAloneSourceControl ∧ ¬PhysicalEinsteinVacuumFieldSelectionAt physicalEinsteinTensorialEnergyComplexControl := by obtain ⟨_, _, source, covariance, tensor⟩ := relativity_covariance_does_not_select_matter_coupling_or_tensorial_energy exact ⟨source, covariance, tensor⟩ example : physicalSchwarzschild1916ExteriorReadout.secondIntegrationConstant = physicalSchwarzschild1916ExteriorReadout.firstIntegrationConstant ^ 3 ∧ ¬PhysicalSchwarzschildExactPointMassExteriorAt physicalSchwarzschildWrongSecondConstantControl ∧ ¬PhysicalSchwarzschildExactPointMassExteriorAt physicalSchwarzschildIncludedPointSourceControl := by obtain ⟨_, continuity, _, constant, source⟩ := point_source_continuity_fixes_second_constant_and_excludes_source exact ⟨continuity, constant, source⟩ example : physicalSchwarzschild1916ExteriorReadout.timeMetricCoefficient = 1 / 2 ∧ physicalSchwarzschild1916ExteriorReadout.radialMetricCoefficient = -2 ∧ ¬PhysicalSchwarzschildExactPointMassExteriorAt physicalSchwarzschildFiniteApproximationUniquenessControl := by obtain ⟨_, _, time, radial, _, _, _, approximate⟩ := exact_exterior_radius_and_coefficients_are_not_finite_approximation exact ⟨time, radial, approximate⟩ example : physicalSchwarzschild1916ExteriorReadout.massParameter < physicalSchwarzschild1916ExteriorReadout.arealRadialCoordinate ∧ ¬PhysicalSchwarzschildExactPointMassExteriorAt physicalSchwarzschildHorizonControl ∧ ¬PhysicalSchwarzschildExactPointMassExteriorAt physicalSchwarzschildMaximalExtensionControl ∧ ¬PhysicalSchwarzschildExactPointMassExteriorAt physicalSchwarzschildBlackHoleControl := by obtain ⟨_, exterior, _, _, _, horizon, extension, blackHole⟩ := exterior_point_mass_chart_does_not_establish_horizon_extension_or_black_hole exact ⟨exterior, horizon, extension, blackHole⟩ end GeneralRelativityVacuumMetricPhysicalReuseAudit namespace TrappedSurfaceIncompletenessPhysicalReuseAudit open PhysicsLogic.Framework.TrappedSurfaceIncompletenessContext.Countermodels example : PhysicalSourceTrappedSurfaceIncompletenessAt .penrose1965NonsphericalCollapse physicalPenrose1965FutureCollapseReadout ∧ PhysicalSourceTrappedSurfaceIncompletenessAt .hawking1965OpenCosmology physicalHawking1965OpenCoveringReadout ∧ physicalPenrose1965FutureCollapseReadout.direction = .future ∧ physicalHawking1965OpenCoveringReadout.direction = .past := by obtain ⟨penrose, _, hawking, _, future, past, _⟩ := trapped_surface_incompleteness_distinguishes_future_collapse_past_cosmology_coverings_and_scope exact ⟨penrose, hawking, future, past⟩ example : physicalPenrose1965FutureCollapseReadout.opticalSecondAreaRootDerivative = -physicalPenrose1965FutureCollapseReadout.opticalAreaRoot * (physicalPenrose1965FutureCollapseReadout.opticalShearSquared + physicalPenrose1965FutureCollapseReadout.opticalRicciFocusing) ∧ ¬PhysicalPenroseFutureCollapseAt physicalPenroseMissingPositiveEnergyControl ∧ ¬PhysicalPenroseFutureCollapseAt physicalPenroseCompactCauchyControl ∧ ¬PhysicalPenroseFutureCollapseAt physicalPenroseActuallyCompactBoundaryControl := by obtain ⟨_, optical, _, _, energy, _, topology, _, _, boundary, _⟩ := future_collapse_requires_optical_focusing_noncompact_topology_and_counterfactual_boundary exact ⟨optical, energy, topology, boundary⟩ example : physicalHawking1965OpenCoveringReadout.inwardNullConvergence = 3 / 2 ∧ physicalHawking1965OpenCoveringReadout.outwardNullConvergence = 13 / 2 ∧ ¬PhysicalHawkingPastCosmologyAt physicalHawkingOneUntrappedNullFamilyControl ∧ ¬PhysicalHawkingPastCosmologyAt physicalHawkingZeroConvergenceMarginControl ∧ ¬PhysicalHawkingPastCosmologyAt physicalHawkingPerturbationDestroysTrappingControl := by obtain ⟨_, _, _, inward, outward, _, _, _, _, family, margin, _, perturbation⟩ := past_cosmology_requires_both_null_families_positive_density_and_strict_margin exact ⟨inward, outward, family, margin, perturbation⟩ example : PhysicalHawkingPastCosmologyAt physicalHawking1965FlatUniverseReadout ∧ PhysicalHawkingPastCosmologyAt physicalHawking1965OpenCoveringReadout ∧ PhysicalHawkingPastCosmologyAt physicalHawking1965MatterFlowReplacementReadout ∧ physicalHawking1965OpenCoveringReadout.originalSpatialSliceCompact = true ∧ physicalHawking1965MatterFlowReplacementReadout.cauchySurfaceAssumed = false ∧ ¬PhysicalHawkingPastCosmologyAt physicalHawkingMissingCoveringDescentControl ∧ ¬PhysicalHawkingPastCosmologyAt physicalHawkingMissingReplacementBridgeControl := by obtain ⟨flat, openUniverse, replacement, compact, _, noCauchy, _, descent, _, _, bridge⟩ := open_covering_and_matter_flow_replacement_require_independent_global_bridges exact ⟨flat, openUniverse, replacement, compact, noCauchy, descent, bridge⟩ example : ¬PhysicalPenroseFutureCollapseAt physicalPenroseCurvatureDivergenceControl ∧ ¬PhysicalPenroseFutureCollapseAt physicalPenroseBlackHoleControl ∧ ¬PhysicalHawkingPastCosmologyAt physicalHawkingFutureIncompletenessControl ∧ ¬PhysicalHawkingPastCosmologyAt physicalHawkingClosedUniverseExtensionControl := by obtain ⟨_, _, curvature, blackHole, _, _, future, _, closed⟩ := directional_incompleteness_does_not_establish_curvature_black_holes_or_closed_cosmology exact ⟨curvature, blackHole, future, closed⟩ end TrappedSurfaceIncompletenessPhysicalReuseAudit namespace NuclearPostCalibrationPhysicalReuseAudit open PhysicsLogic.Framework.PostCalibrationPredictionContext.Countermodels example : PhysicalSourcePostCalibrationAt .kaplanSavageWise1998CoupledPDS physicalKSW1998CoupledPredictionReadout ∧ PhysicalSourcePostCalibrationAt .bedaqueHammerVanKolck1998RunningThreeBody physicalBHvK1998ThreeBodyPredictionReadout ∧ PhysicalSourcePostCalibrationAt .epelbaumGloeckleMeissner1999ChiralPartialWaves physicalEGM1999DWavePredictionReadout := by obtain ⟨ksw, bhvk, egm, _⟩ := nuclear_post_calibration_distinguishes_coupled_mixed_order_running_three_body_and_unfitted_chiral_d_wave exact ⟨ksw, bhvk, egm⟩ example : physicalKSW1998CoupledPredictionReadout.targetCalculatedOrder < physicalKSW1998CoupledPredictionReadout.fitOrder ∧ physicalKSW1998CoupledPredictionReadout.kinematicVariable = .centerOfMassMomentum ∧ physicalKSW1998CoupledPredictionReadout.documentedNumericalFitSector ≠ physicalKSW1998CoupledPredictionReadout.fitSector ∧ ¬PhysicalKSWCoupledPDSPostCalibrationAt physicalKSWLaboratoryEnergySubstitutionControl ∧ ¬PhysicalKSWCoupledPDSPostCalibrationAt physicalKSWConflatedSingletCoupledFitControl ∧ ¬PhysicalKSWCoupledPDSPostCalibrationAt physicalKSWClaimedUniformOrderControl ∧ ¬PhysicalKSWCoupledPDSPostCalibrationAt physicalKSWSeparateCouplingIdentificationControl := by obtain ⟨_, order, _, distinctSectors, _, _, _, _, wrongVariable, wrongSector, uniform, coupling⟩ := coupled_s_wave_fit_does_not_supply_uniform_mixing_order_or_identifiable_couplings exact ⟨order, rfl, distinctSectors, wrongVariable, wrongSector, uniform, coupling⟩ example : physicalBHvK1998ThreeBodyPredictionReadout.effectiveRangePrediction = 57 / 100 * physicalBHvK1998ThreeBodyPredictionReadout.twoBodyScatteringLength ∧ physicalBHvK1998ThreeBodyPredictionReadout.kinematicVariable = .centerOfMassMomentum ∧ ¬PhysicalBHvKRunningThreeBodyPostCalibrationAt physicalBHvKLaboratoryEnergySubstitutionControl ∧ ¬PhysicalBHvKRunningThreeBodyPostCalibrationAt physicalBHvKNonrunningCountertermControl ∧ ¬PhysicalBHvKRunningThreeBodyPostCalibrationAt physicalBHvKUncontrolledPhaseLockControl ∧ ¬PhysicalBHvKRunningThreeBodyPostCalibrationAt physicalBHvKClaimedTritonPredictionControl := by obtain ⟨_, effectiveRange, _, _, _, _, wrongVariable, running, phase, triton⟩ := one_three_body_datum_requires_running_phase_lock_and_does_not_establish_triton_binding exact ⟨effectiveRange, rfl, wrongVariable, running, phase, triton⟩ example : physicalEGM1999DWavePredictionReadout.fittedContactCount = 9 ∧ physicalEGM1999DWavePredictionReadout.targetCalculatedOrder < physicalEGM1999DWavePredictionReadout.firstDWaveContactOrder ∧ physicalEGM1999DWavePredictionReadout.kinematicVariable = .laboratoryEnergy ∧ physicalEGM1999DWavePredictionReadout.fittedKinematicUpperMeV = 100 ∧ physicalEGM1999DWavePredictionReadout.predictionKinematicMeV = 200 ∧ physicalEGM1999DWavePredictionReadout.firstInelasticLabEnergyMeV = 280 ∧ physicalEGM1999DWavePredictionReadout.displayedLabEnergyUpperMeV = 300 ∧ physicalEGM1999DWavePredictionReadout.predictionKinematicMeV < physicalEGM1999DWavePredictionReadout.firstInelasticLabEnergyMeV ∧ physicalEGM1999DWavePredictionReadout.firstInelasticLabEnergyMeV < physicalEGM1999DWavePredictionReadout.displayedLabEnergyUpperMeV ∧ ¬PhysicalEGMChiralDWavePostCalibrationAt physicalEGMCenterOfMassMomentumSubstitutionControl ∧ ¬PhysicalEGMChiralDWavePostCalibrationAt physicalEGMAboveInelasticThresholdControl ∧ ¬PhysicalEGMChiralDWavePostCalibrationAt physicalEGMThresholdConflatedWithDisplayBoundControl ∧ ¬PhysicalEGMChiralDWavePostCalibrationAt physicalEGMOutsideCutoffWindowControl ∧ ¬PhysicalPostCalibrationPredictionAt physicalEGMExactRegulatorIndependenceControl := by obtain ⟨_, contacts, order, _, belowThreshold, thresholdBelowDisplay, _, _, _, _, cutoff, _, wrongVariable, aboveThreshold, wrongThreshold, independence⟩ := unfitted_chiral_d_wave_requires_order_and_finite_regulator_scope exact ⟨contacts, order, rfl, rfl, rfl, rfl, rfl, belowThreshold, thresholdBelowDisplay, wrongVariable, aboveThreshold, wrongThreshold, cutoff, independence⟩ example : physicalKSW1998CoupledPredictionReadout.kinematicVariable ≠ physicalEGM1999DWavePredictionReadout.kinematicVariable ∧ physicalBHvK1998ThreeBodyPredictionReadout.kinematicVariable ≠ physicalEGM1999DWavePredictionReadout.kinematicVariable := by exact ⟨by decide, by decide⟩ example : (physicalPostCalibrationEmpiricalBoundary physicalEGM1999DWavePredictionReadout).HasEmpiricalAgreementAt ∧ ¬PhysicalEGMChiralDWavePostCalibrationAt physicalEGMObservableDStateProbabilityControl ∧ ¬PhysicalEGMChiralDWavePostCalibrationAt physicalEGMDeepStateContaminationControl ∧ ¬PhysicalEGMChiralDWavePostCalibrationAt physicalEGMWrongIndependentObservationControl := by obtain ⟨agreement, _, _, observable, contamination, observation⟩ := chiral_empirical_agreement_does_not_make_d_state_observable_or_deep_state_physical exact ⟨agreement, observable, contamination, observation⟩ end NuclearPostCalibrationPhysicalReuseAudit namespace GappedIncompressibleFluidPhysicalReuseAudit open PhysicsLogic.Framework.GappedIncompressibleFluidContext.Countermodels example : PhysicalSourceGappedFluidAt .laughlin1983PlanarVariationalFluid physicalLaughlin1983PlanarFluidReadout ∧ PhysicalSourceGappedFluidAt .haldane1983SphericalHierarchy physicalHaldane1983SphericalParentReadout ∧ physicalLaughlin1983PlanarFluidReadout.gapCalculatedFromExcitations = true ∧ physicalHaldane1983SphericalParentReadout.parentGapAssumed = true := by obtain ⟨laughlin, haldane, _, _, _, _, _, _, _, calculated, assumed, _, _⟩ := gapped_fluids_distinguish_planar_screening_spherical_monopoles_parent_gaps_hierarchy_and_conditional_plateaus exact ⟨laughlin, haldane, calculated, assumed⟩ example : physicalHaldane1983SphericalParentReadout.parentFluxQuanta = 21 ∧ physicalHaldane1983SphericalParentReadout.monopoleStrength = 21 / 2 ∧ physicalHaldane1983SphericalParentReadout.lowestLandauLevelDegeneracy = 22 ∧ physicalHaldane1983SphericalParentReadout.finiteFillingFactor = 8 / 21 ∧ physicalHaldane1983SphericalParentReadout.thermodynamicFillingFactor = 1 / 3 ∧ physicalHaldane1983SphericalParentReadout.finiteFillingFactor ≠ physicalHaldane1983SphericalParentReadout.thermodynamicFillingFactor := by obtain ⟨parent, flux, strength, degeneracy, finite, thermodynamic, _, _, _, _, _, _, _⟩ := spherical_monopole_flux_shift_requires_assumed_gap_and_distinguishes_finite_filling have distinct : physicalHaldane1983SphericalParentReadout.finiteFillingFactor ≠ physicalHaldane1983SphericalParentReadout.thermodynamicFillingFactor := by norm_num [physicalHaldane1983SphericalParentReadout] exact ⟨flux, strength, degeneracy, finite, thermodynamic, distinct⟩ example : physicalLaughlin1983PlanarFluidReadout.quasiholeCharge = 1 / 3 ∧ physicalLaughlin1983PlanarFluidReadout.quasiparticleCharge = -1 / 3 ∧ physicalLaughlin1983PlanarFluidReadout.pairCreationGap = 47 / 1000 ∧ ¬PhysicalLaughlinPlanarFluidAt physicalLaughlinMissingIndependentFluxChargeControl := by obtain ⟨_, _, hole, particle, gap, _, _, _, independent, _, _, _, _⟩ := planar_plasma_screening_requires_odd_zeros_two_charge_routes_and_positive_gap exact ⟨hole, particle, gap, independent⟩ example : physicalHaldane1983QuasiholeHierarchyReadout.childFluxQuanta = 26 ∧ physicalHaldane1983QuasiholeHierarchyReadout.hierarchyChildThermodynamicFillingFactor = 2 / 7 ∧ physicalHaldane1983QuasiparticleHierarchyReadout.childFluxQuanta = 16 ∧ physicalHaldane1983QuasiparticleHierarchyReadout.hierarchyChildThermodynamicFillingFactor = 2 / 5 ∧ ¬PhysicalHaldaneHierarchyCandidateAt physicalHaldaneWrongHoleHierarchyFillingControl ∧ ¬PhysicalHaldaneHierarchyCandidateAt physicalHaldaneWrongParticleHierarchyFillingControl := by obtain ⟨_, _, holeFlux, holeFilling, particleFlux, particleFilling, _, _, _, wrongHole, _, wrongParticle, _⟩ := opposite_hierarchy_branches_require_even_bosonic_condensation_and_finite_flux_shifts exact ⟨holeFlux, holeFilling, particleFlux, particleFilling, wrongHole, wrongParticle⟩ example : PhysicalGappedFluidHallPlateauAt physicalLaughlin1983PlanarFluidReadout ∧ ¬PhysicalGappedFluidHallPlateauAt physicalHaldane1983SphericalParentReadout ∧ ¬PhysicalGappedFluidHallPlateauAt physicalLaughlinMissingLocalizationControl ∧ ¬PhysicalGappedFluidHallPlateauAt physicalLaughlinMissingConductanceGaugeBridgeControl := by obtain ⟨plateau, noHaldanePlateau, noLocalization, _, noGaugeBridge, _⟩ := hall_plateau_requires_localized_excitation_band_and_conductance_bridge exact ⟨plateau, noHaldanePlateau, noLocalization, noGaugeBridge⟩ example : ¬PhysicalLaughlinPlanarFluidAt physicalLaughlinClaimedExactCoulombGroundStateControl ∧ ¬PhysicalHaldaneSphericalParentAt physicalHaldaneClaimedGenericExactnessControl ∧ ¬PhysicalHaldaneHierarchyCandidateAt physicalHaldaneClaimedChildExistenceControl ∧ ¬PhysicalHaldaneHierarchyCandidateAt physicalHaldaneClaimedOddDenominatorExhaustionControl := by obtain ⟨_, _, _, noExact, _, _, noGeneric, _, noExistence, noExhaustion⟩ := variational_overlap_hard_core_exactness_and_hierarchy_do_not_prove_coulomb_existence exact ⟨noExact, noGeneric, noExistence, noExhaustion⟩ end GappedIncompressibleFluidPhysicalReuseAudit namespace RGEndpointMonotonicityPhysicalReuseAudit open PhysicsLogic.Framework.RGEndpointMonotonicityContext.Countermodels example : PhysicalSourceRGEndpointAt .zamolodchikov1986TwoDimensionalCTheorem physicalZamolodchikov1986LocalCReadout ∧ PhysicalSourceRGEndpointAt .komargodskiSchwimmer2011FourDimensionalATheorem physicalKomargodski2011RelevantDeformationReadout ∧ physicalZamolodchikov1986LocalCReadout.spacetimeDimension = 2 ∧ physicalKomargodski2011RelevantDeformationReadout.spacetimeDimension = 4 ∧ physicalZamolodchikov1986LocalCReadout.betaDirectionalDerivative = -6 := by obtain ⟨zamolodchikov, komargodski, _, _, two, four, derivative, _, _, _, _⟩ := rg_endpoints_distinguish_two_dimensional_c_functions_four_dimensional_dilaton_dispersion_and_open_scope exact ⟨zamolodchikov, komargodski, two, four, derivative⟩ example : physicalKomargodski2011RelevantDeformationReadout.forwardFourDilatonSCoefficient = 1 / 10000 ∧ physicalKomargodski2011RelevantDeformationReadout.imaginaryDispersionIntegrand = 1 / 125 ∧ physicalKomargodski2011RelevantDeformationReadout.crossSectionDispersionIntegrand = 1 / 125 ∧ ¬PhysicalKomargodskiDispersiveAAt physicalKomargodskiMissingOpticalTheoremControl ∧ ¬PhysicalKomargodskiDispersiveAAt physicalKomargodskiWrongOpticalWeightControl ∧ ¬PhysicalKomargodskiDispersiveAAt physicalKomargodskiZeroCrossSectionControl := by obtain ⟨_, coefficient, imaginary, crossSection, _, _, _, _, _, _, _, _, _, optical, weight, zero, _⟩ := relevant_deformation_requires_weak_compensator_and_independent_dispersive_bridges exact ⟨coefficient, imaginary, crossSection, optical, weight, zero⟩ example : PhysicalKomargodskiDispersiveAAt physicalKomargodski2011ModuliFlowReadout ∧ PhysicalKomargodskiDispersiveAAt physicalKomargodski2011RelevantDeformationReadout ∧ physicalKomargodski2011ModuliFlowReadout.physicalNambuGoldstoneDilaton = true ∧ physicalKomargodski2011ModuliFlowReadout.weakSpectatorDilatonIntroduced = false ∧ physicalKomargodski2011RelevantDeformationReadout.weakSpectatorDilatonIntroduced = true := by obtain ⟨moduli, relevant, goldstone, noSpectator, spectator, _, _⟩ := moduli_dilaton_and_relevant_flow_spectator_are_distinct_physical_routes exact ⟨moduli, relevant, goldstone, noSpectator, spectator⟩ example : physicalKomargodski2011FreeMassiveScalarAppendixReadout.pairProductionThreshold = 4 ∧ physicalKomargodski2011FreeMassiveScalarAppendixReadout.piSquaredScaledUltravioletEulerAnomaly = 1 / 5760 ∧ physicalKomargodski2011FreeMassiveScalarAppendixReadout.piSquaredScaledFourDerivativeCoefficient = 1 / 2880 ∧ ¬PhysicalRGFreeScalarAppendixAt physicalKomargodskiFreeScalarWrongThresholdControl := by obtain ⟨_, threshold, anomaly, coefficient, wrongThreshold, _, _⟩ := appendix_scalar_threshold_and_anomaly_coefficient_require_weak_compensator exact ⟨threshold, anomaly, coefficient, wrongThreshold⟩ example : physicalKomargodski2011RelevantDeformationReadout.intermediateRunningValue = 5 / 4 ∧ physicalKomargodski2011RelevantDeformationReadout.cardyIntermediateValue = 4 / 3 ∧ physicalKomargodski2011RelevantDeformationReadout.weylAnomalyUltraviolet = 2 ∧ physicalKomargodski2011RelevantDeformationReadout.weylAnomalyInfrared = 3 ∧ ¬PhysicalKomargodskiDispersiveAAt physicalKomargodskiCardyIntermediateIdentificationControl ∧ ¬PhysicalKomargodskiDispersiveAAt physicalKomargodskiClaimedGlobalGradientControl ∧ ¬PhysicalKomargodskiDispersiveAAt physicalKomargodskiClaimedOddDimensionalAnalogueControl := by obtain ⟨_, running, cardy, ultravioletWeyl, infraredWeyl, noIdentification, noGradient, _, _, noOddAnalogue⟩ := monotone_a_does_not_identify_cardy_running_weyl_anomaly_or_global_gradient exact ⟨running, cardy, ultravioletWeyl, infraredWeyl, noIdentification, noGradient, noOddAnalogue⟩ end RGEndpointMonotonicityPhysicalReuseAudit namespace BoundaryGeneratingFunctionalPhysicalReuseAudit open PhysicsLogic.Framework.BoundaryGeneratingFunctionalContext.Countermodels example : physicalGKP1998MinimumActionReadout.radialCutoff = physicalGKP1998MinimumActionReadout.adsRadius ∧ physicalGKP1998MinimumActionReadout.stringLoopSuppression = 1 / 10000 ∧ physicalGKP1998MinimumActionReadout.alphaPrimeSquaredCorrectionSuppression = 1 / 100 ∧ ¬PhysicalGKPMinimumActionAt physicalGKPRankOneControl ∧ ¬PhysicalGKPMinimumActionAt physicalGKPWrongCutoffSurfaceControl := by obtain ⟨_, cutoff, loops, alphaPrime, rank, _, _, _, wrongCutoff, _, _, _⟩ := gkp_minimum_action_requires_large_n_strong_coupling_regular_bulk_and_cutoff exact ⟨cutoff, loops, alphaPrime, rank, wrongCutoff⟩ example : physicalGKP1998MinimumActionReadout.scalingDimension = 4 ∧ physicalGKP1998MinimumActionReadout.separatedPointScalingExponent = 8 ∧ physicalGKP1998MinimumActionReadout.nonanalyticFluxCoefficient = -1 / 4 ∧ physicalGKP1998MinimumActionReadout.stressTensorCentralCharge = 2500 ∧ ¬PhysicalGKPMinimumActionAt physicalGKPUnseparatedContactTermControl := by obtain ⟨_, dimension, exponent, flux, central, _, _, contact, _⟩ := gkp_massless_scalar_flux_distinguishes_contact_terms_and_dimension_four exact ⟨dimension, exponent, flux, central, contact⟩ example : physicalWitten1998MassiveScalarReadout.classicalSaddleExponent = -2 ∧ physicalWitten1998MassiveScalarReadout.connectedGeneratingFunctional = -2 ∧ physicalWitten1998MassiveScalarReadout.euclideanEffectiveAction = 2 ∧ physicalWitten1998MassiveScalarReadout.connectedFunctionalDerivative = -3 ∧ ¬PhysicalWittenPartitionPrescriptionAt physicalWittenWrongEuclideanExponentControl ∧ ¬PhysicalWittenPartitionPrescriptionAt physicalWittenWrongDerivativeSignControl := by obtain ⟨_, exponent, connected, effective, derivative, _, wrongExponent, _, _, _, _, wrongDerivative, _⟩ := witten_partition_function_requires_negative_euclidean_exponent_and_signed_differentiation exact ⟨exponent, connected, effective, derivative, wrongExponent, wrongDerivative⟩ example : physicalWitten1998MassiveScalarReadout.scalingDimension = 3 ∧ physicalWitten1998MassiveScalarReadout.alternateScalingDimension = 1 ∧ physicalWitten1998MassiveScalarReadout.sourceWeylWeight = 1 ∧ physicalWitten1998MassiveScalarReadout.asymptoticRadialExponent = -1 ∧ ¬PhysicalWittenMassiveScalarAt physicalWittenWrongScalarDimensionRootControl ∧ ¬PhysicalWittenMassiveScalarAt physicalWittenBelowBreitenlohnerFreedmanBoundControl := by obtain ⟨_, _, dimension, alternate, weight, exponent, _, wrongRoot, _, belowBound, _, _, _, _⟩ := witten_massive_scalar_chooses_larger_root_and_correct_boundary_source_weight exact ⟨dimension, alternate, weight, exponent, wrongRoot, belowBound⟩ example : physicalWitten1998HighTemperatureBlackHoleReadout.blackHoleAction = 1 ∧ physicalWitten1998HighTemperatureBlackHoleReadout.thermalAdSAction = 2 ∧ physicalWitten1998LowTemperatureThermalAdSReadout.thermalAdSAction = 1 ∧ physicalWitten1998LowTemperatureThermalAdSReadout.blackHoleAction = 2 ∧ ¬PhysicalWittenThermalSaddleAt physicalWittenMissingCompetingFillingControl ∧ ¬PhysicalWittenThermalSaddleAt physicalWittenClaimedFiniteRankPhaseTransitionControl := by obtain ⟨_, _, highBlackHole, highThermal, lowThermal, lowBlackHole, _, missingFilling, _, _, _, _, finiteRank, _⟩ := witten_thermal_fillings_require_compatible_spin_structure_and_action_competition exact ⟨highBlackHole, highThermal, lowThermal, lowBlackHole, missingFilling, finiteRank⟩ example : physicalWitten1998HighTemperatureBlackHoleReadout.spinStructure = .antiperiodicThermalTrace ∧ physicalWitten1998HighTemperatureBlackHoleReadout.selectedThermalCircleContractible = true ∧ physicalWitten1998HighTemperatureBlackHoleReadout.periodicSpinExtendsOverBlackHole = false ∧ ¬PhysicalWittenThermalSaddleAt physicalWittenPeriodicBlackHoleSpinStructureControl ∧ ¬PhysicalWittenThermalSaddleAt physicalWittenMissingAntiperiodicSpinExtensionControl := by obtain ⟨_, spin, contractible, periodic, wrongSpin, _, missingExtension⟩ := periodic_index_does_not_admit_contractible_black_hole_saddle exact ⟨spin, contractible, periodic, wrongSpin, missingExtension⟩ end BoundaryGeneratingFunctionalPhysicalReuseAudit namespace SubthresholdExcitationStabilityPhysicalReuseAudit open PhysicsLogic.Framework.SubthresholdExcitationStabilityContext.Countermodels example : physicalLandau1941PhononRotonReadout.soundVelocity = 3 / 2 ∧ physicalLandau1941PhononRotonReadout.rotonGap = 2 ∧ physicalLandau1941PhononRotonReadout.rotonEffectiveMass = 4 ∧ physicalLandau1941PhononRotonReadout.rotonMinimumMomentum = 0 ∧ physicalLandau1941PhononRotonReadout.rotonThreshold = 1 ∧ physicalLandau1941PhononRotonReadout.criticalVelocity = 1 ∧ ¬PhysicalLandauPhononRotonAt physicalLandauClaimedDerivedOrderingControl ∧ ¬PhysicalLandauPhononRotonAt physicalLandauModernFiniteMomentumRotonControl := by obtain ⟨_, sound, gap, mass, minimum, threshold, critical, _, _, _, ordering, _, modern, _, _, _, _, _, _⟩ := landau_original_roton_threshold_requires_assumed_ordering_zero_momentum_gap_and_both_branches exact ⟨sound, gap, mass, minimum, threshold, critical, ordering, modern⟩ example : physicalLandau1941PhononRotonReadout.finiteTemperatureNormalFraction = 1 / 4 ∧ physicalLandau1941PhononRotonReadout.finiteTemperatureSuperfluidFraction = 3 / 4 ∧ physicalLandau1941PhononRotonReadout.secondSoundVelocitySquared = 3 / 4 ∧ physicalLandau1941PhononRotonReadout.twoFluidDescriptionIsLiteralSpeciesDivision = false ∧ ¬PhysicalLandauPhononRotonAt physicalLandauLiteralParticleSpeciesControl ∧ ¬PhysicalLandauPhononRotonAt physicalLandauMissingHydrodynamicCalculationControl := by obtain ⟨_, normal, superfluid, _, sound, notSpecies, species, _, hydro, _⟩ := landau_two_motion_statistics_and_second_sound_do_not_divide_atoms_into_species exact ⟨normal, superfluid, sound, notSpecies, species, hydro⟩ example : physicalBogoliubov1947CollectiveQuasiparticleReadout.selectedMomentum = 3 / 2 ∧ physicalBogoliubov1947CollectiveQuasiparticleReadout.molecularKineticEnergy = 9 / 8 ∧ physicalBogoliubov1947CollectiveQuasiparticleReadout.collectiveSpectrumRadicand = 225 / 64 ∧ physicalBogoliubov1947CollectiveQuasiparticleReadout.collectiveExcitationEnergy = 15 / 8 ∧ physicalBogoliubov1947CollectiveQuasiparticleReadout.movingFrameExcitationEnergy = 9 / 8 ∧ ¬PhysicalBogoliubovCollectiveGasAt physicalBogoliubovNegativeInteractionControl ∧ ¬PhysicalBogoliubovCollectiveGasAt physicalBogoliubovNegativeEnergyBranchControl := by obtain ⟨_, momentum, kinetic, radicand, energy, _, moving, _, _, interaction, _, _, _, _, _, negative, _⟩ := bogoliubov_positive_interaction_and_canonical_truncation_produce_single_collective_branch exact ⟨momentum, kinetic, radicand, energy, moving, interaction, negative⟩ example : physicalBogoliubov1947CollectiveQuasiparticleReadout.retainedQuasiparticlesNoninteracting = true ∧ physicalBogoliubov1947CollectiveQuasiparticleReadout.omittedResidualInteractionsPermitEquilibration = true ∧ physicalBogoliubov1947CollectiveQuasiparticleReadout.quasiparticleNumberConserved = false ∧ ¬PhysicalBogoliubovCollectiveGasAt physicalBogoliubovArtificialPhononRotonDivisionControl ∧ ¬PhysicalBogoliubovCollectiveGasAt physicalBogoliubovMissingResidualEquilibrationControl := by obtain ⟨_, _, _, noninteracting, equilibration, number, _, division, residual, _⟩ := bogoliubov_collective_quasiparticles_are_not_atoms_phonon_roton_species_or_number_conserving exact ⟨noninteracting, equilibration, number, division, residual⟩ example : physicalBogoliubov1947CollectiveQuasiparticleReadout.condensateFraction = 9 / 10 ∧ physicalBogoliubov1947CollectiveQuasiparticleReadout.depletionFraction = 1 / 10 ∧ physicalBogoliubov1947CollectiveQuasiparticleReadout.zeroTemperatureSuperfluidFraction = 1 ∧ ¬PhysicalBogoliubovCollectiveGasAt physicalBogoliubovCondensateEqualsSuperfluidDensityControl ∧ ¬PhysicalBogoliubovCollectiveGasAt physicalBogoliubovClaimedExactRealLiquidDerivationControl := by obtain ⟨_, condensate, depletion, superfluid, _, _, _, wrongDensity, _, realLiquid⟩ := bogoliubov_zero_temperature_depletion_distinguishes_condensate_and_superfluid_fractions exact ⟨condensate, depletion, superfluid, wrongDensity, realLiquid⟩ example : physicalLandau1941PhononRotonReadout.allBoundaryVortexAndMultiparticleDissipationAbsent = false ∧ physicalBogoliubov1947CollectiveQuasiparticleReadout.allBoundaryVortexAndMultiparticleDissipationAbsent = false ∧ ¬PhysicalLandauPhononRotonAt physicalLandauClaimedUniversalDissipationControl ∧ ¬PhysicalBogoliubovCollectiveGasAt physicalBogoliubovClaimedUniversalDissipationControl := by obtain ⟨_, _, landau, bogoliubov, landauControl, bogoliubovControl⟩ := selected_superfluid_channels_do_not_eliminate_all_dissipation_mechanisms exact ⟨landau, bogoliubov, landauControl, bogoliubovControl⟩ end SubthresholdExcitationStabilityPhysicalReuseAudit namespace AdiabaticHolonomyPhysicalReuseAudit open PhysicsLogic.Framework.AdiabaticHolonomyContext.ModelSensitivity example : physicalBerry1984UpperTwoLevelReadout.representativeConnection = 3 ∧ physicalBerry1984UpperTwoLevelReadout.gaugeRelatedConnection = 5 ∧ physicalBerry1984UpperTwoLevelReadout.geometricInvariantOverPi = -1 ∧ physicalBerry1984UpperTwoLevelReadout.alternativeDurationGeometricInvariantOverPi = -1 ∧ physicalBerry1984UpperTwoLevelReadout.dynamicalInvariantOverPi = -5 ∧ physicalBerry1984UpperTwoLevelReadout.alternativeDurationDynamicalInvariantOverPi = -10 ∧ ¬PhysicalBerryQuantumHolonomyAt physicalBerryGaugeInvariantConnectionControl ∧ ¬PhysicalBerryQuantumHolonomyAt physicalBerryDurationDependentGeometryControl := by obtain ⟨_, connection, alternate, _, geometric, unchanged, dynamical, changed, _, _, _, gauge, _, _, duration, _, _, _⟩ := berry_gauge_connections_change_while_closed_curvature_phases_and_duration_do_not exact ⟨connection, alternate, geometric, unchanged, dynamical, changed, gauge, duration⟩ example : physicalBerry1984UpperTwoLevelReadout.upperCurvatureVectorCoefficient = 1 / 16 ∧ physicalBerry1984UpperTwoLevelReadout.lowerCurvatureVectorCoefficient = -1 / 16 ∧ physicalBerry1984UpperTwoLevelReadout.geometricInvariantOverPi = -1 ∧ physicalBerry1984LowerTwoLevelReadout.geometricInvariantOverPi = 1 ∧ physicalBerry1984UpperTwoLevelReadout.encirclingPhaseFactor = -1 ∧ ¬PhysicalBerryTwoLevelAt physicalBerryEqualMonopoleChargeControl ∧ ¬PhysicalBerryTwoLevelAt physicalBerryWrongHalfSolidAngleControl := by obtain ⟨_, _, _, _, upper, lower, upperAngle, lowerAngle, sign, _, _, _, _, wrongCharge, _, wrongAngle, _, _⟩ := berry_two_level_monopoles_have_opposite_half_solid_angle_branches exact ⟨upper, lower, upperAngle, lowerAngle, sign, wrongCharge, wrongAngle⟩ example : physicalBerry1984HalfIntegerSpinReadout.spinProjection = 1 / 2 ∧ physicalBerry1984HalfIntegerSpinReadout.encirclingPhaseFactor = -1 ∧ physicalBerry1984IntegerSpinReadout.spinProjection = 1 ∧ physicalBerry1984IntegerSpinReadout.encirclingPhaseFactor = 1 ∧ ¬PhysicalBerrySpinAt physicalBerryHalfIntegerWithoutFermionicSignControl ∧ ¬PhysicalBerrySpinAt physicalBerryIntegerWithFermionicSignControl := by obtain ⟨_, _, half, _, negative, integer, _, positive, _, _, _, wrongHalf, wrongInteger⟩ := berry_spin_monopoles_distinguish_fermionic_and_bosonic_circuit_signs exact ⟨half, negative, integer, positive, wrongHalf, wrongInteger⟩ example : physicalHannay1985ActionAngleReadout.initialActionValue = 2 ∧ physicalHannay1985ActionAngleReadout.finalActionValue = 2 ∧ physicalHannay1985ActionAngleReadout.dynamicalInvariantOverPi = 15 ∧ physicalHannay1985ActionAngleReadout.alternativeDurationDynamicalInvariantOverPi = 30 ∧ physicalHannay1985ActionAngleReadout.geometricInvariantOverPi = 5 / 2 ∧ physicalHannay1985ActionAngleReadout.alternativeDurationGeometricInvariantOverPi = 5 / 2 ∧ ¬PhysicalHannayClassicalAngleAt physicalHannayNonintegrableSystemControl ∧ ¬PhysicalHannayClassicalAngleAt physicalHannayMissingLiouvilleAveragingControl := by obtain ⟨_, initial, final, _, dynamical, alternate, geometric, sameGeometric, nonintegrable, _, _, _, _, _, _, liouville, _, _, _⟩ := hannay_action_invariance_and_duration_independence_require_integrability_and_liouville exact ⟨initial, final, dynamical, alternate, geometric, sameGeometric, nonintegrable, liouville⟩ example : physicalHannay1985QuadraticOscillatorReadout.extraAngleOverPi = 5 / 2 ∧ physicalHannay1985CircularHoopReadout.extraAngleOverPi = -2 ∧ physicalHannay1985DrivenTopReadout.extraAngleOverPi = 3 / 2 ∧ ¬PhysicalHannayQuadraticOscillatorAt physicalHannayWrongEllipseReciprocalControl ∧ ¬PhysicalHannayRotatingHoopAt physicalHannayWrongCircularHoopAngleControl ∧ ¬PhysicalHannayDrivenTopAt physicalHannayWrongDrivenTopSolidAngleControl := by obtain ⟨_, _, _, oscillator, hoop, top, _, reciprocal, _, _, wrongHoop, _, wrongTop⟩ := hannay_oscillator_hoop_and_top_have_distinct_source_specific_geometric_angles exact ⟨oscillator, hoop, top, reciprocal, wrongHoop, wrongTop⟩ example : physicalHannay1985ActionAngleReadout.semiclassicalDerivativeRelationSuggested = true ∧ physicalHannay1985ActionAngleReadout.semiclassicalDerivativeRelationProvedInPaper = false ∧ physicalHannay1985ActionAngleReadout.classicalAngleIdentifiedWithUndifferentiatedQuantumPhase = false ∧ ¬PhysicalHannayClassicalAngleAt physicalHannayClaimedSemiclassicalProofControl ∧ ¬PhysicalHannayClassicalAngleAt physicalHannayQuantumPhaseIdentificationControl := by obtain ⟨_, suggested, unproved, distinct, falseProof, wrongIdentity⟩ := hannay_semiclassical_berry_relation_remains_suggested_not_source_proved exact ⟨suggested, unproved, distinct, falseProof, wrongIdentity⟩ end AdiabaticHolonomyPhysicalReuseAudit namespace ConservationPhysicalReuseAudit open PhysicsLogic.Framework.ConservationContext.Countermodels example : PhysicalNoetherFiniteCurrentAt physicalNoether1918FiniteCurrentReadout ∧ PhysicalNoetherOffShellDivergenceAt physicalNoether1918OffShellDivergenceReadout ∧ physicalNoether1918OffShellDivergenceReadout.finiteEulerContraction = 4 ∧ physicalNoether1918OffShellDivergenceReadout.finiteCurrentDivergence = 4 ∧ ¬PhysicalNoetherFiniteCurrentAt physicalNoetherMissingInvariantIntegralControl ∧ ¬PhysicalNoetherFiniteCurrentAt physicalNoetherMissingBoundaryConditionControl := by obtain ⟨finite, offShell, contraction, divergence, _, missingInvariant, _, _, missingBoundary, _⟩ := finite_global_symmetry_requires_on_shell_equations_and_boundary_flux_control exact ⟨finite, offShell, contraction, divergence, missingInvariant, missingBoundary⟩ example : physicalNoether1918GaugeIdentityReadout.eulerExpressionZero = 2 ∧ physicalNoether1918GaugeIdentityReadout.eulerExpressionDerivative = -2 ∧ physicalNoether1918GaugeIdentityReadout.gaugeIdentityResidual = 0 ∧ physicalNoether1918GaugeIdentityReadout.regionChargeConserved = false ∧ ¬PhysicalNoetherGaugeIdentityAt physicalNoetherMissingIntegrationByPartsControl ∧ ¬PhysicalNoetherGaugeIdentityAt physicalNoetherGaugeIdentityAsGlobalChargeControl := by obtain ⟨_, euler, derivative, residual, noCharge, _, noIntegration, _, wrongCharge, _, _⟩ := arbitrary_function_gauge_symmetry_yields_off_shell_identities_not_global_charges exact ⟨euler, derivative, residual, noCharge, noIntegration, wrongCharge⟩ example : PhysicalNoetherInfiniteParameterAt physicalNoether1918InfiniteParameterReadout ∧ physicalNoether1918InfiniteParameterReadout.arbitraryFunctionCount = 0 ∧ ¬PhysicalNoetherInfiniteParameterAt physicalNoetherInfiniteParametersAsArbitraryFunctionsControl := by obtain ⟨infinite, noFunctions, _, wrongFunctions, _⟩ := infinite_parameter_limit_is_not_arbitrary_function_gauge_symmetry exact ⟨infinite, noFunctions, wrongFunctions⟩ example : physicalNoether1918MixedGroupReadout.finiteAndGaugeRelationsIndependent = true ∧ physicalNoether1918ImproperHilbertEnergyReadout.currentImproper = true ∧ ¬PhysicalNoetherMixedGroupAt physicalNoetherMissingMixedIndependenceControl ∧ ¬PhysicalNoetherImproperHilbertEnergyAt physicalNoetherImproperEnergyMissingGaugeEmbeddingControl := by obtain ⟨_, _, independent, improper, noIndependence, _, _, _, noEmbedding⟩ := mixed_relations_require_independence_and_improper_hilbert_currents_require_gauge_embedding exact ⟨independent, improper, noIndependence, noEmbedding⟩ example : physicalNoether1918OneDimensionalFirstIntegralReadout.firstIntegralZero = 2 ∧ physicalNoether1918OneDimensionalFirstIntegralReadout.firstIntegralOne = 4 ∧ ¬PhysicalNoetherOneDimensionalIntegralAt physicalNoetherFirstIntegralClaimedMultidimensionalControl ∧ ¬PhysicalNoetherOneDimensionalIntegralAt physicalNoetherFirstIntegralsClaimedIndependentControl := by obtain ⟨_, first, second, wrongDimension, wrongIndependence⟩ := one_dimensional_first_integrals_do_not_prove_multidimensional_conservation exact ⟨first, second, wrongDimension, wrongIndependence⟩ example : physicalNoether1918BoundaryLeakageReadout.finiteCurrentDivergence = 0 ∧ physicalNoether1918BoundaryLeakageReadout.initialRegionCharge = 5 ∧ physicalNoether1918BoundaryLeakageReadout.finalRegionCharge = 3 ∧ physicalNoether1918BoundaryLeakageReadout.outwardBoundaryFlux = 2 ∧ ¬PhysicalNoetherFiniteCurrentAt physicalNoether1918BoundaryLeakageReadout := by obtain ⟨_, localDivergence, initial, final, flux, noCharge, _, _⟩ := local_noether_current_conservation_does_not_eliminate_boundary_flux exact ⟨localDivergence, initial, final, flux, noCharge⟩ example : physicalEinstein1916MatterGravityTotalComplexReadout.matterCovariantDivergence = 0 ∧ physicalEinstein1916MatterGravityTotalComplexReadout.matterOrdinaryDivergence = 3 ∧ physicalEinstein1916MatterGravityTotalComplexReadout.connectionExchangeTerm = -3 ∧ ¬PhysicalEinsteinMatterNoExchangeAt physicalEinsteinMatterExchangeClaimedAbsentControl ∧ ¬PhysicalEinsteinMatterNoExchangeAt physicalEinsteinMatterBalanceAsNoetherIdentityControl := by obtain ⟨_, _, covariant, ordinary, connection, _, _, _, wrongExchange, _, _, wrongIdentity⟩ := einstein_covariant_matter_balance_does_not_imply_ordinary_matter_conservation exact ⟨covariant, ordinary, connection, wrongExchange, wrongIdentity⟩ example : physicalEinstein1916MatterGravityTotalComplexReadout.matterOrdinaryDivergence = 3 ∧ physicalEinstein1916MatterGravityTotalComplexReadout.gravitationalPseudotensorDivergence = -3 ∧ physicalEinstein1916MatterGravityTotalComplexReadout.totalComplexOrdinaryDivergence = 0 ∧ ¬PhysicalEinsteinMatterGravityTotalComplexAt physicalEinsteinPseudotensorClaimedCovariantControl ∧ ¬PhysicalEinsteinMatterGravityTotalComplexAt physicalEinsteinClaimedLocalCovariantGravityEnergyControl := by obtain ⟨_, matter, gravity, total, _, _, _, _, _, wrongTensor, _, wrongLocalEnergy, _⟩ := einstein_coordinate_total_complex_cancels_matter_gravity_exchange_without_tensorial_energy exact ⟨matter, gravity, total, wrongTensor, wrongLocalEnergy⟩ end ConservationPhysicalReuseAudit namespace ChargePhysicalReuseAudit open PhysicsLogic.Framework.ChargeAssignmentContext.Countermodels example : physicalLaughlin1983QuasiholeReadout.quasiholeCharge = 1 / 3 ∧ physicalLaughlin1983QuasielectronReadout.selectedSignedCharge = -1 / 3 ∧ ¬PhysicalLaughlinPlasmaScreeningAt physicalLaughlinMissingPlasmaScreeningControl ∧ ¬PhysicalLaughlinPlasmaScreeningAt physicalLaughlinMissingFluxInsertionControl := by obtain ⟨_, _, hole, electron, _, _, noScreening, _, noFlux, _⟩ := laughlin_fractional_charge_requires_screening_flux_insertion_and_opposite_excitation_signs exact ⟨hole, electron, noScreening, noFlux⟩ example : physicalHaldane1983SphericalQuasiholeReadout.sphereMonopoleFluxQuanta = 6 ∧ physicalHaldane1983SphericalQuasiholeReadout.sphereLowestLandauDegeneracy = 7 ∧ physicalHaldane1983SphericalQuasiholeReadout.fixedFieldExcitationMultiplicity = 3 ∧ ¬PhysicalHaldaneSphericalCountingAt physicalHaldaneWrongMonopoleFluxControl ∧ ¬PhysicalHaldaneSphericalCountingAt physicalHaldaneWrongFixedFieldMultiplicityControl := by obtain ⟨_, flux, degeneracy, multiplicity, _, _, wrongFlux, _, wrongMultiplicity⟩ := haldane_spherical_flux_counting_and_laughlin_plasma_screening_are_distinct_charge_routes exact ⟨flux, degeneracy, multiplicity, wrongFlux, wrongMultiplicity⟩ example : physicalArovas1984BerryQuasiholeReadout.berryEncirclingPhaseOverTwoPi = 1 ∧ physicalArovas1984BerryQuasiholeReadout.berryExchangePhaseOverPi = 1 / 3 ∧ ¬PhysicalArovasBerryChargeAt physicalArovasMissingExchangeComparisonControl ∧ ¬PhysicalArovasBerryChargeAt physicalArovasWrongExchangeStatisticsControl := by obtain ⟨_, encircling, _, exchange, _, _, noComparison, wrongStatistics⟩ := arovas_berry_charge_and_exchange_statistics_require_distinct_loop_calculations exact ⟨encircling, exchange, noComparison, wrongStatistics⟩ example : physicalWitten1982CartanChiralChargeReadout.towerChiralCharge = 2 ∧ physicalWitten1982CartanChiralChargeReadout.discreteChiralGroupOrder = 6 ∧ physicalWitten1982CartanChiralChargeReadout.discreteChiralBreakingEstablished = false ∧ ¬PhysicalWittenCartanChiralChargeAt physicalWittenChiralBreakingClaimedProvedControl ∧ ¬PhysicalWittenCartanChiralChargeAt physicalWittenInfiniteVolumeVacuaClaimedIdentifiedControl := by obtain ⟨_, charge, group, _, _, unproved, _, _, _, wrongProof, wrongVacua⟩ := witten_cartan_chiral_charges_do_not_prove_infinite_volume_symmetry_breaking exact ⟨charge, group, unproved, wrongProof, wrongVacua⟩ example : physicalSeibergWitten1994MonopoleReadout.magneticCharge = 1 ∧ physicalSeibergWitten1994DyonReadout.electricCharge = -1 ∧ physicalSeibergWitten1994MonopoleReadout.diracPairing = 1 ∧ physicalSeibergWitten1994MonopoleReadout.physicalThetaShiftedElectricCharge = 1 / 2 ∧ physicalSeibergWitten1994MonopoleReadout.transformedPhysicalElectricCharge = 1 / 2 ∧ ¬PhysicalSeibergWittenBPSChargeAt physicalSeibergWittenChargeLabelWithoutParticleControl ∧ ¬PhysicalSeibergWittenBPSChargeAt physicalSeibergWittenThetaRelabelingChangesChargeControl := by obtain ⟨_, _, magnetic, _, dyon, pairing, before, after, _, _, _, noParticle, _, changed, _⟩ := seiberg_witten_dyonic_lattice_flux_periods_and_theta_relabeling_remain_distinct exact ⟨magnetic, dyon, pairing, before, after, noParticle, changed⟩ example : physicalPolchinski1995RamondRamondBraneReadout.braneSpatialDimension = 2 ∧ physicalPolchinski1995RamondRamondBraneReadout.dualBraneSpatialDimension = 4 ∧ physicalPolchinski1995RamondRamondBraneReadout.normalizedRamondRamondDiracPairing = 1 ∧ ¬PhysicalPolchinskiRamondRamondChargeAt physicalPolchinskiNonminimalDiracQuantumControl ∧ ¬PhysicalPolchinskiRamondRamondChargeAt physicalPolchinskiProductAloneFixesIndividualChargesControl := by obtain ⟨_, electricDimension, magneticDimension, _, pairing, _, _, _, _, wrongQuantum, wrongNormalization⟩ := polchinski_ramond_ramond_dirac_unit_does_not_alone_fix_individual_normalizations exact ⟨electricDimension, magneticDimension, pairing, wrongQuantum, wrongNormalization⟩ example : physicalMaldacena1997D3FiveFormChargeReadout.braneCount = 4 ∧ physicalMaldacena1997D3FiveFormChargeReadout.fiveFormFluxUnits = 4 ∧ physicalMaldacena1997D3FiveFormChargeReadout.harmonicCoefficientOverFourPi = 2 ∧ ¬PhysicalMaldacenaD3FiveFormChargeAt physicalMaldacenaWrongFiveFormFluxControl ∧ ¬PhysicalMaldacenaD3FiveFormChargeAt physicalMaldacenaWrongRadiusFluxRelationControl := by obtain ⟨_, branes, flux, _, coefficient, _, wrongFlux, _, _, wrongRadius⟩ := maldacena_d3_radius_tracks_integer_self_dual_five_form_flux exact ⟨branes, flux, coefficient, wrongFlux, wrongRadius⟩ example : physicalGeneralizedGlobal2014ElectricWilsonLineReadout.chargeFormDegree = 1 ∧ physicalGeneralizedGlobal2014ElectricWilsonLineReadout.topologicalDefectSupportDimension = 2 ∧ physicalGeneralizedGlobal2014ElectricWilsonLineReadout.symmetryActionSign = -1 ∧ ¬PhysicalGeneralizedGlobalElectricLineAt physicalGeneralizedGlobalZeroFormChargeControl ∧ ¬PhysicalGeneralizedGlobalElectricLineAt physicalGeneralizedGlobalBothAnomalousOneFormsGaugeableControl := by obtain ⟨_, _, degree, defect, _, sign, zeroForm, _, _, _, _, _, _, bothGaugeable⟩ := higher_form_wilson_and_thooft_charges_require_linking_topology_and_anomaly_scope exact ⟨degree, defect, sign, zeroForm, bothGaugeable⟩ example : physicalGeneralizedGlobal2014MatterScreenedWilsonLineReadout.matterElectricCharge = 2 ∧ physicalGeneralizedGlobal2014MatterScreenedWilsonLineReadout.residualElectricOneFormOrder = 2 ∧ physicalGeneralizedGlobal2014NongenuineAttachedLineReadout.chargedLineGenuine = false ∧ ¬PhysicalGeneralizedGlobalScreenedElectricLineAt physicalGeneralizedGlobalMatterFailsToScreenControl ∧ ¬PhysicalGeneralizedGlobalNongenuineLineAt physicalGeneralizedGlobalAttachedLineClaimedGenuineControl := by obtain ⟨_, _, matter, residual, nongenuine, noScreening, _, genuine, _⟩ := dynamical_matter_screens_electric_one_form_symmetry_and_attached_lines_are_not_genuine exact ⟨matter, residual, nongenuine, noScreening, genuine⟩ end ChargePhysicalReuseAudit namespace BrokenSymmetryModeConversionPhysicalReuseAudit open PhysicsLogic.Framework.BrokenSymmetryModeConversionContext.Countermodels example : physicalNambu1960NeutralCollectivePairReadout.quasiparticleEnergySquared = 13 ∧ physicalNambu1960NeutralCollectivePairReadout.quasiparticleGap = 4 ∧ physicalNambu1960NeutralCollectivePairReadout.chargeWardVertexZeroOne = -4 ∧ physicalNambu1960NeutralCollectivePairReadout.chargeWardVertexOneZero = 4 ∧ ¬PhysicalNambuWardResponseAt physicalNambuCommutedWardOffDiagonalControl ∧ ¬PhysicalNambuWardResponseAt physicalNambuAllOrdersGaugeResponseControl := by obtain ⟨_, _, energy, gap, left, right, _, _, _, _, _, _, _, wrongWard, allOrders⟩ := nambu_nonperturbative_pairing_and_matrix_ward_vertex_preserve_linear_gauge_response exact ⟨energy, gap, left, right, wrongWard, allOrders⟩ example : physicalNambu1960NeutralCollectivePairReadout.bareLongitudinalResponse = 5 ∧ physicalNambu1960NeutralCollectivePairReadout.collectiveLongitudinalResponse = -5 ∧ physicalNambu1960NeutralCollectivePairReadout.correctedLongitudinalResponse = 0 ∧ physicalNambu1960NeutralCollectivePairReadout.correctedTransverseResponse = 7 ∧ ¬PhysicalNambuLongitudinalTransverseResponseAt physicalNambuMissingLongitudinalCancellationControl ∧ ¬PhysicalNambuLongitudinalTransverseResponseAt physicalNambuErasedTransverseMeissnerResponseControl := by obtain ⟨_, bare, collective, corrected, transverse, noCancellation, noTransverse⟩ := nambu_collective_longitudinal_cancellation_preserves_transverse_meissner_response exact ⟨bare, collective, corrected, transverse, noCancellation, noTransverse⟩ example : physicalNambu1960NeutralCollectivePairReadout.neutralPhaseVelocitySquared = 3 ∧ physicalNambu1960NeutralCollectivePairReadout.neutralCollectiveFrequencySquared = 12 ∧ physicalNambu1960NeutralCollectivePairReadout.zeroMomentumPairFrequencySquared = 0 ∧ physicalNambu1960CoulombConvertedPlasmaReadout.nambuPlasmaFrequencySquared = 4 ∧ ¬PhysicalNambuCoulombPlasmaAt physicalNambuMissingPolarizationResummationControl ∧ ¬PhysicalNambuCoulombPlasmaAt physicalNambuUnconvertedGoldstoneSurvivesControl := by obtain ⟨_, _, velocity, frequency, zero, plasma, _, _, _, noPolarization, _, survivingGoldstone⟩ := nambu_neutral_pair_dispersion_and_coulomb_plasma_frequency_belong_to_distinct_spectra exact ⟨velocity, frequency, zero, plasma, noPolarization, survivingGoldstone⟩ example : physicalNambu1960CoulombConvertedPlasmaReadout.quasiparticleCoreChargeBefore = 1 ∧ physicalNambu1960CoulombConvertedPlasmaReadout.quasiparticleCoreChargeAfter = 1 / 2 ∧ physicalNambu1960CoulombConvertedPlasmaReadout.quasiparticleCloudChargeAfter = 1 / 2 ∧ physicalNambu1960CoulombConvertedPlasmaReadout.totalQuasiparticleChargeAfter = 1 ∧ ¬PhysicalNambuCoreCloudChargeAt physicalNambuCoreChargeClaimedIndependentlyConservedControl ∧ ¬PhysicalNambuCoreCloudChargeAt physicalNambuWrongTotalCoreCloudChargeControl := by obtain ⟨_, coreBefore, coreAfter, cloudAfter, total, independentCore, wrongTotal⟩ := nambu_quasiparticle_core_and_cloud_exchange_charge_without_total_loss exact ⟨coreBefore, coreAfter, cloudAfter, total, independentCore, wrongTotal⟩ example : physicalAnderson1963NormalConductorPlasmaReadout.andersonResponseCoefficient = 4 ∧ physicalAnderson1963NormalConductorPlasmaReadout.andersonPoleMassSquaredOverPi = 16 ∧ physicalAnderson1963NormalConductorPlasmaReadout.andersonSelfConsistentDenominatorOverPi = 20 ∧ physicalAnderson1963NormalConductorPlasmaReadout.andersonDressedFieldRatio = 1 / 5 ∧ physicalAnderson1963NormalConductorPlasmaReadout.totalLongWavelengthModeCount = 3 ∧ physicalAnderson1963NormalConductorPlasmaReadout.brokenMatterSymmetryEstablished = false ∧ ¬PhysicalAndersonNormalConductorAt physicalAndersonMissingInducedGaugeFieldControl ∧ ¬PhysicalAndersonNormalConductorAt physicalAndersonNormalConductorClaimedBrokenControl := by obtain ⟨_, coefficient, pole, denominator, ratio, count, unbroken, _, _, _, missingField, _, _, wrongBreaking⟩ := anderson_normal_conductor_self_consistency_produces_three_gapped_plasma_modes exact ⟨coefficient, pole, denominator, ratio, count, unbroken, missingField, wrongBreaking⟩ example : physicalAnderson1963SuperconductingGoldstoneReadout.representationStateNotGaugeInvariant = true ∧ physicalAnderson1963SuperconductingGoldstoneReadout.observableGaugeInvariancePreserved = true ∧ physicalAnderson1963SuperconductingGoldstoneReadout.zeroModeBelongsToUncoupledSpectrum = true ∧ physicalAnderson1963SuperconductingGoldstoneReadout.zeroModeBelongsToInteractingSpectrum = false ∧ ¬PhysicalAndersonSuperconductingGoldstoneAt physicalAndersonMissingSuperconductingGoldstoneControl ∧ ¬PhysicalAndersonSuperconductingGoldstoneAt physicalAndersonPhysicalGaugeRedundancyBrokenControl := by obtain ⟨_, representation, observable, uncoupled, interacting, noGoldstone, _, brokenGauge⟩ := anderson_superconducting_goldstone_conversion_preserves_exact_observable_gauge_symmetry exact ⟨representation, observable, uncoupled, interacting, noGoldstone, brokenGauge⟩ example : physicalAnderson1963NormalConductorPlasmaReadout.longitudinalFiniteMomentumFrequencySquaredOverPi = 20 ∧ physicalAnderson1963NormalConductorPlasmaReadout.transverseFiniteMomentumFrequencySquaredOverPi = 25 ∧ physicalAnderson1963NormalConductorPlasmaReadout.relativisticMassiveVectorAnalogySuggested = true ∧ physicalAnderson1963NormalConductorPlasmaReadout.relativisticMassiveVectorTheoryEstablished = false ∧ ¬PhysicalAndersonConductorResponseAt physicalAndersonEqualFiniteMomentumDispersionsControl ∧ ¬PhysicalAndersonConductorResponseAt physicalAndersonNonAbelianExtensionClaimedControl ∧ ¬PhysicalAndersonConductorResponseAt physicalAndersonBaryonModelClaimedControl := by obtain ⟨_, longitudinal, transverse, analogy, unproved, equalDispersion, _, nonAbelian, baryon⟩ := anderson_massive_vector_language_is_a_nonrelativistic_analogy_not_nonabelian_theory exact ⟨longitudinal, transverse, analogy, unproved, equalDispersion, nonAbelian, baryon⟩ end BrokenSymmetryModeConversionPhysicalReuseAudit namespace SpectralGapClusteringPhysicalReuseAudit open PhysicsLogic.Framework.SpectralGapClusteringContext.Countermodels example : physicalHastingsKoma2006BosonicExponentialReadout.groundSectorGap = 2 ∧ physicalHastingsKoma2006BosonicExponentialReadout.finiteVolumeGroundSplittingAtSmallVolume = 1 / 4 ∧ physicalHastingsKoma2006BosonicExponentialReadout.finiteVolumeGroundSplittingAtLargeVolume = 1 / 8 ∧ physicalHastingsKoma2006BosonicExponentialReadout.limitingGroundSectorSplitting = 0 ∧ ¬PhysicalHastingsUniformGroundSectorAt physicalHastingsMissingGapControl ∧ ¬PhysicalHastingsUniformGroundSectorAt physicalHastingsMissingSupportChainControl := by obtain ⟨_, gap, small, large, limiting, _, missingGap, _, _, _, _, missingPropagation⟩ := hastings_quasi_degenerate_ground_sector_requires_uniform_gap_vanishing_splitting_and_propagation exact ⟨gap, small, large, limiting, missingGap, missingPropagation⟩ example : physicalHastingsKoma2006BosonicPowerLawReadout.effectivePowerExponent = 1 ∧ physicalHastingsKoma2006BosonicExponentialReadout.effectiveExponentialExponent = 1 / 2 ∧ physicalHastingsKoma2006BosonicPowerLawReadout.selectedDistanceScale = 5 ∧ physicalHastingsKoma2006BosonicExponentialReadout.selectedDistanceScale = 4 ∧ ¬PhysicalHastingsInteractionRateAt physicalHastingsWrongPowerLawEffectiveExponentControl ∧ ¬PhysicalHastingsInteractionRateAt physicalHastingsPowerLawClaimedExponentialControl := by obtain ⟨_, _, algebraic, exponential, algebraicScale, exponentialScale, wrongPower, _, wrongTail⟩ := hastings_interaction_tail_class_fixes_renormalized_algebraic_or_exponential_rate exact ⟨algebraic, exponential, algebraicScale, exponentialScale, wrongPower, wrongTail⟩ example : physicalHastingsKoma2006BosonicExponentialReadout.groundSectorCorrectionValue = 4 ∧ physicalHastingsKoma2006BosonicExponentialReadout.rawCorrelationValue = 5 ∧ physicalHastingsKoma2006BosonicExponentialReadout.correctedCorrelationValue = 1 ∧ physicalHastingsKoma2006BosonicExponentialReadout.unsubtractedCorrelationDecayEstablished = false ∧ ¬PhysicalHastingsBosonicCorrectedAt physicalHastingsWrongBosonicProjectionSignControl ∧ ¬PhysicalHastingsBosonicCorrectedAt physicalHastingsGapAloneClaimedClusteringControl := by obtain ⟨_, _, _, correction, raw, corrected, unsubtracted, wrongSign, _, gapAlone, _⟩ := hastings_bosonic_ground_projection_subtraction_does_not_prove_unsubtracted_decay exact ⟨correction, raw, corrected, unsubtracted, wrongSign, gapAlone⟩ example : physicalHastingsKoma2006FermionicProjectionCancellationReadout.groundProjectionLeftValue = 3 ∧ physicalHastingsKoma2006FermionicProjectionCancellationReadout.groundProjectionRightValue = 3 ∧ physicalHastingsKoma2006FermionicProjectionCancellationReadout.groundSectorCorrectionValue = 0 ∧ ¬PhysicalHastingsFermionicProjectionCancellationAt physicalHastingsFermionicPlusCorrectionControl ∧ ¬PhysicalHastingsFermionicProjectionCancellationAt physicalHastingsMissingGroundTraceIdentityControl := by obtain ⟨_, left, right, correction, _, wrongSign, missingIdentity, _⟩ := hastings_fermionic_projection_subtraction_cancels_only_with_ground_trace_identity exact ⟨left, right, correction, wrongSign, missingIdentity⟩ example : physicalHastingsKoma2006SelfSimilarBosonicReadout.groundSectorDegeneracy = 2 ∧ physicalHastingsKoma2006SelfSimilarBosonicReadout.distantSelfSimilarCopyCount = 4 ∧ physicalHastingsKoma2006SelfSimilarBosonicReadout.groundSectorCorrectionValue = 0 ∧ ¬PhysicalHastingsSelfSimilarBosonicAt physicalHastingsMissingPriorWeakBosonicDecayControl ∧ ¬PhysicalHastingsSelfSimilarBosonicAt physicalHastingsWrongSelfSimilarCopyCountControl := by obtain ⟨_, degeneracy, copies, correction, _, _, missingWeakDecay, wrongCopies, _⟩ := hastings_self_similarity_requires_prior_weak_decay_and_q_squared_distant_copies exact ⟨degeneracy, copies, correction, missingWeakDecay, wrongCopies⟩ example : physicalHastingsKoma2006GrowingDegeneracyReadout.finiteVolumeGroundDegeneracy = 4 ∧ physicalHastingsKoma2006GrowingDegeneracyReadout.finiteVolumeSiteCount = 16 ∧ physicalHastingsKoma2006GrowingDegeneracyReadout.finiteVolumeDegeneracyDensity = 1 / 4 ∧ ¬PhysicalHastingsGrowingGroundDegeneracyAt physicalHastingsUncontrolledGrowingDegeneracyControl ∧ ¬PhysicalHastingsGrowingGroundDegeneracyAt physicalHastingsMissingUniformFiniteVolumeBoundsControl := by obtain ⟨_, degeneracy, volume, density, _, uncontrolled, nonuniform⟩ := hastings_growing_ground_degeneracy_requires_controlled_volume_density_and_uniform_bounds exact ⟨degeneracy, volume, density, uncontrolled, nonuniform⟩ example : physicalNachtergaeleSims2006MetricImaginaryTimeReadout.onsiteDimensionWeightFactor = 16 ∧ physicalNachtergaeleSims2006MetricImaginaryTimeReadout.metricClusteringExponent = 1 ∧ physicalNachtergaeleSims2006MetricImaginaryTimeReadout.coneNetExponent = -10 ∧ physicalNachtergaeleSims2006MetricImaginaryTimeReadout.separatedCommutatorPrefactor = 24 ∧ ¬PhysicalNachtergaeleMetricPropagationAt physicalNachtergaeleMissingOnsiteWeightControl ∧ ¬PhysicalNachtergaeleMetricPropagationAt physicalNachtergaeleSharpLightConeControl := by obtain ⟨_, onsite, _, exponent, _, _, cone, prefactor, missingWeight, _, _, _, sharpCone, _, _⟩ := nachtergaele_metric_commutator_bound_retains_onsite_weight_and_exponential_leakage exact ⟨onsite, exponent, cone, prefactor, missingWeight, sharpCone⟩ example : physicalNachtergaeleSims2006MetricImaginaryTimeReadout.equalTimeDecayExponent = 4 ∧ physicalNachtergaeleSims2006MetricImaginaryTimeReadout.imaginaryTimeDecayExponent = 17 / 4 ∧ ¬PhysicalNachtergaeleProjectedImaginaryClusteringAt physicalNachtergaeleMissingAdjointProjectionControl ∧ ¬PhysicalNachtergaeleProjectedImaginaryClusteringAt physicalNachtergaeleOutsideImaginaryTimeWindowControl ∧ ¬PhysicalNachtergaeleProjectedImaginaryClusteringAt physicalNachtergaeleComplexTimeCommutatorClaimedControl := by obtain ⟨_, _, _, equalTime, imaginaryTime, _, missingAdjoint, outsideWindow, _, _, complexTime, _⟩ := nachtergaele_imaginary_time_clustering_requires_both_projections_and_a_bounded_window exact ⟨equalTime, imaginaryTime, missingAdjoint, outsideWindow, complexTime⟩ example : physicalNachtergaeleSims2006ConnectedEqualTimeReadout.rawCorrelationValue = 7 ∧ physicalNachtergaeleSims2006ConnectedEqualTimeReadout.groundSectorCorrectionValue = 6 ∧ physicalNachtergaeleSims2006ConnectedEqualTimeReadout.correctedCorrelationValue = 1 ∧ ¬PhysicalNachtergaeleNondegenerateConnectedAt physicalNachtergaeleConnectedClaimedUnsubtractedControl ∧ ¬PhysicalNachtergaeleNondegenerateConnectedAt physicalNachtergaeleMissingCenteredObservableControl := by obtain ⟨_, raw, correction, connected, _, _, wrongRaw, missingCentering⟩ := nachtergaele_unique_ground_state_controls_connected_not_raw_correlations exact ⟨raw, correction, connected, wrongRaw, missingCentering⟩ end SpectralGapClusteringPhysicalReuseAudit namespace ConstraintConsistencyPhysicalReuseAudit open PhysicsLogic.Framework.ConstraintConsistencyCompletionContext.Countermodels example : physicalDirac1950IteratedConsistencyReadout.weakTotalHamiltonianValue = 7 ∧ physicalDirac1950IteratedConsistencyReadout.shellTotalHamiltonianValue = 13 ∧ physicalDirac1950IteratedConsistencyReadout.weakConstraintPoissonBracketValue = 2 ∧ ¬PhysicalDiracHamiltonianAt physicalDiracWeakPoissonSubstitutionControl ∧ ¬PhysicalDiracHamiltonianAt physicalDiracPhysicalGaugeIdentificationClaimedControl := by obtain ⟨_, weak, strong, _, _, bracket, _, substitution, _, identification⟩ := dirac_weak_surface_hamiltonian_does_not_authorize_strong_poisson_substitution exact ⟨weak, strong, bracket, substitution, identification⟩ example : physicalDirac1950IteratedConsistencyReadout.primaryConstraintCount = 3 ∧ physicalDirac1950IteratedConsistencyReadout.secondaryConstraintCount = 1 ∧ physicalDirac1950IteratedConsistencyReadout.arbitraryMultiplierCount = 1 ∧ physicalDirac1950IteratedConsistencyReadout.determinedMultiplierCount = 2 ∧ ¬PhysicalDiracConsistencyCoreAt physicalDiracFirstClassMultiplierFixedControl ∧ ¬PhysicalDiracConsistencyCoreAt physicalDiracSecondClassMultiplierArbitraryControl := by obtain ⟨_, primary, secondary, _, arbitrary, determined, _, _, _, _, fixedFirstClass, arbitrarySecondClass⟩ := dirac_iterated_consistency_distinguishes_secondary_constraints_contradictions_and_multipliers exact ⟨primary, secondary, arbitrary, determined, fixedFirstClass, arbitrarySecondClass⟩ example : physicalDirac1950SecondClassBracketReadout.secondClassMatrixDeterminant = 4 ∧ physicalDirac1950SecondClassBracketReadout.inverseSecondClassMatrixZeroOne = -1 / 2 ∧ physicalDirac1950SecondClassBracketReadout.inverseSecondClassMatrixOneZero = 1 / 2 ∧ physicalDirac1950SecondClassBracketReadout.modifiedObservableBracket = 4 ∧ ¬PhysicalDiracSecondClassBracketAt physicalDiracSingularSecondClassMatrixControl ∧ ¬PhysicalDiracSecondClassBracketAt physicalDiracWrongModifiedBracketCorrectionControl := by obtain ⟨_, determinant, inverseLeft, inverseRight, _, _, modified, _, singular, _, wrongCorrection, _, _⟩ := dirac_second_class_reduction_requires_inverse_constraint_matrix_and_changes_brackets exact ⟨determinant, inverseLeft, inverseRight, modified, singular, wrongCorrection⟩ example : physicalDirac1950InstantFormReadout.relativisticRequiredFreedomCount = 4 ∧ physicalDirac1950PointFormReadout.relativisticRequiredFreedomCount = 4 ∧ physicalDirac1950FrontFormReadout.relativisticRequiredFreedomCount = 3 ∧ physicalDirac1950SpacelikeSurfaceReadout.infiniteRelativisticFreedomFamily = true ∧ physicalDirac1950InstantFormReadout.relativisticForm ≠ physicalDirac1950PointFormReadout.relativisticForm ∧ ¬PhysicalDiracRelativisticFormAt physicalDiracWrongFrontFreedomCountControl := by obtain ⟨_, _, _, _, instant, point, front, infinite, distinct, _, _, wrongFront, _⟩ := dirac_relativistic_forms_distinguish_instant_point_front_and_infinite_surface_freedoms exact ⟨instant, point, front, infinite, distinct, wrongFront⟩ example : physicalDirac1950ConditionalQuantizationReadout.secondClassRemovedBeforeQuantization = true ∧ physicalDirac1950ConditionalQuantizationReadout.operatorOrderingResolved = false ∧ physicalDirac1950ConditionalQuantizationReadout.quantumConstraintAlgebraConsistencyProved = false ∧ ¬PhysicalDiracConditionalQuantizationAt physicalDiracOperatorOrderingClaimedResolvedControl ∧ ¬PhysicalDiracConditionalQuantizationAt physicalDiracMultiplierOperatorIntroducedControl := by obtain ⟨_, removed, _, ordering, consistency, wrongOrdering, _, multiplier, _⟩ := dirac_first_class_quantization_keeps_operator_ordering_and_full_consistency_open exact ⟨removed, ordering, consistency, wrongOrdering, multiplier⟩ example : physicalAndersonBergmann1951UnsaturatedDescriptorBoundReadout.descriptorConstraintUpperBound = 2 ∧ physicalAndersonBergmann1951UnsaturatedDescriptorBoundReadout.completedConstraintCount = 1 ∧ physicalAndersonBergmann1951ElectromagneticReadout.completedConstraintCount = 2 ∧ ¬PhysicalAndersonBergmannInvariantCompletionAt physicalAndersonBergmannGeneratorClosureWithoutPreservationControl := by obtain ⟨_, _, _, upper, unsaturated, electromagnetic, _, noPreservation⟩ := anderson_bergmann_descriptor_order_bounds_do_not_fix_exact_constraint_counts exact ⟨upper, unsaturated, electromagnetic, noPreservation⟩ example : physicalAndersonBergmann1951ElectromagneticReadout.electromagneticGeneratorOffSurface = 21 ∧ physicalAndersonBergmann1951ElectromagneticReadout.electromagneticGeneratorOnPrimarySurface = 6 ∧ physicalAndersonBergmann1951ElectromagneticReadout.electromagneticGeneratorOnCompletedSurface = 0 ∧ ¬PhysicalAndersonBergmannElectromagneticAt physicalAndersonBergmannMissingDescriptorDerivativeControl ∧ ¬PhysicalAndersonBergmannElectromagneticAt physicalAndersonBergmannElectromagneticSecondaryOmittedControl := by obtain ⟨_, _, _, offSurface, primarySurface, completeSurface, _, noDerivative, _, _, noSecondary⟩ := anderson_bergmann_electromagnetic_generator_needs_primary_derivative_and_secondary_gauss_law exact ⟨offSurface, primarySurface, completeSurface, noDerivative, noSecondary⟩ example : physicalAndersonBergmann1951CoordinateReadout.completedConstraintCount = 8 ∧ physicalAndersonBergmann1951CoordinateReadout.coordinateVelocityContributionOffSurface = 6 ∧ physicalAndersonBergmann1951CoordinateReadout.coordinateVelocityContributionOnPrimarySurface = 0 ∧ physicalAndersonBergmann1951CoordinateReadout.coordinateGeneratorOffSurface = -14 ∧ ¬PhysicalAndersonBergmannCoordinateAt physicalAndersonBergmannCoordinateVelocityTermErasedControl ∧ ¬PhysicalAndersonBergmannCoordinateAt physicalAndersonBergmannCoordinateVelocityTermSurvivesOnSurfaceControl := by obtain ⟨_, _, _, completed, offSurface, onSurface, generator, _, _, erased, survives, _, _⟩ := anderson_bergmann_coordinate_generator_retains_eight_constraints_and_off_surface_velocity exact ⟨completed, offSurface, onSurface, generator, erased, survives⟩ example : physicalAndersonBergmann1951QualifiedInitialValueReadout.canonicalSolutionCount = 1 ∧ physicalAndersonBergmann1951QualifiedInitialValueReadout.coordinateChangeAwayFromInitialSurfaceChangesHamiltonian = true ∧ physicalAndersonBergmann1951QualifiedInitialValueReadout.covariantQuantizationReady = false ∧ ¬PhysicalAndersonBergmannQualifiedInitialValueAt physicalAndersonBergmannCoordinateChangeKeepsHamiltonianControl ∧ ¬PhysicalAndersonBergmannQualifiedInitialValueAt physicalAndersonBergmannQuantizationClaimedReadyControl := by obtain ⟨_, _, _, unique, changed, notReady, _, unchanged, ready, _⟩ := anderson_bergmann_fixed_hamiltonian_uniqueness_does_not_resolve_singular_region_quantization exact ⟨unique, changed, notReady, unchanged, ready⟩ end ConstraintConsistencyPhysicalReuseAudit namespace TrappedSurfaceStabilityPhysicalReuseAudit open PhysicsLogic.Framework.TrappedSurfaceStabilityContext.Countermodels example : physicalPenrose1965NonsphericalStabilityReadout.schwarzschildMass = 1 ∧ physicalPenrose1965NonsphericalStabilityReadout.schwarzschildSurfaceRadius = 1 ∧ physicalPenrose1965NonsphericalStabilityReadout.schwarzschildRadialFactor = -1 ∧ ¬PhysicalPenroseNonsphericalStabilityAt physicalPenroseExteriorSurfaceStabilityControl ∧ ¬PhysicalPenroseNonsphericalStabilityAt physicalPenroseExtremalKerrStabilityControl := by obtain ⟨_, mass, radius, factor, _, _, exterior, _, extremal⟩ := penrose_nonspherical_stability_requires_mass_dependent_interior_and_subextremal_reference exact ⟨mass, radius, factor, exterior, extremal⟩ example : physicalPenrose1965NonsphericalStabilityReadout.referenceInwardExpansion = -2 ∧ physicalPenrose1965NonsphericalStabilityReadout.referenceOutwardExpansion = -1 ∧ physicalPenrose1965NonsphericalStabilityReadout.targetOutwardExpansion = -3 / 4 ∧ ¬PhysicalCommonTrappedStabilityAt physicalPenroseOneNullFamilyStabilityControl ∧ ¬PhysicalCommonTrappedStabilityAt physicalPenroseMarginalReferenceStabilityControl := by obtain ⟨_, inward, outward, _, targetOutward, oneFamily, marginal, _, _, _⟩ := strict_trapping_stability_requires_both_null_families_and_open_margin exact ⟨inward, outward, targetOutward, oneFamily, marginal⟩ example : physicalHawking1965FlatCosmologyStabilityReadout.matterDensity = 12 ∧ physicalHawking1965OpenCosmologyStabilityReadout.matterDensity = 9 ∧ physicalHawking1965OpenCosmologyStabilityReadout.radialRoot = 5 / 3 ∧ physicalHawking1965OpenCosmologyStabilityReadout.radialConvergenceTerm = 5 / 2 ∧ ¬PhysicalHawkingFriedmannLargeSphereAt physicalHawkingWrongFriedmannDensityStabilityControl ∧ ¬PhysicalHawkingFriedmannLargeSphereAt physicalHawkingWrongRadialRootStabilityControl := by obtain ⟨_, _, _, flatDensity, _, openDensity, radialRoot, radialTerm, _, _, friedmann, wrongRoot⟩ := hawking_flat_and_open_large_spheres_retain_distinct_friedmann_and_radial_data exact ⟨flatDensity, openDensity, radialRoot, radialTerm, friedmann, wrongRoot⟩ example : physicalHawking1965OpenCosmologyStabilityReadout.direction = .past ∧ physicalHawking1965OpenCosmologyStabilityReadout.referenceInwardConvergence = 3 / 2 ∧ physicalHawking1965OpenCosmologyStabilityReadout.referenceOutwardConvergence = 13 / 2 ∧ ¬PhysicalHawkingFriedmannLargeSphereAt physicalHawkingFutureDirectionStabilityControl ∧ ¬PhysicalHawkingFriedmannLargeSphereAt physicalHawkingCollapsedNullFamilySignsStabilityControl := by obtain ⟨_, direction, _, inward, outward, future, collapsedSigns, _, _⟩ := hawking_past_null_families_require_opposite_radial_signs_and_a_large_sphere exact ⟨direction, inward, outward, future, collapsedSigns⟩ example : physicalHawking1965OpenCosmologyStabilityReadout.strictConvergenceMargin = 1 ∧ physicalHawking1965OpenCosmologyStabilityReadout.perturbationBound = 1 / 4 ∧ physicalHawking1965OpenCosmologyStabilityReadout.targetExactlyHomogeneous = false ∧ ¬PhysicalHawkingFriedmannLargeSphereAt physicalHawkingExcessivePerturbationStabilityControl ∧ ¬PhysicalHawkingFriedmannLargeSphereAt physicalHawkingExactTargetHomogeneityStabilityControl := by obtain ⟨_, margin, bound, _, _, inhomogeneous, _, excessive, _, exact⟩ := hawking_finite_convergence_margin_preserves_irregular_but_controlled_target exact ⟨margin, bound, inhomogeneous, excessive, exact⟩ example : physicalHawking1965CompactHyperbolicCoverStabilityReadout.originalSpatialSliceCompact = true ∧ physicalHawking1965CompactHyperbolicCoverStabilityReadout.noncompactCoveringAvailable = true ∧ ¬PhysicalTrappedStabilityGlobalHandoffAt physicalHawkingMissingCoveringStabilityControl ∧ ¬PhysicalTrappedStabilityGlobalHandoffAt physicalHawkingMissingCoveringDescentStabilityControl := by obtain ⟨_, _, compact, cover, _, missingCover, missingDescent, _⟩ := compact_hyperbolic_stability_handoff_requires_noncompact_cover_and_descent exact ⟨compact, cover, missingCover, missingDescent⟩ example : physicalHawking1965MatterFlowStabilityReadout.cauchySurfaceAvailable = false ∧ physicalHawking1965MatterFlowStabilityReadout.uniqueTimelikeMatterFlow = true ∧ ¬PhysicalTrappedStabilityGlobalHandoffAt physicalHawkingMissingMatterFlowStabilityControl ∧ ¬PhysicalTrappedStabilityGlobalHandoffAt physicalHawkingMissingGlobalReplacementBridgeStabilityControl := by obtain ⟨_, _, noCauchy, flow, _, missingFlow, _, missingBridge⟩ := hawking_matter_flow_handoff_requires_global_conditions_beyond_local_stability exact ⟨noCauchy, flow, missingFlow, missingBridge⟩ example : PhysicalPenroseNonsphericalStabilityAt physicalPenroseLocalWithoutGlobalStabilityReadout ∧ ¬PhysicalTrappedStabilityGlobalHandoffAt physicalPenroseLocalWithoutGlobalStabilityReadout ∧ ¬PhysicalPenroseNonsphericalStabilityAt physicalPenroseIncompletenessFromStabilityControl ∧ ¬PhysicalPenroseNonsphericalStabilityAt physicalPenroseCurvatureFromStabilityControl := by obtain ⟨stableLocal, noHandoff, _, noIncompleteness, noCurvature, _⟩ := stable_local_trapping_does_not_supply_global_handoff_or_incompleteness exact ⟨stableLocal, noHandoff, noIncompleteness, noCurvature⟩ end TrappedSurfaceStabilityPhysicalReuseAudit namespace FinitePartitionNonvanishingPhysicalReuseAudit open PhysicsLogic.Framework.FinitePartitionNonvanishingContext.Countermodels open PhysicsLogic.Framework.FiniteBinaryConfigurationGraph example : sourceSingleEdgeGraph.hardCorePartitionValue (fun _ => (1 : ℚ)) = 3 ∧ unconstrainedOccupationValue (fun _ : Fin 2 => (1 : ℚ)) = 4 := single_edge_independent_set_sum_excludes_adjacent_occupation example : sourceSingleEdgeGraph.hardCorePartitionValue (fun _ => (1 : ℚ)) ≠ 0 ∧ sourceSingleEdgeGraph.hardCorePartitionValue sourceSingleEdgeCancellationActivity = 0 := single_edge_constant_nonvanishing_does_not_control_multivariate_sum example : sourceSingleEdgeGraph.unpinnedIsingPartitionValue (1 / 2 : ℚ) = 3 ∧ sourceSingleEdgeGraph.pinnedIsingPartitionValue sourceSingleEdgeFirstSpinPinned (1 / 2 : ℚ) = 3 / 2 ∧ sourceSingleEdgeGraph.pinnedIsingPartitionValue sourceSingleEdgeOppositeSpinsPinned (1 / 2 : ℚ) = 1 / 2 := single_edge_ising_sum_retains_disagreement_weights_and_pinning example : unpinnedVertexCount (fun _ : Fin 2 => none) = 2 ∧ unpinnedVertexCount sourceSingleEdgeFirstSpinPinned = 1 ∧ unpinnedVertexCount sourceSingleEdgeOppositeSpinsPinned = 0 := source_pinning_changes_the_actual_induction_complexity example : physicalPetersRegts2017HardCoreReadout.hardCoreParentPartition = sourceSingleEdgeGraph.hardCorePartitionValue (fun _ => physicalPetersRegts2017HardCoreReadout.selectedRealParameter) ∧ sourceSingleEdgeGraph.unpinnedIsingPartitionValue physicalLiuSinclairSrivastava2018FerromagneticReadout.selectedRealParameter = 3 := by obtain ⟨_, hardCore, _, _, _, ising, _, _, _⟩ := physical_source_partition_models_compute_actual_configuration_sums exact ⟨hardCore, ising⟩ example : physicalPetersRegts2017HardCoreReadout.maximumDegree = 3 ∧ physicalPetersRegts2017HardCoreReadout.hardCoreTreeThreshold = 4 ∧ physicalPetersRegts2017HardCoreReadout.selectedRealParameter = 1 ∧ ¬PhysicalHardCoreFiniteNonvanishingAt physicalHardCoreWrongThresholdControl ∧ ¬PhysicalHardCoreFiniteNonvanishingAt physicalHardCoreThresholdEndpointControl := by obtain ⟨_, degree, threshold, activity, _, _, wrongThreshold, endpoint⟩ := hard_core_tree_threshold_is_degree_dependent_and_excludes_its_endpoint exact ⟨degree, threshold, activity, wrongThreshold, endpoint⟩ example : physicalPetersRegts2017HardCoreReadout.hardCoreParentPartition = 3 ∧ physicalPetersRegts2017HardCoreReadout.hardCoreOccupationRatio = 1 / 2 ∧ ¬PhysicalHardCoreFiniteNonvanishingAt physicalHardCoreWrongVertexSplitControl ∧ ¬PhysicalHardCoreFiniteNonvanishingAt physicalHardCoreWrongOccupationRatioControl := by obtain ⟨_, parent, _, _, ratio, wrongSplit, wrongRatio⟩ := hard_core_vertex_split_and_occupation_ratio_require_separate_nonzero_subpartitions exact ⟨parent, ratio, wrongSplit, wrongRatio⟩ example : physicalPetersRegts2017HardCoreReadout.internalRecurrenceArity = 2 ∧ physicalPetersRegts2017HardCoreReadout.finalRecurrenceArity = 3 ∧ physicalPetersRegts2017HardCoreReadout.hardCoreInternalRecurrenceValue = 1 / 4 ∧ physicalPetersRegts2017HardCoreReadout.hardCoreFinalRecurrenceValue = 1 / 8 ∧ ¬PhysicalHardCoreFiniteNonvanishingAt physicalHardCoreWrongFinalArityControl := by obtain ⟨_, internalArity, finalArity, internalValue, finalValue, _, wrongFinal, _⟩ := hard_core_internal_and_final_root_recurrences_have_distinct_degree_bounds exact ⟨internalArity, finalArity, internalValue, finalValue, wrongFinal⟩ example : physicalPetersRegts2017HardCoreReadout.literalMultivariateProductDomainZeroFree = false ∧ physicalPetersRegts2017HardCoreReadout.counterexampleRootRatio = -1 ∧ physicalPetersRegts2017HardCoreReadout.counterexamplePartitionValue = 0 ∧ ¬PhysicalHardCoreFiniteNonvanishingAt physicalHardCoreCollapsedMultivariateDomainsControl := by obtain ⟨_, _, literal, ratio, partition, collapsed, _, _⟩ := hard_core_controlled_multivariate_tube_does_not_imply_literal_product_domain exact ⟨literal, ratio, partition, collapsed⟩ example : physicalLiuSinclairSrivastava2018FerromagneticReadout.isingUniquenessLowerEndpoint = 1 / 3 ∧ physicalLiuSinclairSrivastava2018FerromagneticReadout.isingUniquenessUpperEndpoint = 3 ∧ physicalLiuSinclairSrivastava2018FerromagneticReadout.selectedRealParameter = 1 / 2 ∧ physicalLiuSinclairSrivastava2018AntiferromagneticReadout.selectedRealParameter = 2 ∧ physicalLiuSinclairSrivastava2018InfiniteTemperatureReadout.selectedRealParameter = 1 ∧ ¬PhysicalIsingPinnedFiniteNonvanishingAt physicalIsingOutsideUniquenessIntervalControl := by obtain ⟨_, _, _, lower, upper, ferro, anti, infinite, _, _, outside, _, _⟩ := ising_tree_uniqueness_distinguishes_ferromagnetic_antiferromagnetic_and_infinite_temperature exact ⟨lower, upper, ferro, anti, infinite, outside⟩ example : physicalLiuSinclairSrivastava2018FerromagneticReadout.zeroKind = .fisherInteraction ∧ physicalLiuSinclairSrivastava2018FerromagneticReadout.zeroExternalFieldFixed = true ∧ ¬PhysicalIsingPinnedFiniteNonvanishingAt physicalIsingLeeYangFieldControl := by obtain ⟨_, fisher, field, leeYang, _⟩ := ising_fisher_interaction_zero_freeness_is_not_lee_yang_field_zero_freeness exact ⟨fisher, field, leeYang⟩ example : physicalLiuSinclairSrivastava2018FerromagneticReadout.pinnedPlusContribution = 2 ∧ physicalLiuSinclairSrivastava2018FerromagneticReadout.pinnedMinusContribution = 3 ∧ physicalLiuSinclairSrivastava2018FerromagneticReadout.pinnedParentPartition = 5 ∧ ¬PhysicalIsingPinnedFiniteNonvanishingAt physicalIsingPinnedCancellationControl := by obtain ⟨_, plus, minus, _, parent, _, _, _, cancellation, _⟩ := pinned_ising_induction_requires_both_contributions_and_ratio_cancellation_exclusion exact ⟨plus, minus, parent, cancellation⟩ example : physicalLiuSinclairSrivastava2018FerromagneticReadout.isingRecurrenceKernelValue = 5 / 4 ∧ physicalLiuSinclairSrivastava2018FerromagneticReadout.isingRecurrenceValue = 5 / 8 ∧ ¬PhysicalIsingPinnedFiniteNonvanishingAt physicalIsingWrongSpinOffsetControl ∧ ¬PhysicalIsingPinnedFiniteNonvanishingAt physicalIsingWrongFinalArityControl := by obtain ⟨_, _, _, kernel, output, _, spin, _, finalArity, _⟩ := pinned_ising_self_avoiding_walk_retains_spin_offsets_and_distinct_root_arity exact ⟨kernel, output, spin, finalArity⟩ example : ¬PhysicalExternalApproximationAt physicalPetersRegts2017HardCoreReadout ∧ PhysicalExternalApproximationAt physicalHardCoreExternallyBridgedApproximationReadout ∧ PhysicalExternalApproximationAt physicalIsingExternallyBridgedApproximationReadout ∧ ¬PhysicalExternalApproximationAt physicalIsingAntiferromagneticApproximationClaimControl := by obtain ⟨hardCore, _, bridgedHardCore, _, _, bridgedIsing, anti⟩ := deterministic_approximation_requires_a_separate_source_scoped_algorithmic_bridge exact ⟨hardCore, bridgedHardCore, bridgedIsing, anti⟩ end FinitePartitionNonvanishingPhysicalReuseAudit namespace AdiabaticTransportPhysicalReuseAudit open PhysicsLogic.Framework.AdiabaticCycleTransportContext.Countermodels example : physicalLaughlin1981IdealHallReadout.planckConstant = 4 ∧ physicalLaughlin1981IdealHallReadout.speedOfLight = 3 ∧ physicalLaughlin1981IdealHallReadout.electronCharge = 2 ∧ physicalLaughlin1981IdealHallReadout.fluxQuantum = 6 ∧ ¬PhysicalLaughlinHallFluxTransportAt physicalLaughlinWrongFluxQuantumControl := by obtain ⟨_, planck, light, charge, flux, _, wrongFlux, _, _⟩ := laughlin_flux_quantum_gauge_return_and_integer_transfer_are_independent exact ⟨planck, light, charge, flux, wrongFlux⟩ example : physicalLaughlin1981IdealHallReadout.idealLandauIndex = 2 ∧ physicalLaughlin1981DirtyInteractingHallReadout.dirtyHallIndex = 1 ∧ physicalLaughlin1981DirtyZeroHallReadout.dirtyHallIndex = 0 ∧ physicalLaughlin1981DirtyInteractingHallReadout.totalDensityOfStatesGap = 0 ∧ ¬PhysicalLaughlinHallFluxTransportAt physicalLaughlinMissingMobilityGapControl := by obtain ⟨_, _, ideal, dirty, zero, _, densityGap, noMobility, _, _, _⟩ := dirty_hall_mobility_gap_allows_changed_or_zero_integer_without_density_gap exact ⟨ideal, dirty, zero, densityGap, noMobility⟩ example : physicalLaughlin1981ThermalizedEdgeReadout.edgeGap = 0 ∧ physicalLaughlin1981ThermalizedEdgeReadout.edgeThermalizationOccurs = true ∧ ¬PhysicalLaughlinHallFluxTransportAt physicalLaughlinMissingEdgeEquilibrationControl := by obtain ⟨_, gap, thermalized, _, missing, _⟩ := thermalized_gapless_hall_edges_preserve_bulk_quantization_conditionally exact ⟨gap, thermalized, missing⟩ example : physicalThouless1983FiniteTorusReadout.finiteTorusTransferredParticles = 1 / 2 ∧ physicalThouless1983FiniteTorusReadout.infinitePeriodicSystemSelected = false ∧ ¬PhysicalThoulessFiniteTorusAt physicalThoulessFiniteTorusClaimedQuantizedControl := by obtain ⟨_, transfer, finite, _, quantized, _⟩ := finite_torus_adiabatic_following_does_not_imply_integer_particle_transport exact ⟨transfer, finite, quantized⟩ example : physicalThouless1983InfiniteFilledBandReadout.transportIndex = 2 ∧ physicalThouless1983InfiniteFilledBandReadout.firstChernIndex = 2 ∧ physicalThouless1983InfiniteFilledBandReadout.nodeFluxIndex = 2 ∧ ¬PhysicalThoulessInfiniteFilledBandAt physicalThoulessMissingFermiGapControl ∧ ¬PhysicalThoulessInfiniteFilledBandAt physicalThoulessWrongChernIndexControl := by obtain ⟨_, index, chern, nodes, _, _, gap, _, _, wrongChern, _⟩ := infinite_filled_band_transport_requires_gap_chern_class_and_node_flux exact ⟨index, chern, nodes, gap, wrongChern⟩ example : physicalThouless1983CompetingPeriodsReadout.commonPeriod = 6 ∧ physicalThouless1983CompetingPeriodsReadout.commonCellParticleCount = 7 ∧ ¬PhysicalThoulessCompetingPeriodsAt physicalThoulessWrongCommonPeriodControl ∧ ¬PhysicalThoulessCompetingPeriodsAt physicalThoulessPhaseGapClosureControl := by obtain ⟨_, _, _, period, count, wrongPeriod, _, closed, _⟩ := competing_period_gap_labels_require_commensurability_and_an_open_all_phase_gap exact ⟨period, count, wrongPeriod, closed⟩ example : physicalThouless1983GapClosedHalfCycleReadout.gapClosedHalfCycleTransport = 1 / 2 ∧ ¬PhysicalThoulessGapClosedHalfTransportAt physicalThoulessGapClosedIndexClaimedInvariantControl := by obtain ⟨_, _, _, half, invariant, _⟩ := phase_dependent_gap_closure_permits_half_integer_cycle_transport exact ⟨half, invariant⟩ example : physicalThouless1983TypeAAlternatingSolitonReadout.typeASolitonCharge = 1 / 2 ∧ physicalThouless1983TypeBSuccessiveSolitonReadout.typeBSolitonCharge = 3 / 2 ∧ ¬PhysicalThoulessTypeBSolitonAt physicalThoulessTypeBZeroChainChargeControl := by obtain ⟨_, _, typeA, _, _⟩ := alternating_type_a_solitons_need_not_have_integer_charge obtain ⟨_, _, _, typeB, noChainCharge, _, _⟩ := successive_type_b_fractional_solitons_require_nonzero_chain_charge exact ⟨typeA, typeB, noChainCharge⟩ end AdiabaticTransportPhysicalReuseAudit namespace ValidatedNumericalPredictionPhysicalReuseAudit open PhysicsLogic.Framework.ValidatedNumericalPredictionContext.Countermodels example : PhysicalFPUQuadraticModalReturnAt physicalFermiPastaUlam1955QuadraticModalReturnReadout ∧ PhysicalFPUNonquadraticModeExchangeAt physicalFermiPastaUlam1955CubicModeExchangeReadout ∧ PhysicalFPUNonquadraticModeExchangeAt physicalFermiPastaUlam1955BrokenLinearModeExchangeReadout ∧ ¬PhysicalFPUQuadraticModalReturnAt physicalFPUWrongForceFamilyControl := by obtain ⟨quadratic, cubic, broken, _, _, _, wrongFamily⟩ := fpu_quadratic_cubic_and_broken_linear_forces_are_distinct exact ⟨quadratic, cubic, broken, wrongFamily⟩ example : physicalFermiPastaUlam1955QuadraticModalReturnReadout.firstModeRelativeTolerance = 1 / 100 ∧ ¬PhysicalFPUQuadraticModalReturnAt physicalFPUMissingFixedEndpointsControl ∧ ¬PhysicalFPUQuadraticModalReturnAt physicalFPUWrongModeToleranceControl := by obtain ⟨_, _, tolerance, _, endpoints, _, wrongTolerance⟩ := fpu_quadratic_modal_return_requires_fixed_endpoints_and_one_percent_control exact ⟨tolerance, endpoints, wrongTolerance⟩ example : physicalFermiPastaUlam1955QuadraticModalReturnReadout.reportedEquipartitionDeficit = 1 ∧ physicalFermiPastaUlam1955QuadraticModalReturnReadout.laterEquipartitionDeficit = 0 ∧ physicalFermiPastaUlam1955QuadraticModalReturnReadout.asymptoticThermalizationExcluded = false := by obtain ⟨_, current, later, excluded⟩ := fpu_selected_nonequipartition_permits_later_thermalization exact ⟨current, later, excluded⟩ example : physicalZabuskyKruskal1965PeriodicSolitonReadout.spatialPeriod = 2 ∧ ¬PhysicalZabuskyKruskalSolitonTrainAt physicalZKWrongSpatialPeriodControl ∧ ¬PhysicalZabuskyKruskalSolitonTrainAt physicalZKMissingCosineInitialProfileControl := by obtain ⟨_, period, _, wrongPeriod, missingCosine, _⟩ := zk_periodic_initial_condition_requires_the_cosine_phase_and_spatial_period exact ⟨period, wrongPeriod, missingCosine⟩ example : physicalZabuskyKruskal1965PeriodicSolitonReadout.discreteMomentumDrift = 0 ∧ physicalZabuskyKruskal1965PeriodicSolitonReadout.numericalEnergyDrift = 1 / 100000 ∧ physicalZabuskyKruskal1965PeriodicSolitonReadout.numericalEnergyDrift ≠ 0 ∧ ¬PhysicalZabuskyKruskalSolitonTrainAt physicalZKEnergyCheckFailedControl := by obtain ⟨_, momentum, energy, nonzero, failed⟩ := zk_exact_momentum_conservation_does_not_make_energy_conservation_exact exact ⟨momentum, energy, nonzero, failed⟩ example : physicalZabuskyKruskal1965PeriodicSolitonReadout.pulseCount = 8 ∧ physicalZabuskyKruskal1965PeriodicSolitonReadout.comparedPulseCount = 7 ∧ ¬PhysicalZabuskyKruskalSolitonTrainAt physicalZKMissingCollisionPhaseShiftControl := by obtain ⟨_, pulses, compared, _, _, _, _, missingShift⟩ := zk_soliton_width_velocity_and_collision_identity_require_independent_checks exact ⟨pulses, compared, missingShift⟩ example : PhysicalCreutzSU2HeatBathUpdateAt physicalCreutzUnconvergedFiniteRunControl ∧ ¬PhysicalCreutzFiniteHeatBathEquilibriumAt physicalCreutzUnconvergedFiniteRunControl := by obtain ⟨update, _, unconverged⟩ := creutz_detailed_balance_does_not_by_itself_establish_finite_run_equilibrium exact ⟨update, unconverged⟩ example : physicalCreutz1980HeatBathPlaquetteReadout.gaugeColorCount = 2 ∧ physicalCreutz1980HeatBathPlaquetteReadout.spacetimeDimension = 4 ∧ ¬PhysicalCreutzSU2HeatBathUpdateAt physicalCreutzRestrictedGroupSupportControl ∧ ¬PhysicalCreutzSU2HeatBathUpdateAt physicalCreutzFailedConditionalBoltzmannControl := by obtain ⟨_, colors, dimension, _, _, _, support, boltzmann, _⟩ := creutz_heat_bath_requires_full_su2_support_and_conditional_boltzmann_sampling exact ⟨colors, dimension, support, boltzmann⟩ example : physicalCreutz1980FiniteWilsonLoopFitReadout.selectedLoopSide = 5 ∧ physicalCreutz1980FiniteWilsonLoopFitReadout.selectedLoopArea = 25 ∧ ¬PhysicalCreutzFiniteWilsonLoopFitAt physicalCreutzGaugeVariantWilsonLoopControl ∧ ¬PhysicalCreutzFiniteWilsonLoopFitAt physicalCreutzUncontrolledLoopSideControl := by obtain ⟨_, side, area, _, gauge, size, _, _⟩ := creutz_finite_wilson_loop_fit_requires_gauge_invariance_and_size_control exact ⟨side, area, gauge, size⟩ example : physicalCreutz1980FiniteWilsonLoopFitReadout.selectedSampleMean ≠ physicalCreutz1980FiniteWilsonLoopFitReadout.exactPathIntegralValue ∧ PhysicalCreutzFiniteWilsonLoopFitAt physicalCreutzExactFiniteSampleControl := by obtain ⟨_, _, _, unequal, exactControl, _⟩ := creutz_sample_mean_and_error_do_not_equal_the_exact_path_integral exact ⟨unequal, exactControl⟩ example : physicalCreutz1980FiniteWilsonLoopFitReadout.fittedStringTension = 1 / 2 ∧ ¬PhysicalCreutzFiniteWilsonLoopFitAt physicalCreutzContinuumConfinementClaimedControl ∧ ¬PhysicalCreutzFiniteWilsonLoopFitAt physicalCreutzContinuumMassGapClaimedControl := by obtain ⟨_, tension, _, _, _, confinement, massGap, _⟩ := creutz_positive_finite_cutoff_string_tension_does_not_prove_continuum_confinement exact ⟨tension, confinement, massGap⟩ end ValidatedNumericalPredictionPhysicalReuseAudit namespace FractionalJosephsonPhysicalReuseAudit open PhysicsLogic.Framework.ParityResolvedFractionalJosephsonContext.Countermodels example : physicalKitaev2001FiniteWireParityBranchReadout.bulkGap = 2 ∧ physicalKitaev2001FiniteWireParityBranchReadout.finiteWireEnergySplitting = 1 / 100 ∧ ¬PhysicalKitaevFiniteWireFractionalJosephsonAt physicalKitaevZeroFiniteTunnelingControl := by obtain ⟨_, gap, _, splitting, _, _, zeroTunneling, _⟩ := kitaev_finite_wire_bulk_gap_allows_exponentially_split_avoided_crossing exact ⟨gap, splitting, zeroTunneling⟩ example : physicalKitaev2001FiniteWireParityBranchReadout.evenPhaseCycleEigenvalue = 1 ∧ physicalKitaev2001FiniteWireParityBranchReadout.oddPhaseCycleEigenvalue = -1 ∧ ¬PhysicalKitaevFiniteWireFractionalJosephsonAt physicalKitaevWrongOddParityCycleControl := by obtain ⟨_, even, odd, _, _, _, wrongOdd⟩ := kitaev_parity_cycle_requires_open_ends_localization_and_opposite_eigenvalues exact ⟨even, odd, wrongOdd⟩ example : physicalFuKane2009ProtectedHelicalJunctionReadout.localHamiltonianPhasePeriod = 2 ∧ physicalFuKane2009ProtectedHelicalJunctionReadout.fixedSectorCurrentPhasePeriod = 4 ∧ ¬PhysicalFuKaneProtectedHelicalJosephsonAt physicalJosephsonWrongFixedSectorPeriodControl := by obtain ⟨_, _, _, _, localPeriod, branch, wrongPeriod⟩ := two_pi_local_return_and_four_pi_fixed_parity_current_remain_distinct exact ⟨localPeriod, branch, wrongPeriod⟩ example : physicalFuKane2009ProtectedHelicalJunctionReadout.paritySectorCount = 2 ∧ physicalFuKane2009ConventionalPointContactReadout.paritySectorCount = 4 ∧ ¬PhysicalFuKaneProtectedHelicalJosephsonAt physicalFuKaneOrdinaryWireClaimedHelicalControl := by obtain ⟨_, _, helical, conventional, _, _, wrongPlatform, _⟩ := fu_kane_helical_edge_differs_from_a_spin_degenerate_point_contact exact ⟨helical, conventional, wrongPlatform⟩ example : physicalFuKane2009ProtectedHelicalJunctionReadout.electricChargeLocallyConserved = false ∧ ¬PhysicalFuKaneProtectedHelicalJosephsonAt physicalJosephsonMissingParityConservationControl := by obtain ⟨_, _, _, charge, missingParity, _, _⟩ := fu_kane_crossing_protection_requires_local_parity_not_charge_conservation exact ⟨charge, missingParity⟩ example : physicalFuKane2009MagneticInterfaceReadout.magneticInterfaceGap = 1 ∧ physicalFuKane2009CriticalInterfaceReadout.magneticInterfaceGap = 0 ∧ ¬PhysicalFuKaneCriticalMagneticInterfaceAt physicalFuKaneCriticalInterfaceClaimedGappedControl := by obtain ⟨_, _, _, _, interfaceGap, _, _⟩ := fu_kane_magnetic_interface_requires_mass_domain_wall_and_printed_minimum_gap obtain ⟨_, _, _, criticalGap, _, incorrectlyGapped⟩ := fu_kane_mass_equality_closes_the_interface_gap_without_a_localized_mode exact ⟨interfaceGap, criticalGap, incorrectlyGapped⟩ example : physicalFuKane2009ThermalTelegraphNoiseReadout.paritySwitchingRate = 1 / 100 ∧ physicalFuKane2009VariableRangeHoppingNoiseReadout.paritySwitchingRate = 1 / 200 ∧ physicalFuKane2009ThermalTelegraphNoiseReadout.zeroFrequencyNoise = 100 ∧ physicalFuKane2009VariableRangeHoppingNoiseReadout.zeroFrequencyNoise = 200 := by obtain ⟨_, _, thermalRate, hoppingRate, thermalNoise, hoppingNoise, _, _, _, _⟩ := fu_kane_thermal_and_hopping_parity_switches_are_distinct_source_mechanisms exact ⟨thermalRate, hoppingRate, thermalNoise, hoppingNoise⟩ example : physicalFuKane2009ThermalTelegraphNoiseReadout.fixedSectorCurrentPhasePeriod = 4 ∧ physicalFuKane2009ThermalizedEquilibriumReadout.equilibriumCurrentPhasePeriod = 2 ∧ ¬PhysicalFuKaneLongTimeEquilibriumCurrentAt physicalFuKaneWrongEquilibriumPeriodControl := by obtain ⟨_, _, fixed, equilibrium, _, _, _, wrongEquilibrium, _⟩ := fu_kane_finite_temperature_switching_restores_two_pi_equilibrium_current exact ⟨fixed, equilibrium, wrongEquilibrium⟩ example : physicalFuKane2009GappedFractionalACReadout.fractionalACFrequency = 1 / 5 ∧ physicalFuKane2009GappedFractionalACReadout.conventionalACFrequency = 2 / 5 ∧ ¬PhysicalFuKaneGappedFractionalACAt physicalFuKaneACMissingContinuumGapControl := by obtain ⟨_, _, _, _, fractional, conventional, noGap, _, _, _⟩ := fu_kane_fractional_ac_frequency_requires_switching_bias_and_continuum_window exact ⟨fractional, conventional, noGap⟩ example : physicalFuKane2009ThermalTelegraphNoiseReadout.telegraphNoisePredicted = true ∧ physicalFuKane2009ThermalTelegraphNoiseReadout.telegraphNoiseExperimentObserved = false ∧ ¬PhysicalFuKaneShortTimeTelegraphNoiseAt physicalFuKaneTelegraphNoiseClaimedObservedControl := by obtain ⟨_, predicted, unobserved, falseClaim⟩ := fu_kane_telegraph_noise_is_predicted_not_experimentally_observed exact ⟨predicted, unobserved, falseClaim⟩ end FractionalJosephsonPhysicalReuseAudit namespace CanonicalGeneratorPhysicalReuseAudit open PhysicsLogic.Framework.CanonicalGeneratorBracketClosureContext.Countermodels example : physicalDirac1950WeakConstraintHamiltonianReadout.weakConstraintSurfaceValue = 0 ∧ physicalDirac1950WeakConstraintHamiltonianReadout.poissonBracketAfterPrematureSubstitution = 1 ∧ ¬PhysicalDiracWeakConstraintHamiltonianAt physicalDiracWeakConstraintSubstitutedStronglyControl := by obtain ⟨_, surface, _, bracket, rejected, _⟩ := dirac_weak_surface_equality_does_not_allow_strong_poisson_substitution exact ⟨surface, bracket, rejected⟩ example : physicalDirac1950WeakConstraintHamiltonianReadout.primaryConstraintCount = 3 ∧ physicalDirac1950WeakConstraintHamiltonianReadout.secondaryConstraintCount = 1 ∧ ¬PhysicalDiracWeakConstraintHamiltonianAt physicalDiracUnpreservedConstraintControl := by obtain ⟨_, primary, secondary, _, _, notPreserved⟩ := dirac_primary_constraints_require_secondary_consistency_and_preservation exact ⟨primary, secondary, notPreserved⟩ example : physicalDirac1950WeakConstraintHamiltonianReadout.firstClassMultiplierValue = 2 ∧ physicalDirac1950WeakConstraintHamiltonianReadout.alternateFirstClassMultiplierValue = 3 ∧ physicalDirac1950WeakConstraintHamiltonianReadout.secondClassMultiplierValue = 0 := by obtain ⟨_, first, alternate, second, _, _, _, _⟩ := dirac_first_class_arbitrariness_and_second_class_fixed_multipliers_differ exact ⟨first, alternate, second⟩ example : physicalDirac1950SecondClassModifiedBracketReadout.modifiedBracketConstraintResidual = 0 ∧ ¬PhysicalDiracSecondClassModifiedBracketAt physicalDiracSingularSecondClassMatrixControl := by obtain ⟨_, _, _, _, modified, singular, _⟩ := dirac_second_class_elimination_requires_an_invertible_modified_bracket exact ⟨modified, singular⟩ example : physicalDirac1950InstantFormReadout.relativisticFreedomCount = 4 ∧ physicalDirac1950PointFormReadout.relativisticFreedomCount = 4 ∧ physicalDirac1950FrontFormReadout.relativisticFreedomCount = 3 ∧ physicalDirac1950InstantFormReadout.relativisticForm ≠ physicalDirac1950PointFormReadout.relativisticForm := by obtain ⟨_, _, _, instant, point, front, distinct, _, _⟩ := dirac_instant_point_and_front_forms_have_distinct_scoped_freedom_counts exact ⟨instant, point, front, distinct⟩ example : physicalDirac1950WeakConstraintHamiltonianReadout.derivationOrder ≠ physicalAndersonBergmann1951ElectromagneticReadout.derivationOrder := by exact dirac_and_anderson_bergmann_require_opposite_constraint_generator_orders.2.2.1 example : physicalAndersonBergmann1951InvariantGeneratorOnlyReadout.preservedConstraintCount = 0 ∧ physicalAndersonBergmann1951InvariantGeneratorOnlyReadout.hamiltonianBracketCompatibilityEstablished = false := by obtain ⟨_, _, preservation, compatibility⟩ := anderson_bergmann_generator_closure_does_not_supply_hamiltonian_completion exact ⟨preservation, compatibility⟩ example : physicalAndersonBergmann1951ElectromagneticReadout.primaryConstraintCount = 1 ∧ physicalAndersonBergmann1951ElectromagneticReadout.secondaryConstraintCount = 1 ∧ physicalAndersonBergmann1951ElectromagneticReadout.completedConstraintCount = 2 ∧ physicalAndersonBergmann1951ElectromagneticReadout.generatorDensityValue = 5 ∧ ¬PhysicalAndersonBergmannElectromagneticConstraintsAt physicalAndersonBergmannPrimaryOnlyElectromagnetismControl := by obtain ⟨_, primary, secondary, total, density, _, _, primaryOnly, _, _⟩ := anderson_bergmann_electromagnetism_has_distinct_primary_and_gauss_constraints exact ⟨primary, secondary, total, density, primaryOnly⟩ example : physicalAndersonBergmann1951CoordinateInvariantReadout.primaryConstraintCount = 4 ∧ physicalAndersonBergmann1951CoordinateInvariantReadout.secondaryConstraintCount = 4 ∧ physicalAndersonBergmann1951CoordinateInvariantReadout.completedConstraintCount = 8 := by obtain ⟨_, primary, secondary, total, _, _⟩ := anderson_bergmann_coordinate_covariance_has_four_primary_and_four_secondary_constraints exact ⟨primary, secondary, total⟩ example : physicalAndersonBergmann1951CoordinateInvariantReadout.coordinateVelocityContributionOnSurface = 0 ∧ physicalAndersonBergmann1951CoordinateInvariantReadout.coordinateVelocityContributionOffSurface = 5 ∧ ¬PhysicalAndersonBergmannCoordinateConstraintsAt physicalAndersonBergmannCoordinateVelocityErasedControl := by obtain ⟨_, surface, offSurface, _, _, erased, _, _⟩ := anderson_bergmann_coordinate_generator_keeps_primary_proportional_velocity_terms exact ⟨surface, offSurface, erased⟩ example : physicalAndersonBergmann1951InvariantGeneratorOnlyReadout.hamiltonianBracketWitnessCount = 0 ∧ physicalAndersonBergmann1951ElectromagneticReadout.hamiltonianBracketWitnessCount = 1 ∧ ¬PhysicalAndersonBergmannElectromagneticConstraintsAt physicalAndersonBergmannHamiltonianCompatibilityErasedControl := by obtain ⟨_, _, before, after, rejected⟩ := anderson_bergmann_hamiltonian_bracket_closure_is_a_separate_derivation exact ⟨before, after, rejected⟩ example : physicalAndersonBergmann1951CoordinateInvariantReadout.physicalGaugeStateIdentificationEstablished = false ∧ physicalAndersonBergmann1951CoordinateInvariantReadout.covariantQuantizationEstablished = false ∧ ¬PhysicalAndersonBergmannCoordinateConstraintsAt physicalAndersonBergmannSingularRegionQuantizationClaimedControl := by obtain ⟨_, _, gauge, _, quantization, _, rejected⟩ := anderson_bergmann_completed_generators_leave_gauge_identification_and_quantization_open exact ⟨gauge, quantization, rejected⟩ end CanonicalGeneratorPhysicalReuseAudit namespace PlanarExchangePhysicalReuseAudit open PhysicsLogic.Framework.PlanarExchangeStatisticsContext.Countermodels example : physicalWilczek1982BosonicCompositeReadout.anomalousAngularMomentum = 1 / 4 ∧ physicalWilczek1982BosonicCompositeReadout.oneAnyonAngularMomentum = 5 / 4 ∧ physicalWilczek1982BosonicCompositeReadout.relativeAngularMomentum = 1 / 2 := by obtain ⟨_, anomalous, one, relative, _, _⟩ := wilczek_charge_flux_fraction_shifts_one_and_two_particle_angular_momentum exact ⟨anomalous, one, relative⟩ example : physicalWilczek1982BosonicCompositeReadout.exteriorMagneticFieldVanishes = true ∧ physicalWilczek1982BosonicCompositeReadout.singularGaugeTransformationMultivalued = true ∧ ¬PhysicalWilczekSingularGaugeCompositeAt physicalWilczekSingularGaugeClaimedSingleValuedControl := by obtain ⟨_, outside, singular, _, notSingleValued, _⟩ := wilczek_singular_gauge_retains_zero_exterior_field_and_multivalued_boundary exact ⟨outside, singular, notSingleValued⟩ example : physicalWilczek1982BosonicCompositeReadout.exchangePhaseInPiUnits = 1 / 2 ∧ physicalWilczek1982FermionicCompositeReadout.exchangePhaseInPiUnits = 3 / 2 ∧ ¬PhysicalWilczekFermionicCompositeAt physicalWilczekFermionicExchangeSignErasedControl := by obtain ⟨_, _, bosonic, fermionic, _, sign, _⟩ := wilczek_underlying_fermion_adds_a_distinct_exchange_sign_and_spin exact ⟨bosonic, fermionic, sign⟩ example : physicalWilczek1982HarmonicWellReadout.twoAnyonLowestEnergy = 5 / 2 ∧ physicalWilczek1982HarmonicWellReadout.independentOneAnyonPairEnergy = 2 ∧ ¬PhysicalWilczekSeparatedHarmonicWellAt physicalWilczekNaiveIndependentHarmonicSpectrumControl := by obtain ⟨_, _, _, coupled, independent, notIndependent⟩ := wilczek_two_anyon_harmonic_spectrum_is_not_an_independent_particle_sum exact ⟨coupled, independent, notIndependent⟩ example : physicalWilczek1982ManyAnyonReadout.universalCoverRequiredForManyAnyons = true ∧ physicalWilczek1982ManyAnyonReadout.completeManyAnyonSolutionEstablished = false ∧ ¬PhysicalWilczekUnresolvedManyAnyonScopeAt physicalWilczekManyAnyonSolutionClaimedControl := by obtain ⟨_, cover, unsolved, _, notSolved, _⟩ := wilczek_universal_cover_does_not_solve_many_anyon_statistical_mechanics exact ⟨cover, unsolved, notSolved⟩ example : physicalArovas1984FractionalHallReadout.quasiholeCharge = 1 / 3 ∧ physicalArovas1984FractionalHallReadout.quasiparticleCharge = -(1 / 3) ∧ physicalArovas1984FractionalHallReadout.asymptoticScreeningCorrection = 0 ∧ ¬PhysicalArovasHallQuasiholeAt physicalArovasQuasiparticleWrongChargeSignControl := by obtain ⟨_, _, hole, particle, _, screening, _, _, _, wrongSign⟩ := arovas_screened_adiabatic_density_distinguishes_opposite_fractional_charges exact ⟨hole, particle, screening, wrongSign⟩ example : physicalArovas1984FractionalHallReadout.secondHoleOutsidePhaseInPiUnits = -4 ∧ physicalArovas1984FractionalHallReadout.secondHoleInsidePhaseInPiUnits = -(10 / 3) ∧ physicalArovas1984FractionalHallReadout.statisticalEncirclingPhaseInPiUnits = 2 / 3 ∧ physicalArovas1984FractionalHallReadout.exchangePhaseInPiUnits = 1 / 3 := by obtain ⟨_, _, outside, inside, winding, exchange, _, _, _⟩ := arovas_inside_outside_comparison_separates_background_and_mutual_statistics exact ⟨outside, inside, winding, exchange⟩ example : physicalArovas1984IntegerHallReadout.exchangePhaseInPiUnits = 1 ∧ physicalArovas1984IntegerHallReadout.fractionalStatisticsClassified = false ∧ ¬PhysicalArovasIntegerHallExchangeAt physicalArovasIntegerHallClaimedFractionalControl := by obtain ⟨_, _, _, integer, notFractional, falseClaim⟩ := arovas_integer_filling_recovers_fermionic_not_fractional_exchange exact ⟨integer, notFractional, falseClaim⟩ example : physicalArovas1984StatisticalPotentialReadout.statisticalPotentialX = -(4 / 25) ∧ physicalArovas1984StatisticalPotentialReadout.statisticalPotentialY = 3 / 25 ∧ physicalArovas1984StatisticalPotentialReadout.fermionicStatisticalFluxInPiUnits = -4 ∧ ¬PhysicalArovasStatisticalPotentialAt physicalArovasStatisticalGaugeDynamicsClaimedControl := by obtain ⟨_, x, y, _, fermionic, _, notDynamical, _, _⟩ := arovas_statistical_vector_potential_is_transverse_fictitious_and_convention_dependent exact ⟨x, y, fermionic, notDynamical⟩ example : physicalArovas1984PhysicalFluxReadout.physicalFluxInFluxQuanta = 3 / 2 ∧ physicalArovas1984PhysicalFluxReadout.accumulatedPhysicalFluxCharge = 1 / 2 ∧ ¬PhysicalArovasPhysicalFluxResponseAt physicalArovasPhysicalFluxRequiredIntegralControl := by obtain ⟨_, flux, charge, _, _, notIntegral, _⟩ := arovas_physical_flux_response_is_separate_and_not_integrally_quantized exact ⟨flux, charge, notIntegral⟩ example : physicalArovas1984ThirdQuasiholeReadout.exchangePhaseDependsOnPathWithThirdQuasihole = true ∧ physicalArovas1984ThirdQuasiholeReadout.pairPermutationClassificationSufficient = false ∧ ¬PhysicalArovasThirdQuasiholePathDependenceAt physicalArovasArbitraryCorrectionsClaimedProvedControl := by obtain ⟨_, path, permutation, _, _, _, _, _, corrections⟩ := arovas_third_quasihole_invalidates_pair_permutation_only_classification exact ⟨path, permutation, corrections⟩ example : physicalWilczek1982BosonicCompositeReadout.mechanism ≠ physicalArovas1984FractionalHallReadout.mechanism ∧ ¬PhysicalArovasHallQuasiholeAt physicalArovasPhysicalAttachedFluxRequiredControl := by obtain ⟨_, _, mechanisms, _, _, _, attached, _, _, _⟩ := charge_flux_and_hall_berry_exchange_are_distinct_source_mechanisms exact ⟨mechanisms, attached⟩ end PlanarExchangePhysicalReuseAudit namespace LatticeConfinementPhysicalReuseAudit open PhysicsLogic.Framework.LoopAreaConfinementContext.Countermodels example : physicalWilson1974StrongSurfaceReadout.minimalSurfacePlaquetteCount = 12 ∧ physicalWilson1974StrongSurfaceReadout.leadingInverseCouplingPower = 12 ∧ ¬PhysicalWilsonStrongCouplingAreaAt physicalWilsonLeadingContributionClaimedOnePlaquetteControl := by obtain ⟨_, plaquettes, power, _, _, _, wrongPower, _⟩ := wilson_minimal_surface_tiling_fixes_the_first_nonzero_strong_coupling_order exact ⟨plaquettes, power, wrongPower⟩ example : physicalWilson1974StrongSurfaceReadout.partitionFunctionNormalization = 2 ∧ physicalWilson1974StrongSurfaceReadout.disconnectedVacuumSurfacesCancelAgainstNormalization = true ∧ ¬PhysicalWilsonStrongCouplingAreaAt physicalWilsonDisconnectedVacuumSurfaceKeptControl := by obtain ⟨_, normalization, _, canceled, disconnected, _⟩ := wilson_partition_normalization_cancels_disconnected_vacuum_surfaces exact ⟨normalization, canceled, disconnected⟩ example : physicalWilson1974StrongSurfaceReadout.correctionExponentAreaPerimeterFormEstablished = false ∧ ¬PhysicalWilsonExpectedCorrectionAt physicalWilsonCorrectionExpectationClaimedProvedControl := by obtain ⟨_, _, _, _, unproved, falseClaim⟩ := wilson_area_correction_is_expected_not_established exact ⟨unproved, falseClaim⟩ example : physicalWilson1974StrongSurfaceReadout.dynamicalQuarkVacuumLoopsNeglected = true ∧ physicalWilson1974DynamicalScreeningReadout.dynamicalMatterScreensStaticString = true ∧ ¬PhysicalWilsonDynamicalMatterScreeningAt physicalWilsonStringBreakingClaimedImpossibleControl := by obtain ⟨_, _, quenched, screened, _, _, impossible, _⟩ := wilson_quenched_confinement_does_not_rule_out_dynamical_string_screening exact ⟨quenched, screened, impossible⟩ example : physicalWilson1974StrongSurfaceReadout.gaugeFixingRequired = false ∧ ¬PhysicalRegulatedGaugeLoopSetupAt physicalLoopGaugeInvarianceErasedControl := by obtain ⟨_, _, gaugeFixing, _, _, invariant⟩ := finite_cutoff_gauge_invariance_requires_separate_area_law_admissibility exact ⟨gaugeFixing, invariant⟩ example : physicalCreutz1980FiniteLoopFitReadout.latticeSideSites = 10 ∧ physicalCreutz1980FiniteLoopFitReadout.latticeSiteCount = 10000 ∧ physicalCreutz1980FiniteLoopFitReadout.fluctuationSampleCount = 5 ∧ ¬PhysicalCreutzValidatedFiniteAreaFitAt physicalCreutzHeatBathNotFullGroupControl := by obtain ⟨_, side, sites, _, samples, _, _, _, _, fullGroup, _, _, _⟩ := creutz_full_group_heat_bath_retains_independent_starts_and_finite_sample_error exact ⟨side, sites, samples, fullGroup⟩ example : physicalCreutz1980FiniteLoopFitReadout.fitPerimeterCoefficient = 1 / 20 ∧ physicalCreutz1980FiniteLoopFitReadout.fitAreaCoefficient = 1 / 8 ∧ physicalCreutz1980FiniteLoopFitReadout.fittedNegativeLogExpectation = 23 / 10 ∧ ¬PhysicalCreutzValidatedFiniteAreaFitAt physicalCreutzPerimeterFitTermErasedControl := by obtain ⟨_, _, perimeter, area, exponent, missingPerimeter, _, _⟩ := creutz_square_loop_fit_separates_constant_perimeter_and_area_terms exact ⟨perimeter, area, exponent, missingPerimeter⟩ example : physicalCreutz1980FiniteLoopFitReadout.latticeSpacing = 1 / 2 ∧ physicalCreutz1980FiniteLoopFitReadout.stringTension = 1 / 2 ∧ physicalCreutz1980FiniteLoopFitReadout.fitAreaCoefficient = 1 / 8 ∧ physicalCreutz1980FiniteLoopFitReadout.pureAreaExponent = 2 := by obtain ⟨_, spacing, tension, coefficient, _, exponent, _⟩ := creutz_area_fit_identifies_dimensionless_lattice_string_tension exact ⟨spacing, tension, coefficient, exponent⟩ example : physicalCreutz1980WeakCouplingFitReadout.latticeBeta = 3 ∧ physicalCreutz1980WeakCouplingFitReadout.accurateWeakCouplingTensionDetermination = false ∧ ¬PhysicalCreutzWeakCouplingFitLimitationAt physicalCreutzWeakAreaFitClaimedAccurateControl := by obtain ⟨_, coupling, _, _, unreliable, falselyAccurate, _⟩ := creutz_weak_coupling_perimeter_dominance_blocks_accurate_area_extraction exact ⟨coupling, unreliable, falselyAccurate⟩ example : physicalCreutz1980ContinuumEvidenceReadout.continuumLimitTaken = false ∧ physicalCreutz1980ContinuumEvidenceReadout.continuumConfinementEstablished = false ∧ ¬PhysicalCreutzContinuumEvidenceAt physicalCreutzContinuumConfinementClaimedControl := by obtain ⟨_, _, _, _, cutoff, confinement, _, _, _, falseClaim, _⟩ := creutz_crossover_and_scaling_evidence_do_not_prove_continuum_confinement exact ⟨cutoff, confinement, falseClaim⟩ example : physicalWilson1974StrongSurfaceReadout.derivation ≠ physicalCreutz1980FiniteLoopFitReadout.derivation ∧ physicalWilson1974StrongSurfaceReadout.gaugeGroup ≠ physicalCreutz1980FiniteLoopFitReadout.gaugeGroup := by obtain ⟨_, _, derivations, groups, _, _, _⟩ := strong_surface_expansion_and_su2_sampling_are_distinct_regulated_mechanisms exact ⟨derivations, groups⟩ example : physicalCreutz1980ContinuumEvidenceReadout.finiteCouplingPhaseTransitionObserved = false ∧ physicalCreutz1980ContinuumEvidenceReadout.confinementAndAsymptoticFreedomCoexistenceProved = false ∧ ¬PhysicalCreutzContinuumEvidenceAt physicalCreutzCoexistenceClaimedProvedControl := by obtain ⟨_, _, noTransition, _, _, _, coexistence, _, _, _, falseProof⟩ := creutz_crossover_and_scaling_evidence_do_not_prove_continuum_confinement exact ⟨noTransition, coexistence, falseProof⟩ end LatticeConfinementPhysicalReuseAudit namespace FiniteRegulatorGaugePhysicalReuseAudit open PhysicsLogic.Framework.FiniteRegulatorGaugeInvarianceContext.Countermodels example : physicalWilson1974CompactGaugeReadout.originalGaugePotential = 3 ∧ physicalWilson1974CompactGaugeReadout.transformedGaugePotential = 2 ∧ physicalWilson1974CompactGaugeReadout.originalPlaquettePhase = 1 ∧ physicalWilson1974CompactGaugeReadout.transformedPlaquettePhase = 1 ∧ ¬PhysicalWilsonEndpointPlaquetteGaugeAt physicalWilsonEndpointShiftErasedControl := by obtain ⟨_, before, after, originalPlaquette, transformedPlaquette, wrongEndpoint, _, _⟩ := wilson_endpoint_shift_changes_links_but_cancels_around_plaquettes exact ⟨before, after, originalPlaquette, transformedPlaquette, wrongEndpoint⟩ example : physicalWilson1974CompactGaugeReadout.latticeSpacing = 1 / 2 ∧ physicalWilson1974CompactGaugeReadout.gaugeCoupling = 2 ∧ physicalWilson1974CompactGaugeReadout.originalLinkAngle = 3 ∧ ¬PhysicalWilsonEndpointPlaquetteGaugeAt physicalWilsonLinkRescalingErasedControl := by obtain ⟨_, spacing, coupling, angle, erased⟩ := wilson_angular_link_retains_finite_spacing_coupling_rescaling exact ⟨spacing, coupling, angle, erased⟩ example : ¬PhysicalWilsonEndpointPlaquetteGaugeAt physicalWilsonPlaquetteOrientationErasedControl ∧ ¬PhysicalWilsonEndpointPlaquetteGaugeAt physicalWilsonPlaquetteInvarianceErasedControl := by obtain ⟨_, _, _, _, _, _, orientation, invariance⟩ := wilson_endpoint_shift_changes_links_but_cancels_around_plaquettes exact ⟨orientation, invariance⟩ example : physicalCreutz1980SU2GaugeReadout.reverseImaginary = -1 ∧ physicalCreutz1980SU2GaugeReadout.transformedLinkReal = -1 ∧ ¬PhysicalCreutzSU2EndpointGaugeAt physicalCreutzReverseInverseWrongControl := by obtain ⟨_, reverse, transformed, _, wrongInverse, _⟩ := creutz_su2_endpoint_multiplication_requires_reversed_link_inverse exact ⟨reverse, transformed, wrongInverse⟩ example : ¬PhysicalCreutzSU2EndpointGaugeAt physicalCreutzGaugeGroupWrongControl ∧ ¬PhysicalCreutzSU2EndpointGaugeAt physicalCreutzEndpointMultiplicationWrongControl := by obtain ⟨_, _, _, wrongGroup, _, wrongMultiplication⟩ := creutz_su2_endpoint_multiplication_requires_reversed_link_inverse exact ⟨wrongGroup, wrongMultiplication⟩ example : physicalCreutz1980SU2GaugeReadout.openTraceBefore = 0 ∧ physicalCreutz1980SU2GaugeReadout.openTraceAfter = -2 ∧ physicalCreutz1980SU2GaugeReadout.closedTraceBefore = 0 ∧ physicalCreutz1980SU2GaugeReadout.closedTraceAfter = 0 ∧ ¬PhysicalFiniteGaugeInvariantAt physicalOpenLinkClaimedInvariantControl := by obtain ⟨_, before, after, closedBefore, closedAfter, _, wrongOpen⟩ := creutz_closed_trace_is_invariant_but_open_link_trace_is_not exact ⟨before, after, closedBefore, closedAfter, wrongOpen⟩ example : PhysicalWilsonEndpointPlaquetteGaugeAt physicalWilson1974CompactGaugeReadout ∧ PhysicalCreutzSU2EndpointGaugeAt physicalCreutz1980SU2GaugeReadout ∧ physicalCreutz1980SU2GaugeReadout.actionBefore = 1 ∧ physicalCreutz1980SU2GaugeReadout.actionAfter = 1 ∧ ¬PhysicalFiniteGaugeInvariantAt physicalMissingFiniteCutoffControl := by obtain ⟨wilson, creutz, before, after, missingCutoff, _, _, _⟩ := compact_haar_invariance_needs_no_gauge_fixing_or_open_link_invariance exact ⟨wilson, creutz, before, after, missingCutoff⟩ example : ¬PhysicalFiniteGaugeInvariantAt physicalNoncompactHaarControl ∧ ¬PhysicalFiniteGaugeInvariantAt physicalRequiredGaugeFixingControl ∧ ¬PhysicalCreutzSU2EndpointGaugeAt physicalCreutzActionInvarianceWrongControl := by obtain ⟨_, _, _, _, _, noncompact, fixing, wrongAction⟩ := compact_haar_invariance_needs_no_gauge_fixing_or_open_link_invariance exact ⟨noncompact, fixing, wrongAction⟩ example : physicalCreutz1980SU2GaugeReadout.unfixedIterations = 20 ∧ physicalCreutz1980SU2GaugeReadout.gaugeFixedIterations = 40 ∧ ¬PhysicalCreutzUnfixedSamplingAt physicalCreutzGaugeFixingFasterControl := by obtain ⟨_, unfixed, fixed, _, faster⟩ := creutz_optional_gauge_fixing_slows_conditional_heat_bath_sampling exact ⟨unfixed, fixed, faster⟩ example : ¬PhysicalCreutzUnfixedSamplingAt physicalCreutzGaugeFixingUsedControl := by obtain ⟨_, _, _, fixingUsed, _⟩ := creutz_optional_gauge_fixing_slows_conditional_heat_bath_sampling exact fixingUsed example : physicalCreutz1980SU2GaugeReadout.orbitRepresentatives = 2 ∧ physicalIncompleteFiniteGaugeOrbitReadout.physicalValueRepresentativeIndependent = false ∧ ¬PhysicalIncompleteFiniteGaugeOrbitAt physicalIncompleteOrbitClaimedPhysicalControl := by obtain ⟨_, _, representatives, incomplete, _, _, wrongPhysical, _⟩ := representative_independence_requires_complete_selected_gauge_orbit exact ⟨representatives, incomplete, wrongPhysical⟩ example : ¬PhysicalFiniteGaugeSelectedOrbitAt physicalOrbitRepresentativeMissingControl ∧ ¬PhysicalFiniteGaugeSelectedOrbitAt physicalOrbitCompletenessMissingControl := by obtain ⟨_, _, _, _, missingRepresentative, missingCompleteness, _, _⟩ := representative_independence_requires_complete_selected_gauge_orbit exact ⟨missingRepresentative, missingCompleteness⟩ example : physicalCreutz1980SU2GaugeReadout.globalOrbitClassificationProved = false ∧ ¬PhysicalFiniteGaugeSelectedOrbitAt physicalSelectedOrbitClaimedGlobalControl := by obtain ⟨_, _, _, _, _, _, _, claimedGlobal⟩ := representative_independence_requires_complete_selected_gauge_orbit exact ⟨rfl, claimedGlobal⟩ example : physicalWilson1974CompactGaugeReadout.gaugeGroup ≠ physicalCreutz1980SU2GaugeReadout.gaugeGroup ∧ physicalWilson1974CompactGaugeReadout.mechanism ≠ physicalCreutz1980SU2GaugeReadout.mechanism := by obtain ⟨_, _, groups, mechanisms⟩ := additive_u1_and_multiplicative_su2_transformations_remain_distinct exact ⟨groups, mechanisms⟩ example : physicalCreutz1980SU2GaugeReadout.areaLawDerived = false ∧ physicalCreutz1980SU2GaugeReadout.continuumLorentzInvarianceEstablished = false ∧ physicalCreutz1980SU2GaugeReadout.continuumExistenceProved = false := by obtain ⟨_, area, lorentz, continuum, _, _⟩ := finite_regulated_gauge_symmetry_does_not_prove_area_law_or_continuum exact ⟨area, lorentz, continuum⟩ end FiniteRegulatorGaugePhysicalReuseAudit namespace GeneralizedEntropyPhysicalReuseAudit open PhysicsLogic.Framework.GeneralizedEntropyBalanceContext.Countermodels example : PhysicalControlledGeneralizedEntropyBalanceAt physicalBekenstein1973CaptureReadout ∧ ¬PhysicalControlledGeneralizedEntropyBalanceAt physicalExteriorEntropyAccountMissingControl ∧ ¬PhysicalControlledGeneralizedEntropyBalanceAt physicalHorizonEntropyAccountMissingControl := by obtain ⟨controlled, exterior, horizon, _, _⟩ := generalized_entropy_requires_defined_accounts_horizon_exchange_and_statistical_scope exact ⟨controlled, exterior, horizon⟩ example : ¬PhysicalControlledGeneralizedEntropyBalanceAt physicalHorizonEntropyExchangeMissingControl ∧ ¬PhysicalControlledGeneralizedEntropyBalanceAt physicalStatisticalScopeErasedControl := by obtain ⟨_, _, _, exchange, statistical⟩ := generalized_entropy_requires_defined_accounts_horizon_exchange_and_statistical_scope exact ⟨exchange, statistical⟩ example : physicalBekenstein1973CaptureReadout.particleRestMass = 1 ∧ physicalBekenstein1973CaptureReadout.particleProperRadius = 1 ∧ physicalBekenstein1973CaptureReadout.minimumRationalizedAreaIncrease = 2 ∧ ¬PhysicalBekensteinCaptureBalanceAt physicalBekensteinRadiusErasedControl := by obtain ⟨_, mass, radius, area, radiusErased, _⟩ := bekenstein_finite_radius_capture_fixes_minimum_rationalized_area_increase exact ⟨mass, radius, area, radiusErased⟩ example : physicalBekenstein1973CaptureReadout.areaEntropyCoefficient = (1 / 2 : Real) * Real.log 2 ∧ physicalBekenstein1973CaptureReadout.areaCoefficientExactlyEstablished = false ∧ ¬PhysicalBekensteinCaptureBalanceAt physicalBekensteinCoefficientClaimedExactControl := by obtain ⟨_, coefficient, _, approximate, _, claimedExact, _⟩ := bekenstein_one_bit_matching_is_proposed_not_exact_quarter_normalization exact ⟨coefficient, approximate, claimedExact⟩ example : physicalBekenstein1973CaptureReadout.oneBitEntropy = Real.log 2 ∧ ¬PhysicalBekensteinCaptureBalanceAt physicalBekensteinAreaConventionWrongControl ∧ ¬PhysicalBekensteinCaptureBalanceAt physicalBekensteinOneBitErasedControl := by obtain ⟨_, _, bit, _, wrongConvention, _, erased⟩ := bekenstein_one_bit_matching_is_proposed_not_exact_quarter_normalization exact ⟨bit, wrongConvention, erased⟩ example : physicalBekenstein1973CaptureReadout.characteristicTemperature = 1 ∧ physicalBekenstein1973CaptureReadout.radiationTemperature = 0 ∧ ¬PhysicalBekensteinCaptureBalanceAt physicalBekensteinCharacteristicClaimedRadiatingControl := by obtain ⟨_, characteristic, radiating, _, wrongTemperature⟩ := bekenstein_characteristic_temperature_is_not_a_radiating_temperature exact ⟨characteristic, radiating, wrongTemperature⟩ example : physicalBekenstein1973CaptureReadout.exteriorEntropyChange = -Real.log 2 ∧ physicalBekenstein1973CaptureReadout.blackHoleEntropyChange = Real.log 2 ∧ physicalBekenstein1973CaptureReadout.generalizedEntropyChange = 0 := by obtain ⟨_, exterior, horizon, total, _⟩ := bekenstein_capture_lowers_exterior_entropy_while_horizon_entropy_compensates exact ⟨exterior, horizon, total⟩ example : physicalBekenstein1973FluctuationReadout.generalizedEntropyChange = -Real.log 2 ∧ ¬PhysicalBekensteinFluctuationLimitationAt physicalFluctuationClaimedNegligibleControl ∧ ¬PhysicalControlledGeneralizedEntropyBalanceAt physicalGeneralizedLawClaimedAbsoluteControl := by obtain ⟨_, fluctuation, claimedNegligible, _, absolute⟩ := bekenstein_fluctuation_dominated_process_is_outside_statistical_generalized_law exact ⟨fluctuation, claimedNegligible, absolute⟩ example : physicalHawking1975EmissionReadout.surfaceGravity = 2 * Real.pi ∧ physicalHawking1975EmissionReadout.radiationTemperature = 1 ∧ ¬PhysicalHawkingThermalEmissionBalanceAt physicalHawkingTemperatureNormalizationWrongControl := by obtain ⟨_, gravity, temperature, wrongNormalization⟩ := hawking_surface_gravity_fixes_physical_radiation_temperature exact ⟨gravity, temperature, wrongNormalization⟩ example : physicalHawking1975EmissionReadout.greybodyAbsorptionFraction = 1 / 2 ∧ physicalHawking1975EmissionReadout.thermalOccupation = (1 / 2 : Real) / (Real.exp 1 - 1) ∧ ¬PhysicalHawkingThermalEmissionBalanceAt physicalHawkingGreybodyErasedControl := by obtain ⟨_, greybody, occupation, erased, _⟩ := hawking_greybody_factor_modifies_bosonic_thermal_occupation exact ⟨greybody, occupation, erased⟩ example : ¬PhysicalHawkingThermalEmissionBalanceAt physicalHawkingOccupationWrongControl := by obtain ⟨_, _, _, _, wrongOccupation⟩ := hawking_greybody_factor_modifies_bosonic_thermal_occupation exact wrongOccupation example : physicalHawking1975EmissionReadout.horizonAreaBefore = 8 ∧ physicalHawking1975EmissionReadout.horizonAreaAfter = 4 ∧ physicalHawking1975EmissionReadout.blackHoleEntropyChange = -1 ∧ physicalHawking1975EmissionReadout.inwardEnergyFlux = -1 := by obtain ⟨_, before, after, entropy, inward, _, _, _, _⟩ := hawking_quarter_area_entropy_decreases_under_negative_horizon_flux exact ⟨before, after, entropy, inward⟩ example : ¬PhysicalHawkingThermalEmissionBalanceAt physicalHawkingAreaCoefficientWrongControl ∧ ¬PhysicalHawkingThermalEmissionBalanceAt physicalHawkingAreaIncreaseClaimedControl ∧ ¬PhysicalHawkingThermalEmissionBalanceAt physicalHawkingClassicalAreaTheoremMisappliedControl := by obtain ⟨_, _, _, _, _, coefficient, area, _, classical⟩ := hawking_quarter_area_entropy_decreases_under_negative_horizon_flux exact ⟨coefficient, area, classical⟩ example : physicalHawking1975EmissionReadout.exteriorEntropyChange = 2 ∧ physicalHawking1975EmissionReadout.blackHoleEntropyChange = -1 ∧ physicalHawking1975EmissionReadout.generalizedEntropyChange = 1 ∧ ¬PhysicalHawkingThermalEmissionBalanceAt physicalHawkingExteriorEntropySignWrongControl := by obtain ⟨_, exterior, horizon, total, wrongSign⟩ := hawking_exterior_radiation_entropy_compensates_shrinking_horizon exact ⟨exterior, horizon, total, wrongSign⟩ example : physicalHawking1975PlanckEndpointReadout.massAfter = 1 ∧ physicalHawking1975PlanckEndpointReadout.evaporationEndpointExpected = true ∧ physicalHawking1975PlanckEndpointReadout.planckEndpointResolved = false ∧ ¬PhysicalHawkingPlanckEndpointLimitationAt physicalHawkingEvaporationClaimedProvedControl := by obtain ⟨_, mass, unresolved, expected, _, _, proof⟩ := hawking_semiclassical_endpoint_breakdown_does_not_resolve_evaporation exact ⟨mass, expected, unresolved, proof⟩ example : ¬PhysicalHawkingPlanckEndpointLimitationAt physicalHawkingPlanckRegimeClaimedControlledControl ∧ ¬PhysicalHawkingPlanckEndpointLimitationAt physicalHawkingPlanckEndpointClaimedResolvedControl := by obtain ⟨_, _, _, _, controlled, resolved, _⟩ := hawking_semiclassical_endpoint_breakdown_does_not_resolve_evaporation exact ⟨controlled, resolved⟩ example : ¬PhysicalHawkingThermalEmissionBalanceAt physicalHawkingPairPictureClaimedDerivationControl ∧ ¬PhysicalControlledGeneralizedEntropyBalanceAt physicalGeneralizedEntropyUnitarityClaimedControl ∧ ¬PhysicalControlledGeneralizedEntropyBalanceAt physicalGeneralizedEntropyMicrostatesClaimedControl := by obtain ⟨_, pairPicture, unitarity, microstates⟩ := horizon_pair_picture_and_unitary_endpoint_do_not_follow_from_thermal_balance exact ⟨pairPicture, unitarity, microstates⟩ example : physicalBekenstein1973CaptureReadout.mechanism ≠ physicalHawking1975EmissionReadout.mechanism ∧ physicalBekenstein1973CaptureReadout.areaConvention ≠ physicalHawking1975EmissionReadout.areaConvention ∧ physicalBekenstein1973CaptureReadout.exteriorEntropyChange < 0 ∧ 0 < physicalHawking1975EmissionReadout.exteriorEntropyChange := by obtain ⟨_, _, mechanism, convention, capture, radiation⟩ := capture_information_and_steady_thermal_emission_are_distinct_balance_mechanisms exact ⟨mechanism, convention, capture, radiation⟩ end GeneralizedEntropyPhysicalReuseAudit namespace QuantumErrorRecoveryOriginalPaperCompositionAudit example {CodeTypes : PhysicsLogic.Papers.CalderbankShor1996.CalderbankShorTypes} {ReplicationTypes : PhysicsLogic.Papers.WoottersZurek1982.WoottersZurekTypes} (recovery : PhysicsLogic.Papers.CalderbankShor1996.QuantumCodeFramework CodeTypes) (replication : PhysicsLogic.Papers.WoottersZurek1982.WoottersZurekFramework ReplicationTypes) (logicalMatches : CodeTypes.LogicalState -> ReplicationTypes.State -> Prop) (distributed : CodeTypes.QuantumCode -> CodeTypes.LogicalState -> CodeTypes.EncodedState -> Prop) (encodedCopy : CodeTypes.EncodedState -> ReplicationTypes.State -> Prop) (recoveredCopy : CodeTypes.Recovery -> CodeTypes.Recovery -> CodeTypes.Error -> CodeTypes.EncodedState -> ReplicationTypes.State -> Prop) (codeSetup : recovery.QuantumCodeSetup) (codeCalculation : recovery.TwoBasisCodeCalculation) (decoherence : recovery.DeclaredSupportedDecoherence) (bitCalculation : recovery.BitSyndromeCalculation) (phaseCalculation : recovery.PhaseSyndromeCalculation) (recoveryCalculation : recovery.ArbitraryDecoherenceRecoveryCalculation) (replicationSetup : replication.PolarizationAmplifierSetup) (outputUnique : replication.UnitaryOutputUniqueness) (linearCalculation : replication.LinearSuperpositionOutputCalculation) (apparatusCases : replication.ApparatusRecordCaseCalculation) (requiredClone : replication.RequiredCloneOutputCalculation) (outputMismatch : replication.OutputMismatchCalculation) (stateAlignment : logicalMatches recovery.logicalState replication.sharedReplicationModel.superposedState) (distributedEncoding : distributed recovery.quantumCode recovery.logicalState recovery.encodedState) (encodedNoCopy : ¬encodedCopy recovery.encodedState replication.sharedReplicationModel.superposedState) (recoveredNoCopy : ¬recoveredCopy recovery.bitRecovery recovery.phaseRecovery recovery.selectedDecoherence recovery.encodedState replication.sharedReplicationModel.superposedState) : (recovery.nestedCodeNoCloningRecoveryHandoff replication.toLinearStateReplicationContext logicalMatches distributed encodedCopy recoveredCopy) |>.SourceSensitiveNoCloningExactRecoveryCompositionAt recovery.toExactQuantumErrorRecoveryContext replication.toLinearStateReplicationContext .nestedCodesFourierBasisDecodingAndAncillaFactorization recovery.calderbankShorRecoveryDerivationObtains .polarizationLinearityAndCloneMismatch replication.WoottersZurekReplicationDerivationObtains .nestedDualCodeDistributedEncodingAndAncillaOnlySyndromes (recovery.calderbankShorEncodingCompatibilityDerivationObtains (recovery.nestedCodeNoCloningRecoveryHandoff replication.toLinearStateReplicationContext logicalMatches distributed encodedCopy recoveredCopy)) := recovery |>.nested_code_recovery_and_independent_no_cloning_realize_source_sensitive_compatibility replication.toLinearStateReplicationContext logicalMatches distributed encodedCopy recoveredCopy PhysicsLogic.Papers.WoottersZurek1982.WoottersZurekReplicationDerivation.polarizationLinearityAndCloneMismatch replication.WoottersZurekReplicationDerivationObtains (replication |>.polarization_argument_exhibits_nonvacuous_shared_replication_obstruction replicationSetup outputUnique linearCalculation apparatusCases requiredClone outputMismatch) (replication |>.polarization_source_inputs_realize_structural_replication_obstruction replicationSetup outputUnique linearCalculation apparatusCases requiredClone outputMismatch) codeSetup codeCalculation decoherence bitCalculation phaseCalculation recoveryCalculation stateAlignment distributedEncoding encodedNoCopy recoveredNoCopy example {CodeTypes : PhysicsLogic.Papers.Steane1996.SteaneTypes} {ReplicationTypes : PhysicsLogic.Papers.Dieks1982.DieksTypes} (recovery : PhysicsLogic.Papers.Steane1996.QuantumComputerFramework CodeTypes) (replication : PhysicsLogic.Papers.Dieks1982.DieksFramework ReplicationTypes) (logicalMatches : CodeTypes.LogicalState -> ReplicationTypes.State -> Prop) (distributed : CodeTypes.QuantumCode -> CodeTypes.LogicalState -> CodeTypes.EncodedState -> Prop) (encodedCopy : CodeTypes.EncodedState -> ReplicationTypes.State -> Prop) (recoveredCopy : CodeTypes.Recovery -> CodeTypes.Recovery -> CodeTypes.Error -> CodeTypes.EncodedState -> ReplicationTypes.State -> Prop) (codeSetup : recovery.CodeConstructionSetup) (constructionCalculation : recovery.CosetCodeConstructionCalculation) (defection : recovery.DeclaredSupportedDefection) (basisOneCalculation : recovery.BasisOneRecoveryCalculation) (basisTwoCalculation : recovery.BasisTwoRecoveryCalculation) (restorationCalculation : recovery.ExactRestorationCalculation) (replicationSetup : replication.XBasisMultiplierSetup) (outputUnique : replication.UnitaryOutputUniqueness) (linearCalculation : replication.ZStateLinearityCalculation) (requiredClone : replication.RequiredZCloneCalculation) (outputMismatch : replication.LinearAndRequiredBurstMismatchCalculation) (stateAlignment : logicalMatches recovery.logicalState replication.sharedReplicationModel.superposedState) (distributedEncoding : distributed recovery.quantumCode recovery.logicalState recovery.encodedState) (encodedNoCopy : ¬encodedCopy recovery.encodedState replication.sharedReplicationModel.superposedState) (recoveredNoCopy : ¬recoveredCopy recovery.basisOneRecovery recovery.basisTwoRecovery recovery.selectedDefection recovery.encodedState replication.sharedReplicationModel.superposedState) : (recovery.cosetCodeNoCloningRecoveryHandoff replication.toLinearStateReplicationContext logicalMatches distributed encodedCopy recoveredCopy) |>.SourceSensitiveNoCloningExactRecoveryCompositionAt recovery.toExactQuantumErrorRecoveryContext replication.toLinearStateReplicationContext .cosetParityChecksTwoBasisCorrectionAndFinalDisentanglement recovery.steaneRecoveryDerivationObtains .multiplierLinearityAndRequiredBurstMismatch replication.DieksReplicationDerivationObtains .cosetParityEncodingAndApparatusOnlySyndromes (recovery.steaneEncodingCompatibilityDerivationObtains (recovery.cosetCodeNoCloningRecoveryHandoff replication.toLinearStateReplicationContext logicalMatches distributed encodedCopy recoveredCopy)) := recovery |>.coset_recovery_and_independent_no_cloning_realize_source_sensitive_compatibility replication.toLinearStateReplicationContext logicalMatches distributed encodedCopy recoveredCopy PhysicsLogic.Papers.Dieks1982.DieksReplicationDerivation.multiplierLinearityAndRequiredBurstMismatch replication.DieksReplicationDerivationObtains (replication |>.multiplier_argument_exhibits_nonvacuous_shared_replication_obstruction replicationSetup outputUnique linearCalculation requiredClone outputMismatch) (replication |>.multiplier_source_inputs_realize_structural_replication_obstruction replicationSetup outputUnique linearCalculation requiredClone outputMismatch) codeSetup constructionCalculation defection basisOneCalculation basisTwoCalculation restorationCalculation stateAlignment distributedEncoding encodedNoCopy recoveredNoCopy open PhysicsLogic.Framework.NoCloningExactRecoveryHandoff.Countermodels example : physicalNoCloningDistributedRecoveryHandoff |>.NonvacuousNoCloningExactRecoveryCompositionAt := distributed_exact_recovery_and_no_cloning_have_nonvacuous_composition example : (independentlyCopiedEncodedStateHandoff.recoveryModel |>.NonvacuousExactQuantumErrorRecoveryAt) /\ (independentlyCopiedEncodedStateHandoff.replicationModel |>.NonvacuousLinearStateReplicationObstructionAt) /\ (independentlyCopiedEncodedStateHandoff |>.encodedStateContainsIndependentCopyAt .nestedBlock .polarizedSuperposition) /\ ¬(independentlyCopiedEncodedStateHandoff |>.SelectedNoCloningExactRecoveryCompatibilityAt) := exact_recovery_and_no_cloning_do_not_license_independent_encoded_copy example : (physicalNoCloningDistributedRecoveryHandoff |>.NonvacuousNoCloningExactRecoveryCompositionAt) /\ (PhysicsLogic.Framework.ExactQuantumErrorRecoveryContext.Countermodels.physicalExactRecoveryModel |>.DirectSourceMechanisticExactRecoveryExplanationAt () availableNoCloningRecoveryChildDerivationObtains) /\ (PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.physicalPolarizationReplicationModel |>.DirectSourceStructuralLinearReplicationObstructionExplanationAt () availableNoCloningRecoveryChildDerivationObtains) /\ ¬(physicalNoCloningDistributedRecoveryHandoff |>.SourceSensitiveNoCloningExactRecoveryCompositionAt PhysicsLogic.Framework.ExactQuantumErrorRecoveryContext.Countermodels.physicalExactRecoveryContext PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels.physicalPolarizationReplicationContext () availableNoCloningRecoveryChildDerivationObtains () availableNoCloningRecoveryChildDerivationObtains .absent missingNoCloningRecoveryHandoffDerivationObtains) := completed_no_cloning_recovery_and_child_sources_without_encoding_derivation_have_no_source_sensitive_composition end QuantumErrorRecoveryOriginalPaperCompositionAudit namespace QuantumErrorRecoveryNoCloningPhysicalReuseAudit open PhysicsLogic.Framework.ExactQuantumErrorRecoveryContext.Countermodels example : NonvacuousNoCloningCompatibleExactRecoveryAt physicalCalderbankNoCloningCompatibleRecoveryHandoff ∧ NonvacuousNoCloningCompatibleExactRecoveryAt physicalSteaneNoCloningCompatibleRecoveryHandoff := ⟨physical_calderbank_no_cloning_compatible_recovery_is_nonvacuous, physical_steane_no_cloning_compatible_recovery_is_nonvacuous⟩ example : ¬physicalCalderbankNoCloningCompatibleRecoveryHandoff.replicationModel.ArbitraryStateReplicator := physical_calderbank_no_cloning_compatible_recovery_is_nonvacuous.2.1.2.2.2.2 example : physicalCalderbankNoCloningCompatibleRecoveryHandoff.replicationModel.NonvacuousLinearStateReplicationObstructionAt ∧ physicalCalderbankNoCloningCompatibleRecoveryHandoff.encodingSpreadsUnknownState = true ∧ physicalCalderbankNoCloningCompatibleRecoveryHandoff.recoveryProducesUniversalClone = false := composed_quantum_recovery_retains_no_cloning_and_noncopying_encoding physicalCalderbankNoCloningCompatibleRecoveryHandoff physical_calderbank_no_cloning_compatible_recovery_is_nonvacuous example : ¬NonvacuousNoCloningCompatibleExactRecoveryAt physicalDuplicatingRecoveryHandoff ∧ ¬NonvacuousNoCloningCompatibleExactRecoveryAt physicalRevealingRecoveryHandoff := by obtain ⟨_, _, _, _, duplicated, revealed⟩ := no_cloning_obstruction_coexists_with_exact_encoded_recovery exact ⟨duplicated, revealed⟩ example : physicalCalderbankShor1996CodeReadout.nestedCodeInclusion = true ∧ physicalCalderbankShor1996CodeReadout.dualCodeOrthogonality = true := by obtain ⟨_, nested, dual, _, _⟩ := nested_css_inclusion_and_dual_orthogonality_are_indispensable exact ⟨nested, dual⟩ example : ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecNestedInclusionMissingControl ∧ ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecDualOrthogonalityMissingControl := by obtain ⟨_, _, _, nested, dual⟩ := nested_css_inclusion_and_dual_orthogonality_are_indispensable exact ⟨nested, dual⟩ example : physicalCalderbankShor1996CodeReadout.blockLength = 7 ∧ physicalCalderbankShor1996CodeReadout.logicalQubitCount = 1 ∧ physicalCalderbankShor1996CodeReadout.containingCodeDimension = 4 ∧ physicalCalderbankShor1996CodeReadout.subcodeDimension = 3 ∧ physicalCalderbankShor1996CodeReadout.computationalCodewordTermCount = 16 := by obtain ⟨_, length, logical, containing, nested, terms, _, _⟩ := nested_css_dimensions_dual_cosets_and_fourier_codewords_are_distinct exact ⟨length, logical, containing, nested, terms⟩ example : ¬PhysicalCalderbankNestedDualCodeAt physicalQecNestedSubcodeDimensionWrongControl ∧ ¬PhysicalCalderbankNestedDualCodeAt physicalQecFourierBasisErasedControl := by obtain ⟨_, _, _, _, _, _, dimension, fourier⟩ := nested_css_dimensions_dual_cosets_and_fourier_codewords_are_distinct exact ⟨dimension, fourier⟩ example : physicalCalderbankShor1996CodeReadout.logicalCosetCount = 2 ∧ physicalCalderbankShor1996CodeReadout.ambientHilbertDimension = 128 ∧ physicalCalderbankShor1996CodeReadout.quantumCodeRate = (1 / 7 : ℚ) := by obtain ⟨_, cosets, dimension, rate, _, _, _, _⟩ := seven_qubit_code_retains_two_logical_cosets_and_one_seventh_rate exact ⟨cosets, dimension, rate⟩ example : ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecLogicalCosetCountWrongControl ∧ ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecLogicalDimensionWrongControl := by obtain ⟨_, _, _, _, cosets, logical, _, _⟩ := seven_qubit_code_retains_two_logical_cosets_and_one_seventh_rate exact ⟨cosets, logical⟩ example : ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecPhysicalDimensionWrongControl ∧ ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecCodeRateWrongControl := by obtain ⟨_, _, _, _, _, _, physical, rate⟩ := seven_qubit_code_retains_two_logical_cosets_and_one_seventh_rate exact ⟨physical, rate⟩ example : physicalCalderbankShor1996CodeReadout.firstMinimumDistance = 3 ∧ physicalCalderbankShor1996CodeReadout.correctableQubitCount = 1 := by obtain ⟨_, distance, errors, _, _, _⟩ := distance_three_corrects_one_supported_coherent_qubit_not_two exact ⟨distance, errors⟩ example : ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecTooManyErrorsClaimedControl ∧ ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecDistanceConditionWrongControl ∧ ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecUnboundedErrorsClaimedControl := by obtain ⟨_, _, _, errors, distance, unbounded⟩ := distance_three_corrects_one_supported_coherent_qubit_not_two exact ⟨errors, distance, unbounded⟩ example : physicalCalderbankShor1996CodeReadout.amplitudeSyndromeBits = 3 ∧ physicalCalderbankShor1996CodeReadout.phaseSyndromeBits = 3 ∧ physicalCalderbankShor1996CodeReadout.amplitudeSyndromeCount = 8 ∧ physicalCalderbankShor1996CodeReadout.phaseSyndromeCount = 8 := by obtain ⟨_, amplitude, phase, _, _, _, _⟩ := complementary_syndromes_identify_errors_without_learning_logical_state exact ⟨amplitude, phase, rfl, rfl⟩ example : ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecAmplitudeSyndromeRevealsStateControl ∧ ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecPhaseSyndromeRevealsStateControl := by obtain ⟨_, _, _, amplitude, phase, _, _⟩ := complementary_syndromes_identify_errors_without_learning_logical_state exact ⟨amplitude, phase⟩ example : ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecAmplitudeRecoveryMissingControl ∧ ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecPhaseRecoveryMissingControl := by obtain ⟨_, _, _, _, _, amplitude, phase⟩ := complementary_syndromes_identify_errors_without_learning_logical_state exact ⟨amplitude, phase⟩ example : physicalCalderbankShor1996CodeReadout.environmentMayInitiallyBeEntangled = true ∧ physicalCalderbankShor1996CodeReadout.recoveredStateIndependentOfEnvironment = true ∧ physicalCalderbankShor1996CodeReadout.unknownStateDuplicated = false := by obtain ⟨_, initially, decoupled, noClone, _, _, _⟩ := supported_recovery_decouples_environment_without_universal_cloning exact ⟨initially, decoupled, noClone⟩ example : ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecEnvironmentNotDecoupledControl ∧ ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecUnknownStateClonedControl ∧ ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecNoCloningForbidsRecoveryControl := by obtain ⟨_, _, _, _, environment, cloned, prohibited⟩ := supported_recovery_decouples_environment_without_universal_cloning exact ⟨environment, cloned, prohibited⟩ example : physicalSteane1996CodeReadout.firstClassicalCodeDimension = 4 ∧ physicalSteane1996CodeReadout.secondClassicalCodeDimension = 4 ∧ physicalSteane1996CodeReadout.computationalCodewordTermCount = 8 := by obtain ⟨_, first, second, terms, _, _⟩ := steane_related_code_dimensions_and_complementary_coset_terms_are_distinct exact ⟨first, second, terms⟩ example : ¬PhysicalSteaneDualCosetCodeAt physicalQecSteaneDimensionIdentityWrongControl ∧ ¬PhysicalSteaneDualCosetCodeAt physicalQecSteaneCosetTermsWrongControl := by obtain ⟨_, _, _, _, identity, terms⟩ := steane_related_code_dimensions_and_complementary_coset_terms_are_distinct exact ⟨identity, terms⟩ example : physicalSteane1996CodeReadout.theoremFiveCosetConstructionUsed = true ∧ physicalSteane1996CodeReadout.theoremSixEnvironmentDisentanglementUsed = true := by obtain ⟨_, fifth, sixth, _, _⟩ := steane_two_basis_theorems_restore_environment_coupled_logical_states exact ⟨fifth, sixth⟩ example : ¬PhysicalSteaneDualCosetCodeAt physicalQecSteaneTheoremFiveErasedControl ∧ ¬PhysicalSteaneDualCosetCodeAt physicalQecSteaneTheoremSixErasedControl := by obtain ⟨_, _, _, fifth, sixth⟩ := steane_two_basis_theorems_restore_environment_coupled_logical_states exact ⟨fifth, sixth⟩ example : physicalSteane1996CodeReadout.unequalDistanceExistenceConjectured = true ∧ physicalSteane1996CodeReadout.unequalDistanceExistenceProved = false ∧ physicalSteane1996CodeReadout.asymptoticEqualDistanceRateEstablished = false := by obtain ⟨_, conjectured, unproved, conditional, _, _⟩ := steane_unequal_distance_existence_remains_conditional exact ⟨conjectured, unproved, conditional⟩ example : ¬PhysicalSteaneDualCosetCodeAt physicalQecSteaneUnequalDistanceClaimedProvedControl ∧ ¬PhysicalSteaneDualCosetCodeAt physicalQecSteaneConditionalRateClaimedEstablishedControl := by obtain ⟨_, _, _, _, proved, rate⟩ := steane_unequal_distance_existence_remains_conditional exact ⟨proved, rate⟩ example : physicalCalderbankShor1996CodeReadout.faultTolerantComputationEstablished = false ∧ physicalCalderbankShor1996CodeReadout.channelCapacitySaturated = false ∧ ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecFaultToleranceClaimedControl ∧ ¬PhysicalBoundedNoCloningCompatibleRecoveryAt physicalQecChannelCapacityClaimedControl := by obtain ⟨_, faultTolerance, capacity, faultControl, capacityControl⟩ := finite_exact_recovery_does_not_establish_fault_tolerance_or_capacity exact ⟨faultTolerance, capacity, faultControl, capacityControl⟩ example : physicalCalderbankShor1996CodeReadout.construction ≠ physicalSteane1996CodeReadout.construction ∧ physicalCalderbankShor1996CodeReadout.computationalCodewordTermCount = 16 ∧ physicalSteane1996CodeReadout.computationalCodewordTermCount = 8 := by obtain ⟨_, _, construction, calderbank, steane⟩ := nested_fourier_and_steane_coset_parity_recovery_are_distinct_mechanisms exact ⟨construction, calderbank, steane⟩ example : physicalQuantumCodePossibility.TentativeAt physicalQuantumCodeEstablishment .unequalDistanceExistenceUnconditionallyProved .steane1996CosetParityChecks := by obtain ⟨_, _, _, _, _, unequal, _, _⟩ := exact_quantum_recovery_source_overclaims_remain_epistemically_open exact unequal example : physicalQuantumCodePossibility.TentativeAt physicalQuantumCodeEstablishment .noCloningForbidsQuantumErrorCorrection .calderbankShor1996NestedDualCodes := by obtain ⟨noCloning, _, _, _, _, _, _, _⟩ := exact_quantum_recovery_source_overclaims_remain_epistemically_open exact noCloning example : physicalCalderbankPrintedHammingCodewords.length = 16 ∧ physicalCalderbankPrintedHammingCodewords.Nodup := physical_calderbank_printed_hamming_table_has_sixteen_distinct_words example : physicalCssParityCheckRow .calderbankShor1996NestedDualCodes .first ≠ physicalCssParityCheckRow .steane1996CosetParityChecks .first := physical_css_source_parity_checks_are_genuinely_distinct example (word : PhysicalSevenBitWord) : physicalCssSyndrome .steane1996CosetParityChecks (physicalCalderbankToSteaneCoordinates word) = physicalCssSyndrome .calderbankShor1996NestedDualCodes word := physical_css_coordinate_permutation_identifies_source_syndromes word example (source : PhysicalQuantumCodeSource) (leftIndex rightIndex : PhysicalCssDualWordIndex) : physicalSevenBitDotParity (physicalCssDualWord source leftIndex) (physicalCssDualWord source rightIndex) = false := physical_css_simplex_words_are_mutually_orthogonal source leftIndex rightIndex example (source : PhysicalQuantumCodeSource) (index : PhysicalCssDualWordIndex) (error : PhysicalSevenQubitError) : physicalCssSyndrome source (physicalApplySevenQubitError (physicalCssLogicalCosetWord source .zero index) error) = physicalCssSyndrome source (physicalApplySevenQubitError (physicalCssLogicalCosetWord source .one index) error) := by rw [physical_css_syndrome_reveals_error_but_not_logical_coset, physical_css_syndrome_reveals_error_but_not_logical_coset] example (source : PhysicalQuantumCodeSource) (logical : PhysicalCssLogicalCoset) (index : PhysicalCssDualWordIndex) (error : PhysicalSevenQubitError) : physicalCssSyndromeRecovery source (physicalApplySevenQubitError (physicalCssLogicalCosetWord source logical index) error) = physicalCssLogicalCosetWord source logical index := physical_css_single_qubit_recovery_preserves_every_logical_coset source logical index error example : physicalCssSyndromeRecovery .calderbankShor1996NestedDualCodes (physicalCssFirstTwoQubitErrorWord .calderbankShor1996NestedDualCodes) ≠ physicalCssLogicalCosetWord .calderbankShor1996NestedDualCodes .zero .zero ∧ physicalCssSyndromeRecovery .steane1996CosetParityChecks (physicalCssFirstTwoQubitErrorWord .steane1996CosetParityChecks) ≠ physicalCssLogicalCosetWord .steane1996CosetParityChecks .zero .zero := ⟨physical_css_two_qubit_error_is_not_logically_restored .calderbankShor1996NestedDualCodes, physical_css_two_qubit_error_is_not_logically_restored .steane1996CosetParityChecks⟩ example : ¬PhysicalSourceCssCodeRealizationAt physicalQecFalseParityMatrixControl ∧ ¬PhysicalSourceCssCodeRealizationAt physicalQecSteaneFalseParityMatrixControl := ⟨actual_css_parity_matrix_cannot_be_replaced_by_boolean_certification.2.2, actual_steane_parity_matrix_cannot_be_replaced_by_boolean_certification.2.2⟩ example : ¬PhysicalSourceCssCodeRealizationAt physicalQecFalseAmplitudeSyndromeRecordControl ∧ ¬PhysicalSourceCssCodeRealizationAt physicalQecSteaneFalseAmplitudeSyndromeRecordControl := ⟨actual_css_syndrome_record_cannot_be_replaced_by_privacy_flag.2, actual_steane_syndrome_record_cannot_be_replaced_by_privacy_flag.2⟩ example : physicalCalderbankSignedLogicalOverlap .zero .one = 0 ∧ physicalSteane1996CodeReadout.logicalZeroOddSectorCoefficient = 0 ∧ physicalSteane1996CodeReadout.logicalOneEvenSectorCoefficient = 0 := ⟨physical_calderbank_signed_logical_states_are_normalized_and_orthogonal.2.2, rfl, rfl⟩ example : ¬PhysicalCalderbankNestedDualCodeAt physicalQecCalderbankLogicalPhaseErasedControl ∧ ¬PhysicalSteaneDualCosetCodeAt physicalQecSteaneLogicalCosetsMixedControl := ⟨calderbank_fourier_logical_phase_cannot_be_erased.2, steane_complementary_logical_cosets_cannot_be_mixed.2⟩ example : PhysicalCalderbankNestedDualCodeAt physicalCalderbankShor1996CodeReadout ∧ PhysicalSteaneDualCosetCodeAt physicalSteane1996CodeReadout := ⟨PhysicsLogic.Papers.CalderbankShor1996.QuantumCodeFramework.original_hamming_code_realizes_signed_private_bounded_recovery.1, PhysicsLogic.Papers.Steane1996.QuantumComputerFramework.original_simplex_code_realizes_private_complementary_coset_recovery.1⟩ end QuantumErrorRecoveryNoCloningPhysicalReuseAudit namespace NonequilibriumOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.NonequilibriumWorkIdentityContext.Countermodels example : physicalFluctuatingWorkModel.NonvacuousNonequilibriumWorkIdentityAndBoundAt := physical_fluctuating_work_model_is_nonvacuous example : PhysicalJarzynskiIsolatedHamiltonianAt physicalJarzynskiIsolatedWorkReadout ∧ PhysicalJarzynskiWeakReservoirAt physicalJarzynskiWeakReservoirWorkReadout ∧ PhysicalJarzynskiThermostatAt physicalJarzynskiThermostatWorkReadout := ⟨physical_jarzynski_isolated_fluctuating_work_is_nonvacuous, physical_jarzynski_weak_reservoir_fluctuating_work_is_nonvacuous, physical_jarzynski_thermostat_fluctuating_work_is_nonvacuous⟩ example : physicalJarzynskiIsolatedWorkReadout.forwardLowWorkProbability = (1 / 3 : Real) ∧ physicalJarzynskiIsolatedWorkReadout.forwardHighWorkProbability = (2 / 3 : Real) ∧ physicalJarzynskiIsolatedWorkReadout.lowBoltzmannWeight = 2 ∧ physicalJarzynskiIsolatedWorkReadout.highBoltzmannWeight = (1 / 2 : Real) := by exact ⟨rfl, rfl, rfl, rfl⟩ example : physicalJarzynskiIsolatedWorkReadout.lowWork < physicalJarzynskiIsolatedWorkReadout.freeEnergyDifference ∧ physicalJarzynskiIsolatedWorkReadout.freeEnergyDifference < physicalJarzynskiIsolatedWorkReadout.meanWork ∧ physicalJarzynskiIsolatedWorkReadout.exponentialWorkAverage = 1 := by obtain ⟨_, negative, mean, average, _, _⟩ := exact_work_identity_allows_negative_trajectories_with_positive_mean exact ⟨negative, mean, average⟩ example : ¬PhysicalEstablishedFiniteTimeWorkIdentityAt physicalNegativeWorkErasedControl ∧ ¬PhysicalEstablishedFiniteTimeWorkIdentityAt physicalPathwiseSecondLawClaimedControl := by obtain ⟨_, _, _, _, negative, pathwise⟩ := exact_work_identity_allows_negative_trajectories_with_positive_mean exact ⟨negative, pathwise⟩ example : ¬PhysicalEstablishedFiniteTimeWorkIdentityAt physicalNoncanonicalFiniteTimeWorkControl ∧ ¬PhysicalTwoTrajectoryWorkStatisticsAt physicalUnnormalizedWorkEnsembleControl := by obtain ⟨_, canonical, normalized, _, _⟩ := finite_time_work_identity_requires_preparation_normalization_and_mean exact ⟨canonical, normalized⟩ example : ¬PhysicalTwoTrajectoryWorkStatisticsAt physicalWrongMeanWorkControl ∧ ¬PhysicalEstablishedFiniteTimeWorkIdentityAt physicalArbitraryInitialEnsembleClaimedControl := by obtain ⟨_, _, _, mean, initial⟩ := finite_time_work_identity_requires_preparation_normalization_and_mean exact ⟨mean, initial⟩ example : physicalJarzynskiIsolatedWorkReadout.physicalHeatReservoirPresent = false ∧ ¬PhysicalJarzynskiIsolatedHamiltonianAt physicalLiouvilleTransportErasedControl ∧ ¬PhysicalJarzynskiIsolatedHamiltonianAt physicalTrajectoryEnergyBalanceErasedControl := by obtain ⟨_, reservoir, liouville, energy⟩ := isolated_hamiltonian_route_requires_liouville_and_energy_balance exact ⟨reservoir, liouville, energy⟩ example : PhysicalFullReservoirPartitionRatioAt physicalStronglyCoupledReservoirControl ∧ physicalStronglyCoupledReservoirControl.fullPartitionRatioIdentityExact = true ∧ physicalStronglyCoupledReservoirControl.subsystemFreeEnergyIdentificationEstablished = false := by obtain ⟨_, totalRatio, exactTotal, noSubsystem, _⟩ := full_reservoir_partition_ratio_does_not_identify_strong_coupling_free_energy exact ⟨totalRatio, exactTotal, noSubsystem⟩ example : ¬PhysicalJarzynskiWeakReservoirAt physicalStronglyCoupledReservoirControl := by obtain ⟨_, _, _, _, strongCoupling⟩ := full_reservoir_partition_ratio_does_not_identify_strong_coupling_free_energy exact strongCoupling example : ¬PhysicalJarzynskiWeakReservoirAt physicalUndersizedReservoirControl ∧ ¬PhysicalFullReservoirPartitionRatioAt physicalFullPartitionIdentityErasedControl := by obtain ⟨_, reservoir, ratio⟩ := weak_physical_reservoir_route_requires_size_and_full_canonical_identity exact ⟨reservoir, ratio⟩ example : physicalJarzynskiThermostatWorkReadout.thermostatWeakCouplingRequired = false ∧ physicalJarzynskiThermostatWorkReadout.physicalHeatReservoirPresent = false := by obtain ⟨_, coupling, reservoir, _, _⟩ := nose_hoover_route_retains_compressibility_without_weak_coupling exact ⟨coupling, reservoir⟩ example : ¬PhysicalJarzynskiThermostatAt physicalThermostatCompressibilityErasedControl ∧ ¬PhysicalJarzynskiThermostatAt physicalThermostatWeakCouplingForcedControl := by obtain ⟨_, _, _, compressibility, coupling⟩ := nose_hoover_route_retains_compressibility_without_weak_coupling exact ⟨compressibility, coupling⟩ example : physicalJarzynskiIsolatedWorkReadout.mechanism ≠ physicalJarzynskiWeakReservoirWorkReadout.mechanism ∧ physicalJarzynskiWeakReservoirWorkReadout.mechanism ≠ physicalJarzynskiThermostatWorkReadout.mechanism ∧ physicalJarzynskiIsolatedWorkReadout.mechanism ≠ physicalJarzynskiThermostatWorkReadout.mechanism := by obtain ⟨_, _, _, isolatedReservoir, reservoirThermostat, isolatedThermostat⟩ := jarzynski_hamiltonian_reservoir_and_thermostat_routes_are_distinct exact ⟨isolatedReservoir, reservoirThermostat, isolatedThermostat⟩ example : physicalCrooksEquilibriumWorkReadout.microscopicPathReversibility = true ∧ physicalCrooksEquilibriumWorkReadout.instantaneousDetailedBalanceDuringDriving = false := by obtain ⟨_, reversibility, noBalance, _, _⟩ := crooks_path_reversibility_does_not_require_driven_detailed_balance exact ⟨reversibility, noBalance⟩ example : ¬PhysicalCrooksFiniteTimePathFluctuationAt physicalCrooksMicroscopicReversibilityErasedControl ∧ ¬PhysicalCrooksEquilibriumForwardReverseAt physicalCrooksDetailedBalanceConflationControl := by obtain ⟨_, _, _, reversibility, balance⟩ := crooks_path_reversibility_does_not_require_driven_detailed_balance exact ⟨reversibility, balance⟩ example : PhysicalCrooksPositiveNormalizedDrivenTransitionKernelAt physicalCrooksEquilibriumWorkReadout ∧ physicalCrooksSelectedClockwiseProbabilityCurrent physicalCrooksEquilibriumWorkReadout = (1 / 9 : Real) ∧ ¬PhysicalCrooksSameProtocolStateDetailedBalanceAt physicalCrooksEquilibriumWorkReadout := by obtain ⟨_, kernel, _, _, _, _, current, _, notDetailed⟩ := crooks_driven_path_reversal_retains_normalized_stationary_circulation exact ⟨kernel, current, notDetailed⟩ example : PhysicalCrooksEquilibriumForwardReverseAt physicalCrooksBalancedDetailedBalanceControl ∧ PhysicalCrooksSameProtocolStateDetailedBalanceAt physicalCrooksBalancedDetailedBalanceControl := by obtain ⟨process, _, detailed, _⟩ := crooks_microscopic_path_reversibility_allows_detailed_balance exact ⟨process, detailed⟩ example : PhysicalCrooksFiniteTimePathFluctuationAt physicalCrooksDetailedBalanceConflationControl ∧ ¬PhysicalCrooksStateDetailedBalanceReportAccurateAt physicalCrooksDetailedBalanceConflationControl := by obtain ⟨fluctuation, notAccurate, _⟩ := crooks_boolean_balance_report_cannot_override_transition_probabilities exact ⟨fluctuation, notAccurate⟩ example : physicalCrooksEquilibriumWorkReadout.reverseProtocolDistinct = true ∧ physicalCrooksEquilibriumWorkReadout.forwardReverseEndpointExchange = true := by obtain ⟨_, distinct, endpoints, _, _⟩ := crooks_reverse_equilibrium_and_path_support_are_not_optional exact ⟨distinct, endpoints⟩ example : ¬PhysicalCrooksEquilibriumForwardReverseAt physicalCrooksReverseEndpointErasedControl ∧ ¬PhysicalCrooksFiniteTimePathFluctuationAt physicalCrooksReverseSupportErasedControl := by obtain ⟨_, _, _, endpoint, support⟩ := crooks_reverse_equilibrium_and_path_support_are_not_optional exact ⟨endpoint, support⟩ example : physicalCrooksEquilibriumWorkReadout.systemInformationContributionRetained = true ∧ physicalCrooksEquilibriumWorkReadout.bathEntropyContributionRetained = true := by obtain ⟨_, information, bath, _, _, _⟩ := crooks_entropy_retains_system_information_bath_and_reversal_oddness exact ⟨information, bath⟩ example : ¬PhysicalCrooksFiniteTimePathFluctuationAt physicalCrooksInformationEntropyErasedControl ∧ ¬PhysicalCrooksFiniteTimePathFluctuationAt physicalCrooksBathEntropyErasedControl ∧ ¬PhysicalCrooksFiniteTimePathFluctuationAt physicalCrooksEntropyOddnessErasedControl := by obtain ⟨_, _, _, information, bath, oddness⟩ := crooks_entropy_retains_system_information_bath_and_reversal_oddness exact ⟨information, bath, oddness⟩ example : physicalCrooksEquilibriumWorkReadout.forwardLowWorkProbability / physicalCrooksEquilibriumWorkReadout.reversePartnerOfLowProbability = (1 / 2 : Real) ∧ physicalCrooksEquilibriumWorkReadout.forwardHighWorkProbability / physicalCrooksEquilibriumWorkReadout.reversePartnerOfHighProbability = 2 := by obtain ⟨_, low, high, _, _⟩ := crooks_forward_reverse_ratios_distinguish_negative_and_positive_work exact ⟨low, high⟩ example : physicalCrooksEquilibriumWorkReadout.lowEntropyProduction < 0 ∧ 0 < physicalCrooksEquilibriumWorkReadout.highEntropyProduction := by obtain ⟨_, _, _, negative, positive⟩ := crooks_forward_reverse_ratios_distinguish_negative_and_positive_work exact ⟨negative, positive⟩ example : physicalCrooksSymmetricSteadyStateReadout.finiteTimeHeatOnlyRelationExact = false ∧ physicalCrooksSymmetricSteadyStateReadout.longTimeHeatApproximationControlled = true := by obtain ⟨_, notExact, controlled, _, _, _⟩ := symmetric_steady_entropy_relation_does_not_make_finite_time_heat_exact exact ⟨notExact, controlled⟩ example : ¬PhysicalCrooksSymmetricSteadyStateAt physicalCrooksSteadyTimeSymmetryErasedControl ∧ ¬PhysicalCrooksSymmetricSteadyStateAt physicalCrooksFiniteTimeHeatClaimedExactControl ∧ ¬PhysicalCrooksSymmetricSteadyStateAt physicalCrooksShortTimeHeatClaimedControl := by obtain ⟨_, _, _, symmetry, exact, time⟩ := symmetric_steady_entropy_relation_does_not_make_finite_time_heat_exact exact ⟨symmetry, exact, time⟩ example : ¬PhysicalEstablishedFiniteTimeWorkIdentityAt physicalCrooksSymmetricSteadyStateReadout ∧ ¬physicalSteadyStateFluctuationModel.NonvacuousNonequilibriumWorkIdentityAndBoundAt := by obtain ⟨_, notCanonical, notWorkIdentity⟩ := crooks_steady_state_is_not_a_canonical_jarzynski_work_preparation exact ⟨notCanonical, notWorkIdentity⟩ example : physicalCrooksEquilibriumWorkReadout.mechanism ≠ physicalCrooksSymmetricSteadyStateReadout.mechanism ∧ physicalCrooksEquilibriumWorkReadout.reverseProtocolDistinct = true ∧ physicalCrooksSymmetricSteadyStateReadout.reverseProtocolDistinct = false := by obtain ⟨_, _, mechanisms, forwardReverse, steady⟩ := crooks_equilibrium_switching_and_symmetric_steady_state_are_distinct exact ⟨mechanisms, forwardReverse, steady⟩ example : ¬PhysicalEstablishedFiniteTimeWorkIdentityAt physicalFiniteSampleExactFreeEnergyClaimedControl ∧ ¬PhysicalEstablishedFiniteTimeWorkIdentityAt physicalQuantumWorkExtensionClaimedControl := by obtain ⟨_, finiteSample, quantum⟩ := finite_time_work_identity_does_not_establish_finite_sample_or_quantum_claims exact ⟨finiteSample, quantum⟩ example : physicalNonequilibriumWorkPossibility.TentativeAt physicalNonequilibriumWorkEstablishment .strongCouplingSubsystemFreeEnergyIdentity .jarzynski1997HamiltonianSwitching := by obtain ⟨_, strongCoupling, _, _, _, _, _, _⟩ := nonequilibrium_work_source_overclaims_remain_epistemically_open exact strongCoupling example : physicalNonequilibriumWorkPossibility.TentativeAt physicalNonequilibriumWorkEstablishment .exactFiniteTimeHeatOnlyFluctuationRelation .crooks1999ForwardReverseFluctuation := by obtain ⟨_, _, _, finiteTimeHeat, _, _, _, _⟩ := nonequilibrium_work_source_overclaims_remain_epistemically_open exact finiteTimeHeat end NonequilibriumOriginalSourcePhysicalReuseAudit namespace DensityFunctionalOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.DensityFunctionalContext.Countermodels example : physicalGroundDensityModel.NonvacuousGroundDensityVariationalAt ∧ physicalKohnShamInteractingDensityModel.NonvacuousGroundDensityVariationalAt ∧ physicalKohnShamAuxiliaryDensityModel.NonvacuousGroundDensityVariationalAt := ⟨physical_interacting_ground_density_variational_is_nonvacuous, physical_kohn_sham_interacting_density_variational_is_nonvacuous, physical_kohn_sham_auxiliary_density_variational_is_nonvacuous⟩ example : PhysicalHKNondegeneratePotentialEquivalenceAt physicalHohenbergKohn1964DensityReadout ∧ PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnSham1965ExactOrbitalReadout ∧ PhysicalKohnShamControlledLocalApproximationAt physicalKohnSham1965LocalApproximationReadout := ⟨physical_hohenberg_kohn_nondegenerate_potential_route_is_nonvacuous, physical_kohn_sham_auxiliary_orbital_route_is_nonvacuous, physical_kohn_sham_controlled_local_approximation_is_nonvacuous⟩ example : physicalHohenbergKohn1964DensityReadout.potentialAdditiveShift = 5 ∧ physicalHohenbergKohn1964DensityReadout.shiftedGroundEnergyValue = 11 := by obtain ⟨_, shift, energy, _, _, _⟩ := hk_potential_equivalence_requires_nondegeneracy_and_additive_constant exact ⟨shift, energy⟩ example : ¬PhysicalHKNondegeneratePotentialEquivalenceAt physicalHKDegenerateGroundControl ∧ ¬PhysicalHKNondegeneratePotentialEquivalenceAt physicalHKPotentialConstantErasedControl := by obtain ⟨_, _, _, degenerate, constant, _⟩ := hk_potential_equivalence_requires_nondegeneracy_and_additive_constant exact ⟨degenerate, constant⟩ example : ¬PhysicalHKNondegeneratePotentialEquivalenceAt physicalHKWrongPotentialShiftEnergyControl := by obtain ⟨_, _, _, _, _, energy⟩ := hk_potential_equivalence_requires_nondegeneracy_and_additive_constant exact energy example : physicalHohenbergKohn1964DensityReadout.inequivalentPotentialsShareDensity = false := by obtain ⟨_, shares, _, _, _⟩ := hk_cross_variational_reductio_excludes_inequivalent_equal_density exact shares example : ¬PhysicalHKNondegeneratePotentialEquivalenceAt physicalHKInequivalentPotentialSharesDensityControl ∧ ¬PhysicalHKNondegeneratePotentialEquivalenceAt physicalHKCrossInequalityErasedControl ∧ ¬PhysicalHKNondegeneratePotentialEquivalenceAt physicalHKReductioErasedControl := by obtain ⟨_, _, sameDensity, inequalities, reductio⟩ := hk_cross_variational_reductio_excludes_inequivalent_equal_density exact ⟨sameDensity, inequalities, reductio⟩ example : physicalHohenbergKohn1964DensityReadout.fixedParticleNumber = 2 ∧ physicalHohenbergKohn1964DensityReadout.everyPositiveNormalizedDensityRepresentable = false := by obtain ⟨_, number, representability, _, _, _⟩ := hk_variation_is_fixed_sector_and_potential_representability_is_open exact ⟨number, representability⟩ example : ¬PhysicalFixedSectorDensityMinimumAt physicalHKWrongParticleSectorControl ∧ ¬PhysicalFixedSectorDensityMinimumAt physicalHKUnrepresentableTrialControl ∧ ¬PhysicalFixedSectorDensityMinimumAt physicalHKUniversalRepresentabilityClaimedControl := by obtain ⟨_, _, _, particleNumber, trial, universal⟩ := hk_variation_is_fixed_sector_and_potential_representability_is_open exact ⟨particleNumber, trial, universal⟩ example : physicalHohenbergKohn1964DensityReadout.universalInternalIndependentOfExternalPotential = true ∧ physicalHohenbergKohn1964DensityReadout.exactUniversalInternalFunctionalExplicitlyKnown = false := by obtain ⟨_, independent, unknown, _⟩ := universal_internal_functional_is_not_an_explicitly_known_formula exact ⟨independent, unknown⟩ example : ¬PhysicalFixedSectorDensityMinimumAt physicalHKExplicitUniversalFunctionalClaimedControl := by obtain ⟨_, _, _, claimed⟩ := universal_internal_functional_is_not_an_explicitly_known_formula exact claimed example : physicalKohnSham1965ExactOrbitalReadout.selectedGroundState ≠ physicalKohnSham1965ExactOrbitalReadout.auxiliaryGroundState ∧ physicalKohnSham1965ExactOrbitalReadout.auxiliaryDensity = physicalKohnSham1965ExactOrbitalReadout.selectedDensity := by obtain ⟨_, distinct, density, _, _, _⟩ := kohn_sham_auxiliary_density_matches_interacting_density_not_wavefunction exact ⟨distinct, density⟩ example : ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamAuxiliaryInteractionConflatedControl ∧ ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamDensityMismatchControl ∧ ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamDeterminantConflatedControl := by obtain ⟨_, _, _, interaction, density, determinant⟩ := kohn_sham_auxiliary_density_matches_interacting_density_not_wavefunction exact ⟨interaction, density, determinant⟩ example : physicalKohnShamInteractingDensityModel.density = physicalKohnShamAuxiliaryDensityModel.density ∧ physicalKohnShamInteractingDensityModel.state ≠ physicalKohnShamAuxiliaryDensityModel.state := ⟨rfl, by decide⟩ example : physicalKohnSham1965ExactOrbitalReadout.occupiedOrbitalCount = 2 := by obtain ⟨_, occupancy, _, _⟩ := kohn_sham_imported_minimum_and_orbital_occupancy_are_indispensable exact occupancy example : ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamInteractingMinimumErasedControl ∧ ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamOrbitalOccupancyWrongControl := by obtain ⟨_, _, minimum, occupancy⟩ := kohn_sham_imported_minimum_and_orbital_occupancy_are_indispensable exact ⟨minimum, occupancy⟩ example : physicalKohnSham1965ExactOrbitalReadout.noninteractingKineticEnergyValue = 3 ∧ physicalKohnSham1965ExactOrbitalReadout.hartreeEnergyValue = 2 ∧ physicalKohnSham1965ExactOrbitalReadout.exchangeCorrelationEnergyValue = -5 := by obtain ⟨_, kinetic, hartree, exchange, _, _⟩ := kohn_sham_energy_decomposition_retains_hartree_kinetic_and_exchange exact ⟨kinetic, hartree, exchange⟩ example : ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamHartreeErasedControl ∧ ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamExchangeEnergyDecompositionWrongControl := by obtain ⟨_, _, _, _, hartree, exchange⟩ := kohn_sham_energy_decomposition_retains_hartree_kinetic_and_exchange exact ⟨hartree, exchange⟩ example : physicalKohnSham1965ExactOrbitalReadout.correctLocalExchangePotentialValue = -2 ∧ physicalKohnSham1965ExactOrbitalReadout.slaterExchangePotentialValue = -3 := by obtain ⟨_, correct, slater, _, _, _⟩ := kohn_sham_self_consistency_and_exchange_derivative_are_independent exact ⟨correct, slater⟩ example : ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamSelfConsistencyErasedControl ∧ ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamExchangeCorrelationDerivativeErasedControl ∧ ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamSlaterExchangeFactorWrongControl := by obtain ⟨_, _, _, consistency, derivative, factor⟩ := kohn_sham_self_consistency_and_exchange_derivative_are_independent exact ⟨consistency, derivative, factor⟩ example : physicalKohnSham1965ExactOrbitalReadout.exactExchangeCorrelationFunctionalKnown = false ∧ physicalKohnSham1965ExactOrbitalReadout.orbitalEigenvaluesAreInteractingExcitations = false := by obtain ⟨_, exchange, excitations, _, _⟩ := exact_auxiliary_orbitals_do_not_supply_exact_exchange_or_excitation_spectrum exact ⟨exchange, excitations⟩ example : ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamExactExchangeCorrelationClaimedControl ∧ ¬PhysicalKohnShamAuxiliaryOrbitalAt physicalKohnShamOrbitalExcitationsClaimedControl := by obtain ⟨_, _, _, exchange, excitations⟩ := exact_auxiliary_orbitals_do_not_supply_exact_exchange_or_excitation_spectrum exact ⟨exchange, excitations⟩ example : physicalKohnSham1965LocalApproximationReadout.localDensityApproximationApplied = true ∧ physicalKohnSham1965LocalApproximationReadout.localDensityApproximationUniversallyExact = false := by obtain ⟨_, applied, exact, _, _⟩ := local_exchange_correlation_approximation_has_controlled_nonuniversal_scope exact ⟨applied, exact⟩ example : ¬PhysicalKohnShamControlledLocalApproximationAt physicalKohnShamLocalApproximationExactControl ∧ ¬PhysicalKohnShamControlledLocalApproximationAt physicalKohnShamLocalScopeErasedControl := by obtain ⟨_, _, _, exact, scope⟩ := local_exchange_correlation_approximation_has_controlled_nonuniversal_scope exact ⟨exact, scope⟩ example : physicalKohnSham1965LocalApproximationReadout.finiteBoundaryEffectsAbsent = false ∧ physicalKohnSham1965LocalApproximationReadout.gradientExpansionCapturesFriedelOscillations = false := by obtain ⟨_, boundary, friedel, _, _⟩ := local_gradient_approximation_does_not_erase_boundary_or_friedel_errors exact ⟨boundary, friedel⟩ example : ¬PhysicalKohnShamControlledLocalApproximationAt physicalKohnShamBoundaryErrorsErasedControl ∧ ¬PhysicalKohnShamControlledLocalApproximationAt physicalKohnShamFriedelGradientClaimedControl := by obtain ⟨_, _, _, boundary, friedel⟩ := local_gradient_approximation_does_not_erase_boundary_or_friedel_errors exact ⟨boundary, friedel⟩ example : physicalHohenbergKohn1964DensityReadout.source ≠ physicalKohnSham1965ExactOrbitalReadout.source ∧ physicalHohenbergKohn1964DensityReadout.mechanism ≠ physicalKohnSham1965ExactOrbitalReadout.mechanism ∧ physicalKohnSham1965ExactOrbitalReadout.mechanism ≠ physicalKohnSham1965LocalApproximationReadout.mechanism := by obtain ⟨_, _, _, sources, interacting, localApproximation⟩ := interacting_reductio_and_auxiliary_orbital_routes_are_distinct exact ⟨sources, interacting, localApproximation⟩ example : physicalDensityPossibility.TentativeAt physicalDensityEstablishment .unrestrictedPositiveDensityRepresentability .hohenbergKohn1964InteractingVariational ∧ physicalDensityPossibility.TentativeAt physicalDensityEstablishment .degenerateGroundStateArgument .hohenbergKohn1964InteractingVariational := by obtain ⟨representability, degeneracy, _, _, _, _, _, _, _⟩ := density_functional_source_overclaims_remain_epistemically_open exact ⟨representability, degeneracy⟩ example : physicalDensityPossibility.TentativeAt physicalDensityEstablishment .potentialIdentifiedWithoutAdditiveConstant .hohenbergKohn1964InteractingVariational := by obtain ⟨_, _, additive, _, _, _, _, _, _⟩ := density_functional_source_overclaims_remain_epistemically_open exact additive example : physicalDensityPossibility.TentativeAt physicalDensityEstablishment .exactExchangeCorrelationFunctionalKnown .kohnSham1965AuxiliaryOrbital ∧ physicalDensityPossibility.TentativeAt physicalDensityEstablishment .localDensityApproximationUniversallyExact .kohnSham1965AuxiliaryOrbital := by obtain ⟨_, _, _, exchange, localApproximation, _, _, _, _⟩ := density_functional_source_overclaims_remain_epistemically_open exact ⟨exchange, localApproximation⟩ example : physicalDensityPossibility.TentativeAt physicalDensityEstablishment .gradientExpansionRecoversFriedelOscillations .kohnSham1965AuxiliaryOrbital ∧ physicalDensityPossibility.TentativeAt physicalDensityEstablishment .auxiliaryEigenvaluesAreExactInteractingExcitations .kohnSham1965AuxiliaryOrbital := by obtain ⟨_, _, _, _, _, friedel, excitations, _, _⟩ := density_functional_source_overclaims_remain_epistemically_open exact ⟨friedel, excitations⟩ example : physicalDensityPossibility.TentativeAt physicalDensityEstablishment .auxiliaryDeterminantEqualsInteractingGroundState .kohnSham1965AuxiliaryOrbital ∧ physicalDensityPossibility.TentativeAt physicalDensityEstablishment .finiteBoundaryErrorsAbsent .kohnSham1965AuxiliaryOrbital := by obtain ⟨_, _, _, _, _, _, _, determinant, boundary⟩ := density_functional_source_overclaims_remain_epistemically_open exact ⟨determinant, boundary⟩ end DensityFunctionalOriginalSourcePhysicalReuseAudit namespace NuclearOriginalSourceShellPhysicalReuseAudit open PhysicsLogic.Framework.SpinOrbitShellClosureContext.Countermodels example : physicalMayerShellModel.NonvacuousSpinOrbitShellClosureAt ∧ physicalHaxelJensenSuessShellModel.NonvacuousSpinOrbitShellClosureAt := ⟨physical_mayer_shell_model_is_nonvacuous, physical_haxel_jensen_suess_shell_model_is_nonvacuous⟩ example : PhysicalMayerSquareWellShellAt physicalMayer1949SquareWellReadout ∧ PhysicalHaxelJensenSuessOscillatorShellAt physicalHaxelJensenSuess1949OscillatorReadout ∧ PhysicalMayerShellEndIsomerismAt physicalMayer1949SquareWellReadout := ⟨physical_mayer_square_well_shell_route_is_nonvacuous, physical_haxel_jensen_suess_oscillator_route_is_nonvacuous, physical_mayer_shell_end_isomerism_is_nonvacuous⟩ example : physicalMayer1949SquareWellReadout.source ≠ physicalHaxelJensenSuess1949OscillatorReadout.source ∧ physicalMayer1949SquareWellReadout.mechanism ≠ physicalHaxelJensenSuess1949OscillatorReadout.mechanism ∧ physicalMayer1949SquareWellReadout.centralPotential ≠ physicalHaxelJensenSuess1949OscillatorReadout.centralPotential := by obtain ⟨_, _, sources, mechanisms, potentials, _, _⟩ := square_well_and_oscillator_shell_source_routes_are_distinct exact ⟨sources, mechanisms, potentials⟩ example : ¬PhysicalMayerSquareWellShellAt physicalMayerOscillatorSubstitutionControl ∧ ¬PhysicalHaxelJensenSuessOscillatorShellAt physicalHaxelJensenSuessSquareWellSubstitutionControl := by obtain ⟨_, _, _, _, _, mayer, oscillator⟩ := square_well_and_oscillator_shell_source_routes_are_distinct exact ⟨mayer, oscillator⟩ example : physicalMayer1949SquareWellReadout.selectedPartnerCapacity = 8 ∧ ¬PhysicalEstablishedNuclearShellAt physicalMayerWrongPartnerQuantumNumberControl ∧ ¬PhysicalEstablishedNuclearShellAt physicalMayerWrongPartnerCapacityControl := by obtain ⟨_, capacity, quantum, wrongCapacity⟩ := high_j_partner_rule_and_state_capacity_are_indispensable exact ⟨capacity, quantum, wrongCapacity⟩ example : PhysicalMayerSquareWellShellAt physicalMayer1949SquareWellReadout ∧ ¬PhysicalEstablishedNuclearShellAt physicalMayerHighJReorderingErasedControl := by obtain ⟨established, reordered⟩ := high_j_reordering_is_not_supplied_by_named_shell_splitting exact ⟨established, reordered⟩ example : physicalMayer1949SquareWellReadout.radialNodeNoCrossingRetained = true ∧ physicalMayer1949SquareWellReadout.exactWithinShellTableOrderingClaimed = false := by obtain ⟨_, radial, ordered, _, _⟩ := mayer_square_well_retains_radial_order_without_exact_table_order exact ⟨radial, ordered⟩ example : ¬PhysicalMayerSquareWellShellAt physicalMayerRadialNodeOrderingErasedControl ∧ ¬PhysicalMayerSquareWellShellAt physicalMayerExactTableOrderingClaimedControl := by obtain ⟨_, _, _, radial, ordered⟩ := mayer_square_well_retains_radial_order_without_exact_table_order exact ⟨radial, ordered⟩ example : physicalMayer1949SquareWellReadout.calculatedClosures = [2, 8, 20, 28, 50, 82, 126] ∧ physicalMayer1949SquareWellReadout.observedClosurePattern = [28, 50, 82, 126] := by obtain ⟨calculated, observed, _⟩ := mayer_complete_pattern_retains_early_and_high_j_closures exact ⟨calculated, observed⟩ example : ¬PhysicalMayerSquareWellShellAt physicalMayerLowClosurePatternErasedControl := by obtain ⟨_, _, earlyClosures⟩ := mayer_complete_pattern_retains_early_and_high_j_closures exact earlyClosures example : physicalMayer1949SquareWellReadout.oddNucleusExceptionsRetained = true ∧ physicalMayer1949SquareWellReadout.allOddNucleusAssignmentsAgree = false := by obtain ⟨_, exceptions, agreement, _⟩ := mayer_odd_nucleus_agreement_retains_reported_exceptions exact ⟨exceptions, agreement⟩ example : ¬PhysicalMayerSquareWellShellAt physicalMayerOddNucleusExceptionsErasedControl := by obtain ⟨_, _, _, exceptions⟩ := mayer_odd_nucleus_agreement_retains_reported_exceptions exact exceptions example : physicalMayer1949SquareWellReadout.knownOddNucleusUpperBound = 83 ∧ physicalMayer1949SquareWellReadout.examinedHeavyNucleusCases < physicalMayer1949SquareWellReadout.availableHeavyNucleusCases := by obtain ⟨boundary, incomplete, _, _⟩ := mayer_known_evidence_boundary_does_not_complete_heavy_nucleus_data exact ⟨boundary, incomplete⟩ example : ¬PhysicalMayerSquareWellShellAt physicalMayerEvidenceBoundaryErasedControl ∧ ¬PhysicalMayerSquareWellShellAt physicalMayerHeavyEvidenceDeclaredCompleteControl := by obtain ⟨_, _, boundary, complete⟩ := mayer_known_evidence_boundary_does_not_complete_heavy_nucleus_data exact ⟨boundary, complete⟩ example : physicalMayer1949SquareWellReadout.isomerHighJTwiceAngularMomentum = 11 ∧ physicalMayer1949SquareWellReadout.neighboringTwiceAngularMomentum = 1 ∧ physicalMayer1949SquareWellReadout.highJParity ≠ physicalMayer1949SquareWellReadout.neighboringParity := by obtain ⟨_, spin, neighboring, parity, _, _, _⟩ := mayer_shell_end_isomerism_requires_spin_parity_and_adjacency exact ⟨spin, neighboring, parity⟩ example : ¬PhysicalMayerShellEndIsomerismAt physicalMayerIsomerSpinConflatedControl ∧ ¬PhysicalMayerShellEndIsomerismAt physicalMayerIsomerParityErasedControl ∧ ¬PhysicalMayerShellEndIsomerismAt physicalMayerIsomerAdjacencyErasedControl := by obtain ⟨_, _, _, _, spin, parity, adjacency⟩ := mayer_shell_end_isomerism_requires_spin_parity_and_adjacency exact ⟨spin, parity, adjacency⟩ example : physicalHaxelJensenSuess1949OscillatorReadout.oscillatorQuantumNumber = 4 ∧ physicalHaxelJensenSuess1949OscillatorReadout.oscillatorCoordinates = [2, 1, 1] := by obtain ⟨number, coordinates, _, _⟩ := hjs_oscillator_coordinates_require_the_three_axis_quantum_sum exact ⟨number, coordinates⟩ example : ¬PhysicalHaxelJensenSuessOscillatorShellAt physicalHaxelJensenSuessCoordinateSumWrongControl ∧ ¬PhysicalHaxelJensenSuessOscillatorShellAt physicalHaxelJensenSuessAnharmonicSplittingErasedControl := by obtain ⟨_, _, coordinates, anharmonic⟩ := hjs_oscillator_coordinates_require_the_three_axis_quantum_sum exact ⟨coordinates, anharmonic⟩ example : physicalHaxelJensenSuess1949OscillatorReadout.completedOscillatorGroupQuantumNumber = 3 ∧ physicalHaxelJensenSuess1949OscillatorReadout.markedHighJSourceGroupQuantumNumber = 4 ∧ physicalHaxelJensenSuess1949OscillatorReadout.markedHighJDestinationGroupQuantumNumber = 3 := by obtain ⟨completed, sourceGroup, destination, _⟩ := hjs_next_group_high_j_partner_moves_into_preceding_shell exact ⟨completed, sourceGroup, destination⟩ example : ¬PhysicalHaxelJensenSuessOscillatorShellAt physicalHaxelJensenSuessHighJDestinationWrongControl := by obtain ⟨_, _, _, destination⟩ := hjs_next_group_high_j_partner_moves_into_preceding_shell exact destination example : physicalHaxelJensenSuess1949OscillatorReadout.completedOscillatorGroupCounts = [8, 20, 40, 70, 112] ∧ physicalHaxelJensenSuess1949OscillatorReadout.highJPartnerCapacities = [6, 8, 10, 12, 14] ∧ physicalHaxelJensenSuess1949OscillatorReadout.calculatedClosures = [14, 28, 50, 82, 126] := by obtain ⟨groups, capacities, closures, _, _⟩ := hjs_oscillator_arithmetic_produces_fourteen_and_higher_closures exact ⟨groups, capacities, closures⟩ example : ¬PhysicalHaxelJensenSuessOscillatorShellAt physicalHaxelJensenSuessFourteenClosureErasedControl ∧ ¬PhysicalHaxelJensenSuessOscillatorShellAt physicalHaxelJensenSuessLastCapacityWrongControl := by obtain ⟨_, _, _, fourteen, capacity⟩ := hjs_oscillator_arithmetic_produces_fourteen_and_higher_closures exact ⟨fourteen, capacity⟩ example : physicalHaxelJensenSuess1949OscillatorReadout.selectedCompletedGroupCount + physicalHaxelJensenSuess1949OscillatorReadout.selectedPartnerCapacity = physicalHaxelJensenSuess1949OscillatorReadout.selectedClosure := rfl example : physicalHaxelJensenSuess1949OscillatorReadout.nuclearForceRangePreferenceRetained = true ∧ physicalHaxelJensenSuess1949OscillatorReadout.exactNuclearForceRangeMeasured = false := by obtain ⟨preference, measured, _, _⟩ := hjs_nuclear_force_range_preference_is_not_an_exact_measurement exact ⟨preference, measured⟩ example : ¬PhysicalHaxelJensenSuessOscillatorShellAt physicalHaxelJensenSuessForceRangePreferenceErasedControl ∧ ¬PhysicalEstablishedNuclearShellAt physicalHaxelJensenSuessExactForceRangeClaimedControl := by obtain ⟨_, _, preference, measured⟩ := hjs_nuclear_force_range_preference_is_not_an_exact_measurement exact ⟨preference, measured⟩ example : physicalHaxelJensenSuess1949OscillatorReadout.detailedNuclearAccountDeferred = true ∧ physicalHaxelJensenSuess1949OscillatorReadout.shellEndIsomerismExplained = false := by obtain ⟨details, isomer, _, _⟩ := hjs_deferred_details_do_not_import_mayer_isomerism exact ⟨details, isomer⟩ example : ¬PhysicalHaxelJensenSuessOscillatorShellAt physicalHaxelJensenSuessDetailedAccountClaimedControl ∧ ¬PhysicalHaxelJensenSuessOscillatorShellAt physicalHaxelJensenSuessMayerIsomerismImportedControl := by obtain ⟨_, _, details, isomer⟩ := hjs_deferred_details_do_not_import_mayer_isomerism exact ⟨details, isomer⟩ example : physicalMayer1949SquareWellReadout.highJArithmeticClosures = [28, 50, 82, 126] ∧ physicalHaxelJensenSuess1949OscillatorReadout.highJArithmeticClosures = [14, 28, 50, 82, 126] ∧ physicalMayer1949SquareWellReadout.calculatedClosures ≠ physicalHaxelJensenSuess1949OscillatorReadout.calculatedClosures := by obtain ⟨mayer, oscillator, distinct, _⟩ := common_high_magic_numbers_do_not_identify_complete_source_patterns exact ⟨mayer, oscillator, distinct⟩ example : ¬PhysicalHaxelJensenSuessOscillatorShellAt physicalHaxelJensenSuessUniversalFourteenClaimedControl := by obtain ⟨_, _, _, universal⟩ := common_high_magic_numbers_do_not_identify_complete_source_patterns exact universal example : ¬PhysicalEstablishedNuclearShellAt physicalMayerEmpiricalComparisonErasedControl ∧ ¬PhysicalEstablishedNuclearShellAt physicalMayerInterpretiveBridgeErasedControl := by obtain ⟨_, empirical, interpretation⟩ := shell_explanation_requires_empirical_and_interpretive_bridges exact ⟨empirical, interpretation⟩ example : physicalMayer1949SquareWellReadout.residualInteractionEnergyValue = 1 ∧ physicalMayer1949SquareWellReadout.exactNuclearManyBodyDynamicsEstablished = false := by obtain ⟨residual, manyBody, _, _⟩ := one_body_shell_success_does_not_establish_exact_many_body_dynamics exact ⟨residual, manyBody⟩ example : ¬PhysicalEstablishedNuclearShellAt physicalMayerExactManyBodyClaimedControl ∧ ¬PhysicalEstablishedNuclearShellAt physicalMayerUniversalParticleExactnessClaimedControl := by obtain ⟨_, _, manyBody, universal⟩ := one_body_shell_success_does_not_establish_exact_many_body_dynamics exact ⟨manyBody, universal⟩ example : ¬PhysicalEstablishedNuclearShellAt physicalMayerMicroscopicForceClaimedControl ∧ ¬PhysicalEstablishedNuclearShellAt physicalMayerUniquePotentialClaimedControl := by obtain ⟨_, _, force, potential⟩ := shell_sources_do_not_derive_microscopic_force_or_unique_potential exact ⟨force, potential⟩ example : physicalNuclearShellPossibility.TentativeAt physicalNuclearShellEstablishment .exceptionFreeOddNucleusAssignments .mayer1949SquareWell ∧ physicalNuclearShellPossibility.TentativeAt physicalNuclearShellEstablishment .completeHeavyNucleusEvidence .mayer1949SquareWell := by obtain ⟨exceptions, evidence, _, _, _, _, _, _, _⟩ := nuclear_shell_source_overclaims_remain_epistemically_open exact ⟨exceptions, evidence⟩ example : physicalNuclearShellPossibility.TentativeAt physicalNuclearShellEstablishment .exactNuclearManyBodyDynamics .mayer1949SquareWell ∧ physicalNuclearShellPossibility.TentativeAt physicalNuclearShellEstablishment .microscopicSpinOrbitForceDerived .haxelJensenSuess1949Oscillator := by obtain ⟨_, _, manyBody, force, _, _, _, _, _⟩ := nuclear_shell_source_overclaims_remain_epistemically_open exact ⟨manyBody, force⟩ example : physicalNuclearShellPossibility.TentativeAt physicalNuclearShellEstablishment .uniquelyEstablishedCentralPotential .haxelJensenSuess1949Oscillator ∧ physicalNuclearShellPossibility.TentativeAt physicalNuclearShellEstablishment .exactWithinShellTableOrdering .mayer1949SquareWell := by obtain ⟨_, _, _, _, potential, ordering, _, _, _⟩ := nuclear_shell_source_overclaims_remain_epistemically_open exact ⟨potential, ordering⟩ example : physicalNuclearShellPossibility.TentativeAt physicalNuclearShellEstablishment .universalFourteenMagicNumber .haxelJensenSuess1949Oscillator ∧ physicalNuclearShellPossibility.TentativeAt physicalNuclearShellEstablishment .exactlyMeasuredNuclearForceRange .haxelJensenSuess1949Oscillator := by obtain ⟨_, _, _, _, _, _, fourteen, forceRange, _⟩ := nuclear_shell_source_overclaims_remain_epistemically_open exact ⟨fourteen, forceRange⟩ example : physicalNuclearShellPossibility.TentativeAt physicalNuclearShellEstablishment .universalIndependentParticleExactness .mayer1949SquareWell := by obtain ⟨_, _, _, _, _, _, _, _, particle⟩ := nuclear_shell_source_overclaims_remain_epistemically_open exact particle end NuclearOriginalSourceShellPhysicalReuseAudit namespace CosmologyOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.HomogeneousCosmicExpansionContext.Countermodels noncomputable section example : physicalFriedmannExpansionModel.NonvacuousHomogeneousCosmicExpansionAt ∧ lemaitreExpansionControlModel.NonvacuousHomogeneousCosmicExpansionAt := ⟨physical_friedmann_selected_expansion_is_nonvacuous, physical_lemaitre_selected_expansion_is_nonvacuous⟩ example : PhysicalFriedmannVariableCurvatureAt physicalFriedmann1922CurvatureReadout ∧ PhysicalFriedmannFinitePeriodIdentificationAt physicalFriedmannFinitePeriodReadout ∧ PhysicalFriedmannUnboundedCurvaturePeriodicityAt physicalFriedmannUnboundedPeriodReadout := ⟨physical_friedmann_variable_curvature_route_is_nonvacuous, physical_friedmann_finite_period_reading_is_nonvacuous, physical_friedmann_unbounded_period_reading_is_nonvacuous⟩ example : PhysicalLemaitreConstantMassRedshiftAt physicalLemaitre1927NebularReadout ∧ PhysicalLemaitreQualifiedNebularCalibrationAt physicalLemaitre1927NebularReadout ∧ PhysicalLemaitreScopedPressureAndCausationAt physicalLemaitre1927NebularReadout := ⟨physical_lemaitre_constant_mass_redshift_is_nonvacuous, physical_lemaitre_qualified_nebular_calibration_is_nonvacuous, physical_lemaitre_scoped_pressure_and_causation_are_nonvacuous⟩ example : physicalFriedmann1922CurvatureReadout.einsteinMatterDensityValue = 1 ∧ physicalFriedmann1922CurvatureReadout.deSitterMatterDensityValue = 0 := by obtain ⟨_, matter, vacuum, _, _⟩ := friedmann_stationary_matter_and_vacuum_worlds_are_distinct exact ⟨matter, vacuum⟩ example : ¬PhysicalFriedmannVariableCurvatureAt physicalFriedmannEinsteinMatterErasedControl ∧ ¬PhysicalFriedmannVariableCurvatureAt physicalFriedmannDeSitterMatterConflatedControl := by obtain ⟨_, _, _, matter, vacuum⟩ := friedmann_stationary_matter_and_vacuum_worlds_are_distinct exact ⟨matter, vacuum⟩ example : physicalFriedmann1922CurvatureReadout.orthogonalTimeSlicingChosenForCalculation = true ∧ physicalFriedmann1922CurvatureReadout.orthogonalSlicingPhysicallyJustified = false := by obtain ⟨chosen, unjustified, _⟩ := friedmann_orthogonal_slicing_is_calculational_not_physical exact ⟨chosen, unjustified⟩ example : ¬PhysicalFriedmannVariableCurvatureAt physicalFriedmannSlicingClaimedPhysicalControl := by obtain ⟨_, _, slicing⟩ := friedmann_orthogonal_slicing_is_calculational_not_physical exact slicing example : physicalFriedmann1922CurvatureReadout.lowerTurningRadiusValue = 1 ∧ physicalFriedmann1922CurvatureReadout.upperTurningRadiusValue = 2 ∧ physicalTurningPolynomial physicalFriedmann1922CurvatureReadout physicalFriedmann1922CurvatureReadout.forbiddenBetweenRootRadiusValue < 0 := by obtain ⟨lower, upper, forbidden, _, _, _⟩ := friedmann_two_positive_turning_roots_exclude_intermediate_radius exact ⟨lower, upper, forbidden⟩ example : ¬PhysicalFriedmannVariableCurvatureAt physicalFriedmannCriticalParameterWrongControl ∧ ¬PhysicalFriedmannVariableCurvatureAt physicalFriedmannLowerTurningRootWrongControl ∧ ¬PhysicalFriedmannVariableCurvatureAt physicalFriedmannForbiddenIntervalClaimedAllowedControl := by obtain ⟨_, _, _, critical, turning, forbidden⟩ := friedmann_two_positive_turning_roots_exclude_intermediate_radius exact ⟨critical, turning, forbidden⟩ example : physicalFriedmann1922CurvatureReadout.firstMonotoneOriginRadiusValue = 0 ∧ physicalFriedmann1922CurvatureReadout.secondMonotoneOriginRadiusValue = 2 := by obtain ⟨zero, positive, _, _⟩ := friedmann_monotone_worlds_have_distinct_zero_and_positive_origins exact ⟨zero, positive⟩ example : ¬PhysicalFriedmannVariableCurvatureAt physicalFriedmannFirstMonotoneOriginConflatedControl ∧ ¬PhysicalFriedmannVariableCurvatureAt physicalFriedmannSecondMonotoneOriginConflatedControl := by obtain ⟨_, _, first, second⟩ := friedmann_monotone_worlds_have_distinct_zero_and_positive_origins exact ⟨first, second⟩ example : physicalFriedmann1922CurvatureReadout.periodicMinimumRadiusValue = 0 ∧ physicalFriedmann1922CurvatureReadout.periodicMaximumRadiusValue = 1 ∧ physicalFriedmann1922CurvatureReadout.periodicBranchGloballyExpanding = false := by obtain ⟨minimum, maximum, notExpanding, _, _⟩ := friedmann_periodic_branch_does_not_expand_at_all_times exact ⟨minimum, maximum, notExpanding⟩ example : ¬PhysicalFriedmannVariableCurvatureAt physicalFriedmannPeriodicMaximumWrongControl ∧ ¬PhysicalFriedmannVariableCurvatureAt physicalFriedmannPeriodicAlwaysExpandingClaimedControl := by obtain ⟨_, _, _, maximum, expansion⟩ := friedmann_periodic_branch_does_not_expand_at_all_times exact ⟨maximum, expansion⟩ example : physicalFriedmannFinitePeriodReadout.temporalReading ≠ physicalFriedmannUnboundedPeriodReadout.temporalReading := by obtain ⟨_, _, distinct, _, _⟩ := friedmann_finite_and_unbounded_periodic_time_readings_are_distinct exact distinct example : ¬PhysicalFriedmannFinitePeriodIdentificationAt physicalFriedmannFiniteReadingClaimedUnboundedControl ∧ ¬PhysicalFriedmannUnboundedCurvaturePeriodicityAt physicalFriedmannUnboundedReadingClaimedFiniteControl := by obtain ⟨_, _, _, finite, unbounded⟩ := friedmann_finite_and_unbounded_periodic_time_readings_are_distinct exact ⟨finite, unbounded⟩ example : physicalFriedmann1922CurvatureReadout.illustrativeSolarMassCount = 5 * 10 ^ (21 : Nat) ∧ physicalFriedmann1922CurvatureReadout.illustrativeWorldPeriodYears = 10 ^ (10 : Nat) ∧ physicalFriedmann1922CurvatureReadout.actualWorldUniquelyDetermined = false := by obtain ⟨mass, years, unknown, _, _, _⟩ := friedmann_illustrative_period_does_not_determine_the_actual_universe exact ⟨mass, years, unknown⟩ example : ¬PhysicalFriedmannVariableCurvatureAt physicalFriedmannIllustrativePeriodClaimedMeasuredControl ∧ ¬PhysicalSourceEstablishedCosmicExpansionAt physicalFriedmannActualWorldClaimedDeterminedControl ∧ ¬PhysicalSourceEstablishedCosmicExpansionAt physicalFriedmannCosmologicalConstantClaimedMeasuredControl := by obtain ⟨_, _, _, period, world, constant⟩ := friedmann_illustrative_period_does_not_determine_the_actual_universe exact ⟨period, world, constant⟩ example : physicalLemaitre1927NebularReadout.asymptoticPastRadiusValue = 1 ∧ physicalLemaitre1927NebularReadout.zeroRadiusInitialOriginClaimed = false := by obtain ⟨_, radius, nonzero, _, _⟩ := lemmaitre_constant_mass_expands_from_nonzero_asymptotic_radius exact ⟨radius, nonzero⟩ example : ¬PhysicalLemaitreConstantMassRedshiftAt physicalLemaitreConstantMassErasedControl ∧ ¬PhysicalLemaitreConstantMassRedshiftAt physicalLemaitreZeroRadiusOriginClaimedControl := by obtain ⟨_, _, _, mass, origin⟩ := lemmaitre_constant_mass_expands_from_nonzero_asymptotic_radius exact ⟨mass, origin⟩ example : physicalLemaitre1927NebularReadout.distinguishedSpatialCenterPresent = false ∧ ¬PhysicalLemaitreConstantMassRedshiftAt physicalLemaitreSpuriousCenterRetainedControl := by obtain ⟨center, spurious⟩ := lemmaitre_homogeneous_coordinates_exclude_a_spurious_cosmic_center exact ⟨center, spurious⟩ example : physicalLemaitre1927NebularReadout.observedLightPeriodValue / physicalLemaitre1927NebularReadout.emittedLightPeriodValue = physicalLemaitre1927NebularReadout.laterRadiusValue / physicalLemaitre1927NebularReadout.earlierRadiusValue ∧ physicalLemaitre1927NebularReadout.exactUniversalVelocityDistanceLawEstablished = false := by obtain ⟨ratio, notUniversal, _, _, _⟩ := lemmaitre_exact_radius_redshift_is_distinct_from_nearby_approximation exact ⟨ratio, notUniversal⟩ example : ¬PhysicalLemaitreConstantMassRedshiftAt physicalLemaitreExactRedshiftRatioWrongControl ∧ ¬PhysicalLemaitreConstantMassRedshiftAt physicalLemaitreNearbyVelocityApproximationErasedControl ∧ ¬PhysicalLemaitreConstantMassRedshiftAt physicalLemaitreUniversalLinearLawClaimedControl := by obtain ⟨_, _, exactRatio, nearby, universal⟩ := lemmaitre_exact_radius_redshift_is_distinct_from_nearby_approximation exact ⟨exactRatio, nearby, universal⟩ example : physicalLemaitre1927NebularReadout.radialVelocityCatalogCount = 43 ∧ physicalLemaitre1927NebularReadout.calibratedNebulaCount = 42 := by obtain ⟨catalog, sample, _⟩ := lemmaitre_uses_forty_two_nebulae_from_a_forty_three_object_catalog exact ⟨catalog, sample⟩ example : ¬PhysicalLemaitreQualifiedNebularCalibrationAt physicalLemaitreFortyThreeCatalogConflatedWithSampleControl := by obtain ⟨_, _, sample⟩ := lemmaitre_uses_forty_two_nebulae_from_a_forty_three_object_catalog exact sample example : physicalLemaitre1927NebularReadout.weightedExpansionRateKilometersPerSecondPerMegaparsec = 625 ∧ physicalLemaitre1927NebularReadout.unweightedExpansionRateKilometersPerSecondPerMegaparsec = 575 ∧ physicalLemaitre1927NebularReadout.averageNebularVelocityKilometersPerSecond = 600 := by obtain ⟨weighted, unweighted, velocity, _, _⟩ := lemmaitre_weighted_and_unweighted_rates_remain_distinct_estimates exact ⟨weighted, unweighted, velocity⟩ example : ¬PhysicalLemaitreQualifiedNebularCalibrationAt physicalLemaitreWeightedAndUnweightedRateConflatedControl ∧ ¬PhysicalSourceEstablishedCosmicExpansionAt physicalLemaitreExactExpansionRateClaimedControl := by obtain ⟨_, _, _, weighted, exactRate⟩ := lemmaitre_weighted_and_unweighted_rates_remain_distinct_estimates exact ⟨weighted, exactRate⟩ example : physicalLemaitre1927NebularReadout.nebularPeculiarVelocityKilometersPerSecond = 300 ∧ physicalLemaitre1927NebularReadout.lowerRelativeDistanceFactor = (2 / 5 : Real) ∧ physicalLemaitre1927NebularReadout.upperRelativeDistanceFactor = (5 / 2 : Real) := by obtain ⟨peculiar, lower, upper, _, _⟩ := lemmaitre_peculiar_velocities_and_distance_errors_are_indispensable exact ⟨peculiar, lower, upper⟩ example : ¬PhysicalLemaitreQualifiedNebularCalibrationAt physicalLemaitrePeculiarVelocityErasedControl ∧ ¬PhysicalLemaitreQualifiedNebularCalibrationAt physicalLemaitreDistanceUncertaintyErasedControl := by obtain ⟨_, _, _, peculiar, distance⟩ := lemmaitre_peculiar_velocities_and_distance_errors_are_indispensable exact ⟨peculiar, distance⟩ example : physicalLemaitre1927NebularReadout.generalRadiationPressureValue = 10 ∧ physicalLemaitre1927NebularReadout.astronomicalPressureValue = 1 ∧ physicalLemaitre1927NebularReadout.pressureZeroInAllRegimes = false := by obtain ⟨general, astronomical, scopedPressure, _⟩ := lemmaitre_astronomical_pressure_approximation_is_not_global_zero exact ⟨general, astronomical, scopedPressure⟩ example : ¬PhysicalLemaitreScopedPressureAndCausationAt physicalLemaitrePressureEverywhereZeroClaimedControl := by obtain ⟨_, _, _, pressure⟩ := lemmaitre_astronomical_pressure_approximation_is_not_global_zero exact pressure example : physicalLemaitre1927NebularReadout.radiationExpansionWorkValue = 20 ∧ physicalLemaitre1927NebularReadout.radiationCauseSuggested = true ∧ physicalLemaitre1927NebularReadout.radiationCauseDerived = false := by obtain ⟨work, suggested, notDerived, _, _⟩ := lemmaitre_radiation_work_suggests_but_does_not_derive_expansion_cause exact ⟨work, suggested, notDerived⟩ example : ¬PhysicalLemaitreScopedPressureAndCausationAt physicalLemaitreRadiationCauseClaimedDerivedControl ∧ ¬PhysicalLemaitreScopedPressureAndCausationAt physicalLemaitreFullRadiusHistoryClaimedControl := by obtain ⟨_, _, _, cause, history⟩ := lemmaitre_radiation_work_suggests_but_does_not_derive_expansion_cause exact ⟨cause, history⟩ example : physicalFriedmann1922CurvatureReadout.source ≠ physicalLemaitre1927NebularReadout.source ∧ physicalFriedmann1922CurvatureReadout.mechanism ≠ physicalLemaitre1927NebularReadout.mechanism ∧ physicalFriedmann1922CurvatureReadout.selectedBranch ≠ physicalLemaitre1927NebularReadout.selectedBranch := by obtain ⟨_, _, sources, mechanisms, branches⟩ := variable_curvature_and_nebular_calibration_are_distinct_source_routes exact ⟨sources, mechanisms, branches⟩ example : physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .actualWorldUniquelyDetermined .friedmann1922VariableCurvature ∧ physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .cosmologicalConstantExactlyMeasured .friedmann1922VariableCurvature := by obtain ⟨world, constant, _, _, _, _, _, _, _, _, _, _, _⟩ := cosmological_expansion_source_overclaims_remain_epistemically_open exact ⟨world, constant⟩ example : physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .illustrativeWorldPeriodEmpiricallyMeasured .friedmann1922VariableCurvature ∧ physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .allWorldBranchesExpandMonotonically .friedmann1922VariableCurvature := by obtain ⟨_, _, illustration, monotonic, _, _, _, _, _, _, _, _, _⟩ := cosmological_expansion_source_overclaims_remain_epistemically_open exact ⟨illustration, monotonic⟩ example : physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .periodIdentificationAndUnboundedTimeTogether .friedmann1922VariableCurvature ∧ physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .lemaitreExpansionStartsAtZeroRadius .lemaitre1927NebularExpansion := by obtain ⟨_, _, _, _, period, origin, _, _, _, _, _, _, _⟩ := cosmological_expansion_source_overclaims_remain_epistemically_open exact ⟨period, origin⟩ example : physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .exactUniversalRecessionDistanceLaw .lemaitre1927NebularExpansion ∧ physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .exactUniversalExpansionRate .lemaitre1927NebularExpansion := by obtain ⟨_, _, _, _, _, _, law, rate, _, _, _, _, _⟩ := cosmological_expansion_source_overclaims_remain_epistemically_open exact ⟨law, rate⟩ example : physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .allCosmicPressureVanishes .lemaitre1927NebularExpansion ∧ physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .radiationEstablishedAsExpansionCause .lemaitre1927NebularExpansion := by obtain ⟨_, _, _, _, _, _, _, _, pressure, radiation, _, _, _⟩ := cosmological_expansion_source_overclaims_remain_epistemically_open exact ⟨pressure, radiation⟩ example : physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .uniqueGlobalRadiusHistoryFromLocalLaw .lemaitre1927NebularExpansion ∧ physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .exactObservationalDistances .lemaitre1927NebularExpansion ∧ physicalCosmologyPossibility.TentativeAt physicalCosmologyEstablishment .spatiallyDistinguishedCosmicCenter .lemaitre1927NebularExpansion := by obtain ⟨_, _, _, _, _, _, _, _, _, _, history, distances, center⟩ := cosmological_expansion_source_overclaims_remain_epistemically_open exact ⟨history, distances, center⟩ end end CosmologyOriginalSourcePhysicalReuseAudit namespace EntanglementOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.BipartiteEntanglementContext.Countermodels example : PhysicalCKWPureThreeQubitTangleAt physicalCKWBellPairReadout := physical_ckw_bell_pair_with_spectator_is_nonvacuous.1 example : physicalCKWBellPairReadout.pairABTangleValue = 1 ∧ physicalCKWBellPairReadout.pairACTangleValue = 0 ∧ physicalCKWBellPairReadout.residualThreeTangleValue = 0 := physical_ckw_bell_pair_with_spectator_is_nonvacuous.2 example : PhysicalCKWPureThreeQubitTangleAt physicalCKWGHZReadout := physical_ckw_ghz_collective_tangle_is_nonvacuous.1 example : physicalCKWGHZReadout.pairABTangleValue = 0 ∧ physicalCKWGHZReadout.pairACTangleValue = 0 ∧ physicalCKWGHZReadout.residualThreeTangleValue = 1 := physical_ckw_ghz_collective_tangle_is_nonvacuous.2 example : physicalCKWGHZReadout.firstPairModel.ClassicallyCorrelatedAt ∧ physicalCKWGHZReadout.firstPairModel.SeparableAt ∧ ¬physicalCKWGHZReadout.firstPairModel.EntangledAt := by obtain ⟨classical, _, separable, _, notEntangled, _⟩ := ckw_ghz_pairs_are_classical_separable_not_pairwise_entangled exact ⟨classical, separable, notEntangled⟩ example : physicalCKWGHZReadout.secondPairModel.ClassicallyCorrelatedAt ∧ physicalCKWGHZReadout.secondPairModel.SeparableAt := by obtain ⟨_, classical, _, separable, _, _⟩ := ckw_ghz_pairs_are_classical_separable_not_pairwise_entangled exact ⟨classical, separable⟩ example : PhysicalCKWPureThreeQubitTangleAt physicalCKWSingleExcitationReadout ∧ physicalCKWSingleExcitationReadout.pairABTangleValue = (4 / 9 : Real) ∧ physicalCKWSingleExcitationReadout.pairACTangleValue = (4 / 9 : Real) := by obtain ⟨route, firstPair, secondPair, _, _⟩ := physical_ckw_single_excitation_saturation_is_nonvacuous exact ⟨route, firstPair, secondPair⟩ example : physicalCKWSingleExcitationReadout.pairABTangleValue + physicalCKWSingleExcitationReadout.pairACTangleValue = physicalCKWSingleExcitationReadout.focusParentTangleValue ∧ physicalCKWSingleExcitationReadout.residualThreeTangleValue = 0 := by obtain ⟨saturation, residual, _⟩ := ckw_single_excitation_exhausts_parent_tangle_without_residual exact ⟨saturation, residual⟩ example : ¬PhysicalCKWPureThreeQubitTangleAt physicalCKWEqualPartnersControl ∧ ¬PhysicalCKWPureThreeQubitTangleAt physicalCKWParentMarginalMismatchControl := ckw_pure_tangle_requires_three_distinct_qubit_roles.2 example : physicalCKWSingleExcitationReadout.pairABTangleValue = (4 / 9 : Real) ∧ physicalCKWSingleExcitationReadout.pairABConcurrenceSquaredValue = (4 / 9 : Real) ∧ ¬PhysicalCKWPureThreeQubitTangleAt physicalCKWConcurrenceSquareErasedControl := ckw_tangle_retains_squared_concurrence_not_concurrence example : physicalCKWGHZReadout.residualThreeTangleValue = physicalCKWGHZReadout.alternateFocusResidualValue ∧ ¬PhysicalCKWPureThreeQubitTangleAt physicalCKWNegativeResidualControl ∧ ¬PhysicalCKWPureThreeQubitTangleAt physicalCKWResidualPermutationBrokenControl := ckw_residual_tangle_must_be_nonnegative_and_focus_invariant example : PhysicalCKWMixedConvexRoofTangleAt physicalCKWMixedConvexRoofReadout ∧ physicalCKWMixedConvexRoofReadout.parentStatePure = false ∧ physicalCKWMixedConvexRoofReadout.parentTangleDefined = false := ⟨physical_ckw_mixed_convex_roof_is_nonvacuous.1, physical_ckw_mixed_convex_roof_is_nonvacuous.2.1, physical_ckw_mixed_convex_roof_is_nonvacuous.2.2.1⟩ example : ¬PhysicalCKWMixedConvexRoofTangleAt physicalCKWMixedPureParentConflationControl ∧ ¬PhysicalCKWMixedConvexRoofTangleAt physicalCKWMixedConvexityErasedControl ∧ ¬PhysicalCKWMixedConvexRoofTangleAt physicalCKWMixedOptimalDecompositionErasedControl := ckw_mixed_monogamy_needs_convex_roof_and_pair_tangle_convexity.2 example : physicalCKWFormationCounterexampleReadout.pairABTangleValue + physicalCKWFormationCounterexampleReadout.pairACTangleValue = physicalCKWFormationCounterexampleReadout.focusParentTangleValue := ckw_entanglement_of_formation_fails_the_tangle_sharing_bound.1 example : physicalCKWFormationCounterexampleReadout.formationFocusParentValue < physicalCKWFormationCounterexampleReadout.formationABValue + physicalCKWFormationCounterexampleReadout.formationACValue := ckw_entanglement_of_formation_fails_the_tangle_sharing_bound.2 example : physicalEntanglementPossibility.TentativeAt physicalEntanglementEstablishment .arbitraryQubitMonogamy .coffmanKunduWootters2000 ∧ physicalEntanglementPossibility.TentativeAt physicalEntanglementEstablishment .higherDimensionalSpinFlipMonogamy .coffmanKunduWootters2000 := by obtain ⟨qubits, dimensions, _, _⟩ := ckw_arbitrary_qubit_conjecture_and_higher_dimensions_stay_open exact ⟨qubits, dimensions⟩ example : ¬PhysicalCKWPureThreeQubitTangleAt physicalCKWUniversalQubitConjectureClaimedControl ∧ ¬PhysicalCKWPureThreeQubitTangleAt physicalCKWHigherDimensionalSpinFlipClaimedControl := by obtain ⟨_, _, qubits, dimensions⟩ := ckw_arbitrary_qubit_conjecture_and_higher_dimensions_stay_open exact ⟨qubits, dimensions⟩ example : PhysicalAMPSOldHolePagePurificationAt physicalAMPSOldHoleReadout := physical_amps_old_hole_page_purification_is_nonvacuous.1 example : physicalAMPSOldHoleReadout.earlyLateEntropyValue < physicalAMPSOldHoleReadout.earlyRadiationEntropyValue := physical_amps_old_hole_page_purification_is_nonvacuous.2 example : PhysicalAMPSSmoothHorizonVacuumAt physicalAMPSOldHoleReadout := physical_amps_smooth_horizon_vacuum_is_nonvacuous.1 example : physicalAMPSOldHoleReadout.lateInteriorEntropyValue = 0 ∧ physicalAMPSOldHoleReadout.earlyLateInteriorEntropyValue = physicalAMPSOldHoleReadout.earlyRadiationEntropyValue := physical_amps_smooth_horizon_vacuum_is_nonvacuous.2 example : physicalAMPSOldHoleReadout.firstPairModel.firstSubsystem ≠ physicalAMPSOldHoleReadout.firstPairModel.secondSubsystem ∧ ¬PhysicalAMPSOldHolePagePurificationAt physicalAMPSEarlyLateRolesConflatedControl ∧ ¬PhysicalAMPSOldHolePagePurificationAt physicalAMPSYoungHolePurificationClaimedControl := amps_old_hole_requires_distinct_early_radiation_and_late_mode.2 example : ¬PhysicalAMPSSmoothHorizonVacuumAt physicalAMPSInteriorPartnerErasedControl ∧ ¬PhysicalAMPSSmoothHorizonVacuumAt physicalAMPSNoDramaAbandonedReadout := amps_no_drama_requires_a_pure_late_mode_interior_pair.2 example : PhysicalAMPSOldHolePagePurificationAt physicalAMPSOldHoleReadout ∧ PhysicalAMPSSmoothHorizonVacuumAt physicalAMPSOldHoleReadout ∧ ¬PhysicalAMPSStrongSubadditivityAt physicalAMPSOldHoleReadout := by obtain ⟨oldHole, smooth, _, contradiction⟩ := amps_old_hole_vacuum_and_purification_violate_strong_subadditivity exact ⟨oldHole, smooth, contradiction⟩ example (readout : PhysicalSourceEntanglementReadout) (postulates : PhysicalAMPSConditionalIncompatibilityAt readout) : False := amps_strong_subadditivity_makes_old_hole_postulates_incompatible readout postulates example : PhysicalAMPSOldHolePagePurificationAt physicalAMPSNoDramaAbandonedReadout ∧ PhysicalAMPSStrongSubadditivityAt physicalAMPSNoDramaAbandonedReadout ∧ physicalAMPSNoDramaAbandonedReadout.postulateOneUnitaryEvaporation = true ∧ physicalAMPSNoDramaAbandonedReadout.postulateFourSmoothInfallingHorizon = false := physical_amps_abandoning_no_drama_preserves_unitary_purification example : PhysicalAMPSSmoothHorizonVacuumAt physicalAMPSUnitarityAbandonedReadout ∧ PhysicalAMPSStrongSubadditivityAt physicalAMPSUnitarityAbandonedReadout ∧ physicalAMPSUnitarityAbandonedReadout.postulateOneUnitaryEvaporation = false ∧ physicalAMPSUnitarityAbandonedReadout.postulateFourSmoothInfallingHorizon = true := physical_amps_abandoning_unitarity_preserves_smooth_horizon example : 0 < physicalAMPSOldHoleReadout.outgoingLateModeEntropyValue ∧ physicalAMPSLateThermalEntropyErasedControl.outgoingLateModeEntropyValue = 0 := by obtain ⟨positive, _, erased, _⟩ := amps_positive_late_mode_entropy_and_ssa_are_independent_inputs exact ⟨positive, erased⟩ example : physicalAMPSOldHoleReadout.strongSubadditivityApplicable = true ∧ physicalAMPSSSAApplicabilityErasedControl.strongSubadditivityApplicable = false := by obtain ⟨_, retained, _, erased⟩ := amps_positive_late_mode_entropy_and_ssa_are_independent_inputs exact ⟨retained, erased⟩ example : physicalAMPSOldHoleReadout.actualEarlyRadiationMeasurementPerformed = false ∧ physicalAMPSOldHoleReadout.possibleEarlyRadiationMeasurementSuffices = true := by obtain ⟨notPerformed, sufficient, _, _⟩ := amps_possible_early_measurement_does_not_require_its_performance exact ⟨notPerformed, sufficient⟩ example : physicalAMPSQualifiedMiningReadout.highAngularMomentumMiningAssumed = true ∧ physicalAMPSQualifiedMiningReadout.miningRecordIncludedInLateRadiation = true := by obtain ⟨mining, record, _, _⟩ := amps_high_partial_wave_extension_requires_mining_record_in_late_radiation exact ⟨mining, record⟩ example : physicalAMPSMiningRecordErasedControl.highAngularMomentumMiningAssumed = true ∧ physicalAMPSMiningRecordErasedControl.miningRecordIncludedInLateRadiation = false := by obtain ⟨_, _, mining, record⟩ := amps_high_partial_wave_extension_requires_mining_record_in_late_radiation exact ⟨mining, record⟩ example : physicalFirewallAlternativeSelection.OpenAlternativePairAt .infallingFirewall .macroscopicExteriorDynamics .almheiriMarolfPolchinskiSully2012 := amps_firewall_and_exterior_dynamics_are_open_distinct_alternatives.1 example : physicalEntanglementPossibility.TentativeAt physicalEntanglementEstablishment .firewallResolution .almheiriMarolfPolchinskiSully2012 ∧ physicalEntanglementPossibility.TentativeAt physicalEntanglementEstablishment .novelExteriorDynamicsResolution .almheiriMarolfPolchinskiSully2012 := amps_firewall_and_exterior_dynamics_are_open_distinct_alternatives.2 example : ¬physicalEntanglementEstablishment.establishedAt .universalFirewallEstablished .almheiriMarolfPolchinskiSully2012 ∧ ¬physicalEntanglementEstablishment.establishedAt .uniquelySelectedPostulateFailure .almheiriMarolfPolchinskiSully2012 := by obtain ⟨_, _, universal, unique, _, _, _⟩ := entanglement_source_overclaims_remain_epistemically_unestablished exact ⟨universal, unique⟩ example : ¬physicalEntanglementEstablishment.establishedAt .youngHolePagePurification .almheiriMarolfPolchinskiSully2012 ∧ ¬physicalEntanglementEstablishment.establishedAt .rindlerHorizonOldHoleArgument .almheiriMarolfPolchinskiSully2012 ∧ ¬physicalEntanglementEstablishment.establishedAt .exteriorObserverNecessarilySeesFirewall .almheiriMarolfPolchinskiSully2012 := by obtain ⟨_, _, _, _, young, rindler, exterior⟩ := entanglement_source_overclaims_remain_epistemically_unestablished exact ⟨young, rindler, exterior⟩ end EntanglementOriginalSourcePhysicalReuseAudit namespace CKWPrintedResidualPolynomialReuseAudit open PhysicsLogic.Papers.CoffmanKunduWootters2000 open ThreeQubitCoefficientVector open DistributedEntanglementFramework.Countermodels example (coefficients : ThreeQubitCoefficientVector) : coefficients.swapFirstSecond.residualPolynomial = coefficients.residualPolynomial ∧ coefficients.swapSecondThird.residualPolynomial = coefficients.residualPolynomial := printed_residual_polynomial_is_permutation_invariant coefficients example (alpha beta gamma : Complex) : singleExcitationABTangle alpha beta + singleExcitationACTangle alpha gamma = singleExcitationAWithBCTangle alpha beta gamma := single_excitation_family_saturates_printed_pairwise_tangle_bound alpha beta gamma example : normalizedGHZCoefficients.normSquared = 1 ∧ 4 * normalizedGHZCoefficients.residualPolynomialMagnitude = 1 := by obtain ⟨normalized, _, _, _, _, residual⟩ := original_normalized_ghz_state_has_unit_residual_tangle exact ⟨normalized, residual⟩ example : planarProductCoefficients.diagonalPolynomial = 1 / 8 ∧ planarProductCoefficients.planarPolynomial = 1 / 16 ∧ planarProductCoefficients.residualPolynomial = 0 := by obtain ⟨_, diagonal, planar, _, canceled⟩ := planar_product_state_cancels_diagonal_and_planar_contributions exact ⟨diagonal, planar, canceled⟩ example : tetrahedralGHZCoefficients.tetrahedralPolynomial = 1 / 16 ∧ tetrahedralGHZCoefficients.residualPolynomial = 1 / 4 := by obtain ⟨_, _, _, tetrahedral, polynomial, _⟩ := tetrahedral_ghz_state_requires_the_printed_factor_four exact ⟨tetrahedral, polynomial⟩ example : phaseReversedTetrahedralCoefficients.residualPolynomial = -(1 / 4) ∧ phaseReversedTetrahedralCoefficients.residualPolynomialMagnitude = 1 / 4 := by obtain ⟨_, signed, absolute⟩ := phase_reversed_tetrahedron_requires_absolute_polynomial_value exact ⟨signed, absolute⟩ example : physicalTetrahedralResidualReadout.SourceFormulaAt ∧ physicalPlanarProductResidualReadout.SourceFormulaAt ∧ physicalPhaseReversedResidualReadout.SourceFormulaAt := ⟨tetrahedral_original_residual_formula_is_nonvacuous, planar_product_original_residual_formula_is_nonvacuous, phase_reversed_original_residual_formula_is_nonvacuous⟩ example : ¬namedResidualWithoutPrintedTetrahedron.SourceFormulaAt ∧ ¬zeroFullDeterminantMistakenForRestrictedProduct.SourceFormulaAt ∧ ¬planarCancellationErasedResidual.SourceFormulaAt ∧ ¬negativeSignedResidualWithoutAbsoluteValue.SourceFormulaAt := ⟨named_unit_residual_does_not_supply_printed_tetrahedral_coefficient.2, full_zero_determinant_does_not_supply_restricted_eigenvalue_product.2.2, planar_coefficient_support_does_not_remove_destructive_cancellation.2.2.2, negative_signed_polynomial_does_not_supply_a_negative_tangle.2.2⟩ example {T : CKWTypes} (framework : DistributedEntanglementFramework T) (calculation : framework.ResidualPolynomialCalculation) : framework.residualTangle framework.selectedPureState = 4 * framework.residualTangleFormulaReadout.coefficientVector.residualPolynomialMagnitude := (framework.residual_calculation_retains_printed_polynomial_and_qubit_symmetry calculation).1 example {T : CKWTypes} (framework : DistributedEntanglementFramework T) (calculation : framework.ResidualPolynomialCalculation) : framework.residualTanglePermutationInvariant framework.selectedPureState := framework.residual_calculation_establishes_permutation_invariance calculation end CKWPrintedResidualPolynomialReuseAudit namespace BrownianDiffusionOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.RandomDisplacementDiffusionContext.Countermodels example : PhysicalEinsteinSymmetricTransitionAt physicalEinstein1905TransitionReadout := physical_einstein_symmetric_gaussian_diffusion_is_nonvacuous.1 example : physicalEinstein1905TransitionReadout.meanSquareDisplacementValue = 16 := physical_einstein_symmetric_gaussian_diffusion_is_nonvacuous.2 example : PhysicalEinsteinOsmoticStokesBalanceAt physicalEinstein1905OsmoticReadout := physical_einstein_osmotic_stokes_balance_is_nonvacuous.1 example : physicalEinstein1905OsmoticReadout.diffusionCoefficientValue = 2 := physical_einstein_osmotic_stokes_balance_is_nonvacuous.2 example : physicalEinstein1905TransitionReadout.stepVarianceValue = 2 * physicalEinstein1905TransitionReadout.diffusionCoefficientValue * physicalEinstein1905TransitionReadout.stepDurationValue := einstein_symmetric_independent_steps_retain_the_factor_two_variance.1 example : ¬PhysicalEinsteinSymmetricTransitionAt physicalEinsteinSymmetryErasedControl ∧ ¬PhysicalEinsteinSymmetricTransitionAt physicalEinsteinIndependenceErasedControl ∧ ¬PhysicalEinsteinSymmetricTransitionAt physicalEinsteinFactorTwoErasedControl := einstein_symmetric_independent_steps_retain_the_factor_two_variance.2 example : physicalEinstein1905TransitionReadout.physicalSpatialDimension = 3 ∧ physicalEinstein1905TransitionReadout.observedCoordinateDimension = 1 := by obtain ⟨space, observed, _, _⟩ := einstein_gaussian_is_a_one_coordinate_point_source_prediction exact ⟨space, observed⟩ example : ¬PhysicalEinsteinSymmetricTransitionAt physicalEinsteinThreeDimensionalProjectionClaimedControl ∧ ¬PhysicalEinsteinSymmetricTransitionAt physicalEinsteinPointSourceErasedControl := by obtain ⟨_, _, projection, pointSource⟩ := einstein_gaussian_is_a_one_coordinate_point_source_prediction exact ⟨projection, pointSource⟩ example : ¬PhysicalEinsteinOsmoticStokesBalanceAt physicalEinsteinOsmoticBalanceErasedControl ∧ ¬PhysicalEinsteinOsmoticStokesBalanceAt physicalEinsteinHydrodynamicRegimeErasedControl ∧ ¬PhysicalEinsteinOsmoticStokesBalanceAt physicalEinsteinMolecularCountErasedControl := einstein_osmotic_drag_and_molecular_count_are_independent_inputs.2 example : PhysicalSmoluchowskiPersistentFlightAt physicalSmoluchowski1906SmallParticleReadout := physical_smoluchowski_persistent_flight_is_nonvacuous.1 example : physicalSmoluchowski1906SmallParticleReadout.selectedMassDisplacementSquaredValue = 16 := physical_smoluchowski_persistent_flight_is_nonvacuous.2 example : physicalSmoluchowski1906SmallParticleReadout.finiteCollisionImbalanceRetained = true ∧ physicalSmoluchowski1906SmallParticleReadout.collisionImpulsesCancelExactly = false := ⟨smoluchowski_finite_imbalance_is_not_exact_impulse_cancellation.1, smoluchowski_finite_imbalance_is_not_exact_impulse_cancellation.2.1⟩ example : ¬PhysicalSmoluchowskiPersistentFlightAt physicalSmoluchowskiCollisionCancellationClaimedControl := smoluchowski_finite_imbalance_is_not_exact_impulse_cancellation.2.2 example : physicalSmoluchowski1906SmallParticleReadout.particleMassValue ≠ physicalSmoluchowski1906SmallParticleReadout.comparisonParticleMassValue ∧ physicalSmoluchowski1906SmallParticleReadout.selectedMassDisplacementSquaredValue = physicalSmoluchowski1906SmallParticleReadout.comparisonMassDisplacementSquaredValue := by obtain ⟨distinctMasses, equalSpread, _, _, _⟩ := smoluchowski_persistence_and_equipartition_cancel_particle_mass exact ⟨distinctMasses, equalSpread⟩ example : ¬PhysicalSmoluchowskiPersistentFlightAt physicalSmoluchowskiPersistenceErasedControl ∧ ¬PhysicalSmoluchowskiPersistentFlightAt physicalSmoluchowskiEquipartitionErasedControl ∧ ¬PhysicalSmoluchowskiPersistentFlightAt physicalSmoluchowskiMassCancellationErasedControl := by obtain ⟨_, _, persistence, equipartition, mass⟩ := smoluchowski_persistence_and_equipartition_cancel_particle_mass exact ⟨persistence, equipartition, mass⟩ example : ¬PhysicalSmoluchowskiPersistentFlightAt physicalSmoluchowskiExactMicroscopicPathClaimedControl := smoluchowski_persistent_flight_is_a_controlled_not_exact_model.2 example : PhysicalSmoluchowskiLargeParticleRelaxationAt physicalSmoluchowski1906LargeParticleReadout := physical_smoluchowski_large_particle_relaxation_is_nonvacuous.1 example : physicalSmoluchowski1906LargeParticleReadout.einsteinComparisonSquaredValue = 27 ∧ physicalSmoluchowski1906LargeParticleReadout.smoluchowskiComparisonSquaredValue = 64 := physical_smoluchowski_large_particle_relaxation_is_nonvacuous.2 example : 64 * physicalSmoluchowski1906LargeParticleReadout.einsteinComparisonSquaredValue = 27 * physicalSmoluchowski1906LargeParticleReadout.smoluchowskiComparisonSquaredValue ∧ physicalSmoluchowski1906LargeParticleReadout.einsteinComparisonSquaredValue < physicalSmoluchowski1906LargeParticleReadout.smoluchowskiComparisonSquaredValue := ⟨smoluchowski_large_particle_prefactor_differs_from_einstein.1, smoluchowski_large_particle_prefactor_differs_from_einstein.2.1⟩ example : ¬PhysicalSmoluchowskiLargeParticleRelaxationAt physicalSmoluchowskiRelaxationErasedControl ∧ ¬PhysicalSmoluchowskiLargeParticleRelaxationAt physicalSmoluchowskiEinsteinCoefficientConflatedControl := smoluchowski_large_particle_prefactor_differs_from_einstein.2.2 example : PhysicalLLMComovingHubbleNoiseAt physicalLLM1993ComovingDiffusionReadout := physical_llm_comoving_hubble_noise_is_nonvacuous.1 example : physicalLLM1993ComovingDiffusionReadout.diffusionCoefficientValue = 2 := physical_llm_comoving_hubble_noise_is_nonvacuous.2 example : physicalLLM1993ComovingDiffusionReadout.hubbleStepAmplitudeValue = 1 ∧ physicalLLM1993ComovingDiffusionReadout.whiteNoiseAmplitudeValue = 2 ∧ physicalLLM1993ComovingDiffusionReadout.varianceGrowthRateValue = 4 ∧ physicalLLM1993ComovingDiffusionReadout.diffusionCoefficientValue = 2 := by obtain ⟨step, noise, variance, diffusion, _, _⟩ := llm_hubble_kick_noise_and_diffusion_have_distinct_coefficients exact ⟨step, noise, variance, diffusion⟩ example : ¬PhysicalLLMComovingHubbleNoiseAt physicalLLMHubbleStepAmplitudeErasedControl ∧ ¬PhysicalLLMComovingHubbleNoiseAt physicalLLMVarianceFactorTwoErasedControl := by obtain ⟨_, _, _, _, step, variance⟩ := llm_hubble_kick_noise_and_diffusion_have_distinct_coefficients exact ⟨step, variance⟩ example : ¬PhysicalLLMComovingHubbleNoiseAt physicalLLMItoConventionSubstitutedControl ∧ ¬PhysicalLLMComovingHubbleNoiseAt physicalLLMComovingAndPhysicalMeasureConflatedControl := llm_stratonovich_convention_and_comoving_measure_are_independent.2 example : PhysicalLLMComovingHubbleNoiseAt physicalLLM1993ComovingDiffusionReadout ∧ ¬PhysicalLLMPhysicalVolumeBranchingAt physicalLLM1993ComovingDiffusionReadout := ⟨llm_comoving_diffusion_does_not_automatically_branch.1, llm_comoving_diffusion_does_not_automatically_branch.2.1⟩ example : PhysicalLLMPhysicalVolumeBranchingAt physicalLLM1993VolumeBranchingReadout := physical_llm_physical_volume_branching_is_nonvacuous.1 example : physicalLLM1993VolumeBranchingReadout.physicalVolumeBranchingContributionValue = 24 := physical_llm_physical_volume_branching_is_nonvacuous.2 example : physicalLLM1993VolumeBranchingReadout.comovingDensityDerivativeValue = 5 ∧ physicalLLM1993VolumeBranchingReadout.physicalVolumeDensityDerivativeValue = 29 := by obtain ⟨_, comoving, physical⟩ := llm_physical_volume_reproduction_adds_three_hubble_density exact ⟨comoving, physical⟩ example : ¬PhysicalLLMPhysicalVolumeBranchingAt physicalLLMClockIndependenceClaimedControl ∧ ¬PhysicalLLMComovingHubbleNoiseAt physicalLLMPlanckBoundaryResolvedControl := ⟨llm_clock_independence_and_planck_boundary_remain_unestablished.1, llm_clock_independence_and_planck_boundary_remain_unestablished.2.1⟩ example : physicalDiffusionPossibility.TentativeAt physicalDiffusionEstablishment .clockIndependentStationaryVolumeDistribution .lindeLindeMezhlumian1993StochasticInflation ∧ physicalDiffusionPossibility.TentativeAt physicalDiffusionEstablishment .resolvedPlanckBoundaryDynamics .lindeLindeMezhlumian1993StochasticInflation := llm_clock_independence_and_planck_boundary_remain_unestablished.2.2 example : physicalEinstein1905TransitionReadout.source ≠ physicalSmoluchowski1906SmallParticleReadout.source ∧ physicalSmoluchowski1906SmallParticleReadout.source ≠ physicalLLM1993ComovingDiffusionReadout.source := ⟨symmetric_transitions_persistent_flights_and_hubble_noise_are_distinct.1, symmetric_transitions_persistent_flights_and_hubble_noise_are_distinct.2.1⟩ example : ¬physicalDiffusionEstablishment.establishedAt .exactCollisionImpulseCancellation .smoluchowski1906PersistentCollision ∧ ¬physicalDiffusionEstablishment.establishedAt .exactSmoluchowskiEinsteinCoefficientAgreement .smoluchowski1906PersistentCollision ∧ ¬physicalDiffusionEstablishment.establishedAt .ordinaryComovingDiffusionAutomaticallyBranches .lindeLindeMezhlumian1993StochasticInflation := diffusion_source_overclaims_remain_epistemically_open.2.2 end BrownianDiffusionOriginalSourcePhysicalReuseAudit namespace EquilibriumSamplingOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.EquilibriumSamplingContext.Countermodels example : PhysicalMetropolisFinitePeriodicSetupAt physicalMetropolis1953SamplingReadout := physical_metropolis_finite_periodic_canonical_sampling_is_nonvacuous.1 example : physicalMetropolis1953SamplingReadout.finiteParticleCount = 224 := physical_metropolis_finite_periodic_canonical_sampling_is_nonvacuous.2 example : physicalMetropolis1953SamplingReadout.physicalSpatialDimension = 2 := metropolis_original_rigid_disk_run_has_224_particles_in_two_dimensions.2.1 example : PhysicalMetropolisAcceptanceBalanceAt physicalMetropolis1953SamplingReadout := physical_metropolis_boltzmann_accept_reject_balance_is_nonvacuous.1 example : physicalMetropolis1953SamplingReadout.downhillAcceptanceProbability = 1 := physical_metropolis_boltzmann_accept_reject_balance_is_nonvacuous.2 example : 0 < physicalMetropolis1953SamplingReadout.uphillAcceptanceProbability ∧ physicalMetropolis1953SamplingReadout.uphillAcceptanceProbability < 1 := ⟨metropolis_uphill_and_downhill_acceptance_branches_are_distinct.2.1, metropolis_uphill_and_downhill_acceptance_branches_are_distinct.2.2.1⟩ example : physicalMetropolis1953SamplingReadout.uphillAcceptanceProbability = Real.exp (-(physicalMetropolis1953SamplingReadout.interactionEnergyUphill - physicalMetropolis1953SamplingReadout.interactionEnergyOld) / physicalMetropolis1953SamplingReadout.uphillThermalEnergy) := by obtain ⟨_, _, _, _, _, _, _, _, _, _, factor, _⟩ := physical_metropolis_boltzmann_accept_reject_balance_is_nonvacuous.1 exact factor example : ¬PhysicalMetropolisAcceptanceBalanceAt physicalMetropolisUphillAcceptedAlwaysControl := metropolis_uphill_and_downhill_acceptance_branches_are_distinct.2.2.2 example : physicalMetropolis1953SamplingReadout.forwardProposalProbability = physicalMetropolis1953SamplingReadout.reverseProposalProbability := metropolis_pairwise_balance_requires_symmetric_reversible_proposal.1 example : ¬PhysicalMetropolisAcceptanceBalanceAt physicalMetropolisProposalAsymmetryControl := metropolis_pairwise_balance_requires_symmetric_reversible_proposal.2 example : PhysicalMetropolisRejectedProposalAccountingAt physicalMetropolis1953SamplingReadout := physical_metropolis_rejected_old_state_accounting_is_nonvacuous.1 example : physicalMetropolis1953SamplingReadout.actualPostRejectionConfiguration = .oldParticleConfiguration := physical_metropolis_rejected_old_state_accounting_is_nonvacuous.2 example : physicalMetropolis1953SamplingReadout.rejectedConfigurationCountContribution = 0 ∧ physicalMetropolis1953SamplingReadout.oldConfigurationRepeatedCountContribution = 1 := ⟨metropolis_rejected_trial_must_repeat_the_old_configuration.2.1, metropolis_rejected_trial_must_repeat_the_old_configuration.2.2.1⟩ example : ¬PhysicalMetropolisRejectedProposalAccountingAt physicalMetropolisRejectedStateDroppedControl ∧ ¬PhysicalMetropolisRejectedProposalAccountingAt physicalMetropolisRejectedConfigurationCountedControl := metropolis_rejected_trial_must_repeat_the_old_configuration.2.2.2 example : ¬PhysicalMetropolisFinitePeriodicSetupAt physicalMetropolisNoPeriodicWrapControl ∧ ¬PhysicalMetropolisFinitePeriodicSetupAt physicalMetropolisNoConfigurationAccessibilityControl := metropolis_periodic_wrap_and_accessibility_are_independent example : ¬PhysicalMetropolisFinitePeriodicSetupAt physicalMetropolisZeroParticleControl ∧ ¬PhysicalMetropolisFinitePeriodicSetupAt physicalMetropolisThreeDimensionalRunClaimedControl := metropolis_original_rigid_disk_run_has_224_particles_in_two_dimensions.2.2 example : PhysicalMetropolisFiniteNumericalEvidenceAt physicalMetropolis1953SamplingReadout := physical_metropolis_finite_numerical_evidence_is_nonvacuous.1 example : physicalMetropolis1953SamplingReadout.productionCyclesLowerBound = 48 ∧ physicalMetropolis1953SamplingReadout.productionCyclesUpperBound = 64 ∧ physicalMetropolis1953SamplingReadout.measurementCycleSpacing = 16 := ⟨metropolis_thermalization_and_production_cycles_are_distinct.2.1, metropolis_thermalization_and_production_cycles_are_distinct.2.2.1, metropolis_thermalization_and_production_cycles_are_distinct.2.2.2.1⟩ example : ¬PhysicalMetropolisFiniteNumericalEvidenceAt physicalMetropolisWrongProductionCycleCountControl := metropolis_thermalization_and_production_cycles_are_distinct.2.2.2.2 example : ¬PhysicalMetropolisFiniteNumericalEvidenceAt physicalMetropolisPhaseInterfaceErasedControl ∧ ¬PhysicalMetropolisFiniteNumericalEvidenceAt physicalMetropolisExactThermodynamicLimitClaimedControl := metropolis_finite_cycles_do_not_erase_phase_interfaces_or_limits.2.2 example : PhysicalCreutzSU2HeatBathAt physicalCreutz1980HeatBathReadout := physical_creutz_pure_su2_full_group_heat_bath_is_nonvacuous.1 example : physicalCreutz1980HeatBathReadout.gaugeMatrixDimension = 2 := physical_creutz_pure_su2_full_group_heat_bath_is_nonvacuous.2 example : physicalCreutz1980HeatBathReadout.latticeSpacetimeDimension = 4 ∧ physicalCreutz1980HeatBathReadout.gaugeGeneratorCount = 3 := ⟨creutz_finite_su2_lattices_are_not_continuum_quark_theory.1, creutz_finite_su2_lattices_are_not_continuum_quark_theory.2.2.1⟩ example : ¬PhysicalCreutzSU2HeatBathAt physicalCreutzMetropolisProposalConflatedControl ∧ ¬PhysicalCreutzSU2HeatBathAt physicalCreutzFullGroupErasedControl := creutz_single_link_heat_bath_is_not_metropolis_accept_reject.2.2 example : ¬PhysicalCreutzSU2HeatBathAt physicalCreutzConditionalWeightErasedControl ∧ ¬PhysicalCreutzSU2HeatBathAt physicalCreutzHaarMeasureErasedControl := creutz_conditional_update_requires_invariant_group_measure.2 example : physicalCreutz1980HeatBathReadout.gaugeFixingRequired = false ∧ ¬PhysicalCreutzSU2HeatBathAt physicalCreutzGaugeFixingRequiredControl := creutz_gauge_fixing_is_not_required_for_heat_bath_sampling example : PhysicalCreutzIndependentStartDiagnosisAt physicalCreutz1980HeatBathReadout := physical_creutz_independent_start_diagnosis_is_nonvacuous.1 example : physicalCreutz1980HeatBathReadout.empiricalThermalizationSweepCount = 20 := physical_creutz_independent_start_diagnosis_is_nonvacuous.2 example : physicalCreutz1980HeatBathReadout.orderedGaugeConfiguration ≠ physicalCreutz1980HeatBathReadout.randomizedGaugeConfiguration ∧ physicalCreutz1980HeatBathReadout.selectedFiniteRunExactlyConverged = false := ⟨creutz_ordered_and_random_starts_diagnose_not_prove_equilibrium.1, creutz_ordered_and_random_starts_diagnose_not_prove_equilibrium.2.2.1⟩ example : ¬PhysicalCreutzIndependentStartDiagnosisAt physicalCreutzStartComparisonErasedControl ∧ ¬PhysicalCreutzIndependentStartDiagnosisAt physicalCreutzExactFiniteRunClaimedControl := creutz_ordered_and_random_starts_diagnose_not_prove_equilibrium.2.2.2 example : PhysicalCreutzFiniteGaugeMeasurementAt physicalCreutz1980HeatBathReadout := physical_creutz_finite_loop_measurement_is_nonvacuous.1 example : physicalCreutz1980HeatBathReadout.sampledLoopSideLength = 5 := physical_creutz_finite_loop_measurement_is_nonvacuous.2 example : physicalCreutz1980HeatBathReadout.displayedIterationCount = 30 ∧ physicalCreutz1980HeatBathReadout.empiricalThermalizationSweepCount = 20 ∧ physicalCreutz1980HeatBathReadout.postEquilibrationFluctuationIterationCount = 5 := ⟨creutz_thermalization_and_five_iteration_error_are_distinct.1, creutz_thermalization_and_five_iteration_error_are_distinct.2.1, creutz_thermalization_and_five_iteration_error_are_distinct.2.2.1⟩ example : ¬PhysicalCreutzFiniteGaugeMeasurementAt physicalCreutzZeroStatisticalErrorControl := creutz_thermalization_and_five_iteration_error_are_distinct.2.2.2.2 example : ¬PhysicalCreutzSU2HeatBathAt physicalCreutzWrongLatticeDimensionControl ∧ ¬PhysicalCreutzSU2HeatBathAt physicalCreutzWrongGaugeMatrixDimensionControl := ⟨creutz_finite_su2_lattices_are_not_continuum_quark_theory.2.2.2.1, creutz_finite_su2_lattices_are_not_continuum_quark_theory.2.2.2.2.1⟩ example : ¬PhysicalCreutzFiniteGaugeMeasurementAt physicalCreutzContinuumConfinementClaimedControl ∧ ¬PhysicalCreutzFiniteGaugeMeasurementAt physicalCreutzDynamicalQuarksClaimedControl := creutz_finite_su2_lattices_are_not_continuum_quark_theory.2.2.2.2.2 example : physicalMetropolis1953SamplingReadout.source ≠ physicalCreutz1980HeatBathReadout.source ∧ physicalMetropolis1953SamplingReadout.mechanism ≠ physicalCreutz1980HeatBathReadout.mechanism := ⟨particle_accept_reject_and_compact_link_heat_bath_are_distinct.1, particle_accept_reject_and_compact_link_heat_bath_are_distinct.2.1⟩ example : physicalSamplingPossibility.TentativeAt physicalSamplingEstablishment .universalFiniteMixingTime .metropolisEtAl1953RigidDisk ∧ physicalSamplingPossibility.TentativeAt physicalSamplingEstablishment .thermodynamicLimitEquationOfState .metropolisEtAl1953RigidDisk := ⟨equilibrium_sampling_source_overclaims_remain_epistemically_open.1, equilibrium_sampling_source_overclaims_remain_epistemically_open.2.1⟩ example : ¬physicalSamplingEstablishment.establishedAt .exactFiniteRunEquilibrium .metropolisEtAl1953RigidDisk ∧ ¬physicalSamplingEstablishment.establishedAt .zeroMonteCarloStatisticalError .creutz1980PureSU2 := ⟨equilibrium_sampling_source_overclaims_remain_epistemically_open.2.2.2.1, equilibrium_sampling_source_overclaims_remain_epistemically_open.2.2.2.2.1⟩ end EquilibriumSamplingOriginalSourcePhysicalReuseAudit namespace StellarMassLimitOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.SelfGravitatingMassLimitContext.Countermodels example : PhysicalChandrasekharCompositionAt physicalChandrasekhar1931WhiteDwarfReadout := physical_chandrasekhar_ideal_white_dwarf_composition_is_nonvacuous.1 example : physicalChandrasekhar1931WhiteDwarfReadout.molecularWeightValue = (5 / 2 : Real) := physical_chandrasekhar_ideal_white_dwarf_composition_is_nonvacuous.2 example : physicalChandrasekhar1931WhiteDwarfReadout.electronDensityThresholdValue < physicalChandrasekhar1931WhiteDwarfReadout.electronNumberDensityValue := chandrasekhar_relativistic_density_threshold_depends_on_composition.2.1 example : ¬PhysicalChandrasekharCompositionAt physicalChandrasekharCompositionErasedControl ∧ ¬PhysicalChandrasekharCompositionAt physicalChandrasekharRelativisticThresholdErasedControl := chandrasekhar_relativistic_density_threshold_depends_on_composition.2.2 example : PhysicalChandrasekharIndexThreePolytropeAt physicalChandrasekhar1931WhiteDwarfReadout := physical_chandrasekhar_index_three_polytrope_is_nonvacuous.1 example : physicalChandrasekhar1931WhiteDwarfReadout.polytropicIndexValue = 3 := physical_chandrasekhar_index_three_polytrope_is_nonvacuous.2 example : physicalChandrasekhar1931WhiteDwarfReadout.equationOfStateExponentValue = (4 / 3 : Real) := chandrasekhar_four_thirds_pressure_requires_index_three_polytrope.1 example : ¬PhysicalChandrasekharIndexThreePolytropeAt physicalChandrasekharWrongPolytropicIndexControl ∧ ¬PhysicalChandrasekharIndexThreePolytropeAt physicalChandrasekharNonrelativisticExponentControl := chandrasekhar_four_thirds_pressure_requires_index_three_polytrope.2.2 example : PhysicalChandrasekharReportedMassAt physicalChandrasekhar1931WhiteDwarfReadout := physical_chandrasekhar_published_ideal_mass_is_nonvacuous.1 example : physicalChandrasekhar1931WhiteDwarfReadout.reportedMassInSolarUnitsValue = (91 / 100 : Real) := physical_chandrasekhar_published_ideal_mass_is_nonvacuous.2 example : physicalChandrasekhar1931WhiteDwarfReadout.exactMassInSolarUnitsValue = (911 / 1000 : Real) := chandrasekhar_original_mass_is_point_nine_one_not_modern_textbook_mass.2.1 example : ¬PhysicalChandrasekharReportedMassAt physicalChandrasekharTextbookMassSubstitutedControl ∧ ¬PhysicalChandrasekharReportedMassAt physicalChandrasekharExactSolarConversionClaimedControl := chandrasekhar_original_mass_is_point_nine_one_not_modern_textbook_mass.2.2 example : physicalChandrasekhar1931WhiteDwarfReadout.uniformDensityApproximationRetained = false := chandrasekhar_polytropic_mass_does_not_validate_uniform_density.1 example : physicalChandrasekhar1931WhiteDwarfReadout.reportedMassCoefficientValue < physicalChandrasekhar1931WhiteDwarfReadout.uniformDensityComparisonMassCoefficientValue := chandrasekhar_polytropic_mass_does_not_validate_uniform_density.2.1 example : ¬PhysicalChandrasekharIndexThreePolytropeAt physicalChandrasekharUniformDensityClaimedControl ∧ ¬PhysicalChandrasekharReportedMassAt physicalChandrasekharUniformStonerMassConflatedControl := chandrasekhar_polytropic_mass_does_not_validate_uniform_density.2.2 example : physicalChandrasekhar1931WhiteDwarfReadout.uniquePolytropicMassCalculated = true ∧ physicalChandrasekhar1931WhiteDwarfReadout.maximumMassInterpretationRetained = true := ⟨chandrasekhar_unique_mass_needs_separate_maximum_interpretation.1, chandrasekhar_unique_mass_needs_separate_maximum_interpretation.2.1⟩ example : ¬PhysicalChandrasekharIndexThreePolytropeAt physicalChandrasekharUpperLimitInterpretationErasedControl := chandrasekhar_unique_mass_needs_separate_maximum_interpretation.2.2 example : PhysicalOppenheimerVolkoffColdNeutronAt physicalOppenheimerVolkoff1939NeutronCoreReadout := physical_oppenheimer_volkoff_relativistic_neutron_core_is_nonvacuous.1 example : physicalOppenheimerVolkoff1939NeutronCoreReadout.regularInteriorEnclosedMassValue = 1 := physical_oppenheimer_volkoff_relativistic_neutron_core_is_nonvacuous.2 example : physicalOppenheimerVolkoff1939NeutronCoreReadout.generalRelativisticHydrostaticsRetained = true ∧ physicalOppenheimerVolkoff1939NeutronCoreReadout.interparticleForcesNeglected = true := ⟨oppenheimer_volkoff_requires_cold_noninteracting_relativistic_matter.1, oppenheimer_volkoff_requires_cold_noninteracting_relativistic_matter.2.1⟩ example : ¬PhysicalOppenheimerVolkoffColdNeutronAt physicalOppenheimerVolkoffNewtonianGravityControl ∧ ¬PhysicalOppenheimerVolkoffColdNeutronAt physicalOppenheimerVolkoffInteractingMatterConflatedControl := oppenheimer_volkoff_requires_cold_noninteracting_relativistic_matter.2.2 example : ¬PhysicalOppenheimerVolkoffColdNeutronAt physicalOppenheimerVolkoffMassDerivativeErasedControl ∧ ¬PhysicalOppenheimerVolkoffColdNeutronAt physicalOppenheimerVolkoffHydrostaticPressureErasedControl := oppenheimer_volkoff_coupled_mass_and_pressure_equations_are_independent.2 example : physicalOppenheimerVolkoff1939NeutronCoreReadout.centralEnclosedMassValue = 0 ∧ physicalOppenheimerVolkoff1939NeutronCoreReadout.boundaryPressureValue = 0 := ⟨oppenheimer_volkoff_regular_center_and_zero_pressure_surface_are_distinct.1, oppenheimer_volkoff_regular_center_and_zero_pressure_surface_are_distinct.2.1⟩ example : ¬PhysicalOppenheimerVolkoffColdNeutronAt physicalOppenheimerVolkoffCentralMassNonzeroControl ∧ ¬PhysicalOppenheimerVolkoffColdNeutronAt physicalOppenheimerVolkoffSurfacePressureNonzeroControl := oppenheimer_volkoff_regular_center_and_zero_pressure_surface_are_distinct.2.2 example : PhysicalOppenheimerVolkoffTurningPointAt physicalOppenheimerVolkoff1939NeutronCoreReadout := physical_oppenheimer_volkoff_numerical_turning_point_is_nonvacuous.1 example : physicalOppenheimerVolkoff1939NeutronCoreReadout.numericalMaximumMassSolarValue = (71 / 100 : Real) := physical_oppenheimer_volkoff_numerical_turning_point_is_nonvacuous.2 example : physicalOppenheimerVolkoff1939NeutronCoreReadout.approximateMaximumMassSolarValue = (7 / 10 : Real) ∧ physicalOppenheimerVolkoff1939NeutronCoreReadout.maximumMassRoundingResidualValue = (1 / 100 : Real) := ⟨oppenheimer_volkoff_table_mass_and_rounded_prose_are_distinct.2.1, oppenheimer_volkoff_table_mass_and_rounded_prose_are_distinct.2.2.1⟩ example : ¬PhysicalOppenheimerVolkoffTurningPointAt physicalOppenheimerVolkoffTableAndRoundedMassConflatedControl := oppenheimer_volkoff_table_mass_and_rounded_prose_are_distinct.2.2.2 example : physicalOppenheimerVolkoff1939NeutronCoreReadout.maximumMassRadiusKilometersValue = (95 / 10 : Real) ∧ physicalOppenheimerVolkoff1939NeutronCoreReadout.maximumRestMassSolarValue = (781 / 1000 : Real) := ⟨oppenheimer_volkoff_maximum_radius_and_baryonic_mass_are_retained.1, oppenheimer_volkoff_maximum_radius_and_baryonic_mass_are_retained.2.1⟩ example : ¬PhysicalOppenheimerVolkoffTurningPointAt physicalOppenheimerVolkoffMaximumRadiusErasedControl ∧ ¬PhysicalOppenheimerVolkoffTurningPointAt physicalOppenheimerVolkoffBaryonicMassConflatedControl := oppenheimer_volkoff_maximum_radius_and_baryonic_mass_are_retained.2.2 example : physicalOppenheimerVolkoff1939NeutronCoreReadout.lowMassEquilibriumCount = 1 ∧ physicalOppenheimerVolkoff1939NeutronCoreReadout.intermediateMassEquilibriumCount = 2 ∧ physicalOppenheimerVolkoff1939NeutronCoreReadout.supercriticalEquilibriumCount = 0 := ⟨oppenheimer_volkoff_turning_point_keeps_one_two_and_zero_static_branches.1, oppenheimer_volkoff_turning_point_keeps_one_two_and_zero_static_branches.2.1, oppenheimer_volkoff_turning_point_keeps_one_two_and_zero_static_branches.2.2.1⟩ example : ¬PhysicalOppenheimerVolkoffTurningPointAt physicalOppenheimerVolkoffTurningPointErasedControl := oppenheimer_volkoff_turning_point_keeps_one_two_and_zero_static_branches.2.2.2 example : physicalOppenheimerVolkoff1939NeutronCoreReadout.lowDensityBranchStable = true ∧ physicalOppenheimerVolkoff1939NeutronCoreReadout.condensedBranchStable = false := ⟨oppenheimer_volkoff_free_energy_stability_needs_continuity.1, oppenheimer_volkoff_free_energy_stability_needs_continuity.2.1⟩ example : ¬PhysicalOppenheimerVolkoffTurningPointAt physicalOppenheimerVolkoffFreeEnergyContinuityErasedControl ∧ ¬PhysicalOppenheimerVolkoffTurningPointAt physicalOppenheimerVolkoffCondensedBranchStableControl := oppenheimer_volkoff_free_energy_stability_needs_continuity.2.2 example : PhysicalOppenheimerVolkoffHighDensityScopeAt physicalOppenheimerVolkoff1939NeutronCoreReadout := physical_oppenheimer_volkoff_high_density_scope_is_nonvacuous.1 example : physicalOppenheimerVolkoff1939NeutronCoreReadout.maximumMassRadiusKilometersValue = (95 / 10 : Real) := physical_oppenheimer_volkoff_high_density_scope_is_nonvacuous.2 example : physicalOppenheimerVolkoff1939NeutronCoreReadout.nuclearDensityValue < physicalOppenheimerVolkoff1939NeutronCoreReadout.criticalCentralDensityValue := oppenheimer_volkoff_above_nuclear_density_requires_scope_qualification.1 example : ¬PhysicalOppenheimerVolkoffHighDensityScopeAt physicalOppenheimerVolkoffRepulsiveCorrectionErasedControl ∧ ¬PhysicalOppenheimerVolkoffHighDensityScopeAt physicalOppenheimerVolkoffUniversalActualMatterLimitControl := oppenheimer_volkoff_above_nuclear_density_requires_scope_qualification.2.2 example : physicalOppenheimerVolkoff1939NeutronCoreReadout.noStaticEquilibriumAboveIdealBound = true ∧ physicalOppenheimerVolkoff1939NeutronCoreReadout.finiteTimeCollapseEstablished = false ∧ physicalOppenheimerVolkoff1939NeutronCoreReadout.horizonFormationEstablished = false := ⟨oppenheimer_volkoff_no_static_solution_does_not_prove_horizon.1, oppenheimer_volkoff_no_static_solution_does_not_prove_horizon.2.1, oppenheimer_volkoff_no_static_solution_does_not_prove_horizon.2.2.1⟩ example : ¬PhysicalOppenheimerVolkoffHighDensityScopeAt physicalOppenheimerVolkoffFiniteTimeCollapseClaimedControl ∧ ¬PhysicalOppenheimerVolkoffHighDensityScopeAt physicalOppenheimerVolkoffHorizonClaimedControl := oppenheimer_volkoff_no_static_solution_does_not_prove_horizon.2.2.2 example : physicalChandrasekhar1931WhiteDwarfReadout.source ≠ physicalOppenheimerVolkoff1939NeutronCoreReadout.source ∧ physicalChandrasekhar1931WhiteDwarfReadout.mechanism ≠ physicalOppenheimerVolkoff1939NeutronCoreReadout.mechanism := ⟨electron_polytrope_and_relativistic_neutron_turning_point_are_distinct.1, electron_polytrope_and_relativistic_neutron_turning_point_are_distinct.2.1⟩ example : physicalStellarMassPossibility.TentativeAt physicalStellarMassEstablishment .modernCompositionAdjustedWhiteDwarfMaximum .chandrasekhar1931IdealWhiteDwarf ∧ physicalStellarMassPossibility.TentativeAt physicalStellarMassEstablishment .realisticInteractingNeutronMaximum .oppenheimerVolkoff1939IdealNeutronCore := ⟨stellar_mass_source_overclaims_remain_epistemically_open.1, stellar_mass_source_overclaims_remain_epistemically_open.2.1⟩ example : physicalStellarMassPossibility.TentativeAt physicalStellarMassEstablishment .finiteTimeDynamicalCollapse .oppenheimerVolkoff1939IdealNeutronCore ∧ physicalStellarMassPossibility.TentativeAt physicalStellarMassEstablishment .blackHoleOrHorizonFormation .oppenheimerVolkoff1939IdealNeutronCore := ⟨stellar_mass_source_overclaims_remain_epistemically_open.2.2.1, stellar_mass_source_overclaims_remain_epistemically_open.2.2.2.1⟩ example : ¬physicalStellarMassEstablishment.establishedAt .sourceReportsOnePointFourSolarMasses .chandrasekhar1931IdealWhiteDwarf ∧ ¬physicalStellarMassEstablishment.establishedAt .equationOfStateIndependentNeutronMaximum .oppenheimerVolkoff1939IdealNeutronCore := stellar_mass_source_overclaims_remain_epistemically_open.2.2.2.2 end StellarMassLimitOriginalSourcePhysicalReuseAudit namespace DefectUnbindingOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.DefectUnbindingTransitionContext.Countermodels example : PhysicalKTScalarVortexDielectricAt physicalKosterlitzThouless1973ScalarVortexReadout := physical_kt_scalar_vortex_dielectric_route_is_nonvacuous.1 example : physicalKosterlitzThouless1973ScalarVortexReadout.totalScalarCharge = 0 := physical_kt_scalar_vortex_dielectric_route_is_nonvacuous.2 example : physicalKosterlitzThouless1973ScalarVortexReadout.elementaryPositiveCharge = 1 ∧ physicalKosterlitzThouless1973ScalarVortexReadout.elementaryNegativeCharge = -1 := ⟨kt_scalar_neutrality_core_cutoff_and_dilute_pairs_are_independent.1, kt_scalar_neutrality_core_cutoff_and_dilute_pairs_are_independent.2.1⟩ example : ¬PhysicalKTScalarVortexDielectricAt physicalKTScalarChargeNeutralityErasedControl ∧ ¬PhysicalKTScalarVortexDielectricAt physicalKTCoreCutoffErasedControl := ⟨kt_scalar_neutrality_core_cutoff_and_dilute_pairs_are_independent.2.2.1, kt_scalar_neutrality_core_cutoff_and_dilute_pairs_are_independent.2.2.2.1⟩ example : ¬PhysicalKTScalarVortexDielectricAt physicalKTDilutePairApproximationErasedControl := kt_scalar_neutrality_core_cutoff_and_dilute_pairs_are_independent.2.2.2.2 example : physicalKosterlitzThouless1973ScalarVortexReadout.dielectricScreeningIterationRetained = true ∧ physicalKosterlitzThouless1973ScalarVortexReadout.laterUniversalStiffnessJumpClaimed = false := ⟨kt_iterated_dielectric_screening_does_not_derive_later_universal_jump.1, kt_iterated_dielectric_screening_does_not_derive_later_universal_jump.2.1⟩ example : ¬PhysicalKTScalarVortexDielectricAt physicalKTDielectricIterationErasedControl ∧ ¬PhysicalKTScalarVortexDielectricAt physicalKTLaterUniversalJumpClaimedControl := kt_iterated_dielectric_screening_does_not_derive_later_universal_jump.2.2 example : ¬PhysicalKTScalarVortexDielectricAt physicalKTChargedSuperconductorConflatedControl ∧ ¬PhysicalKTScalarVortexDielectricAt physicalKTVectorBurgersChargeConflatedControl := kt_neutral_vortices_are_not_charged_films_or_burgers_vectors example : PhysicalHNTriangularDislocationHexaticAt physicalHalperinNelson1978TriangularMeltingReadout := physical_hn_triangular_dislocation_hexatic_route_is_nonvacuous.1 example : physicalHalperinNelson1978TriangularMeltingReadout.triangularBurgersOrientationCount = 6 := physical_hn_triangular_dislocation_hexatic_route_is_nonvacuous.2 example : PhysicalHNDisclinationIsotropicAt physicalHalperinNelson1978TriangularMeltingReadout := physical_hn_independent_disclination_transition_is_nonvacuous.1 example : physicalHalperinNelson1978TriangularMeltingReadout.secondTransitionTemperatureValue = 3 := physical_hn_independent_disclination_transition_is_nonvacuous.2 example : physicalHalperinNelson1978TriangularMeltingReadout.angularInteractionWeight = physicalHalperinNelson1978TriangularMeltingReadout.radialInteractionWeight := hn_triangular_dislocations_keep_six_vector_orientations_and_angular_force.2.1 example : ¬PhysicalHNTriangularDislocationHexaticAt physicalHNSquareLatticeSubstitutedControl ∧ ¬PhysicalHNTriangularDislocationHexaticAt physicalHNVectorBurgersOrientationsCollapsedControl := ⟨hn_triangular_dislocations_keep_six_vector_orientations_and_angular_force.2.2.1, hn_triangular_dislocations_keep_six_vector_orientations_and_angular_force.2.2.2.1⟩ example : ¬PhysicalHNTriangularDislocationHexaticAt physicalHNAngularElasticInteractionErasedControl := hn_triangular_dislocations_keep_six_vector_orientations_and_angular_force.2.2.2.2 example : physicalHalperinNelson1978TriangularMeltingReadout.intermediatePrimaryCorrelation = .translationalExponential ∧ physicalHalperinNelson1978TriangularMeltingReadout.intermediateOrientationalCorrelation = .sixfoldOrientationalAlgebraic := ⟨hn_hexatic_loses_translation_while_sixfold_orientation_survives.1, hn_hexatic_loses_translation_while_sixfold_orientation_survives.2.1⟩ example : ¬PhysicalHNTriangularDislocationHexaticAt physicalHNHexaticPhaseErasedControl ∧ ¬PhysicalHNTriangularDislocationHexaticAt physicalHNTranslationalDecayErasedControl := ⟨hn_hexatic_loses_translation_while_sixfold_orientation_survives.2.2.1, hn_hexatic_loses_translation_while_sixfold_orientation_survives.2.2.2.1⟩ example : ¬PhysicalHNTriangularDislocationHexaticAt physicalHNOrientationalOrderDestroyedEarlyControl := hn_hexatic_loses_translation_while_sixfold_orientation_survives.2.2.2.2 example : physicalHalperinNelson1978TriangularMeltingReadout.screenedShearModulusValue = 0 ∧ physicalHalperinNelson1978TriangularMeltingReadout.criticalElasticCouplingValue = 16 * Real.pi := ⟨hn_melting_loses_shear_and_retains_original_sixteen_pi_threshold.1, hn_melting_loses_shear_and_retains_original_sixteen_pi_threshold.2.1⟩ example : ¬PhysicalHNTriangularDislocationHexaticAt physicalHNShearRigidityRetainedAfterMeltingControl ∧ ¬PhysicalHNTriangularDislocationHexaticAt physicalHNCriticalElasticNormalizationErasedControl := hn_melting_loses_shear_and_retains_original_sixteen_pi_threshold.2.2 example : PhysicalHNTriangularDislocationHexaticAt physicalHNDisclinationBridgeErasedControl ∧ ¬PhysicalHNDisclinationIsotropicAt physicalHNDisclinationBridgeErasedControl := ⟨hn_dislocation_transition_does_not_supply_separate_disclination_bridge.1, hn_dislocation_transition_does_not_supply_separate_disclination_bridge.2.1⟩ example : ¬PhysicalHNDisclinationIsotropicAt physicalHNDisclinationTransitionErasedControl ∧ ¬PhysicalHNDisclinationIsotropicAt physicalHNDislocationAndDisclinationConflatedControl := hn_dislocation_transition_does_not_supply_separate_disclination_bridge.2.2 example : PhysicalHNPeriodicSubstrateQualificationAt physicalHNPeriodicSubstrateReadout := hn_periodic_substrate_and_first_order_alternative_remain_distinct.1 example : ¬PhysicalHNTriangularDislocationHexaticAt physicalHNPeriodicSubstrateReadout ∧ ¬PhysicalHNPeriodicSubstrateQualificationAt physicalHNPeriodicSubstrateIgnoredControl := ⟨hn_periodic_substrate_and_first_order_alternative_remain_distinct.2.1, hn_periodic_substrate_and_first_order_alternative_remain_distinct.2.2.1⟩ example : ¬PhysicalHNTriangularDislocationHexaticAt physicalHNFirstOrderAlternativeExcludedControl := hn_periodic_substrate_and_first_order_alternative_remain_distinct.2.2.2 example : PhysicalYoungVectorCoulombScreeningAt physicalYoung1979VectorCoulombGasReadout := physical_young_vector_coulomb_screening_is_nonvacuous.1 example : physicalYoung1979VectorCoulombGasReadout.correlationExponentValue = (3696 / 10000 : Real) := physical_young_vector_coulomb_screening_is_nonvacuous.2 example : PhysicalYoungVectorChargeFusionAt physicalYoung1979VectorCoulombGasReadout := physical_young_vector_charge_fusion_is_nonvacuous.1 example : physicalYoung1979VectorCoulombGasReadout.normalizedQuadraticVectorFusionContributionValue = (1 / 20 : Real) := physical_young_vector_charge_fusion_is_nonvacuous.2 example : physicalYoung1979VectorCoulombGasReadout.triangularBurgersOrientationCount = 6 ∧ physicalYoung1979VectorCoulombGasReadout.squareLatticeIndependentElasticConstantCount = 3 := ⟨young_vector_charge_and_full_triangular_orientations_are_independent.1, young_vector_charge_and_full_triangular_orientations_are_independent.2.1⟩ example : ¬PhysicalYoungVectorCoulombScreeningAt physicalYoungVectorChargesCollapsedToScalarsControl ∧ ¬PhysicalYoungVectorCoulombScreeningAt physicalYoungSixBurgersOrientationsErasedControl := young_vector_charge_and_full_triangular_orientations_are_independent.2.2 example : physicalYoung1979VectorCoulombGasReadout.screenedLongitudinalElasticCouplingValue < physicalYoung1979VectorCoulombGasReadout.longitudinalElasticCouplingValue ∧ physicalYoung1979VectorCoulombGasReadout.screenedTransverseElasticCouplingValue < physicalYoung1979VectorCoulombGasReadout.transverseElasticCouplingValue := ⟨young_radial_angular_longitudinal_and_transverse_screening_are_distinct.2.1, young_radial_angular_longitudinal_and_transverse_screening_are_distinct.2.2.1⟩ example : ¬PhysicalYoungVectorCoulombScreeningAt physicalYoungAngularInteractionErasedControl ∧ ¬PhysicalYoungVectorCoulombScreeningAt physicalYoungLongitudinalScreeningErasedControl ∧ ¬PhysicalYoungVectorCoulombScreeningAt physicalYoungTransverseScreeningErasedControl := young_radial_angular_longitudinal_and_transverse_screening_are_distinct.2.2.2 example : physicalYoung1979VectorCoulombGasReadout.normalizedLinearFugacityContributionValue = (1 / 10 : Real) ∧ physicalYoung1979VectorCoulombGasReadout.normalizedQuadraticVectorFusionContributionValue = (1 / 20 : Real) ∧ physicalYoung1979VectorCoulombGasReadout.normalizedFugacityDerivativeValue = (3 / 20 : Real) := ⟨young_vector_fusion_adds_a_quadratic_fugacity_term.1, young_vector_fusion_adds_a_quadratic_fugacity_term.2.1, young_vector_fusion_adds_a_quadratic_fugacity_term.2.2.1⟩ example : ¬PhysicalYoungVectorChargeFusionAt physicalYoungVectorFusionErasedControl ∧ ¬PhysicalYoungVectorChargeFusionAt physicalYoungScalarRecursionSubstitutedControl := young_vector_fusion_adds_a_quadratic_fugacity_term.2.2.2 example : physicalYoung1979VectorCoulombGasReadout.correlationExponentValue ≠ physicalYoung1979VectorCoulombGasReadout.scalarComparisonCorrelationExponentValue := young_physical_exponent_is_not_scalar_half_or_exact_universal_number.2.1 example : ¬PhysicalYoungVectorCoulombScreeningAt physicalYoungScalarExponentSubstitutedControl ∧ ¬PhysicalYoungVectorCoulombScreeningAt physicalYoungExactUniversalExponentClaimedControl := young_physical_exponent_is_not_scalar_half_or_exact_universal_number.2.2 example : physicalYoung1979VectorCoulombGasReadout.screenedBulkModulusValue < physicalYoung1979VectorCoulombGasReadout.bareBulkModulusValue ∧ physicalYoung1979VectorCoulombGasReadout.vacancyInterstitialBulkContributionResolved = false := ⟨young_elastic_bulk_screening_does_not_compute_vacancy_contribution.1, young_elastic_bulk_screening_does_not_compute_vacancy_contribution.2.1⟩ example : ¬PhysicalYoungVectorCoulombScreeningAt physicalYoungBulkScreeningErasedControl ∧ ¬PhysicalYoungVectorCoulombScreeningAt physicalYoungVacancyBulkCalculationClaimedControl := young_elastic_bulk_screening_does_not_compute_vacancy_contribution.2.2 example : physicalYoung1979VectorCoulombGasReadout.disclinationPairsUnbindAtSecondTransition = false ∧ physicalYoung1979VectorCoulombGasReadout.dislocationCalculationAloneDerivesDisclinationTransition = false := ⟨young_dislocation_screening_does_not_derive_disclination_transition.1, young_dislocation_screening_does_not_derive_disclination_transition.2.1⟩ example : ¬PhysicalYoungVectorCoulombScreeningAt physicalYoungDisclinationDerivedFromDislocationsControl := young_dislocation_screening_does_not_derive_disclination_transition.2.2 example : physicalKosterlitzThouless1973ScalarVortexReadout.source ≠ physicalHalperinNelson1978TriangularMeltingReadout.source ∧ physicalHalperinNelson1978TriangularMeltingReadout.source ≠ physicalYoung1979VectorCoulombGasReadout.source := ⟨scalar_vortex_triangular_hexatic_and_vector_gas_are_distinct_routes.1, scalar_vortex_triangular_hexatic_and_vector_gas_are_distinct_routes.2.1⟩ example : physicalKosterlitzThouless1973ScalarVortexReadout.chargeKind ≠ physicalYoung1979VectorCoulombGasReadout.chargeKind ∧ physicalKosterlitzThouless1973ScalarVortexReadout.model.defect ≠ physicalYoung1979VectorCoulombGasReadout.model.defect := scalar_vortex_triangular_hexatic_and_vector_gas_are_distinct_routes.2.2 example : physicalDefectUnbindingPossibility.TentativeAt physicalDefectUnbindingEstablishment .universalKosterlitzJumpAlreadyDerivedIn1973 .kosterlitzThouless1973ScalarVortex ∧ physicalDefectUnbindingPossibility.TentativeAt physicalDefectUnbindingEstablishment .everyTwoDimensionalCrystalMeltsContinuously .halperinNelson1978TriangularMelting := ⟨defect_unbinding_source_overclaims_remain_epistemically_open.1, defect_unbinding_source_overclaims_remain_epistemically_open.2.1⟩ example : physicalDefectUnbindingPossibility.TentativeAt physicalDefectUnbindingEstablishment .squareLatticeHasTriangularElasticRecursion .young1979VectorCoulombGas ∧ physicalDefectUnbindingPossibility.TentativeAt physicalDefectUnbindingEstablishment .triangularDisclinationIsotropicTransition .young1979VectorCoulombGas := ⟨defect_unbinding_source_overclaims_remain_epistemically_open.2.2.1, defect_unbinding_source_overclaims_remain_epistemically_open.2.2.2.1⟩ example : physicalDefectUnbindingPossibility.TentativeAt physicalDefectUnbindingEstablishment .vacanciesAndInterstitialBulkContributionCalculated .young1979VectorCoulombGas ∧ ¬physicalDefectUnbindingEstablishment.establishedAt .scalarAndVectorRecursionsIdentical .young1979VectorCoulombGas := defect_unbinding_source_overclaims_remain_epistemically_open.2.2.2.2 end DefectUnbindingOriginalSourcePhysicalReuseAudit namespace SuperconductingGapOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.SelfConsistentSpectralGapContext.Countermodels example : PhysicalBCSPhononProjectedPairingAt physicalBCS1957ProjectedPairingReadout := physical_bcs_projected_phonon_pairing_is_nonvacuous.1 example : physicalBCS1957ProjectedPairingReadout.totalPairMomentum = 0 := physical_bcs_projected_phonon_pairing_is_nonvacuous.2 example : PhysicalBCSSingleGapAndPairThresholdAt physicalBCS1957ProjectedPairingReadout := physical_bcs_single_gap_and_pair_threshold_are_nonvacuous.1 example : physicalBCS1957ProjectedPairingReadout.pairBreakingThresholdValue = 2 := physical_bcs_single_gap_and_pair_threshold_are_nonvacuous.2 example : PhysicalBCSConditionalIsotopeScalingAt physicalBCS1957ProjectedPairingReadout := physical_bcs_conditional_isotope_scaling_is_nonvacuous.1 example : physicalBCS1957ProjectedPairingReadout.comparisonIsotopeMassValue = 4 := physical_bcs_conditional_isotope_scaling_is_nonvacuous.2 example : PhysicalBCSCoherenceAndApproximationScopeAt physicalBCS1957ProjectedPairingReadout := physical_bcs_distinct_coherence_cases_are_nonvacuous.1 example : physicalBCS1957ProjectedPairingReadout.coherenceCaseTwoSignValue = -1 := physical_bcs_distinct_coherence_cases_are_nonvacuous.2 example : physicalBCS1957ProjectedPairingReadout.selectedPairMomentum = 1 ∧ physicalBCS1957ProjectedPairingReadout.opposingPairMomentum = -1 ∧ physicalBCS1957ProjectedPairingReadout.totalPairSpin = 0 := ⟨bcs_phonon_attraction_coulomb_screening_and_pair_quantum_numbers_are_distinct.1, bcs_phonon_attraction_coulomb_screening_and_pair_quantum_numbers_are_distinct.2.1, bcs_phonon_attraction_coulomb_screening_and_pair_quantum_numbers_are_distinct.2.2.1⟩ example : ¬PhysicalBCSPhononProjectedPairingAt physicalBCSPhononAttractionErasedControl ∧ ¬PhysicalBCSPhononProjectedPairingAt physicalBCSScreenedCoulombRepulsionErasedControl := ⟨bcs_phonon_attraction_coulomb_screening_and_pair_quantum_numbers_are_distinct.2.2.2.1, bcs_phonon_attraction_coulomb_screening_and_pair_quantum_numbers_are_distinct.2.2.2.2.1⟩ example : ¬PhysicalBCSPhononProjectedPairingAt physicalBCSPairMomentumConflatedControl ∧ ¬PhysicalBCSPhononProjectedPairingAt physicalBCSPairSpinConflatedControl := bcs_phonon_attraction_coulomb_screening_and_pair_quantum_numbers_are_distinct.2.2.2.2.2 example : physicalBCS1957ProjectedPairingReadout.originalFixedParticleNumberProjectionRetained = true ∧ physicalBCS1957ProjectedPairingReadout.auxiliaryProductHasIndefiniteParticleNumber = true := ⟨bcs_original_fixed_number_projection_is_not_the_auxiliary_product.1, bcs_original_fixed_number_projection_is_not_the_auxiliary_product.2.1⟩ example : physicalBCS1957ProjectedPairingReadout.variationalGroundStateExactnessProved = false := bcs_original_fixed_number_projection_is_not_the_auxiliary_product.2.2.1 example : ¬PhysicalBCSPhononProjectedPairingAt physicalBCSFixedNumberProjectionErasedControl ∧ ¬PhysicalBCSPhononProjectedPairingAt physicalBCSAuxiliaryProductConflatedControl := ⟨bcs_original_fixed_number_projection_is_not_the_auxiliary_product.2.2.2.1, bcs_original_fixed_number_projection_is_not_the_auxiliary_product.2.2.2.2.1⟩ example : ¬PhysicalBCSPhononProjectedPairingAt physicalBCSExactVariationalStateClaimedControl := bcs_original_fixed_number_projection_is_not_the_auxiliary_product.2.2.2.2.2 example : physicalBCS1957ProjectedPairingReadout.zeroTemperatureGapValue = 1 ∧ physicalBCS1957ProjectedPairingReadout.pairBreakingThresholdValue = 2 := ⟨bcs_single_quasiparticle_gap_is_half_the_pair_breaking_threshold.1, bcs_single_quasiparticle_gap_is_half_the_pair_breaking_threshold.2.1⟩ example : physicalBCS1957ProjectedPairingReadout.reportedPairBreakingToCriticalRatioValue = (7 / 2 : Real) ∧ physicalBCS1957ProjectedPairingReadout.reportedSingleGapToCriticalRatioValue = (7 / 4 : Real) := ⟨bcs_single_quasiparticle_gap_is_half_the_pair_breaking_threshold.2.2.1, bcs_single_quasiparticle_gap_is_half_the_pair_breaking_threshold.2.2.2.1⟩ example : ¬PhysicalBCSSingleGapAndPairThresholdAt physicalBCSSingleGapAsPairThresholdControl ∧ ¬PhysicalBCSSingleGapAndPairThresholdAt physicalBCSWeakCouplingGapRatioDoubledControl := ⟨bcs_single_quasiparticle_gap_is_half_the_pair_breaking_threshold.2.2.2.2.1, bcs_single_quasiparticle_gap_is_half_the_pair_breaking_threshold.2.2.2.2.2.1⟩ example : ¬PhysicalBCSSingleGapAndPairThresholdAt physicalBCSExactUniversalGapRatioClaimedControl := bcs_single_quasiparticle_gap_is_half_the_pair_breaking_threshold.2.2.2.2.2.2 example : physicalBCS1957ProjectedPairingReadout.referenceIsotopeMassValue = 1 ∧ physicalBCS1957ProjectedPairingReadout.comparisonIsotopeMassValue = 4 := ⟨bcs_isotope_scaling_requires_fixed_electronic_pairing_factor.1, bcs_isotope_scaling_requires_fixed_electronic_pairing_factor.2.1⟩ example : physicalBCS1957ProjectedPairingReadout.referencePhononFrequencyValue = 2 ∧ physicalBCS1957ProjectedPairingReadout.comparisonPhononFrequencyValue = 1 := ⟨bcs_isotope_scaling_requires_fixed_electronic_pairing_factor.2.2.1, bcs_isotope_scaling_requires_fixed_electronic_pairing_factor.2.2.2.1⟩ example : ¬PhysicalBCSConditionalIsotopeScalingAt physicalBCSIsotopeElectronicFactorChangedControl ∧ ¬PhysicalBCSConditionalIsotopeScalingAt physicalBCSUnconditionalIsotopeLawClaimedControl := bcs_isotope_scaling_requires_fixed_electronic_pairing_factor.2.2.2.2 example : physicalBCS1957ProjectedPairingReadout.coherenceCaseOneSignValue = 1 ∧ physicalBCS1957ProjectedPairingReadout.coherenceCaseTwoSignValue = -1 := ⟨bcs_coherence_cases_and_qualified_current_argument_are_distinct.1, bcs_coherence_cases_and_qualified_current_argument_are_distinct.2.1⟩ example : physicalBCS1957ProjectedPairingReadout.exactPersistentCurrentProofSupplied = false := bcs_coherence_cases_and_qualified_current_argument_are_distinct.2.2.1 example : ¬PhysicalBCSCoherenceAndApproximationScopeAt physicalBCSCoherenceCasesCollapsedControl ∧ ¬PhysicalBCSCoherenceAndApproximationScopeAt physicalBCSCoherenceSignsCollapsedControl := ⟨bcs_coherence_cases_and_qualified_current_argument_are_distinct.2.2.2.1, bcs_coherence_cases_and_qualified_current_argument_are_distinct.2.2.2.2.1⟩ example : ¬PhysicalBCSCoherenceAndApproximationScopeAt physicalBCSExactPersistentCurrentClaimedControl ∧ ¬PhysicalBCSCoherenceAndApproximationScopeAt physicalBCSHigherPhononCorrectionsClaimedControl := bcs_coherence_cases_and_qualified_current_argument_are_distinct.2.2.2.2.2 example : physicalBCS1957ProjectedPairingReadout.gapGeneratedAtFinitePerturbativeOrder = false ∧ ¬PhysicalBCSPhononProjectedPairingAt physicalBCSFiniteOrderPerturbativeGapClaimedControl := bcs_nonperturbative_gap_does_not_follow_from_finite_order_pairing example : PhysicalNambuAnomalousQuasiparticleAt physicalNambu1960ConservingGapReadout := physical_nambu_anomalous_quasiparticle_gap_is_nonvacuous.1 example : physicalNambu1960ConservingGapReadout.sourceReportedGapValue = 2 := physical_nambu_anomalous_quasiparticle_gap_is_nonvacuous.2 example : PhysicalNambuMatchedGaugeWardIdentityAt physicalNambu1960ConservingGapReadout := physical_nambu_matched_matrix_ward_identity_is_nonvacuous.1 example : physicalNambu1960ConservingGapReadout.wardVertex01 = -2 := physical_nambu_matched_matrix_ward_identity_is_nonvacuous.2 example : PhysicalNambuNeutralCollectiveModeAt physicalNambu1960ConservingGapReadout := physical_nambu_neutral_collective_pair_is_nonvacuous.1 example : physicalNambu1960ConservingGapReadout.zeroMomentumCollectiveFrequencyValue = 0 := physical_nambu_neutral_collective_pair_is_nonvacuous.2 example : PhysicalNambuCoulombPlasmaShiftAt physicalNambu1960ConservingGapReadout := physical_nambu_coulomb_shifted_plasma_mode_is_nonvacuous.1 example : physicalNambu1960ConservingGapReadout.plasmaFrequencyValue = 2 := physical_nambu_coulomb_shifted_plasma_mode_is_nonvacuous.2 example : physicalNambu1960ConservingGapReadout.nambuMatrixDimension = 2 ∧ physicalNambu1960ConservingGapReadout.offDiagonalPairingSelfEnergyValue = 1 := ⟨nambu_anomalous_gap_requires_diagonal_and_off_diagonal_self_energy.1, nambu_anomalous_gap_requires_diagonal_and_off_diagonal_self_energy.2.1⟩ example : ¬PhysicalNambuAnomalousQuasiparticleAt physicalNambuOffDiagonalPairingErasedControl ∧ ¬PhysicalNambuAnomalousQuasiparticleAt physicalNambuDiagonalSelfEnergyErasedControl := ⟨nambu_anomalous_gap_requires_diagonal_and_off_diagonal_self_energy.2.2.2.1, nambu_anomalous_gap_requires_diagonal_and_off_diagonal_self_energy.2.2.2.2.1⟩ example : ¬PhysicalNambuAnomalousQuasiparticleAt physicalNambuSourceGapConflatedWithSingleGapControl := nambu_anomalous_gap_requires_diagonal_and_off_diagonal_self_energy.2.2.2.2.2 example : physicalNambu1960ConservingGapReadout.trueManyBodyStateHasFixedCharge = true ∧ physicalNambu1960ConservingGapReadout.quasiparticleRepresentativeIsChargeEigenstate = false := ⟨nambu_fixed_charge_physical_state_is_not_charge_eigenstate_quasiparticle.1, nambu_fixed_charge_physical_state_is_not_charge_eigenstate_quasiparticle.2.1⟩ example : physicalNambu1960ConservingGapReadout.quasiparticleParticleWeightValue = (1 / 2 : Real) ∧ physicalNambu1960ConservingGapReadout.quasiparticleHoleWeightValue = (1 / 2 : Real) := ⟨nambu_fixed_charge_physical_state_is_not_charge_eigenstate_quasiparticle.2.2.1, nambu_fixed_charge_physical_state_is_not_charge_eigenstate_quasiparticle.2.2.2.1⟩ example : ¬PhysicalNambuAnomalousQuasiparticleAt physicalNambuRepresentativeChargeEigenstateClaimedControl ∧ ¬PhysicalNambuAnomalousQuasiparticleAt physicalNambuTrueStateFixedChargeErasedControl := nambu_fixed_charge_physical_state_is_not_charge_eigenstate_quasiparticle.2.2.2.2 example : physicalNambu1960ConservingGapReadout.initialPairingComponentOneValue = 1 ∧ physicalNambu1960ConservingGapReadout.rotatedPairingComponentOneValue = 0 ∧ physicalNambu1960ConservingGapReadout.rotatedPairingComponentTwoValue = 1 := ⟨nambu_gauge_phase_rotation_keeps_pairing_radius_but_changes_components.1, nambu_gauge_phase_rotation_keeps_pairing_radius_but_changes_components.2.1, nambu_gauge_phase_rotation_keeps_pairing_radius_but_changes_components.2.2.1⟩ example : ¬PhysicalNambuMatchedGaugeWardIdentityAt physicalNambuGaugePhaseRotationErasedControl := nambu_gauge_phase_rotation_keeps_pairing_radius_but_changes_components.2.2.2 example : physicalNambu1960ConservingGapReadout.wardVertex01 = -2 ∧ physicalNambu1960ConservingGapReadout.wardVertex10 = 2 := ⟨nambu_matrix_ward_identity_retains_noncommuting_off_diagonal_vertex.1, nambu_matrix_ward_identity_retains_noncommuting_off_diagonal_vertex.2.1⟩ example : ¬PhysicalNambuMatchedGaugeWardIdentityAt physicalNambuWardOffDiagonalVertexErasedControl ∧ ¬PhysicalNambuMatchedGaugeWardIdentityAt physicalNambuMatchedVertexKernelErasedControl := nambu_matrix_ward_identity_retains_noncommuting_off_diagonal_vertex.2.2 example : physicalNambu1960ConservingGapReadout.bareLongitudinalKernelValue = 1 ∧ physicalNambu1960ConservingGapReadout.collectiveLongitudinalKernelValue = -1 := ⟨nambu_collective_longitudinal_cancellation_preserves_transverse_response.1, nambu_collective_longitudinal_cancellation_preserves_transverse_response.2.1⟩ example : physicalNambu1960ConservingGapReadout.correctedLongitudinalKernelValue = 0 ∧ physicalNambu1960ConservingGapReadout.correctedTransverseKernelValue = 2 := ⟨nambu_collective_longitudinal_cancellation_preserves_transverse_response.2.2.1, nambu_collective_longitudinal_cancellation_preserves_transverse_response.2.2.2.1⟩ example : ¬PhysicalNambuMatchedGaugeWardIdentityAt physicalNambuBareGaugeInvarianceClaimedControl ∧ ¬PhysicalNambuMatchedGaugeWardIdentityAt physicalNambuCollectiveLongitudinalCancellationErasedControl ∧ ¬PhysicalNambuMatchedGaugeWardIdentityAt physicalNambuTransverseResponseErasedControl := nambu_collective_longitudinal_cancellation_preserves_transverse_response.2.2.2.2 example : physicalNambu1960ConservingGapReadout.zeroTemperatureGapValue = 1 ∧ physicalNambu1960ConservingGapReadout.zeroMomentumCollectiveFrequencyValue = 0 := ⟨nambu_quasiparticle_gap_coexists_with_gapless_collective_spectrum.1, nambu_quasiparticle_gap_coexists_with_gapless_collective_spectrum.2.1⟩ example : ¬PhysicalNambuAnomalousQuasiparticleAt physicalNambuAllExcitationSpectraGappedControl ∧ ¬PhysicalNambuNeutralCollectiveModeAt physicalNambuNeutralCollectiveGapInsertedControl := nambu_quasiparticle_gap_coexists_with_gapless_collective_spectrum.2.2 example : physicalNambu1960ConservingGapReadout.finiteMomentumCollectiveFrequencySquaredValue = 3 ∧ ¬PhysicalNambuNeutralCollectiveModeAt physicalNambuCollectiveDispersionErasedControl := nambu_neutral_pair_dispersion_needs_its_fermi_velocity_coefficient example : physicalNambu1960ConservingGapReadout.zeroMomentumCollectiveFrequencyValue = 0 ∧ physicalNambu1960ConservingGapReadout.plasmaFrequencyValue = 2 := ⟨nambu_coulomb_polarization_shifts_neutral_mode_to_finite_plasma_frequency.1, nambu_coulomb_polarization_shifts_neutral_mode_to_finite_plasma_frequency.2.1⟩ example : ¬PhysicalNambuCoulombPlasmaShiftAt physicalNambuCoulombPolarizationErasedControl ∧ ¬PhysicalNambuCoulombPlasmaShiftAt physicalNambuPlasmaFrequencyConflatedWithNeutralModeControl := nambu_coulomb_polarization_shifts_neutral_mode_to_finite_plasma_frequency.2.2 example : physicalNambu1960ConservingGapReadout.correctedPhysicalResponseGaugeInvariant = true ∧ physicalNambu1960ConservingGapReadout.fullNonlinearGaugeResponseEstablished = false := ⟨nambu_linear_gauge_restoration_does_not_establish_full_nonlinear_response.1, nambu_linear_gauge_restoration_does_not_establish_full_nonlinear_response.2.1⟩ example : ¬PhysicalNambuMatchedGaugeWardIdentityAt physicalNambuFullNonlinearGaugeResponseClaimedControl := nambu_linear_gauge_restoration_does_not_establish_full_nonlinear_response.2.2 example : physicalBCS1957ProjectedPairingReadout.source ≠ physicalNambu1960ConservingGapReadout.source ∧ physicalBCS1957ProjectedPairingReadout.mechanism ≠ physicalNambu1960ConservingGapReadout.mechanism := ⟨bcs_number_projected_pairing_and_nambu_gauge_restoration_are_distinct.1, bcs_number_projected_pairing_and_nambu_gauge_restoration_are_distinct.2.1⟩ example : physicalBCS1957ProjectedPairingReadout.selectedState ≠ physicalNambu1960ConservingGapReadout.selectedState ∧ physicalBCS1957ProjectedPairingReadout.model.description ≠ physicalNambu1960ConservingGapReadout.model.description := bcs_number_projected_pairing_and_nambu_gauge_restoration_are_distinct.2.2 example : physicalSpectralGapPossibility.TentativeAt physicalSpectralGapEstablishment .unconditionalIsotopeLawAcrossArbitraryMetals .bardeenCooperSchrieffer1957ProjectedPairing ∧ physicalSpectralGapPossibility.TentativeAt physicalSpectralGapEstablishment .originalVariationalStateProvedExactEigenstate .bardeenCooperSchrieffer1957ProjectedPairing := ⟨spectral_gap_source_overclaims_remain_epistemically_open.1, spectral_gap_source_overclaims_remain_epistemically_open.2.1⟩ example : physicalSpectralGapPossibility.TentativeAt physicalSpectralGapEstablishment .persistentCurrentExactlyProvedWithoutBarrierAssumption .bardeenCooperSchrieffer1957ProjectedPairing ∧ physicalSpectralGapPossibility.TentativeAt physicalSpectralGapEstablishment .everyPhysicalExcitationSpectrumHasTheSameGap .nambu1960ConservingAnomalousSelfEnergy := ⟨spectral_gap_source_overclaims_remain_epistemically_open.2.2.1, spectral_gap_source_overclaims_remain_epistemically_open.2.2.2.1⟩ example : physicalSpectralGapPossibility.TentativeAt physicalSpectralGapEstablishment .fullNonlinearElectromagneticGaugeResponseProved .nambu1960ConservingAnomalousSelfEnergy ∧ ¬physicalSpectralGapEstablishment.establishedAt .quasiparticleRepresentativeIsChargeEigenstate .nambu1960ConservingAnomalousSelfEnergy := spectral_gap_source_overclaims_remain_epistemically_open.2.2.2.2 end SuperconductingGapOriginalSourcePhysicalReuseAudit namespace CosmologicalCollapseOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.CosmologicalPerturbationCollapseContext.Countermodels example : PhysicalPressSchechterMaximumVarianceCountingAt physicalPressSchechter1974CondensationReadout := physical_press_schechter_maximum_variance_gaussian_counting_is_nonvacuous.1 example : physicalPressSchechter1974CondensationReadout.selectedMassVarianceValue = 4 := physical_press_schechter_maximum_variance_gaussian_counting_is_nonvacuous.2 example : PhysicalPressSchechterAccretionCompletedSpectrumAt physicalPressSchechter1974CondensationReadout := physical_press_schechter_separate_accretion_and_spectrum_are_nonvacuous.1 example : physicalPressSchechter1974CondensationReadout.completedCollapsedMassFractionValue = 2 * physicalPressSchechter1974CondensationReadout.rawGaussianTailFractionValue := physical_press_schechter_separate_accretion_and_spectrum_are_nonvacuous.2 example : PhysicalPressSchechterFiniteBootstrapAndSeedScopeAt physicalPressSchechter1974CondensationReadout := physical_press_schechter_finite_bootstrap_and_approximate_seed_are_nonvacuous.1 example : physicalPressSchechter1974CondensationReadout.maximumVarianceParticleCount = 1000 ∧ physicalPressSchechter1974CondensationReadout.orderedParticleCount = 1000 := physical_press_schechter_finite_bootstrap_and_approximate_seed_are_nonvacuous.2 example : PhysicalWeinbergNonlinearPositiveVacuumTurnaroundAt physicalWeinberg1987ConditionalVacuumBoundReadout := physical_weinberg_nonlinear_positive_vacuum_turnaround_is_nonvacuous.1 example : physicalWeinberg1987ConditionalVacuumBoundReadout.nonlinearVacuumUpperBoundValue = 5000 := physical_weinberg_nonlinear_positive_vacuum_turnaround_is_nonvacuous.2 example : PhysicalWeinbergConditionalAnthropicEnvelopeAt physicalWeinberg1987ConditionalVacuumBoundReadout := physical_weinberg_conditional_anthropic_envelope_is_nonvacuous.1 example : physicalWeinberg1987ConditionalVacuumBoundReadout.observerProbabilityMeasureDerived = false := physical_weinberg_conditional_anthropic_envelope_is_nonvacuous.2 example : PhysicalWeinbergRejectedLinearAndObservationalScopeAt physicalWeinberg1987ConditionalVacuumBoundReadout := physical_weinberg_rejected_linear_estimate_and_observational_scope_are_nonvacuous.1 example : physicalWeinberg1987ConditionalVacuumBoundReadout.nonlinearVacuumUpperBoundValue < physicalWeinberg1987ConditionalVacuumBoundReadout.rejectedLinearVacuumUpperBoundValue := physical_weinberg_rejected_linear_estimate_and_observational_scope_are_nonvacuous.2 example : PhysicalPressSchechterMaximumVarianceCountingAt physicalPressSchechterRawTailWithoutAccretionControl ∧ ¬PhysicalPressSchechterAccretionCompletedSpectrumAt physicalPressSchechterRawTailWithoutAccretionControl := press_schechter_gaussian_tail_does_not_supply_underdense_accretion example : ¬PhysicalPressSchechterAccretionCompletedSpectrumAt physicalPressSchechterErasedAccretionFactorControl := press_schechter_accretion_factor_cannot_be_erased example : physicalPressSchechter1974CondensationReadout.originalSphericalCollapseThresholdValue = physicalPressSchechter1974CondensationReadout.initialExpansionScaleValue / physicalPressSchechter1974CondensationReadout.laterExpansionScaleValue := press_schechter_original_threshold_is_not_modern_delta_c.1 example : ¬PhysicalPressSchechterAccretionCompletedSpectrumAt physicalPressSchechterModernThresholdInsertedControl := press_schechter_original_threshold_is_not_modern_delta_c.2 example : ¬PhysicalPressSchechterMaximumVarianceCountingAt physicalPressSchechterGaussianSymmetryErasedControl := press_schechter_centered_gaussian_symmetry_cannot_be_erased example : ¬PhysicalPressSchechterMaximumVarianceCountingAt physicalPressSchechterVarianceVolumeScalingErasedControl := press_schechter_maximum_variance_volume_scaling_cannot_be_erased example : ¬PhysicalPressSchechterFiniteBootstrapAndSeedScopeAt physicalPressSchechterOrderedCorrectionRetractedControl := press_schechter_ordered_case_correction_cannot_be_retracted example : physicalPressSchechter1974CondensationReadout.boundaryOverlapErrorControlledExactly = false := press_schechter_overlap_accounting_is_not_rigorous.1 example : ¬PhysicalPressSchechterAccretionCompletedSpectrumAt physicalPressSchechterExactOverlapClaimedControl := press_schechter_overlap_accounting_is_not_rigorous.2 example : physicalPressSchechter1974CondensationReadout.numericalBootstrapTentative = true ∧ physicalPressSchechter1974CondensationReadout.numericalInfiniteParticleConvergenceEstablished = false := ⟨press_schechter_finite_nbody_bootstrap_does_not_prove_continuum_convergence.1, press_schechter_finite_nbody_bootstrap_does_not_prove_continuum_convergence.2.1⟩ example : ¬PhysicalPressSchechterFiniteBootstrapAndSeedScopeAt physicalPressSchechterInfiniteSimulationClaimedControl := press_schechter_finite_nbody_bootstrap_does_not_prove_continuum_convergence.2.2 example : physicalPressSchechter1974CondensationReadout.initialIsothermalSeedMassSolarValue = 30000000 ∧ physicalPressSchechter1974CondensationReadout.seedOriginExplained = false := ⟨press_schechter_approximate_seed_mass_does_not_explain_seed_origin.1, press_schechter_approximate_seed_mass_does_not_explain_seed_origin.2.1⟩ example : ¬PhysicalPressSchechterFiniteBootstrapAndSeedScopeAt physicalPressSchechterSeedOriginExplainedControl := press_schechter_approximate_seed_mass_does_not_explain_seed_origin.2.2 example : physicalWeinberg1987ConditionalVacuumBoundReadout.nonlinearCollapseNumeratorValue / physicalWeinberg1987ConditionalVacuumBoundReadout.nonlinearCollapseDenominatorValue = (500 / 729 : Real) := weinberg_exact_nonlinear_coefficient_rejects_linear_substitution.1 example : physicalWeinberg1987ConditionalVacuumBoundReadout.nonlinearCollapseNumeratorValue / physicalWeinberg1987ConditionalVacuumBoundReadout.nonlinearCollapseDenominatorValue = PhysicsLogic.Papers.Weinberg1987.nonlinearCollapseCoefficient := by simpa [PhysicsLogic.Papers.Weinberg1987.nonlinearCollapseCoefficient] using weinberg_exact_nonlinear_coefficient_rejects_linear_substitution.1 example : ¬PhysicalWeinbergNonlinearPositiveVacuumTurnaroundAt physicalWeinbergWrongLinearCoefficientControl := weinberg_exact_nonlinear_coefficient_rejects_linear_substitution.2 example : physicalWeinberg1987ConditionalVacuumBoundReadout.minimumPerturbedMatterDensityValue = 2 * physicalWeinberg1987ConditionalVacuumBoundReadout.vacuumEnergyDensityValue := weinberg_turnaround_requires_twice_vacuum_matter_density.1 example : ¬PhysicalWeinbergNonlinearPositiveVacuumTurnaroundAt physicalWeinbergVacuumTurnaroundMinimumErasedControl := weinberg_turnaround_requires_twice_vacuum_matter_density.2 example : ¬PhysicalWeinbergConditionalAnthropicEnvelopeAt physicalWeinbergMissingVacuumIndependentEnvelopeControl := weinberg_anthropic_bound_requires_vacuum_independent_early_envelope example : physicalWeinberg1987ConditionalVacuumBoundReadout.primaryDistributionRole ≠ physicalWeinberg1987ConditionalVacuumBoundReadout.comparisonDistributionRole := weinberg_perturbation_envelope_is_not_intrinsic_vacuum_prior.1 example : ¬PhysicalWeinbergConditionalAnthropicEnvelopeAt physicalWeinbergPerturbationAndPriorConflatedControl := weinberg_perturbation_envelope_is_not_intrinsic_vacuum_prior.2 example : physicalWeinberg1987ConditionalVacuumBoundReadout.boundStructureNecessaryForObservers = true ∧ physicalWeinberg1987ConditionalVacuumBoundReadout.observerProbabilityMeasureDerived = false := ⟨weinberg_anthropic_condition_does_not_derive_observer_measure.1, weinberg_anthropic_condition_does_not_derive_observer_measure.2.1⟩ example : ¬PhysicalWeinbergConditionalAnthropicEnvelopeAt physicalWeinbergObserverMeasureClaimedControl := weinberg_anthropic_condition_does_not_derive_observer_measure.2.2 example : physicalWeinberg1987ConditionalVacuumBoundReadout.boundStructureNecessaryForObservers = true ∧ physicalWeinberg1987ConditionalVacuumBoundReadout.boundStructureSufficientForObservers = false := ⟨weinberg_bound_structure_is_necessary_not_biologically_sufficient.1, weinberg_bound_structure_is_necessary_not_biologically_sufficient.2.1⟩ example : ¬PhysicalWeinbergConditionalAnthropicEnvelopeAt physicalWeinbergObserversMadeSufficientControl := weinberg_bound_structure_is_necessary_not_biologically_sufficient.2.2 example : physicalWeinberg1987ConditionalVacuumBoundReadout.observationalVacuumDensityValue < physicalWeinberg1987ConditionalVacuumBoundReadout.nonlinearVacuumUpperBoundValue := weinberg_upper_bound_does_not_predict_observed_vacuum_density.1 example : ¬PhysicalWeinbergRejectedLinearAndObservationalScopeAt physicalWeinbergObservedVacuumPredictedControl := weinberg_upper_bound_does_not_predict_observed_vacuum_density.2 example : ¬PhysicalWeinbergRejectedLinearAndObservationalScopeAt physicalWeinbergNegativeVacuumIncludedControl := weinberg_positive_vacuum_argument_does_not_cover_negative_or_recollapsing_cases.1 example : ¬PhysicalWeinbergRejectedLinearAndObservationalScopeAt physicalWeinbergRecollapsingUniverseIncludedControl := weinberg_positive_vacuum_argument_does_not_cover_negative_or_recollapsing_cases.2 example : physicalPressSchechter1974CondensationReadout.source ≠ physicalWeinberg1987ConditionalVacuumBoundReadout.source ∧ physicalPressSchechter1974CondensationReadout.mechanism ≠ physicalWeinberg1987ConditionalVacuumBoundReadout.mechanism := ⟨press_schechter_and_weinberg_collapse_mechanisms_are_physically_distinct.1, press_schechter_and_weinberg_collapse_mechanisms_are_physically_distinct.2.1⟩ example : physicalPressSchechter1974CondensationReadout.primaryDistributionRole ≠ physicalWeinberg1987ConditionalVacuumBoundReadout.primaryDistributionRole ∧ physicalPressSchechter1974CondensationReadout.inferenceRole ≠ physicalWeinberg1987ConditionalVacuumBoundReadout.inferenceRole := press_schechter_and_weinberg_collapse_mechanisms_are_physically_distinct.2.2 example : physicalCosmologicalCollapsePossibility.TentativeAt physicalCosmologicalCollapseEstablishment .rigorousBoundaryOverlapOrCloudAccounting .pressSchechter1974SelfSimilarCondensation ∧ physicalCosmologicalCollapsePossibility.TentativeAt physicalCosmologicalCollapseEstablishment .infiniteParticleNumericalConvergenceProved .pressSchechter1974SelfSimilarCondensation := ⟨cosmological_collapse_source_overclaims_remain_epistemically_open.1, cosmological_collapse_source_overclaims_remain_epistemically_open.2.1⟩ example : physicalCosmologicalCollapsePossibility.TentativeAt physicalCosmologicalCollapseEstablishment .exactGalaxySeedOriginExplained .pressSchechter1974SelfSimilarCondensation ∧ physicalCosmologicalCollapsePossibility.TentativeAt physicalCosmologicalCollapseEstablishment .anthropicUpperBoundPredictsObservedVacuumDensity .weinberg1987PositiveVacuumBound := ⟨cosmological_collapse_source_overclaims_remain_epistemically_open.2.2.1, cosmological_collapse_source_overclaims_remain_epistemically_open.2.2.2.1⟩ example : physicalCosmologicalCollapsePossibility.TentativeAt physicalCosmologicalCollapseEstablishment .observerProbabilityMeasureDerived .weinberg1987PositiveVacuumBound ∧ physicalCosmologicalCollapsePossibility.TentativeAt physicalCosmologicalCollapseEstablishment .negativeVacuumOrRecollapsingCasesCovered .weinberg1987PositiveVacuumBound := cosmological_collapse_source_overclaims_remain_epistemically_open.2.2.2.2 example : PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework.Countermodels.HasGaussianTailCountedAt PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework.Countermodels.rawGaussianTailWithoutDoubling ∧ ¬PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework.Countermodels.HasFactorTwoMassNormalizationAt PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework.Countermodels.rawGaussianTailWithoutDoubling := PhysicsLogic.Papers.PressSchechter1974.PressSchechterFramework.Countermodels.gaussian_tail_counting_does_not_supply_the_factor_two_normalization example : PhysicalPressSchechterFiniteBootstrapAndSeedScopeAt physicalPressSchechter1974CondensationReadout ∧ PhysicalWeinbergRejectedLinearAndObservationalScopeAt physicalWeinberg1987ConditionalVacuumBoundReadout := ⟨cosmological_collapse_sources_distinguish_gaussian_accretion_nonlinear_vacuum_and_anthropic_scope.1, cosmological_collapse_sources_distinguish_gaussian_accretion_nonlinear_vacuum_and_anthropic_scope.2.1⟩ example : physicalPressSchechter1974CondensationReadout.completedCollapsedMassFractionValue = 2 * physicalPressSchechter1974CondensationReadout.rawGaussianTailFractionValue ∧ physicalWeinberg1987ConditionalVacuumBoundReadout.minimumPerturbedMatterDensityValue = 2 * physicalWeinberg1987ConditionalVacuumBoundReadout.vacuumEnergyDensityValue := ⟨cosmological_collapse_sources_distinguish_gaussian_accretion_nonlinear_vacuum_and_anthropic_scope.2.2.1, cosmological_collapse_sources_distinguish_gaussian_accretion_nonlinear_vacuum_and_anthropic_scope.2.2.2.1⟩ example : physicalWeinberg1987ConditionalVacuumBoundReadout.nonlinearVacuumUpperBoundValue < physicalWeinberg1987ConditionalVacuumBoundReadout.rejectedLinearVacuumUpperBoundValue ∧ ¬physicalCosmologicalCollapseEstablishment.establishedAt .observerProbabilityMeasureDerived .weinberg1987PositiveVacuumBound := cosmological_collapse_sources_distinguish_gaussian_accretion_nonlinear_vacuum_and_anthropic_scope.2.2.2.2 end CosmologicalCollapseOriginalSourcePhysicalReuseAudit namespace EffectiveRangeOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.LowEnergyEffectiveRangeContext.Countermodels example : PhysicalBetheExactRadialComparisonAt physicalBethe1949RadialEffectiveRangeReadout := physical_bethe_exact_radial_comparison_is_nonvacuous.1 example : physicalBethe1949RadialEffectiveRangeReadout.tripletEffectiveRangeValue = (39 / 25 : Real) := physical_bethe_exact_radial_comparison_is_nonvacuous.2 example : PhysicalBetheUnitarityAndDeuteronExtractionAt physicalBethe1949RadialEffectiveRangeReadout := physical_bethe_unitarity_and_deuteron_extraction_are_nonvacuous.1 example : physicalBethe1949RadialEffectiveRangeReadout.reportedTripletRangeUncertaintyValue = (13 / 100 : Real) := physical_bethe_unitarity_and_deuteron_extraction_are_nonvacuous.2 example : PhysicalBetheCoulombAndConditionalChargeScopeAt physicalBethe1949RadialEffectiveRangeReadout := physical_bethe_conditional_coulomb_charge_comparison_is_nonvacuous.1 example : physicalBethe1949RadialEffectiveRangeReadout.chargedNeutralDifferenceValue = (1 / 50 : Real) := physical_bethe_conditional_coulomb_charge_comparison_is_nonvacuous.2 example : PhysicalBlattJacksonStationaryQuadraticErrorAt physicalBlattJackson1949StationaryEffectiveRangeReadout := physical_blatt_jackson_stationary_quadratic_error_is_nonvacuous.1 example : physicalBlattJackson1949StationaryEffectiveRangeReadout.firstVariationValue = 0 := physical_blatt_jackson_stationary_quadratic_error_is_nonvacuous.2 example : PhysicalBlattJacksonShapeFamilyAndBoundStateAt physicalBlattJackson1949StationaryEffectiveRangeReadout := physical_blatt_jackson_shape_family_and_deuteron_pole_are_nonvacuous.1 example : physicalBlattJackson1949StationaryEffectiveRangeReadout.conventionalWellShapeCount = 4 := physical_blatt_jackson_shape_family_and_deuteron_pole_are_nonvacuous.2 example : physicalBethe1949RadialEffectiveRangeReadout.omittedFourthOrderPhaseRemainderValue ≠ 0 := effective_range_phase_law_retains_nonzero_shape_sensitive_fourth_order_term.1 example : ¬PhysicalNeutralTwoParameterPhaseLawAt physicalNeutralPhaseFourthOrderErasedControl := effective_range_phase_law_retains_nonzero_shape_sensitive_fourth_order_term.2 example : ¬PhysicalNeutralTwoParameterPhaseLawAt physicalNeutralScatteringLengthSignReversedControl := effective_range_zero_energy_phase_fixes_scattering_length_sign example : ¬PhysicalBetheExactRadialComparisonAt physicalBetheWronskianOverlapErasedControl := bethe_exact_radial_identity_requires_selected_overlap_integral example : physicalBethe1949RadialEffectiveRangeReadout.interiorReplacementErrorValue ≠ 0 := bethe_exact_comparison_does_not_make_interior_replacement_exact.1 example : ¬PhysicalBetheExactRadialComparisonAt physicalBetheExactInteriorReplacementClaimedControl := bethe_exact_comparison_does_not_make_interior_replacement_exact.2 example : physicalBethe1949RadialEffectiveRangeReadout.nextCrossSectionTermNeededNearMaximum = true := bethe_unitarity_maximum_requires_next_cross_section_term.1 example : ¬PhysicalBetheUnitarityAndDeuteronExtractionAt physicalBetheCrossSectionNearMaximumTruncatedControl := bethe_unitarity_maximum_requires_next_cross_section_term.2 example : ¬PhysicalBetheUnitarityAndDeuteronExtractionAt physicalBetheBoundStateRangeReplacementErasedControl := bethe_bound_state_relation_requires_controlled_range_replacement example : physicalBethe1949RadialEffectiveRangeReadout.indirectDeuteronBindingEnergyKeVValue = 2208 ∧ physicalBethe1949RadialEffectiveRangeReadout.directDeuteronBindingEnergyKeVValue = 2237 := ⟨bethe_discrepant_deuteron_binding_measurements_remain_unresolved.1, bethe_discrepant_deuteron_binding_measurements_remain_unresolved.2.1⟩ example : ¬PhysicalBetheUnitarityAndDeuteronExtractionAt physicalBetheDiscrepantBindingMeasurementsResolvedControl := bethe_discrepant_deuteron_binding_measurements_remain_unresolved.2.2 example : physicalBethe1949RadialEffectiveRangeReadout.protonCoulombCorrectionValue ≠ 0 := bethe_proton_channel_requires_nonzero_coulomb_phase_correction.1 example : ¬PhysicalBetheCoulombAndConditionalChargeScopeAt physicalBetheCoulombCorrectionErasedControl := bethe_proton_channel_requires_nonzero_coulomb_phase_correction.2 example : physicalBethe1949RadialEffectiveRangeReadout.approximateChargeIndependenceRetained = true ∧ physicalBethe1949RadialEffectiveRangeReadout.exactChargeIndependenceEstablished = false := ⟨bethe_charge_comparison_is_approximate_not_exact_charge_independence.1, bethe_charge_comparison_is_approximate_not_exact_charge_independence.2.1⟩ example : ¬PhysicalBetheCoulombAndConditionalChargeScopeAt physicalBetheExactChargeIndependenceClaimedControl := bethe_charge_comparison_is_approximate_not_exact_charge_independence.2.2 example : ¬PhysicalBetheCoulombAndConditionalChargeScopeAt physicalBetheBorrowedSingletRangeAssumptionErasedControl := bethe_singlet_range_borrowing_requires_force_equality_assumption example : ¬PhysicalBlattJacksonStationaryQuadraticErrorAt physicalBlattJacksonStationarityErasedControl := blatt_jackson_stationarity_requires_zero_first_variation example : ¬PhysicalBlattJacksonStationaryQuadraticErrorAt physicalBlattJacksonQuadraticTrialErrorErasedControl := blatt_jackson_trial_phase_error_is_quadratic_not_linear example : ¬PhysicalBlattJacksonStationaryQuadraticErrorAt physicalBlattJacksonUpperBoundClaimedControl := blatt_jackson_stationary_scattering_functional_supplies_no_bound_direction.1 example : ¬PhysicalBlattJacksonStationaryQuadraticErrorAt physicalBlattJacksonLowerBoundClaimedControl := blatt_jackson_stationary_scattering_functional_supplies_no_bound_direction.2 example : physicalBlattJackson1949StationaryEffectiveRangeReadout.originalTrialNormalizationValue ≠ physicalBlattJackson1949StationaryEffectiveRangeReadout.rescaledTrialNormalizationValue := blatt_jackson_stationary_functional_is_trial_normalization_independent.1 example : ¬PhysicalBlattJacksonStationaryQuadraticErrorAt physicalBlattJacksonNormalizationDependenceControl := blatt_jackson_stationary_functional_is_trial_normalization_independent.2 example : physicalBlattJackson1949StationaryEffectiveRangeReadout.squareWellEffectiveRangeValue = physicalBlattJackson1949StationaryEffectiveRangeReadout.yukawaEffectiveRangeValue := blatt_jackson_two_parameter_fit_does_not_select_potential_shape.1 example : physicalBlattJackson1949StationaryEffectiveRangeReadout.squareWellIntrinsicRangeValue ≠ physicalBlattJackson1949StationaryEffectiveRangeReadout.yukawaIntrinsicRangeValue := blatt_jackson_two_parameter_fit_does_not_select_potential_shape.2.1 example : ¬PhysicalBlattJacksonShapeFamilyAndBoundStateAt physicalBlattJacksonPotentialShapeIdentifiedControl := blatt_jackson_two_parameter_fit_does_not_select_potential_shape.2.2 example : ¬PhysicalBlattJacksonShapeFamilyAndBoundStateAt physicalBlattJacksonIntrinsicAndEffectiveRangeConflatedControl := blatt_jackson_intrinsic_range_is_not_the_effective_range example : physicalBlattJackson1949StationaryEffectiveRangeReadout.singletRangeUncertaintyValue > physicalBlattJackson1949StationaryEffectiveRangeReadout.acceptedSingletRangeUncertaintyValue := blatt_jackson_triplet_extraction_does_not_determine_singlet_range.1 example : ¬PhysicalBlattJacksonShapeFamilyAndBoundStateAt physicalBlattJacksonSingletRangeDeterminedControl := blatt_jackson_triplet_extraction_does_not_determine_singlet_range.2 example : physicalBlattJackson1949StationaryEffectiveRangeReadout.tensorForceIncluded = false ∧ physicalBlattJackson1949StationaryEffectiveRangeReadout.tensorForceCorrectionsCalculated = false := ⟨blatt_jackson_central_force_calculation_does_not_include_tensor_corrections.1, blatt_jackson_central_force_calculation_does_not_include_tensor_corrections.2.1⟩ example : ¬PhysicalBlattJacksonShapeFamilyAndBoundStateAt physicalBlattJacksonTensorForceComputedControl := blatt_jackson_central_force_calculation_does_not_include_tensor_corrections.2.2 example : ¬PhysicalBlattJacksonShapeFamilyAndBoundStateAt physicalBlattJacksonProtonCoulombChannelClaimedControl := blatt_jackson_neutral_paper_does_not_analyze_proton_coulomb_channel example : physicalBlattJackson1949StationaryEffectiveRangeReadout.interpolationRelativeErrorValue = (1 / 5000 : Real) := blatt_jackson_interpolation_accuracy_is_scoped_to_reported_domain.1 example : ¬PhysicalBlattJacksonShapeFamilyAndBoundStateAt physicalBlattJacksonInterpolationOutsideDomainControl := blatt_jackson_interpolation_accuracy_is_scoped_to_reported_domain.2 example : PhysicalNeutralTwoParameterPhaseLawAt physicalBethe1949RadialEffectiveRangeReadout ∧ PhysicalNeutralTwoParameterPhaseLawAt physicalBlattJackson1949StationaryEffectiveRangeReadout := ⟨bethe_and_blatt_jackson_share_phase_law_but_not_derivation.1, bethe_and_blatt_jackson_share_phase_law_but_not_derivation.2.1⟩ example : physicalBethe1949RadialEffectiveRangeReadout.source ≠ physicalBlattJackson1949StationaryEffectiveRangeReadout.source ∧ physicalBethe1949RadialEffectiveRangeReadout.derivation ≠ physicalBlattJackson1949StationaryEffectiveRangeReadout.derivation := ⟨bethe_and_blatt_jackson_share_phase_law_but_not_derivation.2.2.1, bethe_and_blatt_jackson_share_phase_law_but_not_derivation.2.2.2.1⟩ example : physicalBethe1949RadialEffectiveRangeReadout.selectedWave ≠ physicalBlattJackson1949StationaryEffectiveRangeReadout.selectedWave := bethe_and_blatt_jackson_share_phase_law_but_not_derivation.2.2.2.2 example : physicalEffectiveRangePossibility.TentativeAt physicalEffectiveRangeEstablishment .uniquePotentialShapeFromTwoParameters .bethe1949ExactRadialComparison ∧ physicalEffectiveRangePossibility.TentativeAt physicalEffectiveRangeEstablishment .exactChargeIndependence .bethe1949ExactRadialComparison := ⟨effective_range_source_overclaims_remain_epistemically_open.1, effective_range_source_overclaims_remain_epistemically_open.2.1⟩ example : physicalEffectiveRangePossibility.TentativeAt physicalEffectiveRangeEstablishment .discrepantDeuteronBindingMeasurementsResolved .bethe1949ExactRadialComparison ∧ physicalEffectiveRangePossibility.TentativeAt physicalEffectiveRangeEstablishment .stationaryFunctionalUpperBound .blattJackson1949StationarySchwingerFunctional := ⟨effective_range_source_overclaims_remain_epistemically_open.2.2.1, effective_range_source_overclaims_remain_epistemically_open.2.2.2.1⟩ example : physicalEffectiveRangePossibility.TentativeAt physicalEffectiveRangeEstablishment .stationaryFunctionalLowerBound .blattJackson1949StationarySchwingerFunctional ∧ physicalEffectiveRangePossibility.TentativeAt physicalEffectiveRangeEstablishment .singletRangeDeterminedFromTripletData .blattJackson1949StationarySchwingerFunctional := ⟨effective_range_source_overclaims_remain_epistemically_open.2.2.2.2.1, effective_range_source_overclaims_remain_epistemically_open.2.2.2.2.2.1⟩ example : physicalEffectiveRangePossibility.TentativeAt physicalEffectiveRangeEstablishment .tensorForceEffectsComputed .blattJackson1949StationarySchwingerFunctional := effective_range_source_overclaims_remain_epistemically_open.2.2.2.2.2.2 example : PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.indefinite.StationaryAt (PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.reference 0) ∧ ¬PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.UpperBoundForEveryTrial PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.indefinite (PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.reference 0) ∧ ¬PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.LowerBoundForEveryTrial PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.indefinite (PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.reference 0) := PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework.Countermodels.stationary_functional_does_not_give_a_bound_direction example : PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.LowEnergyTrialControlled ∧ ¬PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.TwoParameterLawExactAtAllOrders := PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.the_controlled_family_is_not_exact_at_all_orders example : PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.PointwiseFreeEquationAndBoundaryValues ∧ ¬PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.discontinuousFreeFamily.RegularZeroEnergyRadialEquation := PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.Controls.pointwise_free_equation_and_boundary_values_do_not_supply_a_regular_radial_wave example : PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.NoGeneralBoundDirection := PhysicsLogic.Papers.BlattJackson1949.SchwingerFunctional.the_variational_template_does_not_supply_a_general_bound -- Pinned Lean elaboration of the dependent submodule carrier. set_option backward.isDefEq.respectTransparency false in example {T : PhysicsLogic.Papers.BlattJackson1949.BlattJacksonTypes} (framework : PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework T) (foundation : framework.EstablishedVariationalFoundation) : (framework.schwingerFunctionalAt framework.selectedMomentum).StationaryAt (framework.standingWaveProfile framework.exactScatteringWave) ∧ (framework.schwingerFunctionalAt framework.selectedMomentum).value (framework.standingWaveProfile framework.exactScatteringWave) = framework.phaseFunctionValue framework.neutronProtonTriplet framework.selectedLowEnergy := framework.established_variational_foundation_identifies_the_selected_phase foundation example : PhysicalBetheCoulombAndConditionalChargeScopeAt physicalBethe1949RadialEffectiveRangeReadout ∧ PhysicalBlattJacksonShapeFamilyAndBoundStateAt physicalBlattJackson1949StationaryEffectiveRangeReadout := ⟨effective_range_sources_distinguish_radial_identity_stationary_error_coulomb_shapes_and_scope.1, effective_range_sources_distinguish_radial_identity_stationary_error_coulomb_shapes_and_scope.2.1⟩ example : physicalBethe1949RadialEffectiveRangeReadout.omittedFourthOrderPhaseRemainderValue ≠ 0 ∧ physicalBlattJackson1949StationaryEffectiveRangeReadout.firstTrialPhaseValue > physicalBlattJackson1949StationaryEffectiveRangeReadout.exactVariationalPhaseValue := ⟨effective_range_sources_distinguish_radial_identity_stationary_error_coulomb_shapes_and_scope.2.2.1, effective_range_sources_distinguish_radial_identity_stationary_error_coulomb_shapes_and_scope.2.2.2.1⟩ example : physicalBlattJackson1949StationaryEffectiveRangeReadout.secondTrialPhaseValue < physicalBlattJackson1949StationaryEffectiveRangeReadout.exactVariationalPhaseValue ∧ ¬physicalEffectiveRangeEstablishment.establishedAt .tensorForceEffectsComputed .blattJackson1949StationarySchwingerFunctional := effective_range_sources_distinguish_radial_identity_stationary_error_coulomb_shapes_and_scope.2.2.2.2 end EffectiveRangeOriginalSourcePhysicalReuseAudit namespace PartitionZeroOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.PartitionZeroPhaseBoundaryContext.Countermodels example : UniformFinitePartitionZeroFreeAt physicalLeeYangBoundaryModel .leeYangAnalyticNeighborhood := physical_yang_lee_analytic_region_is_uniformly_zero_free example : PhysicalYangLeePositiveFinitePolynomialAt physicalYangLee1952PartitionZeroReadout := physical_yang_lee_positive_finite_polynomial_is_nonvacuous.1 example : physicalYangLee1952PartitionZeroReadout.finitePositivePartitionValue = 12 := physical_yang_lee_positive_finite_polynomial_is_nonvacuous.2 example : PhysicalYangLeeFerromagneticCircleAndPinchingAt physicalYangLee1952PartitionZeroReadout := physical_yang_lee_ferromagnetic_circle_and_conjugate_pinching_are_nonvacuous.1 example : physicalYangLee1952PartitionZeroReadout.upperRootImaginaryValue = (4 / 5 : Real) := physical_yang_lee_ferromagnetic_circle_and_conjugate_pinching_are_nonvacuous.2 example : PhysicalYangLeeDimensionalRootDensityAndAnalyticityAt physicalYangLee1952PartitionZeroReadout := physical_yang_lee_dimensional_density_and_analyticity_are_nonvacuous.1 example : physicalYangLee1952PartitionZeroReadout.normalizedTwoDimensionalRootDensityValue = (1 / 2 : Real) := physical_yang_lee_dimensional_density_and_analyticity_are_nonvacuous.2 example : PhysicalBBCKKAssumptionCContourControlAt physicalBBCKK2004ContourPartitionZeroReadout := physical_bbckk_assumption_c_contour_control_is_nonvacuous.1 example : physicalBBCKK2004ContourPartitionZeroReadout.torusVolumeValue = 100 := physical_bbckk_assumption_c_contour_control_is_nonvacuous.2 example : PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKK2004ContourPartitionZeroReadout := physical_bbckk_periodic_phase_sum_and_imported_theorem_are_nonvacuous.1 example : physicalBBCKK2004ContourPartitionZeroReadout.phaseSumRemainderValue = (1 / 10000 : Real) := physical_bbckk_periodic_phase_sum_and_imported_theorem_are_nonvacuous.2 example : PhysicalBBCKKTwoPhaseInterferenceAndZeroLocalizationAt physicalBBCKK2004ContourPartitionZeroReadout := physical_bbckk_two_phase_interference_and_zero_localization_are_nonvacuous.1 example : physicalBBCKK2004ContourPartitionZeroReadout.actualZeroEquationDisplacementValue = (1 / 10000 : Real) := physical_bbckk_two_phase_interference_and_zero_localization_are_nonvacuous.2 example : PhysicalBBCKKMultiplePointSymmetryAndScopeAt physicalBBCKK2004ContourPartitionZeroReadout := physical_bbckk_multiple_point_symmetry_and_scope_are_nonvacuous.1 example : physicalBBCKK2004ContourPartitionZeroReadout.maximumMultiplePointZeroMultiplicityValue = 2 := physical_bbckk_multiple_point_symmetry_and_scope_are_nonvacuous.2 example : ¬PhysicalYangLeePositiveFinitePolynomialAt physicalYangLeePositiveCoefficientErasedControl := yang_lee_positive_finite_polynomial_requires_positive_coefficients example : ¬physicalLeeYangBoundaryModel.finitePartitionZeroAt .latticeGasFiniteVolume .positiveActivityTransition := yang_lee_positive_physical_activity_has_no_finite_partition_zero.1 example : physicalLeeYangBoundaryModel.finitePartitionZeroAt .latticeGasFiniteVolume .upperComplexFiniteZero := yang_lee_positive_physical_activity_has_no_finite_partition_zero.2.1 example : ¬PhysicalYangLeePositiveFinitePolynomialAt physicalYangLeeFinitePhysicalRootClaimedControl := yang_lee_positive_physical_activity_has_no_finite_partition_zero.2.2 example : ¬physicalLeeYangBoundaryModel.physicalControl .upperComplexFiniteZero := yang_lee_complex_roots_are_not_positive_real_physical_controls.1 example : ¬PhysicalYangLeePositiveFinitePolynomialAt physicalYangLeeComplexRootDeclaredPhysicalControl := yang_lee_complex_roots_are_not_positive_real_physical_controls.2 example : ¬PhysicalYangLeeFerromagneticCircleAndPinchingAt physicalYangLeeConjugateRootCollapsedControl := yang_lee_circle_roots_must_form_distinct_complex_conjugate_pairs example : ¬PhysicalYangLeeFerromagneticCircleAndPinchingAt physicalYangLeeUnitCircleErasedControl := yang_lee_ferromagnetic_roots_retain_the_unit_circle example : ¬PhysicalYangLeeFerromagneticCircleAndPinchingAt physicalYangLeeLargerVolumeRootDoesNotApproachControl := yang_lee_pinch_requires_volume_dependent_approach_to_real_activity example : ¬PhysicalYangLeeFerromagneticCircleAndPinchingAt physicalYangLeeFerromagneticSignErasedControl := yang_lee_circle_theorem_requires_ferromagnetic_interaction_sign.1 example : ¬PhysicalYangLeeFerromagneticCircleAndPinchingAt physicalYangLeeAntiferromagneticCircleClaimedControl := yang_lee_circle_theorem_requires_ferromagnetic_interaction_sign.2 example : ¬PhysicalYangLeeFerromagneticCircleAndPinchingAt physicalYangLeeCircleAloneImpliesTransitionControl := yang_lee_unit_circle_location_alone_does_not_force_transition example : ¬PhysicalYangLeeDimensionalRootDensityAndAnalyticityAt physicalYangLeeOneDimensionalGapErasedControl := yang_lee_one_dimensional_arc_gap_prevents_finite_temperature_transition.1 example : ¬PhysicalYangLeeDimensionalRootDensityAndAnalyticityAt physicalYangLeeOneDimensionalTransitionClaimedControl := yang_lee_one_dimensional_arc_gap_prevents_finite_temperature_transition.2 example : physicalYangLee1952PartitionZeroReadout.gasPressureValue = physicalYangLee1952PartitionZeroReadout.liquidPressureValue := yang_lee_two_dimensional_density_jump_requires_nonzero_root_density.1 example : physicalYangLee1952PartitionZeroReadout.gasDensityValue < physicalYangLee1952PartitionZeroReadout.liquidDensityValue := yang_lee_two_dimensional_density_jump_requires_nonzero_root_density.2.1 example : ¬PhysicalYangLeeDimensionalRootDensityAndAnalyticityAt physicalYangLeeTwoDimensionalRootDensityErasedControl := yang_lee_two_dimensional_density_jump_requires_nonzero_root_density.2.2 example : ¬PhysicalYangLeeDimensionalRootDensityAndAnalyticityAt physicalYangLeeDiscontinuousPressureClaimedControl := yang_lee_first_order_transition_keeps_pressure_continuous example : physicalYangLee1952PartitionZeroReadout.positiveAnalyticActivityValue < physicalYangLee1952PartitionZeroReadout.positivePhysicalActivityValue ∧ physicalLeeYangBoundaryModel.physicalControl .positiveActivityAnalytic := ⟨yang_lee_uniform_zero_free_segment_is_distinct_from_transition_point.1, yang_lee_uniform_zero_free_segment_is_distinct_from_transition_point.2.1⟩ example : ¬physicalLeeYangBoundaryModel.finitePartitionZeroAt .latticeGasFiniteVolume .positiveActivityAnalytic ∧ ¬physicalLeeYangBoundaryModel.phaseBoundaryAt .positiveActivityAnalytic := ⟨yang_lee_uniform_zero_free_segment_is_distinct_from_transition_point.2.2.1, yang_lee_uniform_zero_free_segment_is_distinct_from_transition_point.2.2.2.1⟩ example : ¬physicalLeeYangBoundaryModel.contains .leeYangAnalyticNeighborhood .upperComplexFiniteZero ∧ ¬PhysicalYangLeeDimensionalRootDensityAndAnalyticityAt physicalYangLeeAnalyticSegmentAtTransitionCollapsedControl := yang_lee_uniform_zero_free_segment_is_distinct_from_transition_point.2.2.2.2 example : ¬PhysicalYangLeeDimensionalRootDensityAndAnalyticityAt physicalYangLeeDerivativeLimitExchangeUncontrolledControl := yang_lee_analytic_derivative_exchange_needs_uniform_zero_free_region example : ¬PhysicalYangLeeDimensionalRootDensityAndAnalyticityAt physicalYangLeeFullFiniteFieldSolutionClaimedControl := yang_lee_zero_field_results_do_not_solve_full_finite_field_model example : ¬PhysicalBBCKKAssumptionCContourControlAt physicalBBCKKDimensionOneClaimedControl := bbckk_contour_theorem_requires_dimension_at_least_two example : ¬PhysicalBBCKKAssumptionCContourControlAt physicalBBCKKNonperiodicBoundaryClaimedControl := bbckk_contour_theorem_requires_periodic_low_temperature_tori.1 example : ¬PhysicalBBCKKAssumptionCContourControlAt physicalBBCKKHighTemperatureClaimedControl := bbckk_contour_theorem_requires_periodic_low_temperature_tori.2 example : ¬PhysicalBBCKKAssumptionCContourControlAt physicalBBCKKHolomorphicWeightErasedControl := bbckk_assumption_c_requires_holomorphy_peierls_and_nondegeneracy.1 example : ¬PhysicalBBCKKAssumptionCContourControlAt physicalBBCKKPeierlsSuppressionErasedControl := bbckk_assumption_c_requires_holomorphy_peierls_and_nondegeneracy.2.1 example : ¬PhysicalBBCKKAssumptionCContourControlAt physicalBBCKKPairwiseNondegeneracyErasedControl := bbckk_assumption_c_requires_holomorphy_peierls_and_nondegeneracy.2.2 example : ¬PhysicalBBCKKAssumptionCContourControlAt physicalBBCKKMultiplePointConvexityErasedControl := bbckk_multiple_phase_points_require_strict_convexity example : ¬PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKKImportedPreviousTheoremErasedControl := bbckk_zero_localization_requires_imported_previous_theorem example : physicalBBCKK2004ContourPartitionZeroReadout.partitionValueAtInterferenceEquationValue ≠ 0 := bbckk_phase_sum_keeps_nonzero_controlled_remainder.1 example : ¬PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKKPhaseSumRemainderErasedControl := bbckk_phase_sum_keeps_nonzero_controlled_remainder.2 example : physicalBBCKK2004ContourPartitionZeroReadout.finitePhaseWeightSuppressionExponentValue = (20 : Real) ∧ physicalBBCKK2004ContourPartitionZeroReadout.phaseSumRemainderSuppressionExponentValue = (10 : Real) := ⟨bbckk_phase_weight_and_remainder_keep_distinct_source_exponents.1, bbckk_phase_weight_and_remainder_keep_distinct_source_exponents.2.1⟩ example : ¬PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKKFiniteWeightExponentCollapsedControl ∧ ¬PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKKRemainderExponentCollapsedControl := bbckk_phase_weight_and_remainder_keep_distinct_source_exponents.2.2 example : ¬PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKKPhaseSumAloneLocatesZerosClaimedControl := bbckk_phase_sum_alone_does_not_locate_actual_partition_zeros example : ¬PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKKPhaseSumDerivativeControlErasedControl := bbckk_periodic_phase_sum_requires_controlled_derivatives example : ¬PhysicalBBCKKTwoPhaseInterferenceAndZeroLocalizationAt physicalBBCKKEvenInterferenceClaimedControl := bbckk_two_phase_roots_require_odd_destructive_interference example : physicalBBCKK2004ContourPartitionZeroReadout.inverseVolumeScaleValue = (1 / 100 : Real) := bbckk_coexistence_spacing_has_inverse_volume_scale.1 example : ¬PhysicalBBCKKTwoPhaseInterferenceAndZeroLocalizationAt physicalBBCKKInverseVolumeScaleCollapsedControl := bbckk_coexistence_spacing_has_inverse_volume_scale.2 example : ¬PhysicalBBCKKTwoPhaseInterferenceAndZeroLocalizationAt physicalBBCKKTwoPhaseExponentialErrorErasedControl := bbckk_two_phase_localization_requires_exponentially_small_error example : physicalBBCKK2004ContourPartitionZeroReadout.maximumMultiplePointZeroMultiplicityValue = 2 ∧ physicalBBCKK2004ContourPartitionZeroReadout.multiplePointStablePhaseCountValue = 3 := ⟨bbckk_multiple_point_zero_multiplicity_is_stable_phase_count_minus_one.1, bbckk_multiple_point_zero_multiplicity_is_stable_phase_count_minus_one.2.1⟩ example : ¬PhysicalBBCKKMultiplePointSymmetryAndScopeAt physicalBBCKKMultiplicityStableCountClaimedControl := bbckk_multiple_point_zero_multiplicity_is_stable_phase_count_minus_one.2.2 example : physicalBBCKK2004ContourPartitionZeroReadout.twoPhaseExponentialErrorBoundValue < physicalBBCKK2004ContourPartitionZeroReadout.multiplePointErrorBoundValue := bbckk_multiple_point_errors_are_weaker_than_two_phase_errors.1 example : ¬PhysicalBBCKKMultiplePointSymmetryAndScopeAt physicalBBCKKMultiplePointExponentialPrecisionClaimedControl := bbckk_multiple_point_errors_are_weaker_than_two_phase_errors.2 example : physicalBBCKK2004ContourPartitionZeroReadout.perturbedRootSquaredModulusValue ≠ 1 := bbckk_exact_unit_circle_requires_unbroken_plus_minus_symmetry.1 example : ¬PhysicalBBCKKMultiplePointSymmetryAndScopeAt physicalBBCKKPerturbedExactCircleClaimedControl := bbckk_exact_unit_circle_requires_unbroken_plus_minus_symmetry.2 example : physicalBBCKK2004ContourPartitionZeroReadout.blumeCapelZeroCurveCountValue = 2 := bbckk_three_phase_blume_capel_point_has_two_bifurcating_zero_curves.1 example : ¬PhysicalBBCKKMultiplePointSymmetryAndScopeAt physicalBBCKKBlumeCapelZeroCurveCollapsedControl := bbckk_three_phase_blume_capel_point_has_two_bifurcating_zero_curves.2 example : ¬PhysicalBBCKKMultiplePointSymmetryAndScopeAt physicalBBCKKArbitraryBoundaryClaimedControl := bbckk_low_temperature_torus_results_do_not_cover_all_boundaries_or_temperatures.1 example : ¬PhysicalBBCKKMultiplePointSymmetryAndScopeAt physicalBBCKKAllTemperaturePottsClaimedControl := bbckk_low_temperature_torus_results_do_not_cover_all_boundaries_or_temperatures.2 example : PhysicalYangLeeDimensionalRootDensityAndAnalyticityAt physicalYangLee1952PartitionZeroReadout ∧ PhysicalBBCKKMultiplePointSymmetryAndScopeAt physicalBBCKK2004ContourPartitionZeroReadout := ⟨partition_zero_sources_keep_distinct_polynomial_and_contour_derivations.1, partition_zero_sources_keep_distinct_polynomial_and_contour_derivations.2.1⟩ example : physicalYangLee1952PartitionZeroReadout.source ≠ physicalBBCKK2004ContourPartitionZeroReadout.source ∧ physicalYangLee1952PartitionZeroReadout.derivation ≠ physicalBBCKK2004ContourPartitionZeroReadout.derivation := ⟨partition_zero_sources_keep_distinct_polynomial_and_contour_derivations.2.2.1, partition_zero_sources_keep_distinct_polynomial_and_contour_derivations.2.2.2.1⟩ example : physicalYangLee1952PartitionZeroReadout.selectedSystem ≠ physicalBBCKK2004ContourPartitionZeroReadout.selectedSystem := partition_zero_sources_keep_distinct_polynomial_and_contour_derivations.2.2.2.2 example : physicalPartitionZeroPossibility.TentativeAt physicalPartitionZeroEstablishment .circleLocationAloneForcesPhaseTransition .yangLee1952HardCoreGasAndFerromagneticCircle ∧ physicalPartitionZeroPossibility.TentativeAt physicalPartitionZeroEstablishment .antiferromagneticCircleConclusion .yangLee1952HardCoreGasAndFerromagneticCircle := ⟨partition_zero_source_overclaims_remain_epistemically_open.1, partition_zero_source_overclaims_remain_epistemically_open.2.1⟩ example : physicalPartitionZeroPossibility.TentativeAt physicalPartitionZeroEstablishment .exactFiniteFieldTwoDimensionalIsingSolution .yangLee1952HardCoreGasAndFerromagneticCircle ∧ physicalPartitionZeroPossibility.TentativeAt physicalPartitionZeroEstablishment .phaseSumAloneLocatesActualZeros .biskupBorgsChayesKotecky2004LowTemperatureContours := ⟨partition_zero_source_overclaims_remain_epistemically_open.2.2.1, partition_zero_source_overclaims_remain_epistemically_open.2.2.2.1⟩ example : physicalPartitionZeroPossibility.TentativeAt physicalPartitionZeroEstablishment .exponentialPrecisionAtMultiplePoints .biskupBorgsChayesKotecky2004LowTemperatureContours ∧ physicalPartitionZeroPossibility.TentativeAt physicalPartitionZeroEstablishment .exactCircleForSymmetryBrokenPerturbation .biskupBorgsChayesKotecky2004LowTemperatureContours := ⟨partition_zero_source_overclaims_remain_epistemically_open.2.2.2.2.1, partition_zero_source_overclaims_remain_epistemically_open.2.2.2.2.2.1⟩ example : physicalPartitionZeroPossibility.TentativeAt physicalPartitionZeroEstablishment .arbitraryBoundaryConditionsControlled .biskupBorgsChayesKotecky2004LowTemperatureContours ∧ physicalPartitionZeroPossibility.TentativeAt physicalPartitionZeroEstablishment .allTemperaturePottsZerosControlled .biskupBorgsChayesKotecky2004LowTemperatureContours := partition_zero_source_overclaims_remain_epistemically_open.2.2.2.2.2.2 example : PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKK2004ContourPartitionZeroReadout ∧ ¬PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKKBelowTheoremBMinimumSizeControl := ⟨bbckk_phase_sum_requires_original_size_width_subset_and_domain_scope.1, bbckk_phase_sum_requires_original_size_width_subset_and_domain_scope.2.1⟩ example : ¬PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKKAlmostStableWidthCollapsedControl ∧ ¬PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKKAlmostStablePhaseSubsetCollapsedControl := ⟨bbckk_phase_sum_requires_original_size_width_subset_and_domain_scope.2.2.1, bbckk_phase_sum_requires_original_size_width_subset_and_domain_scope.2.2.2.1⟩ example : ¬PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKKAllTorusSizesControlledClaimedControl ∧ ¬PhysicalBBCKKPeriodicPhaseSumAndImportedTheoremAt physicalBBCKKAllComplexControlsControlledClaimedControl := ⟨bbckk_phase_sum_requires_original_size_width_subset_and_domain_scope.2.2.2.2.2.2.1, bbckk_phase_sum_requires_original_size_width_subset_and_domain_scope.2.2.2.2.2.2.2⟩ namespace LeeYangPaperCountermodelReuse open PhysicsLogic.Papers.LeeYang1952.LeeYangFramework.Countermodels example : PossibleTransitionIsLocalized localizedPointWithoutTransition ∧ ¬PhysicalTransitionExists localizedPointWithoutTransition := localization_of_possible_transitions_does_not_prove_existence example : ZerosAccumulateAtPhysicalPoint accumulationWithoutDensityJump ∧ ¬AccumulationSuppliesPhysicalTransition accumulationWithoutDensityJump := accumulation_without_physical_bridge_is_not_yet_a_transition end LeeYangPaperCountermodelReuse namespace BBCKKPaperCountermodelReuse open PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework.Countermodels example : ControlledTwoPhaseSumAvailable controlledPhaseSumWithoutCancellation ∧ ¬PartitionZeroLocated controlledPhaseSumWithoutCancellation := controlled_phase_sum_does_not_itself_locate_zeros example : ComplexZeroIsLocalized localizedNonphysicalControl ∧ ¬ControlIsPhysical localizedNonphysicalControl := complex_zero_localization_does_not_make_every_control_physical example : FiniteZeroLiesOnCircle finiteCircleZeroWithoutDensity ∧ ¬PositiveLimitingZeroDensity finiteCircleZeroWithoutDensity := circle_location_does_not_supply_zero_density example : LowTemperatureControlAvailable lowTemperatureOnlyControl ∧ ¬AllTemperatureControlAvailable lowTemperatureOnlyControl := low_temperature_control_does_not_imply_all_temperature_control example : FinitePhysicalPartitionFamilyAvailable finitePhysicalFamilyBelowTheoremBThreshold ∧ ¬ScopedTheoremBPhaseSumAvailable finitePhysicalFamilyBelowTheoremBThreshold := finite_physical_partition_family_does_not_extend_theorem_b_below_threshold example : AmbientComplexControlAvailable ambientComplexControlWithExcludedPhaseClosure ∧ ¬TheoremBPhaseSubsetDomainAvailable ambientComplexControlWithExcludedPhaseClosure := ambient_complex_control_does_not_supply_almost_stable_subset_domain example : TheoremBSizeQualificationAvailable validSizeWithIncorrectAlmostStableWidth ∧ ¬TheoremBKappaWidthQualificationAvailable validSizeWithIncorrectAlmostStableWidth := valid_torus_size_does_not_fix_the_original_kappa_over_size_width example : ScopedTheoremBPhaseSumAvailable scopedComplexPhaseSumWithoutPhysicalRegularity ∧ ¬FiniteThermodynamicRegularityAvailable scopedComplexPhaseSumWithoutPhysicalRegularity := scoped_complex_phase_sum_does_not_establish_thermodynamic_regularity example : FinitePartitionAnalyticityAvailable analyticPartitionWithVanishingFiniteMetastableWeight ∧ ¬FiniteMetastableWeightNonvanishingAvailable analyticPartitionWithVanishingFiniteMetastableWeight := finite_partition_analyticity_does_not_supply_nonzero_metastable_weights example : PhaseSumRemainderSuppressionAvailable sourceRemainderWithIncorrectFiniteWeightCorrection ∧ ¬FiniteWeightLogCorrectionSuppressionAvailable sourceRemainderWithIncorrectFiniteWeightCorrection := phase_sum_remainder_bound_does_not_supply_finite_weight_correction example : FiniteWeightDerivativeBoundAvailable controlledSinglePhaseWithoutOverlapSeparation ∧ ¬OverlapLogDerivativeNondegeneracyAvailable controlledSinglePhaseWithoutOverlapSeparation := single_phase_derivative_bound_does_not_supply_overlap_nondegeneracy example {T : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroTypes} (framework : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework T) (result : framework.ControlledTorusPhaseSumResult) (phaseSubset : T.Phase → Prop) (size : T.Size) (control : T.Control) (sizeScope : framework.TheoremBSizeAdmissibleAt size) (domainScope : framework.TheoremBAlmostStableSubsetDomainAt phaseSubset size control) : framework.periodicPartitionHasPhaseSumAt framework.selectedModel phaseSubset size control ∧ ∀ derivativeOrder, framework.TheoremBSourceRemainderBoundAt phaseSubset size control derivativeOrder := framework.controlled_phase_sum_requires_original_theorem_b_scope result phaseSubset size control sizeScope domainScope example {T : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroTypes} (framework : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework T) (result : framework.ControlledTorusPhaseSumResult) (size : T.Size) (control : T.Control) (sizeScope : framework.TheoremBSizeAdmissibleAt size) (extendedDomain : framework.contains framework.theoremBExtendedRegion control) : framework.finitePartitionDefinedAt framework.selectedModel size control ∧ framework.finitePartitionAnalyticAt framework.selectedModel size control := framework.controlled_phase_sum_retains_extended_domain_partition_analyticity result size control sizeScope extendedDomain example {T : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroTypes} (framework : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework T) (result : framework.ControlledTorusPhaseSumResult) (phase : T.Phase) (size : T.Size) (control : T.Control) (sizeScope : framework.TheoremBSizeAdmissibleAt size) (almostStable : framework.almostStablePhaseAt phase control (framework.theoremBAlmostStableWidthAt size)) : framework.TheoremBFiniteWeightCorrectionAt phase size control ∧ framework.TheoremBFiniteWeightDerivativeBoundsAt phase size control := framework.controlled_phase_sum_retains_finite_metastable_weight_source_bounds result phase size control sizeScope almostStable example {T : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroTypes} (framework : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework T) (result : framework.ControlledTorusPhaseSumResult) (first second : T.Phase) (size : T.Size) (control : T.Control) (sizeScope : framework.TheoremBSizeAdmissibleAt size) (distinct : first ≠ second) (firstStable : framework.almostStablePhaseAt first control (framework.theoremBAlmostStableWidthAt size)) (secondStable : framework.almostStablePhaseAt second control (framework.theoremBAlmostStableWidthAt size)) : framework.TheoremBFiniteWeightNondegeneracyAt first second size control := framework.controlled_phase_sum_retains_overlapping_phase_nondegeneracy result first second size control sizeScope distinct firstStable secondStable end BBCKKPaperCountermodelReuse example : PhysicalYangLeeDimensionalRootDensityAndAnalyticityAt physicalYangLee1952PartitionZeroReadout ∧ PhysicalBBCKKMultiplePointSymmetryAndScopeAt physicalBBCKK2004ContourPartitionZeroReadout := ⟨partition_zero_sources_distinguish_positive_activity_circle_density_contours_interference_and_scope.1, partition_zero_sources_distinguish_positive_activity_circle_density_contours_interference_and_scope.2.1⟩ example : ¬physicalLeeYangBoundaryModel.finitePartitionZeroAt .latticeGasFiniteVolume .positiveActivityTransition ∧ physicalLeeYangBoundaryModel.finitePartitionZeroAt .latticeGasFiniteVolume .upperComplexFiniteZero := ⟨partition_zero_sources_distinguish_positive_activity_circle_density_contours_interference_and_scope.2.2.1, partition_zero_sources_distinguish_positive_activity_circle_density_contours_interference_and_scope.2.2.2.1⟩ example : physicalYangLee1952PartitionZeroReadout.gasPressureValue = physicalYangLee1952PartitionZeroReadout.liquidPressureValue ∧ physicalYangLee1952PartitionZeroReadout.gasDensityValue < physicalYangLee1952PartitionZeroReadout.liquidDensityValue := ⟨partition_zero_sources_distinguish_positive_activity_circle_density_contours_interference_and_scope.2.2.2.2.1, partition_zero_sources_distinguish_positive_activity_circle_density_contours_interference_and_scope.2.2.2.2.2.1⟩ example : physicalBBCKK2004ContourPartitionZeroReadout.partitionValueAtInterferenceEquationValue ≠ 0 ∧ physicalBBCKK2004ContourPartitionZeroReadout.maximumMultiplePointZeroMultiplicityValue + 1 = physicalBBCKK2004ContourPartitionZeroReadout.multiplePointStablePhaseCountValue := ⟨partition_zero_sources_distinguish_positive_activity_circle_density_contours_interference_and_scope.2.2.2.2.2.2.1, partition_zero_sources_distinguish_positive_activity_circle_density_contours_interference_and_scope.2.2.2.2.2.2.2.1⟩ example : ¬physicalPartitionZeroEstablishment.establishedAt .allTemperaturePottsZerosControlled .biskupBorgsChayesKotecky2004LowTemperatureContours := partition_zero_sources_distinguish_positive_activity_circle_density_contours_interference_and_scope.2.2.2.2.2.2.2.2 end PartitionZeroOriginalSourcePhysicalReuseAudit namespace CriticalCollapseOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.CriticalCollapseScalingCountermodels open PhysicsLogic.Philosophy.EpistemicStatus example : PhysicalCriticalThresholdAndFiniteMassFitAt physicalChoptuik1993ScalarCollapseReadout := physical_choptuik_scalar_threshold_and_finite_mass_fit_are_nonvacuous.1 example : physicalChoptuik1993ScalarCollapseReadout.selectedMassFitResidualValue = (1 / 1000 : Real) := physical_choptuik_scalar_threshold_and_finite_mass_fit_are_nonvacuous.2 example : PhysicalChoptuikScalarDssEvidenceAt physicalChoptuik1993ScalarCollapseReadout := physical_choptuik_scalar_discrete_echo_evidence_is_nonvacuous.1 example : physicalChoptuik1993ScalarCollapseReadout.radialEchoPeriodValue = (86 / 25 : Real) := physical_choptuik_scalar_discrete_echo_evidence_is_nonvacuous.2 example : PhysicalChoptuikHorizonPrintAndUniversalityScopeAt physicalChoptuik1993ScalarCollapseReadout := physical_choptuik_horizon_print_and_universality_scope_are_nonvacuous.1 example : physicalChoptuik1993ScalarCollapseReadout.physicalHorizonMassValue = (1 : Real) := physical_choptuik_horizon_print_and_universality_scope_are_nonvacuous.2 example : PhysicalCriticalThresholdAndFiniteMassFitAt physicalEvansColeman1994RadiationCollapseReadout := physical_evans_radiation_threshold_and_finite_mass_fit_are_nonvacuous.1 example : physicalEvansColeman1994RadiationCollapseReadout.thresholdParameterValue = (509414117 / 500000000 : Real) := physical_evans_radiation_threshold_and_finite_mass_fit_are_nonvacuous.2 example : PhysicalEvansRadiationEquationAndSimilarityAt physicalEvansColeman1994RadiationCollapseReadout := physical_evans_radiation_equation_and_similarity_are_nonvacuous.1 example : physicalEvansColeman1994RadiationCollapseReadout.radiationAdiabaticIndexValue = (4 / 3 : Real) := physical_evans_radiation_equation_and_similarity_are_nonvacuous.2 example : PhysicalEvansSonicRegularityAndLocalProfileAt physicalEvansColeman1994RadiationCollapseReadout := physical_evans_sonic_regularity_and_local_profile_are_nonvacuous.1 example : physicalEvansColeman1994RadiationCollapseReadout.radiationOuterMetricValue = (107 / 100 : Real) := physical_evans_sonic_regularity_and_local_profile_are_nonvacuous.2 example : PhysicalEvansIntermediateFitAndNakednessScopeAt physicalEvansColeman1994RadiationCollapseReadout := physical_evans_intermediate_fit_and_nakedness_scope_are_nonvacuous.1 example : physicalEvansColeman1994RadiationCollapseReadout.radiationExponentValue = (9 / 25 : Real) := physical_evans_intermediate_fit_and_nakedness_scope_are_nonvacuous.2 example : ¬PhysicalCriticalThresholdAndFiniteMassFitAt physicalScalarThresholdCollapsedControl := scalar_threshold_requires_binary_search_and_adaptive_refinement.1 example : ¬PhysicalCriticalThresholdAndFiniteMassFitAt physicalScalarBinaryThresholdSearchErasedControl := scalar_threshold_requires_binary_search_and_adaptive_refinement.2.1 example : ¬PhysicalChoptuikScalarDssEvidenceAt physicalScalarAdaptiveRefinementErasedControl := scalar_threshold_requires_binary_search_and_adaptive_refinement.2.2 example : ¬PhysicalCriticalThresholdAndFiniteMassFitAt physicalScalarExactMassFitClaimedControl := finite_collapse_mass_fit_retains_residual_without_proving_exact_law.1 example : ¬PhysicalCriticalThresholdAndFiniteMassFitAt physicalScalarMassResidualErasedControl := finite_collapse_mass_fit_retains_residual_without_proving_exact_law.2 example : physicalChoptuik1993ScalarCollapseReadout.physicalHorizonMassValue = physicalChoptuik1993ScalarCollapseReadout.apparentHorizonRadiusValue / 2 := scalar_apparent_horizon_requires_half_radius_and_exposes_print_error.1 example : physicalChoptuik1993ScalarCollapseReadout.sourcePrintedHorizonMassValue = 2 * physicalChoptuik1993ScalarCollapseReadout.apparentHorizonRadiusValue := scalar_apparent_horizon_requires_half_radius_and_exposes_print_error.2.1 example : ¬PhysicalChoptuikHorizonPrintAndUniversalityScopeAt physicalScalarHorizonNormalizationReversedControl := scalar_apparent_horizon_requires_half_radius_and_exposes_print_error.2.2 example : ¬PhysicalChoptuikHorizonPrintAndUniversalityScopeAt physicalScalarPrintedHorizonIdentifiedWithPhysicalControl := scalar_printed_horizon_formula_cannot_be_identified_with_physical_mass.1 example : ¬PhysicalChoptuikHorizonPrintAndUniversalityScopeAt physicalScalarPrintedHorizonClaimedCorrectControl := scalar_printed_horizon_formula_cannot_be_identified_with_physical_mass.2 example : physicalChoptuik1993ScalarCollapseReadout.scalarInitialFamilyCount = 4 := scalar_exponent_evidence_retains_seven_fits_and_nonzero_spread.1 example : physicalChoptuik1993ScalarCollapseReadout.scalarExponentFitRowCount = 7 := scalar_exponent_evidence_retains_seven_fits_and_nonzero_spread.2.1 example : physicalChoptuik1993ScalarCollapseReadout.scalarExponentSpreadValue = (7 / 500 : Real) := scalar_exponent_evidence_retains_seven_fits_and_nonzero_spread.2.2.1 example : ¬PhysicalChoptuikScalarDssEvidenceAt physicalScalarSevenFitRowsCollapsedControl := scalar_exponent_evidence_retains_seven_fits_and_nonzero_spread.2.2.2.1 example : ¬PhysicalChoptuikScalarDssEvidenceAt physicalScalarAllExponentFitsIdenticalClaimedControl := scalar_exponent_evidence_retains_seven_fits_and_nonzero_spread.2.2.2.2 example : ¬PhysicalChoptuikHorizonPrintAndUniversalityScopeAt physicalScalarExactUniversalExponentClaimedControl := scalar_exponent_fits_do_not_establish_exact_universality_or_inverse_euler.1 example : ¬PhysicalChoptuikHorizonPrintAndUniversalityScopeAt physicalScalarInverseEulerExponentClaimedControl := scalar_exponent_fits_do_not_establish_exact_universality_or_inverse_euler.2 example : physicalChoptuik1993ScalarCollapseReadout.temporalEchoPeriodValue = (343 / 100 : Real) := scalar_radial_and_temporal_echo_measurements_are_distinct.2.1 example : ¬PhysicalChoptuikScalarDssEvidenceAt physicalScalarEchoMeasurementsIdentifiedControl := scalar_radial_and_temporal_echo_measurements_are_distinct.2.2 example : physicalChoptuik1993ScalarCollapseReadout.observedProfileCycleFractionValue = (3 / 4 : Real) := scalar_partial_profile_cycle_does_not_establish_complete_echo_period.1 example : ¬PhysicalChoptuikScalarDssEvidenceAt physicalScalarPartialCycleClaimedFullControl := scalar_partial_profile_cycle_does_not_establish_complete_echo_period.2 example : ¬PhysicalChoptuikHorizonPrintAndUniversalityScopeAt physicalScalarFamilyMassCoefficientsIdentifiedControl := scalar_cross_family_mass_coefficients_remain_family_dependent example : ¬PhysicalEvansRadiationEquationAndSimilarityAt physicalRadiationMatterIdentifiedWithScalarControl := radiation_collapse_requires_distinct_matter_and_continuous_homothety.1 example : ¬PhysicalEvansRadiationEquationAndSimilarityAt physicalRadiationDiscreteEchoClaimedControl := radiation_collapse_requires_distinct_matter_and_continuous_homothety.2 example : ¬PhysicalEvansRadiationEquationAndSimilarityAt physicalRadiationEquationOfStateCollapsedControl := radiation_fluid_requires_equation_of_state_and_strength_normalization.1 example : ¬PhysicalEvansRadiationEquationAndSimilarityAt physicalRadiationStrengthNormalizationCollapsedControl := radiation_fluid_requires_equation_of_state_and_strength_normalization.2 example : ¬PhysicalEvansRadiationEquationAndSimilarityAt physicalRadiationThresholdValueCollapsedControl := radiation_threshold_retains_measured_critical_strength example : ¬PhysicalEvansSonicRegularityAndLocalProfileAt physicalRadiationCenterRegularityErasedControl := radiation_similarity_requires_center_outer_and_sonic_regularity.1 example : ¬PhysicalEvansSonicRegularityAndLocalProfileAt physicalRadiationOuterRegularityErasedControl := radiation_similarity_requires_center_outer_and_sonic_regularity.2.1 example : ¬PhysicalEvansSonicRegularityAndLocalProfileAt physicalRadiationSonicRegularityErasedControl := radiation_similarity_requires_center_outer_and_sonic_regularity.2.2 example : physicalEvansColeman1994RadiationCollapseReadout.radiationSimilarityExponentValue = (2297 / 2000 : Real) := radiation_homothetic_eigenvalue_requires_nonlinear_regularity.1 example : ¬PhysicalEvansSonicRegularityAndLocalProfileAt physicalRadiationSimilarityExponentCollapsedControl := radiation_homothetic_eigenvalue_requires_nonlinear_regularity.2 example : physicalEvansColeman1994RadiationCollapseReadout.radiationExactLapseExponentValue = (297 / 2297 : Real) := radiation_rounded_lapse_exponent_is_approximate_not_exact.1 example : physicalEvansColeman1994RadiationCollapseReadout.radiationReportedLapseExponentValue = (129 / 1000 : Real) := radiation_rounded_lapse_exponent_is_approximate_not_exact.2.1 example : ¬PhysicalEvansSonicRegularityAndLocalProfileAt physicalRadiationRoundedLapseClaimedExactControl := radiation_rounded_lapse_exponent_is_approximate_not_exact.2.2 example : physicalEvansColeman1994RadiationCollapseReadout.radiationOuterMassRadiusRatioValue = (149 / 2500 : Real) := radiation_self_similar_profile_is_not_asymptotically_flat.2.1 example : ¬PhysicalEvansSonicRegularityAndLocalProfileAt physicalRadiationOuterProfileClaimedFlatControl := radiation_self_similar_profile_is_not_asymptotically_flat.2.2 example : ¬PhysicalEvansSonicRegularityAndLocalProfileAt physicalRadiationAsymptoticMatchingErasedControl := radiation_asymptotic_flatness_requires_external_matching example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationRarefactionErasedControl := radiation_critical_region_requires_rarefaction_and_scale_separation.1 example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationIntermediateScaleCollapsedControl := radiation_critical_region_requires_rarefaction_and_scale_separation.2 example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationExactSimilarityAwayFromThresholdClaimedControl := radiation_exact_similarity_does_not_persist_away_from_threshold example : physicalEvansColeman1994RadiationCollapseReadout.radiationFineGridScaleValue = (1 / 1000 : Real) := radiation_mass_fit_retains_two_resolutions_and_finite_range.1 example : physicalEvansColeman1994RadiationCollapseReadout.radiationCoarseGridScaleValue = (33 / 10000 : Real) := radiation_mass_fit_retains_two_resolutions_and_finite_range.2.1 example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationFiniteGridScalesIdentifiedControl := radiation_mass_fit_retains_two_resolutions_and_finite_range.2.2.1 example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationMassScalingRangeInflatedControl := radiation_mass_fit_retains_two_resolutions_and_finite_range.2.2.2 example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationScalarExponentIdentifiedControl := radiation_and_scalar_mass_exponents_are_similar_but_not_identical.2.2.1 example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationAllMatterUniversalExponentClaimedControl := radiation_and_scalar_mass_exponents_are_similar_but_not_identical.2.2.2 example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationLinearStabilityCalculationClaimedControl := radiation_source_defers_linear_stability_exponent_derivation example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationNakedSingularityClaimedProvedControl := radiation_nakedness_suggestion_does_not_prove_generic_or_visible_claim.1 example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationNakedSingularityClaimedVisibleControl := radiation_nakedness_suggestion_does_not_prove_generic_or_visible_claim.2.1 example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationGenericCensorshipViolationClaimedControl := radiation_nakedness_suggestion_does_not_prove_generic_or_visible_claim.2.2 example : ¬PhysicalEvansIntermediateFitAndNakednessScopeAt physicalRadiationInfiniteRedshiftQualificationErasedControl := radiation_nakedness_requires_infinite_redshift_qualification example : physicalCriticalCollapsePossibility.TentativeAt physicalCriticalCollapseEstablishment .exactUniversalScalarExponentEstablished .choptuik1993AdaptiveMasslessScalar := critical_collapse_source_overclaims_remain_epistemically_open.1 example : physicalCriticalCollapsePossibility.TentativeAt physicalCriticalCollapseEstablishment .scalarExponentEqualsInverseEulerNumber .choptuik1993AdaptiveMasslessScalar := critical_collapse_source_overclaims_remain_epistemically_open.2.1 example : physicalCriticalCollapsePossibility.TentativeAt physicalCriticalCollapseEstablishment .finiteFitsProveExactMassScaling .choptuik1993AdaptiveMasslessScalar := critical_collapse_source_overclaims_remain_epistemically_open.2.2.1 example : physicalCriticalCollapsePossibility.TentativeAt physicalCriticalCollapseEstablishment .scalarAndRadiationShareIdenticalExponent .evansColeman1994SelfSimilarRadiationFluid := critical_collapse_source_overclaims_remain_epistemically_open.2.2.2.1 example : physicalCriticalCollapsePossibility.TentativeAt physicalCriticalCollapseEstablishment .radiationLinearStabilityExponentDerived .evansColeman1994SelfSimilarRadiationFluid := critical_collapse_source_overclaims_remain_epistemically_open.2.2.2.2.1 example : physicalCriticalCollapsePossibility.TentativeAt physicalCriticalCollapseEstablishment .genericCosmicCensorshipViolationEstablished .evansColeman1994SelfSimilarRadiationFluid := critical_collapse_source_overclaims_remain_epistemically_open.2.2.2.2.2.1 example : physicalCriticalCollapsePossibility.TentativeAt physicalCriticalCollapseEstablishment .radiationNakedSingularityVisibleAtInfinity .evansColeman1994SelfSimilarRadiationFluid := critical_collapse_source_overclaims_remain_epistemically_open.2.2.2.2.2.2 example : physicalCriticalCollapseEvidence.supportsAt .sevenScalarLeastSquaresExponentFits .scalarFourFamilyApproximateAgreement .choptuik1993AdaptiveMasslessScalar := critical_collapse_numerical_support_does_not_establish_source_overclaims.1 example : ¬physicalCriticalCollapseEvidence.supportsAt .sevenScalarLeastSquaresExponentFits .exactUniversalScalarExponentEstablished .choptuik1993AdaptiveMasslessScalar := critical_collapse_numerical_support_does_not_establish_source_overclaims.2.1 example : physicalCriticalCollapseEvidence.supportsAt .radiationSonicRegularityCalculation .radiationThreePointRegularityEigenvalue .evansColeman1994SelfSimilarRadiationFluid := critical_collapse_numerical_support_does_not_establish_source_overclaims.2.2.1 example : ¬physicalCriticalCollapseEvidence.supportsAt .radiationSonicRegularityCalculation .radiationLinearStabilityExponentDerived .evansColeman1994SelfSimilarRadiationFluid := critical_collapse_numerical_support_does_not_establish_source_overclaims.2.2.2 example : physicalChoptuik1993ScalarCollapseReadout.source ≠ physicalEvansColeman1994RadiationCollapseReadout.source := critical_collapse_sources_distinguish_scalar_echoes_radiation_homothety_horizon_normalization_and_scope.2.2.1 example : physicalChoptuik1993ScalarCollapseReadout.matter ≠ physicalEvansColeman1994RadiationCollapseReadout.matter := critical_collapse_sources_distinguish_scalar_echoes_radiation_homothety_horizon_normalization_and_scope.2.2.2.1 example : physicalChoptuik1993ScalarCollapseReadout.similarity ≠ physicalEvansColeman1994RadiationCollapseReadout.similarity := critical_collapse_sources_distinguish_scalar_echoes_radiation_homothety_horizon_normalization_and_scope.2.2.2.2.1 example : physicalChoptuik1993ScalarCollapseReadout.physicalHorizonMassValue ≠ physicalChoptuik1993ScalarCollapseReadout.sourcePrintedHorizonMassValue := critical_collapse_sources_distinguish_scalar_echoes_radiation_homothety_horizon_normalization_and_scope.2.2.2.2.2.1 example : physicalChoptuik1993ScalarCollapseReadout.scalarRepresentativeExponentValue ≠ physicalEvansColeman1994RadiationCollapseReadout.radiationExponentValue := critical_collapse_sources_distinguish_scalar_echoes_radiation_homothety_horizon_normalization_and_scope.2.2.2.2.2.2.1 example : physicalEvansColeman1994RadiationCollapseReadout.radiationOuterMetricValue ≠ physicalEvansColeman1994RadiationCollapseReadout.radiationFlatExteriorMetricValue := critical_collapse_sources_distinguish_scalar_echoes_radiation_homothety_horizon_normalization_and_scope.2.2.2.2.2.2.2.1 example : ¬physicalCriticalCollapseEstablishment.establishedAt .genericCosmicCensorshipViolationEstablished .evansColeman1994SelfSimilarRadiationFluid := critical_collapse_sources_distinguish_scalar_echoes_radiation_homothety_horizon_normalization_and_scope.2.2.2.2.2.2.2.2 namespace ChoptuikPaperCountermodelReuse open PhysicsLogic.Papers.Choptuik1993.Countermodels open PhysicsLogic.Papers.Choptuik1993.MassScalingFormulaReadout.Countermodels example : physicallyNormalizedHorizonReadout.ApparentHorizonThresholdAt ∧ ¬physicallyNormalizedHorizonReadout.HorizonMassFormulaAt := ⟨physically_normalized_horizon_retains_scaling_but_refutes_printed_mass.2.1, physically_normalized_horizon_retains_scaling_but_refutes_printed_mass.2.2.2.2⟩ example : HasSelectedNumericalEchoEvidence selectedScalarCriticalEvidence ∧ ¬HasMathematicalDiscreteEchoing selectedScalarCriticalEvidence := numerical_echoing_evidence_does_not_become_mathematical_proof example : HasSelectedNumericalMassScaling selectedScalarCriticalEvidence ∧ ¬HasMathematicalMassScalingLaw selectedScalarCriticalEvidence := numerical_mass_scaling_evidence_does_not_become_mathematical_proof example : ¬HasCrossFamilyExponentAgreement selectedScalarCriticalEvidence ∧ ¬HasUniversalCriticalExponent selectedScalarCriticalEvidence := ⟨one_family_power_law_does_not_establish_universality.2.1, one_family_power_law_does_not_establish_universality.2.2⟩ example : HasVanishingThresholdMass selectedScalarCriticalEvidence ∧ ¬HasEstablishedNakedSingularity selectedScalarCriticalEvidence := vanishing_black_hole_mass_does_not_prove_a_naked_singularity end ChoptuikPaperCountermodelReuse namespace EvansColemanPaperCountermodelReuse open PhysicsLogic.Papers.EvansColeman1994.Countermodels open PhysicsLogic.Papers.EvansColeman1994 example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (localLimit : context.SelectedLocalCriticalAsymptoticsAt) (coordinate : Real) (positive : 0 < coordinate) : Filter.Tendsto (fun tau => 4 * Real.pi * (context.selectedSimilarityProfile.scale * tau ^ context.selectedSimilarityProfile.index * coordinate)^2 * context.selectedCriticalFields.density (-tau) (context.selectedSimilarityProfile.scale * tau ^ context.selectedSimilarityProfile.index * coordinate)) (nhdsWithin 0 (Set.Ioi 0)) (nhds (context.selectedSimilarityProfile.scaledDensity coordinate)) := (context.local_asymptotics_give_the_selected_critical_field_limits localLimit coordinate positive).2 .scaledDensity example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (continuation : context.PreciselyCriticalContinuationExpectation) : context.SelectedLocalCriticalAsymptoticsAt := context.critical_continuation_uses_the_same_local_field_limit continuation example : HasLocalSelfSimilarCriticalRegion radiationCriticalScopeBoundary /\ radiationCriticalScopeBoundary.evolution.radialMetric (-1) 1 ≠ radiationCriticalScopeBoundary.profile.radialMetric (radiationCriticalScopeBoundary.profile.coordinate (-1) 1) := local_profile_convergence_does_not_assert_exact_field_equality example : HasLocalSelfSimilarCriticalRegion radiationCriticalScopeBoundary ∧ ¬HasFlatRadialTailAtComparisonTime radiationCriticalScopeBoundary := a_local_self_similar_critical_region_need_not_have_a_flat_radial_tail example : Intermediate.ObservedAgreement Intermediate.finitelyMatchingEvolution Similarity.Profile.homotheticNonSolution Intermediate.comparisonWindow Intermediate.comparisonSamples {.radialMetric} (fun _ => 0) ∧ ¬ Intermediate.PointwisePersistence Intermediate.finitelyMatchingEvolution Similarity.Profile.homotheticNonSolution := Intermediate.finite_observed_metric_comparison_does_not_establish_full_persistence example : PhysicsLogic.Papers.EvansColeman1994.Singularity.boundedCurvature.RadiationCurvatureIdentity ∧ (∀ properTime, 0 < PhysicsLogic.Papers.EvansColeman1994.Singularity.boundedCurvature.ricciSquare properTime) ∧ ¬ PhysicsLogic.Papers.EvansColeman1994.Singularity.boundedCurvature.CurvatureDiverges := PhysicsLogic.Papers.EvansColeman1994.Singularity.positive_curvature_and_the_radiation_identity_do_not_establish_a_singularity example : HasNumericallyAgreeingMatterExponents radiationCriticalScopeBoundary ∧ ¬HasMathematicallyUniversalMatterExponent radiationCriticalScopeBoundary := numerical_exponent_agreement_does_not_prove_universal_exponent end EvansColemanPaperCountermodelReuse end CriticalCollapseOriginalSourcePhysicalReuseAudit namespace CurvedParticleOriginalSourcePhysicalReuseAudit open PhysicsLogic.Framework.BogoliubovParticleProductionContext.Countermodels open PhysicsLogic.Philosophy.EpistemicStatus example : 1 < Real.exp (1 : Real) := physical_hawking_positive_frequency_exponential_exceeds_one example : (1 / 2 : Real) / (Real.exp 1 + 1) < (1 / 2 : Real) / (Real.exp 1 - 1) := physical_hawking_fermionic_occupation_is_below_bosonic_occupation example : PhysicalCurvedVacuumAndPositiveMixingAt physicalParker1969CosmologicalParticleReadout := physical_parker_positive_cosmological_mode_mixing_is_nonvacuous.1 example : physicalParker1969CosmologicalParticleReadout.creationNormSquareValue = (1 : Real) := physical_parker_positive_cosmological_mode_mixing_is_nonvacuous.2 example : PhysicalParkerCanonicalPairAndDensityAt physicalParker1969CosmologicalParticleReadout := physical_parker_canonical_pair_and_density_are_nonvacuous.1 example : physicalParker1969CosmologicalParticleReadout.particleDensityValue = (1 : Real) := physical_parker_canonical_pair_and_density_are_nonvacuous.2 example : PhysicalParkerAdiabaticSelectionAndConformalScopeAt physicalParker1969CosmologicalParticleReadout := physical_parker_adiabatic_selection_and_conformal_scope_are_nonvacuous.1 example : physicalParker1969CosmologicalParticleReadout.radiationScaleExponentValue = (1 / 2 : Real) := physical_parker_adiabatic_selection_and_conformal_scope_are_nonvacuous.2 example : PhysicalCurvedVacuumAndPositiveMixingAt physicalHawking1975BlackHoleParticleReadout := physical_hawking_positive_collapse_mode_mixing_is_nonvacuous.1 example : physicalHawking1975BlackHoleParticleReadout.creationNormSquareValue = (1 / 2 : Real) / (Real.exp 1 - 1) := physical_hawking_positive_collapse_mode_mixing_is_nonvacuous.2 example : PhysicalHawkingCollapsePacketAndRayTraceAt physicalHawking1975BlackHoleParticleReadout := physical_hawking_collapse_packet_and_horizon_ray_trace_are_nonvacuous.1 example : physicalHawking1975BlackHoleParticleReadout.selectedAdvancedTimeValue = (-1 : Real) := physical_hawking_collapse_packet_and_horizon_ray_trace_are_nonvacuous.2 example : PhysicalHawkingGreybodyBosonAndFermionSpectrumAt physicalHawking1975BlackHoleParticleReadout := physical_hawking_greybody_boson_and_fermion_spectra_are_nonvacuous.1 example : physicalHawking1975BlackHoleParticleReadout.greybodyAbsorptionFractionValue = (1 / 2 : Real) := physical_hawking_greybody_boson_and_fermion_spectra_are_nonvacuous.2 example : PhysicalHawkingRotationFluxAndSemiclassicalScopeAt physicalHawking1975BlackHoleParticleReadout := physical_hawking_rotation_flux_and_semiclassical_scope_are_nonvacuous.1 example : physicalHawking1975BlackHoleParticleReadout.shiftedHorizonFrequencyValue = (-1 / 4 : Real) := physical_hawking_rotation_flux_and_semiclassical_scope_are_nonvacuous.2 example : ¬PhysicalCurvedVacuumAndPositiveMixingAt physicalParkerZeroMixingClaimedProductionControl := zero_bogoliubov_mixing_does_not_create_particles example : ¬PhysicalCurvedVacuumAndPositiveMixingAt physicalParkerPositiveOccupationWithoutInterpretationControl := positive_occupation_without_particle_interpretation_is_not_production example : ¬PhysicalParkerCanonicalPairAndDensityAt physicalParkerCanonicalNormalizationCollapsedControl := parker_canonical_commutator_requires_bosonic_norm_difference example : ¬PhysicalParkerCanonicalPairAndDensityAt physicalParkerOppositeMomentumPairErasedControl := parker_pair_creation_requires_opposite_momenta example : physicalParker1969CosmologicalParticleReadout.stimulatedFinalOccupationValue = (7 : Real) := parker_existing_bosons_stimulate_opposite_momentum_creation.1 example : ¬PhysicalParkerCanonicalPairAndDensityAt physicalParkerBosonicStimulationErasedControl := parker_existing_bosons_stimulate_opposite_momentum_creation.2 example : ¬PhysicalParkerCanonicalPairAndDensityAt physicalParkerExpansionCollapsedToStaticControl := parker_creation_requires_distinct_finite_time_scale_factors example : ¬PhysicalParkerCanonicalPairAndDensityAt physicalParkerDensityNormalizationCollapsedControl := parker_particle_density_requires_radial_momentum_and_volume_normalization example : ¬PhysicalParkerCanonicalPairAndDensityAt physicalParkerFiniteInfiniteVacuumOverlapIdentifiedControl := parker_finite_and_infinite_volume_vacua_do_not_share_overlap example : ¬PhysicalParkerAdiabaticSelectionAndConformalScopeAt physicalParkerAdiabaticInvarianceClaimedExactControl := parker_adiabatic_invariance_does_not_imply_exact_number_conservation example : ¬PhysicalParkerAdiabaticSelectionAndConformalScopeAt physicalParkerMinimizationClaimedCanonicalTheoremControl := parker_particle_selection_requires_an_independent_minimization_postulate.1 example : ¬PhysicalParkerAdiabaticSelectionAndConformalScopeAt physicalParkerNonminimalParticleCandidateSelectedControl := parker_particle_selection_requires_an_independent_minimization_postulate.2 example : ¬PhysicalParkerAdiabaticSelectionAndConformalScopeAt physicalParkerConformalPhotonIdentifiedWithScalarControl := parker_conformal_nonzero_spin_result_does_not_cover_minimal_scalar example : ¬PhysicalParkerAdiabaticSelectionAndConformalScopeAt physicalParkerArbitraryConformalExpansionClaimedProvedControl := parker_statically_bounded_conformal_result_has_no_arbitrary_extension example : physicalParker1969CosmologicalParticleReadout.radiationScaleExponentValue = (1 / 2 : Real) := parker_radiation_and_dust_friedmann_powers_are_distinct.1 example : physicalParker1969CosmologicalParticleReadout.dustScaleExponentValue = (2 / 3 : Real) := parker_radiation_and_dust_friedmann_powers_are_distinct.2.1 example : ¬PhysicalParkerAdiabaticSelectionAndConformalScopeAt physicalParkerRadiationDustExpansionExponentsIdentifiedControl := parker_radiation_and_dust_friedmann_powers_are_distinct.2.2 example : ¬PhysicalParkerAdiabaticSelectionAndConformalScopeAt physicalParkerFiniteMassDustNoCreationClaimedControl := parker_infinite_mass_dust_limit_does_not_cover_finite_mass_species example : ¬PhysicalParkerAdiabaticSelectionAndConformalScopeAt physicalParkerFeedbackClaimedToDeriveEinsteinControl := parker_heuristic_creation_feedback_does_not_derive_einstein_equations example : ¬PhysicalHawkingCollapsePacketAndRayTraceAt physicalHawkingCollapseFormationErasedControl := hawking_particle_creation_requires_time_dependent_collapse.1 example : ¬PhysicalHawkingCollapsePacketAndRayTraceAt physicalHawkingStaticBackgroundClaimedSufficientControl := hawking_particle_creation_requires_time_dependent_collapse.2 example : ¬PhysicalHawkingCollapsePacketAndRayTraceAt physicalHawkingLatePacketErasedControl := hawking_thermal_emission_requires_late_finite_high_frequency_packets.1 example : ¬PhysicalHawkingCollapsePacketAndRayTraceAt physicalHawkingHighFrequencyTraceErasedControl := hawking_thermal_emission_requires_late_finite_high_frequency_packets.2.1 example : ¬PhysicalHawkingCollapsePacketAndRayTraceAt physicalHawkingContinuumDivergenceErasedControl := hawking_thermal_emission_requires_late_finite_high_frequency_packets.2.2 example : ¬PhysicalHawkingCollapsePacketAndRayTraceAt physicalHawkingAffineHorizonSignReversedControl := hawking_horizon_ray_trace_requires_negative_affine_support.1 example : ¬PhysicalHawkingCollapsePacketAndRayTraceAt physicalHawkingLastEscapeSupportErasedControl := hawking_horizon_ray_trace_requires_negative_affine_support.2 example : ¬PhysicalHawkingGreybodyBosonAndFermionSpectrumAt physicalHawkingTemperatureNormalizationCollapsedControl := hawking_temperature_requires_surface_gravity_over_two_pi example : ¬PhysicalHawkingGreybodyBosonAndFermionSpectrumAt physicalHawkingGreybodyErasedControl := hawking_late_spectrum_requires_nontrivial_greybody_absorption.1 example : ¬PhysicalHawkingGreybodyBosonAndFermionSpectrumAt physicalHawkingPerfectBlackbodyClaimedControl := hawking_late_spectrum_requires_nontrivial_greybody_absorption.2 example : ¬PhysicalHawkingGreybodyBosonAndFermionSpectrumAt physicalHawkingCoefficientFluxNormalizationCollapsedControl := hawking_coefficient_norm_difference_equals_absorbed_fraction example : ¬PhysicalHawkingGreybodyBosonAndFermionSpectrumAt physicalHawkingBoseFermiDenominatorIdentifiedControl := hawking_fermionic_flux_reverses_the_bosonic_thermal_sign example : physicalHawking1975BlackHoleParticleReadout.hawkingTemperatureValue < physicalHawking1975BlackHoleParticleReadout.massiveSpeciesRestMassValue := hawking_massive_emission_is_suppressed_below_rest_mass.1 example : ¬PhysicalHawkingGreybodyBosonAndFermionSpectrumAt physicalHawkingMassiveEmissionUnsuppressedControl := hawking_massive_emission_is_suppressed_below_rest_mass.2 example : physicalHawking1975BlackHoleParticleReadout.chemicalPotentialValue = (5 / 4 : Real) := hawking_rotating_charged_frequency_keeps_both_chemical_shifts.1 example : ¬PhysicalHawkingRotationFluxAndSemiclassicalScopeAt physicalHawkingChargeChemicalPotentialErasedControl := hawking_rotating_charged_frequency_keeps_both_chemical_shifts.2.1 example : ¬PhysicalHawkingRotationFluxAndSemiclassicalScopeAt physicalHawkingSuperradiantFrequencySignErasedControl := hawking_rotating_charged_frequency_keeps_both_chemical_shifts.2.2 example : ¬PhysicalHawkingRotationFluxAndSemiclassicalScopeAt physicalHawkingFermionicSuperradianceClaimedControl := hawking_bosonic_superradiance_does_not_extend_to_fermions example : ¬PhysicalHawkingRotationFluxAndSemiclassicalScopeAt physicalHawkingNegativeHorizonFluxErasedControl := hawking_outgoing_energy_requires_negative_horizon_flux example : ¬PhysicalHawkingRotationFluxAndSemiclassicalScopeAt physicalHawkingOutgoingFluxClaimedToFixLocalStressControl := hawking_integrated_flux_does_not_fix_local_renormalized_stress example : ¬PhysicalHawkingRotationFluxAndSemiclassicalScopeAt physicalHawkingPairPictureClaimedThermalDerivationControl := hawking_virtual_pair_picture_is_not_the_thermal_derivation example : ¬PhysicalHawkingRotationFluxAndSemiclassicalScopeAt physicalHawkingQuasiStationaryTimescalesCollapsedControl := hawking_quasi_stationary_approximation_requires_separated_timescales example : ¬PhysicalHawkingRotationFluxAndSemiclassicalScopeAt physicalHawkingPlanckEndpointClaimedSemiclassicalControl := hawking_semiclassical_calculation_does_not_determine_planck_endpoint.1 example : ¬PhysicalHawkingRotationFluxAndSemiclassicalScopeAt physicalHawkingExactQuantumEndpointClaimedControl := hawking_semiclassical_calculation_does_not_determine_planck_endpoint.2 example : physicalCurvedParticlePossibility.TentativeAt physicalCurvedParticleEstablishment .canonicalQuantizationDerivesUniqueFiniteTimeParticles .parker1969ExpandingEuclideanUniverse := curved_particle_source_overclaims_remain_epistemically_open.1 example : physicalCurvedParticlePossibility.TentativeAt physicalCurvedParticleEstablishment .adiabaticInvarianceMeansExactConservation .parker1969ExpandingEuclideanUniverse := curved_particle_source_overclaims_remain_epistemically_open.2.1 example : physicalCurvedParticlePossibility.TentativeAt physicalCurvedParticleEstablishment .conformalPhotonResultCoversMinimalScalar .parker1969ExpandingEuclideanUniverse := curved_particle_source_overclaims_remain_epistemically_open.2.2.1 example : physicalCurvedParticlePossibility.TentativeAt physicalCurvedParticleEstablishment .heuristicFeedbackDerivesEinsteinEquations .parker1969ExpandingEuclideanUniverse := curved_particle_source_overclaims_remain_epistemically_open.2.2.2.1 example : physicalCurvedParticlePossibility.TentativeAt physicalCurvedParticleEstablishment .staticBlackHoleWithoutFormationProducesHawkingFlux .hawking1975BlackHoleFormation := curved_particle_source_overclaims_remain_epistemically_open.2.2.2.2.1 example : physicalCurvedParticlePossibility.TentativeAt physicalCurvedParticleEstablishment .greybodyAbsorptionIsAlwaysUnity .hawking1975BlackHoleFormation := curved_particle_source_overclaims_remain_epistemically_open.2.2.2.2.2.1 example : physicalCurvedParticlePossibility.TentativeAt physicalCurvedParticleEstablishment .bosonsAndFermionsShareThermalDenominator .hawking1975BlackHoleFormation := curved_particle_source_overclaims_remain_epistemically_open.2.2.2.2.2.2.1 example : physicalCurvedParticlePossibility.TentativeAt physicalCurvedParticleEstablishment .fermionsExhibitClassicalSuperradiance .hawking1975BlackHoleFormation := curved_particle_source_overclaims_remain_epistemically_open.2.2.2.2.2.2.2.1 example : physicalCurvedParticlePossibility.TentativeAt physicalCurvedParticleEstablishment .virtualPairPictureDerivesThermalSpectrum .hawking1975BlackHoleFormation := curved_particle_source_overclaims_remain_epistemically_open.2.2.2.2.2.2.2.2.1 example : physicalCurvedParticlePossibility.TentativeAt physicalCurvedParticleEstablishment .outgoingFluxFixesLocalStressTensor .hawking1975BlackHoleFormation := curved_particle_source_overclaims_remain_epistemically_open.2.2.2.2.2.2.2.2.2.1 example : physicalCurvedParticlePossibility.TentativeAt physicalCurvedParticleEstablishment .semiclassicalCalculationDeterminesPlanckEndpoint .hawking1975BlackHoleFormation := curved_particle_source_overclaims_remain_epistemically_open.2.2.2.2.2.2.2.2.2.2 example : physicalCurvedParticleEvidence.supportsAt .canonicalTimeDependentModeTransformation .parkerCanonicalBosonicNormalization .parker1969ExpandingEuclideanUniverse := curved_particle_evidence_does_not_establish_distinct_source_overclaims.1 example : ¬physicalCurvedParticleEvidence.supportsAt .canonicalTimeDependentModeTransformation .canonicalQuantizationDerivesUniqueFiniteTimeParticles .parker1969ExpandingEuclideanUniverse := curved_particle_evidence_does_not_establish_distinct_source_overclaims.2.1 example : physicalCurvedParticleEvidence.supportsAt .greybodyWeightedThermalFlux .hawkingGreybodyWeightedBosonicSpectrum .hawking1975BlackHoleFormation := curved_particle_evidence_does_not_establish_distinct_source_overclaims.2.2.1 example : ¬physicalCurvedParticleEvidence.supportsAt .heuristicPairInterpretation .virtualPairPictureDerivesThermalSpectrum .hawking1975BlackHoleFormation := curved_particle_evidence_does_not_establish_distinct_source_overclaims.2.2.2 example : physicalParker1969CosmologicalParticleReadout.source ≠ physicalHawking1975BlackHoleParticleReadout.source := curved_particle_sources_distinguish_cosmological_pairs_horizon_packets_greybody_statistics_and_scope.2.2.1 example : physicalParker1969CosmologicalParticleReadout.background ≠ physicalHawking1975BlackHoleParticleReadout.background := curved_particle_sources_distinguish_cosmological_pairs_horizon_packets_greybody_statistics_and_scope.2.2.2.1 example : physicalParker1969CosmologicalParticleReadout.mechanism ≠ physicalHawking1975BlackHoleParticleReadout.mechanism := curved_particle_sources_distinguish_cosmological_pairs_horizon_packets_greybody_statistics_and_scope.2.2.2.2.1 example : physicalParker1969CosmologicalParticleReadout.selectedMomentumValue + physicalParker1969CosmologicalParticleReadout.oppositeMomentumValue = 0 := curved_particle_sources_distinguish_cosmological_pairs_horizon_packets_greybody_statistics_and_scope.2.2.2.2.2.1 example : physicalHawking1975BlackHoleParticleReadout.fermionicOccupationValue < physicalHawking1975BlackHoleParticleReadout.bosonicOccupationValue := curved_particle_sources_distinguish_cosmological_pairs_horizon_packets_greybody_statistics_and_scope.2.2.2.2.2.2.2.1 example : physicalHawking1975BlackHoleParticleReadout.shiftedHorizonFrequencyValue < 0 := curved_particle_sources_distinguish_cosmological_pairs_horizon_packets_greybody_statistics_and_scope.2.2.2.2.2.2.2.2.1 example : ¬physicalCurvedParticleEstablishment.establishedAt .semiclassicalCalculationDeterminesPlanckEndpoint .hawking1975BlackHoleFormation := curved_particle_sources_distinguish_cosmological_pairs_horizon_packets_greybody_statistics_and_scope.2.2.2.2.2.2.2.2.2 namespace ParkerPaperCountermodelReuse open PhysicsLogic.Papers.Parker1969.ExpandingUniverseFramework.Countermodels example : BogoliubovExpansionAt modeExpansionWithoutCreation ∧ ¬PhysicalParticleProductionAt modeExpansionWithoutCreation := a_bogoliubov_expansion_does_not_by_itself_establish_particle_creation example : AdiabaticInvarianceAt adiabaticWithoutStrictConservation ∧ ¬StrictParticleNumberConservationAt adiabaticWithoutStrictConservation := adiabatic_invariance_does_not_supply_strict_conservation example : CanonicalQuantizationAt canonicalAndProducedWithoutMinimization ∧ ¬MinimizationSelectedPhysicalParticleAt canonicalAndProducedWithoutMinimization := canonical_quantization_does_not_derive_the_minimization_postulate example : ProducedParticlesAt canonicalAndProducedWithoutMinimization ∧ ¬MinimizationSelectedPhysicalParticleAt canonicalAndProducedWithoutMinimization := produced_particles_do_not_supply_physical_particle_selection example : NonzeroSpinConformalInvarianceAt boundedConformalWithoutScalarOrArbitraryExtension ∧ ¬MinimallyCoupledScalarConformalInvarianceAt boundedConformalWithoutScalarOrArbitraryExtension := nonzero_spin_conformal_invariance_does_not_cover_the_minimal_scalar example : StaticallyBoundedConformalNoCreationAt boundedConformalWithoutScalarOrArbitraryExtension ∧ ¬ArbitraryExpansionConformalNoCreationAt boundedConformalWithoutScalarOrArbitraryExtension := statically_bounded_conformal_no_creation_does_not_supply_arbitrary_expansion_claim example : HeuristicCreationFeedbackAt feedbackWithoutEinsteinEquation ∧ ¬EinsteinFriedmannEquationAt feedbackWithoutEinsteinEquation := heuristic_feedback_does_not_derive_the_einstein_equation end ParkerPaperCountermodelReuse namespace HawkingPaperCountermodelReuse open PhysicsLogic.Papers.Hawking1975.CollapseFramework.Countermodels example : ExponentialHorizonRayTraceAt rayTraceWithoutLatePacket ∧ ¬LateTimeHighFrequencyWavePacketAt rayTraceWithoutLatePacket := exponential_horizon_ray_trace_does_not_supply_late_time_wave_packet example : BogoliubovExpansionAt stationaryExpansionWithoutCreation ∧ ¬OutgoingParticleCreationAt stationaryExpansionWithoutCreation := a_bogoliubov_expansion_does_not_by_itself_establish_creation example : StationaryFinalGeometryAt stationaryExpansionWithoutCreation ∧ ¬CollapseMixingRouteAt stationaryExpansionWithoutCreation := stationary_final_geometry_does_not_remove_the_collapse_route example : HeuristicPairPictureAt pairPictureWithoutThermalCalculation ∧ ¬ThermalOccupationCalculationAt pairPictureWithoutThermalCalculation := the_heuristic_pair_picture_does_not_supply_the_thermal_calculation example : OutgoingThermalFluxAt sameOutgoingFluxWithDifferentLocalStress ∧ ¬LocalStressEnergyFixedAt sameOutgoingFluxWithDifferentLocalStress := outgoing_thermal_flux_does_not_by_itself_fix_local_stress_energy example : QuasiStationaryEvaporationAt quasiStationaryWithoutControlledEndpoint ∧ ¬ControlledClassicalEndpointAt quasiStationaryWithoutControlledEndpoint := quasi_stationary_evaporation_does_not_determine_the_planck_endpoint example : ThermalEmissionAt thermalEmissionWithoutEntropyCompensation ∧ ¬GeneralizedEntropyNondecreaseAt thermalEmissionWithoutEntropyCompensation := thermal_emission_does_not_by_itself_prove_the_generalized_second_law end HawkingPaperCountermodelReuse end CurvedParticleOriginalSourcePhysicalReuseAudit namespace OriginalLocalizationSourcePhysicalReuseAudit open PhysicsLogic.Framework.ResolventDecayLocalizationContext.Countermodels example : PhysicalAndersonRandomLatticeLocatorAt physicalAndersonRandomLocatorReadout ∧ physicalAndersonRandomLocatorReadout.imaginarySelfEnergyValue < 0 := physical_anderson_random_locator_retains_self_energy_sign example : PhysicalAndersonRareResonanceHeavyTailAt physicalAndersonHeavyTailReadout ∧ 0 < physicalAndersonHeavyTailReadout.heavyTailDensityValue := ⟨physical_anderson_rare_resonance_tail_distinguishes_typical_and_mean.1, physical_anderson_rare_resonance_tail_distinguishes_typical_and_mean.2.1⟩ example : PhysicalAndersonHighOrderPathConvergenceAt physicalAndersonHighOrderReadout ∧ PhysicalAndersonInverseCubeBoundaryAt physicalAndersonInverseCubeReadout := ⟨physical_anderson_short_range_paths_and_inverse_cube_boundary_are_distinct.1, physical_anderson_short_range_paths_and_inverse_cube_boundary_are_distinct.2.1⟩ example : PhysicalMacKinnonKramerSampledTwoDimensionalLocalizationAt physicalMacKinnonKramerTwoDimensionalReadout ∧ physicalMacKinnonKramerTwoDimensionalReadout.unsampledWeakerDisorderValue < physicalMacKinnonKramerTwoDimensionalReadout.minimumSampledDisorderValue := physical_mackinnon_kramer_two_dimensional_localization_has_unsampled_disorder example : PhysicalMacKinnonKramerThreeDimensionalCriticalFitAt physicalMacKinnonKramerThreeDimensionalReadout ∧ 0 < physicalMacKinnonKramerThreeDimensionalReadout.criticalDisorderUncertaintyValue := ⟨physical_mackinnon_kramer_three_dimensional_critical_fit_retains_error_bars.1, physical_mackinnon_kramer_three_dimensional_critical_fit_retains_error_bars.2.1⟩ example : ¬PhysicalMacKinnonKramerConditionalTransportAt physicalMacKinnonKramerLandauerBridgeErasedControl := mackinnon_kramer_transmission_landauer_channel_and_scaling_assumptions_are_independent.2.2.1 example : ¬physicalLocalizationEstablishment.establishedAt .finiteBarRecursionProvesInfiniteVolumeLocalization .macKinnonKramer1981 := localization_sources_distinguish_locator_resonances_finite_bars_critical_fits_and_scope.2.2.2.2.2.2.2.2.2 end OriginalLocalizationSourcePhysicalReuseAudit namespace OriginalFocusingSourcePhysicalReuseAudit open PhysicsLogic.Framework.CongruenceFocusing.Countermodels example : PhysicalRaychaudhuriTimelikeVolumeEquationAt physicalRaychaudhuriGeneralVolumeReadout ∧ physicalRaychaudhuriGeneralVolumeReadout.localVolumeFactorValue = 8 := physical_raychaudhuri_local_volume_equation_retains_density_anisotropy_and_spin_signs example : PhysicalRaychaudhuriRotationRepulsionAt physicalRaychaudhuriRotationReadout ∧ 0 < physicalRaychaudhuriRotationReadout.normalizedVolumeAccelerationValue := physical_raychaudhuri_positive_rotation_can_reverse_timelike_focusing example : PhysicalRaychaudhuriAnisotropicTimeComparisonAt physicalRaychaudhuriAnisotropyReadout ∧ PhysicalRaychaudhuriLambdaFreedomAt physicalRaychaudhuriLambdaFreedomReadout := ⟨physical_raychaudhuri_anisotropy_shortens_fixed_lambda_comparison.1, physical_raychaudhuri_anisotropy_shortens_fixed_lambda_comparison.2.1⟩ example : PhysicalPenroseNullOpticalScalarChainAt physicalPenroseOpticalReadout ∧ physicalPenroseOpticalReadout.nullAreaRootSecondDerivativeValue = -4 := ⟨physical_penrose_optical_scalar_chain_retains_area_root_shear_and_ricci.1, physical_penrose_optical_scalar_chain_retains_area_root_shear_and_ricci.2.1⟩ example : PhysicalPenroseNoncompactCauchyReductioAt physicalPenroseGlobalReductioReadout ∧ physicalPenroseGlobalReductioReadout.boundaryMapDegreeValue = physicalPenroseGlobalReductioReadout.cauchyMapDegreeValue := physical_penrose_incompleteness_requires_noncompact_cauchy_and_global_degree example : ¬PhysicalPenroseDirectionalIncompletenessScopeAt physicalPenroseCurvatureBlowupClaimedControl := penrose_null_incompleteness_does_not_promote_timelike_curvature_or_horizon_claims.2.2.1 example : ¬physicalFocusingEstablishment.establishedAt .nullIncompletenessProvesCurvatureBlowup .penrose1965 := focusing_sources_distinguish_timelike_volume_null_optics_global_reductio_and_scope.2.2.2.2.2.2.2.2.2.2 end OriginalFocusingSourcePhysicalReuseAudit namespace OriginalRegulatedFlowSourcePhysicalReuseAudit open PhysicsLogic.Framework.ExactRegulatedScaleFlowContext.Countermodels example : PhysicalWegnerHoughtonSharpShellSetupAt physicalWegnerHoughtonSharpShellReadout ∧ physicalWegnerHoughtonSharpShellReadout.lowerShellBoundaryValue = 9 / 10 := physical_wegner_houghton_sharp_shell_preserves_partition_and_removes_selected_modes example : PhysicalWegnerHoughtonEpsilonExpansionAt physicalWegnerHoughtonEpsilonReadout ∧ physicalWegnerHoughtonEpsilonReadout.anomalousSecondOrderValue = 1 / 5400 := ⟨physical_wegner_houghton_epsilon_expansion_retains_all_four_printed_coefficients.1, physical_wegner_houghton_epsilon_expansion_retains_all_four_printed_coefficients.2.1⟩ example : PhysicalWegnerHoughtonSeparateLargeNAt physicalWegnerHoughtonLargeNReadout ∧ PhysicalWegnerHoughtonProspectiveFiniteRecursionAt physicalWegnerHoughtonFiniteStepReadout := ⟨physical_wegner_houghton_large_n_is_independent_of_prospective_finite_recursion.1, physical_wegner_houghton_large_n_is_independent_of_prospective_finite_recursion.2.1⟩ example : PhysicalWetterichModifiedLegendreTraceFlowAt physicalWetterichTraceFlowReadout ∧ physicalWetterichTraceFlowReadout.averageActionFlowValue = 1 / 2 := physical_wetterich_modified_legendre_flow_requires_full_inverse_propagator example : PhysicalWetterichExactGoldstoneRadialProjectionAt physicalWetterichPotentialProjectionReadout ∧ physicalWetterichPotentialProjectionReadout.goldstoneDenominatorValue = 5 ∧ physicalWetterichPotentialProjectionReadout.radialDenominatorValue = 8 := ⟨physical_wetterich_exact_potential_projection_distinguishes_goldstone_and_radial_modes.1, physical_wetterich_exact_potential_projection_distinguishes_goldstone_and_radial_modes.2.1, physical_wetterich_exact_potential_projection_distinguishes_goldstone_and_radial_modes.2.2.1⟩ example : PhysicalWetterichQualifiedKineticTruncationAt physicalWetterichTruncationReadout ∧ physicalWetterichTruncationReadout.truncatedPotentialFlowValue < physicalWetterichTruncationReadout.exactPotentialFlowValue := physical_wetterich_kinetic_truncation_is_not_exact_or_error_controlled example : ¬PhysicalWetterichZeroMomentumRegulatorQualificationAt physicalWetterichZeroModeLimitNoncommutationClaimedControl := wetterich_truncation_error_zero_mode_limits_regulator_choice_and_equivalence_are_independent.2.2.2.2.1 example : physicalWetterichZeroMomentumReadout.zeroMomentumScaleLimitsCommute = true ∧ (∀ beta : Real, 1 < beta → physicalWetterichZeroMomentumReadout.zeroMomentumScaleLimitNoncommutationEstablishedAtShape beta ∧ physicalWetterichZeroMomentumReadout.zeroMomentumFirstDivergesAtShape beta) := ⟨wetterich_preferred_and_greater_regulator_shapes_have_distinct_limit_orders.2.1, wetterich_preferred_and_greater_regulator_shapes_have_distinct_limit_orders.2.2.2.1⟩ example : ¬physicalRegulatedFlowEstablishment.establishedAt .usefulTruncationSuppliesQuantitativeErrorControl .wetterich1993 := regulated_flow_sources_distinguish_sharp_shell_legendre_trace_endpoints_projections_and_scope.2.2.2.2.2.2.2.2.2.2 end OriginalRegulatedFlowSourcePhysicalReuseAudit namespace OriginalStandardCandleSourcePhysicalReuseAudit open PhysicsLogic.Framework.StandardCandleAccelerationInferenceCountermodels example : PhysicalRiessDiscoveryAndSpectroscopicSelectionAt physicalRiessSpectroscopicSelectionReadout ∧ physicalRiessSpectroscopicSelectionReadout.spectroscopicallyConfirmedCount + physicalRiessSpectroscopicSelectionReadout.excludedUnclassifiedCount = physicalRiessSpectroscopicSelectionReadout.highRedshiftSampleCount := physical_riess_sample_retains_spectroscopic_and_unclassified_object_cuts example : PhysicalRiessMethodDependentFlatFitsAt physicalRiessDualMethodFlatFitReadout ∧ physicalRiessDualMethodFlatFitReadout.vacuumDensityValue < physicalRiessDualMethodFlatFitReadout.alternateVacuumDensityValue := physical_riess_distinct_flat_prior_methods_retain_different_fit_values example : PhysicalPerlmutterInclusiveAndPrimaryFitCutsAt physicalPerlmutterFitCutsReadout ∧ physicalPerlmutterFitCutsReadout.primaryFitCount < physicalPerlmutterFitCutsReadout.inclusiveFitCount := physical_perlmutter_fit_c_preserves_inclusive_outlier_and_reddening_cuts example : PhysicalPerlmutterJointConstraintAndFlatFitAt physicalPerlmutterFlatFitReadout ∧ physicalPerlmutterFlatFitReadout.statisticalUpperUncertaintyValue ≠ physicalPerlmutterFlatFitReadout.systematicUpperUncertaintyValue := physical_perlmutter_flat_fit_separates_statistical_and_systematic_errors example : PhysicalPerlmutterCrossTeamQualifiedComparisonAt physicalPerlmutterCrossTeamComparisonReadout ∧ physicalPerlmutterCrossTeamComparisonReadout.sharedHighRedshiftEventCount = 2 ∧ physicalPerlmutterCrossTeamComparisonReadout.crossTeamStatisticalIndependenceEstablished = false := physical_perlmutter_independent_searches_share_nearby_sample_and_two_distant_events example : ¬PhysicalPerlmutterCrossTeamQualifiedComparisonAt physicalPerlmutterCrossTeamIndependenceClaimedControl := perlmutter_distinct_searches_shared_samples_and_independence_scope_are_independent.2.2.2 example : ¬PhysicalPerlmutterUnquantifiedAlternativesAt physicalPerlmutterGreyDustDeclaredExcludedControl := identified_systematic_budgets_do_not_eliminate_grey_dust_or_evolution.2.2.2.1 example : ¬physicalStandardCandleEstablishment.establishedAt .identifiedErrorBudgetEliminatesUnknownSystematics .perlmutter1999 := standard_candle_source_overclaims_remain_epistemically_unestablished.2.2.2.1 end OriginalStandardCandleSourcePhysicalReuseAudit namespace OriginalRemoteCorrelationSourcePhysicalReuseAudit open PhysicsLogic.Framework.RemoteCorrelationPredictionContext.Countermodels example : PhysicalEPRSeparatedPositionPredictionAt physicalEprPositionPredictionReadout ∧ PhysicalEPRSeparatedMomentumPredictionAt physicalEprMomentumPredictionReadout := ⟨physical_epr_position_and_momentum_predictions_have_distinct_source_values.1, physical_epr_position_and_momentum_predictions_have_distinct_source_values.2.1⟩ example : PhysicalEPRNoncommutingAlternativeMeasurementsAt physicalEprAlternativeMeasurementsReadout ∧ physicalEprAlternativeMeasurementsReadout.bothIncompatibleProceduresActuallyPerformed = false := physical_epr_incompatible_predictions_are_counterfactual_not_joint_measurements example : PhysicalEPRDistributionalContinuumIdealizationAt physicalEprDistributionalContinuumReadout ∧ physicalEprDistributionalContinuumReadout.idealContinuumStateSquareNormalizable = false ∧ physicalEprDistributionalContinuumReadout.finiteResolutionRegularizationSupplied = false := physical_epr_ideal_continuum_prediction_retains_distributional_state_scope example : ¬PhysicalEPRDistributionalContinuumIdealizationAt physicalEprContinuumStateDeclaredNormalizableControl := epr_distributional_state_normalizability_and_finite_resolution_scope_are_independent.2.2.1 example : PhysicalBellOriginalThreeSettingViolationAt physicalBellOriginalInequalityReadout ∧ physicalBellOriginalInequalityReadout.bellDifferenceMagnitudeValue = 1 ∧ physicalBellOriginalInequalityReadout.bellLocalUpperBoundValue = 1 / 2 := physical_bell_original_three_setting_inequality_has_a_strict_quantum_violation example : ¬PhysicalBellLocalResponseAndNormalizedDistributionAt physicalBellSettingLocalityErasedControl := bell_same_axis_anticorrelation_hidden_normalization_and_locality_are_independent.2.2.2.2.1 example : PhysicalBellNonlocalCompletionAndQualifiedInfluenceAt physicalBellNonlocalCompletionReadout ∧ physicalBellNonlocalCompletionReadout.allHiddenVariablesExcluded = false ∧ physicalBellNonlocalCompletionReadout.operationalSuperluminalSignallingDerived = false := physical_bell_nonlocal_completion_does_not_imply_operational_signalling example : ¬physicalRemoteCorrelationEstablishment.establishedAt .bellAllHiddenVariableCompletionsExcluded .bell1964 := remote_correlation_source_overclaims_remain_epistemically_unestablished.2.2.2.1 end OriginalRemoteCorrelationSourcePhysicalReuseAudit namespace OriginalFractionalHallSourcePhysicalReuseAudit open PhysicsLogic.Framework.FractionalHallQuasiholeChargeContext.Countermodels example : PhysicalLaughlinPlasmaPhantomChargeScreeningAt physicalLaughlinPlasmaScreeningReadout ∧ PhysicalLaughlinIndependentFluxInsertionCountingAt physicalLaughlinFluxInsertionReadout := ⟨physical_laughlin_plasma_screening_and_flux_insertion_independently_give_charge.1, physical_laughlin_plasma_screening_and_flux_insertion_independently_give_charge.2.1⟩ example : PhysicalHaldaneSphericalMonopoleFluxAndShiftAt physicalHaldaneSphericalFluxReadout ∧ physicalHaldaneSphericalFluxReadout.finiteSphereFillingValue = 4 / 9 ∧ physicalHaldaneSphericalFluxReadout.thermodynamicParentFillingValue = 1 / 3 := physical_haldane_spherical_flux_retains_monopole_shift_and_finite_size_filling example : PhysicalHaldaneCoordinateExchangeSymmetryAt physicalHaldaneCoordinateExchangeReadout ∧ 0 < physicalHaldaneCoordinateExchangeReadout.encirclingStatisticalPhaseValue := physical_haldane_coordinate_symmetry_does_not_erase_adiabatic_holonomy example : PhysicalHaldaneCoordinateExchangeSymmetryAt physicalHaldaneGeometricPhaseForcedToZeroControl := haldane_coordinate_exchange_symmetry_and_geometric_winding_are_independent.2.2.1 example : physicalHaldaneCoordinateExchangeReadout.encirclingStatisticalPhaseValue ≠ physicalHaldaneGeometricPhaseForcedToZeroControl.encirclingStatisticalPhaseValue := haldane_coordinate_exchange_symmetry_and_geometric_winding_are_independent.2.2.2 example : PhysicalArovasInsideOutsideEncirclingDifferenceAt physicalArovasEncirclingReadout ∧ physicalArovasEncirclingReadout.encirclingStatisticalPhaseValue = 2 * Real.pi / 5 := physical_arovas_inside_outside_comparison_produces_fractional_winding example : PhysicalArovasFractionalExchangeAndStatisticalPotentialAt physicalArovasExchangeAndPotentialReadout ∧ physicalArovasExchangeAndPotentialReadout.exchangePhaseValue = Real.pi / 5 := physical_arovas_exchange_is_half_winding_with_nontrivial_statistical_potential example : ¬PhysicalArovasFractionalExchangeAndStatisticalPotentialAt physicalArovasExchangeIdentifiedWithFullWindingControl := arovas_enclosed_density_full_winding_exchange_and_potential_are_independent.2.2.2.2.1 example : ¬physicalFractionalHallEstablishment.establishedAt .symmetricQuasiholeCoordinatesEliminateBerryExchangePhase .haldane1983 := fractional_hall_source_overclaims_remain_epistemically_unestablished.2.2.2.1 end OriginalFractionalHallSourcePhysicalReuseAudit namespace OriginalLinearResponseSourcePhysicalReuseAudit open PhysicsLogic.Framework.LinearResponseTransportContext.Countermodels example : PhysicalNakanoCausalPulseConvolutionAt physicalNakanoCausalPulseReadout ∧ physicalNakanoCausalPulseReadout.inducedCurrentValue = 6 := physical_nakano_pulse_convolution_retains_actual_retarded_support example : PhysicalNakanoNormalizedThermalKernelAt physicalNakanoThermalKernelReadout ∧ physicalNakanoThermalKernelReadout.thermalAverageValue = 3 / 2 := physical_nakano_thermal_kernel_requires_normalized_positive_trace example : PhysicalNakanoStaticSpectralConductivityAt physicalNakanoStaticSpectralReadout ∧ physicalNakanoStaticSpectralReadout.staticConductivityValue = Real.pi / 2 := physical_nakano_static_conductivity_retains_energy_gap_and_pi_prefactor example : PhysicalKuboQuantumConductivityTensorAt physicalKuboQuantumTensorReadout ∧ physicalKuboQuantumTensorReadout.oscillatorEnergyValue = 2 := physical_kubo_theorem_five_keeps_both_tensor_channels_and_static_limit example : ¬PhysicalNakanoSingleRelaxationApproximationAt physicalNakanoRelaxationAssumptionErasedControl := nakano_gap_quotient_spectral_prefactor_and_approximation_scope_are_independent.2.2.2.2.1 example : ¬PhysicalKuboQuantumConductivityTensorAt physicalKuboAntisymmetricSpectrumErasedControl := kubo_poisson_sign_past_support_and_two_noise_channels_are_independent.2.2.2.2.2.2 example : ¬physicalLinearResponseEstablishment.establishedAt .everyTransportProcessIsMarkovian .kubo1957 := linear_response_source_overclaims_remain_epistemically_unestablished.2.2.1 end OriginalLinearResponseSourcePhysicalReuseAudit namespace OriginalSolitaryWaveSourcePhysicalReuseAudit open PhysicsLogic.Framework.SolitaryTravelingWaveContext.Countermodels example : PhysicalKortewegCapillarityCoefficientAt physicalKortewegCapillarityReadout ∧ physicalKortewegCapillarityReadout.capillarityDispersionCoefficientValue = 3 := physical_korteweg_dispersion_retains_depth_gravity_density_and_capillarity example : PhysicalKortewegPositiveElevationProfileAt physicalKortewegPositivePulseReadout ∧ PhysicalKortewegCapillaryDepressionProfileAt physicalKortewegCapillaryDepressionReadout ∧ physicalKortewegCapillaryDepressionReadout.signedAmplitudeValue < physicalKortewegPositivePulseReadout.signedAmplitudeValue := physical_korteweg_positive_and_capillary_negative_pulses_are_distinct example : PhysicalZabuskyPeriodicCosineProtocolAt physicalZabuskyPeriodicProtocolReadout ∧ PhysicalZabuskyIsolatedPulseRelationsAt physicalZabuskyIsolatedPulseReadout ∧ physicalZabuskyIsolatedPulseReadout.pulseSpeedValue = 1 := physical_zabusky_period_two_cosine_and_isolated_pulse_relations_are_distinct example : PhysicalZabuskyNumericalConservationAt physicalZabuskyConservationReadout ∧ physicalZabuskyConservationReadout.momentumDriftValue = 0 ∧ 0 < physicalZabuskyConservationReadout.energyDriftValue := physical_zabusky_exact_momentum_and_five_digit_energy_checks_are_distinct example : ¬PhysicalZabuskyQualifiedNearRecurrenceAt physicalZabuskyExactStateRecurrenceClaimedControl := zabusky_profile_near_return_interaction_and_thermalization_scope_are_independent.2.1 example : ¬physicalSolitaryWaveEstablishment.establishedAt .nearReturnProvesExactFullStateRecurrence .zabuskyKruskal1965 := solitary_wave_source_overclaims_remain_epistemically_unestablished.2.2.1 end OriginalSolitaryWaveSourcePhysicalReuseAudit namespace OriginalMicrowaveBackgroundSourcePhysicalReuseAudit open PhysicsLogic.Framework.CalibratedResidualBackgroundContext.Countermodels example : PhysicalHornFrequencyAndColdLoadAt physicalPenziasWilsonColdLoadReadout ∧ PhysicalHornAtmosphericOhmicBacklobeAccountingAt physicalPenziasWilsonForegroundAccountingReadout ∧ physicalPenziasWilsonForegroundAccountingReadout.residualTemperatureKelvinValue = 35 / 10 := physical_horn_residual_retains_4080_mhz_and_calibrated_foreground_budget example : PhysicalHornResidualIsotropyAndSeasonScopeAt physicalPenziasWilsonQualifiedIsotropyReadout ∧ PhysicalHornProofNoteTwoFrequencyComparisonAt physicalPenziasWilsonProofNoteReadout ∧ PhysicalHornCompanionCosmologyScopeAt physicalPenziasWilsonCosmologyScopeReadout := physical_horn_isotropy_proof_note_and_companion_cosmology_are_distinct example : PhysicalFirasTableFourForegroundAndResidualAt physicalFirasTableFourResidualReadout ∧ physicalFirasTableFourResidualReadout.selectedTableFourResidualValue = 5 ∧ physicalFirasTableFourResidualReadout.selectedGalacticPoleValue = 4 := physical_firas_table_four_retains_nonzero_residual_and_galactic_foreground example : PhysicalFirasSystematicTemperatureAndDipoleAt physicalFirasTemperatureAndDipoleReadout ∧ PhysicalFirasChemicalAndComptonDistortionLimitsAt physicalFirasDistortionLimitsReadout ∧ physicalFirasTemperatureAndDipoleReadout.confidencePercentValue = 95 := physical_firas_temperature_dipole_and_distortion_limits_retain_confidence_scope example : ¬PhysicalFirasApproximateBlackbodyScopeAt physicalFirasBlackbodyDeclaredExactControl := firas_small_residual_does_not_supply_exact_blackbody_or_exhaustive_foregrounds.2.1 example : ¬physicalMicrowaveBackgroundEstablishment.establishedAt .fiftyPartPerMillionResidualIsExactlyZero .fixsenEtAl1996 := microwave_background_source_overclaims_remain_epistemically_unestablished.2.2.2.1 end OriginalMicrowaveBackgroundSourcePhysicalReuseAudit namespace OriginalParityExperimentSourcePhysicalReuseAudit open PhysicsLogic.Framework.ControlledSymmetryFalsificationCountermodels example : PhysicalWuGammaPolarizationCalibrationAt physicalWuGammaCalibrationReadout ∧ PhysicalWuBetaFieldReversalAndPseudoscalarAt physicalWuBetaFieldReversalReadout ∧ physicalWuBetaFieldReversalReadout.reflectedSpinMomentumPseudoscalarValue = -1 := physical_wu_gamma_calibration_and_beta_field_reversal_are_distinct example : PhysicalWuIndependentUnpolarizedControlsAt physicalWuIndependentControlsReadout ∧ PhysicalWuApproximateAsymmetryCoefficientsAt physicalWuApproximateCoefficientsReadout ∧ physicalWuApproximateCoefficientsReadout.signedCobaltAlphaValue < 0 := physical_wu_unpolarized_controls_and_rough_coefficients_are_independent example : PhysicalGarwinDelayedCoincidenceGateAt physicalGarwinDelayedGateReadout ∧ physicalGarwinDelayedGateReadout.gateDurationMicrosecondsValue = 5 / 4 := physical_garwin_delayed_gate_retains_microsecond_window_and_energy_threshold example : PhysicalGarwinAngularAnalyzerAndPartialPolarizationAt physicalGarwinAngularAnalyzerReadout ∧ physicalGarwinAngularAnalyzerReadout.observedAnalyzerCoefficientValue ≠ physicalGarwinAngularAnalyzerReadout.theoreticalIntrinsicAnalyzerCoefficientValue := physical_garwin_angular_curve_distinguishes_partial_polarization_from_intrinsic_asymmetry example : PhysicalGarwinStoppedPionNullControlAt physicalGarwinStoppedPionControlReadout ∧ PhysicalGarwinFiniteAccuracyMomentAndPrecessionAt physicalGarwinMomentReadout ∧ physicalGarwinMomentReadout.extractedMuonGUncertaintyValue = 1 / 10 := physical_garwin_stopped_pion_null_and_finite_accuracy_moment_are_distinct example : ¬PhysicalGarwinFiniteAccuracyMomentAndPrecessionAt physicalGarwinExactGDeclaredEstablishedControl := garwin_null_control_moment_error_precession_and_future_precision_are_independent.2.2.2.2.1 example : ¬physicalParityExperimentEstablishment.establishedAt .garwinExactGEqualTwoEstablished .garwinLedermanWeinrich1957 := parity_experiment_source_overclaims_remain_epistemically_unestablished.2.2.2.1 end OriginalParityExperimentSourcePhysicalReuseAudit namespace OriginalNoCloningSourcePhysicalReuseAudit open PhysicsLogic.Framework.LinearStateReplicationContext.Countermodels example : physicalDieksSpinReplicationModel.system = .singletSpinElectron ∧ physicalDieksSpinReplicationModel.firstBasisState = .xSpinPlus ∧ physicalDieksSpinReplicationModel.secondBasisState = .xSpinMinus ∧ physicalDieksSpinReplicationModel.superposedState = .zSpinPlus ∧ physicalDieksSpinReplicationContext.EstablishedLinearStateReplicationObstructionAt physicalDieksSpinReplicationModel := physical_dieks_spin_multiplier_retains_orthogonal_basis_and_neutral_obstruction example : PhysicalWoottersNormalizedPolarizationBasisAt physicalWoottersNormalizedBasisReadout ∧ physicalWoottersNormalizedBasisReadout.fortyFiveDegreeAmplitudeValue ^ 2 = 1 / 2 := physical_wootters_normalized_basis_retains_forty_five_degree_example example : PhysicalWoottersEqualApparatusLinearOutputAt physicalWoottersEqualApparatusReadout ∧ PhysicalWoottersDistinctApparatusMixedOutputAt physicalWoottersDistinctApparatusReadout ∧ physicalWoottersDistinctApparatusReadout.reducedPhotonPurityValue < physicalWoottersEqualApparatusReadout.reducedPhotonPurityValue := physical_wootters_equal_and_distinct_apparatus_records_change_photon_purity example : PhysicalWoottersRequiredTensorSquareCrossTermsAt physicalWoottersTensorSquareReadout ∧ physicalWoottersTensorSquareReadout.cloneVerticalHorizontalAmplitudeValue = 12 / 25 ∧ physicalWoottersTensorSquareReadout.linearVerticalHorizontalAmplitudeValue = 0 := physical_wootters_tensor_square_has_cross_terms_absent_from_linear_output example : ¬PhysicalWoottersDistinctApparatusMixedOutputAt physicalWoottersMixedOutputDeclaredPureControl := wootters_apparatus_records_linear_cross_terms_and_mixed_purity_are_independent.2.2.2.2.2 example : PhysicalDieksSpinSuperpositionAndBranchLinearityAt physicalDieksSpinSuperpositionReadout ∧ PhysicalDieksHalfSplitOrthogonalityAndBornStatisticsAt physicalDieksOccupationStatisticsReadout ∧ physicalDieksOccupationStatisticsReadout.halfSplitProbabilityValue = 0 := physical_dieks_linear_branches_give_half_half_zero_occupation_probabilities example : physicalDieksJointCoherenceErasedControl.coherentMultiplierBranchRetained = true ∧ ¬PhysicalDieksJointMultiplierStateAt physicalDieksJointCoherenceErasedControl ∧ ¬PhysicalDieksCoherenceReportAccurateAt physicalDieksJointCoherenceErasedControl := by obtain ⟨_source, _coherence, reported, notState, notAccurate, _notSource, _notChangedState⟩ := dieks_joint_coherence_cannot_be_replaced_by_a_boolean_report exact ⟨reported, notState, notAccurate⟩ example : physicalDieksHalfSplitAliasedOccupationControl.halfSplitOrthogonalityRetained = true ∧ ¬PhysicalDieksOccupationSectorSeparationAt physicalDieksHalfSplitAliasedOccupationControl ∧ ¬PhysicalDieksOrthogonalityReportAccurateAt physicalDieksHalfSplitAliasedOccupationControl := by obtain ⟨_source, reported, notSeparated, notAccurate, _notSource⟩ := dieks_occupation_orthogonality_cannot_be_replaced_by_a_boolean_report exact ⟨reported, notSeparated, notAccurate⟩ example : PhysicalDieksSpinSuperpositionAndBranchLinearityAt physicalDieksEqualMachineRecordsReadout ∧ physicalDieksEqualMachineRecordsReadout.allPlusMachineRecordValue = physicalDieksEqualMachineRecordsReadout.allMinusMachineRecordValue ∧ physicalDieksEqualMachineRecordsReadout.jointBranchCoherenceValue = 1 / 2 := dieks_global_coherence_does_not_require_distinct_machine_records example : PhysicalDieksSpinSuperpositionAndBranchLinearityAt physicalDieksSpinSuperpositionReadout ∧ PhysicalDieksSpinSuperpositionAndBranchLinearityAt physicalDieksNegativeRelativePhaseReadout ∧ physicalDieksSpinSuperpositionReadout.jointBranchCoherenceValue = 1 / 2 ∧ physicalDieksNegativeRelativePhaseReadout.jointBranchCoherenceValue = -1 / 2 := by obtain ⟨positiveSource, negativeSource, _samePlus, _sameMinus, positivePhase, negativePhase⟩ := dieks_opposite_joint_phases_preserve_occupation_probabilities exact ⟨positiveSource, negativeSource, positivePhase, negativePhase⟩ example : PhysicalDieksConditionalNaiveMultiplierPredictionAt physicalDieksNaiveMultiplierReadout ∧ physicalDieksNaiveMultiplierReadout.naiveHalfSplitProbabilityValue = 3 / 8 ∧ physicalDieksNaiveMultiplierReadout.halfSplitProbabilityValue = 0 := physical_dieks_conditional_half_split_differs_from_actual_linear_burst example : ¬PhysicalDieksHalfSplitOrthogonalityAndBornStatisticsAt physicalDieksHalfSplitProbabilityDeclaredPositiveControl := dieks_occupation_orthogonality_and_born_branch_probabilities_are_independent.2.2.2.1 example : ¬PhysicalDieksSameDistributionAndSingleRunNoSignalAt physicalDieksCommutativityDeclaredSufficientControl := dieks_naive_multiplier_remote_distribution_and_linearity_scope_are_independent.2.2.2.2.2.2 example : ¬physicalNoCloningEstablishment.establishedAt .naiveHalfSplitActuallyOccursInLinearBurst .dieks1982 := no_cloning_source_overclaims_remain_epistemically_unestablished.2.2.2.2.1 end OriginalNoCloningSourcePhysicalReuseAudit namespace OriginalAmplitudeProbabilitySourcePhysicalReuseAudit open PhysicsLogic.Framework.AmplitudeOutcomeProbabilityCountermodels example : PhysicalBornNormalizedDiscreteFrequenciesAt physicalBornNormalizedChannelReadout ∧ physicalBornNormalizedChannelReadout.selectedChannelCountValue = 9 := physical_born_discrete_weights_are_normalized_squared_ensemble_frequencies example : PhysicalBornDirectionalSolidAngleWeightAt physicalBornDirectionalWeightReadout ∧ physicalBornDirectionalWeightReadout.directionalWeightValue = 9 / 100 := physical_born_directional_probability_retains_the_solid_angle_element example : PhysicalBornInelasticEnergyAndChannelWeightAt physicalBornInelasticChannelReadout ∧ physicalBornInelasticChannelReadout.finalEnergyValue = 1 := physical_born_inelastic_channel_retains_energy_jump_and_pi_squared_factor example : PhysicalKennardContinuousDensityAndIntervalAt physicalKennardIntervalProbabilityReadout ∧ physicalKennardIntervalProbabilityReadout.positionDensityValue ≠ physicalKennardIntervalProbabilityReadout.intervalProbabilityValue := physical_kennard_probability_density_becomes_probability_only_on_a_region example : PhysicalKennardDualGaussianWidthsAt physicalKennardGaussianWidthReadout ∧ physicalKennardGaussianWidthReadout.positionWidthValue * physicalKennardGaussianWidthReadout.momentumWidthValue = 1 / 2 := physical_kennard_dual_gaussian_widths_saturate_hbar_over_two example : ¬PhysicalBornInelasticEnergyAndChannelWeightAt physicalBornWrongEnergyJumpControl := by obtain ⟨_, _, _, _, _, wrongJump, _⟩ := born_squared_weight_solid_angle_energy_jump_and_channel_factor_are_independent exact wrongJump example : ¬PhysicalKennardContinuousDensityAndIntervalAt physicalKennardDensityAsIntervalProbabilityControl := kennard_density_interval_gaussian_widths_and_delta_regulator_are_independent.2.1 example : ¬physicalProbabilityEstablishment.establishedAt .predictiveStochasticityProvesOnticIndeterminism .born1926 := amplitude_probability_source_overclaims_remain_epistemically_unestablished.2.2.2.1 end OriginalAmplitudeProbabilitySourcePhysicalReuseAudit namespace OriginalMajoranaSourcePhysicalReuseAudit open PhysicsLogic.Framework.BoundaryMajoranaModeContext.Countermodels example : PhysicalKitaevSolvableUnpairedEndpointsAt physicalKitaevSolvableEndpointReadout ∧ physicalKitaevSolvableEndpointReadout.forwardMajoranaCouplingValue = 4 := physical_kitaev_solvable_chain_has_two_opposite_parity_endpoints example : PhysicalKitaevExponentiallySplitFiniteWireAt physicalKitaevFiniteSplittingReadout ∧ 0 < physicalKitaevFiniteSplittingReadout.finiteWireSplittingValue := physical_kitaev_finite_wire_splitting_is_positive_and_exponentially_small example : PhysicalOregCentralGapAndCriticalSurfaceAt physicalOregCentralGapReadout ∧ physicalOregCentralGapReadout.leftCentralGapValue = 2 ∧ physicalOregCentralGapReadout.criticalCentralGapValue = 0 := physical_oreg_central_gap_distinguishes_topological_trivial_and_critical_sides example : PhysicalOregDoubleModePairingSignWallAt physicalOregDoubleModeReadout ∧ physicalOregDoubleModeReadout.signChangingPairingModeCountValue = 2 := physical_oreg_pairing_sign_wall_can_support_two_splittable_modes example : PhysicalFuKaneParityProtectedHalfPhaseBranchAt physicalFuKaneHalfPhaseReadout ∧ physicalFuKaneHalfPhaseReadout.localHamiltonianPeriodPiValue = 2 ∧ physicalFuKaneHalfPhaseReadout.fixedParityBranchPeriodPiValue = 4 := ⟨physical_fu_kane_two_pi_hamiltonian_supports_a_four_pi_fixed_parity_branch.1, rfl, rfl⟩ example : PhysicalFuKaneParitySwitchingEquilibriumAt physicalFuKanePoisonedEquilibriumReadout ∧ physicalFuKanePoisonedEquilibriumReadout.equilibriumPeriodPiValue = 2 := ⟨physical_fu_kane_local_parity_exchange_restores_two_pi_equilibrium.1, rfl⟩ example : ¬PhysicalKitaevExponentiallySplitFiniteWireAt physicalKitaevExactlyDegenerateFiniteWireControl := by obtain ⟨_, _, _, _, notExactlyDegenerate, _, _, _, _⟩ := kitaev_pairing_roots_finite_tunneling_and_oreg_interfaces_are_independent exact notExactlyDegenerate example : ¬PhysicalFuKaneParityProtectedHalfPhaseBranchAt physicalFuKaneHamiltonianDeclaredFourPiControl := by obtain ⟨_, _, _, _, _, _, _, notFourPi, _, _, _⟩ := oreg_mode_count_fu_kane_mass_sign_and_parity_branches_are_independent exact notFourPi example : ¬physicalMajoranaEstablishment.establishedAt .kitaevFiniteWireGroundStatesAreExactlyDegenerate .kitaev2001 := majorana_source_overclaims_remain_epistemically_unestablished.1 end OriginalMajoranaSourcePhysicalReuseAudit namespace OriginalCommutatorSourcePhysicalReuseAudit open PhysicsLogic.Framework.FiniteVelocityCommutatorConeContext.Countermodels example : PhysicalLiebRobinsonFiniteRangeExteriorRayAt physicalLiebRobinsonExteriorReadout ∧ 0 < physicalLiebRobinsonExteriorReadout.exteriorCommutatorValue := physical_lieb_robinson_exterior_bound_allows_a_positive_commutator example : PhysicalNachtergaeleSimsWeightedMetricBoundAt physicalNachtergaeleSimsMetricReadout ∧ physicalNachtergaeleSimsMetricReadout.initialSharpMetricBoundValue = 0 := physical_nachtergaele_sims_weighted_bound_improves_at_separated_supports example : PhysicalNachtergaeleSimsGapAndGroundProjectionAt physicalNachtergaeleSimsClusteringReadout ∧ physicalNachtergaeleSimsClusteringReadout.clusteringDecayRateValue = 1 / 3 := physical_nachtergaele_sims_clustering_requires_gap_and_both_ground_projections example : PhysicalHastingsKomaExponentialInteractionAt physicalHastingsKomaExponentialReadout ∧ PhysicalHastingsKomaPowerLawInteractionAt physicalHastingsKomaPowerLawReadout ∧ physicalHastingsKomaPowerLawReadout.sharedConeModel = none := physical_hastings_koma_exponential_and_power_law_tails_remain_distinct example : PhysicalHastingsKomaOddFermionicChannelAt physicalHastingsKomaOddFermionReadout ∧ physicalHastingsKomaOddFermionReadout.channel = .oddFermionicAnticommutator := physical_hastings_koma_odd_modes_use_an_anticommutator example : ¬PhysicalHastingsKomaPowerLawInteractionAt physicalHastingsKomaPowerLawDeclaredExponentialControl := hastings_koma_interaction_tails_statistics_and_ground_corrections_are_independent.2.1 example : ¬PhysicalNachtergaeleSimsGapAndGroundProjectionAt physicalNachtergaeleSimsMissingGroundProjectionControl := by obtain ⟨_, _, _, _, _, _, _, missingProjection⟩ := lieb_robinson_interaction_control_weighted_bounds_and_projection_are_independent exact missingProjection example : ¬physicalCommutatorEstablishment.establishedAt .exponentialExteriorBoundMeansExactCommutatorVanishing .liebRobinson1972 := commutator_source_overclaims_remain_epistemically_unestablished.1 end OriginalCommutatorSourcePhysicalReuseAudit namespace OriginalCoarseGrainingSourcePhysicalReuseAudit open PhysicsLogic.Framework.CoarseGrainedScalingContext.Countermodels example : PhysicalKadanoffSeparatedCriticalCellAt physicalKadanoffSeparatedCellReadout ∧ physicalKadanoffSeparatedCellReadout.blockSiteCountValue = 16 := physical_kadanoff_cell_has_sixteen_sites_between_lattice_and_coherence_scales example : PhysicalKadanoffNeighborFreeEnergyAndCorrelationAt physicalKadanoffFreeEnergyReadout ∧ physicalKadanoffFreeEnergyReadout.microscopicFreeEnergyValue = 3 / 4 := physical_kadanoff_free_energy_retains_cell_volume_neighbor_term_and_correlation example : PhysicalKadanoffLogarithmicRegularCompensationAt physicalKadanoffLogarithmicReadout ∧ physicalKadanoffLogarithmicReadout.aboveCriticalLogAmplitudeValue ≠ physicalKadanoffLogarithmicReadout.belowCriticalLogAmplitudeValue := physical_kadanoff_logarithmic_cancellation_does_not_force_equal_amplitudes example : PhysicalFNSTransverseIncompressibleProjectorAt physicalFNSTransverseProjectorReadout ∧ physicalFNSTransverseProjectorReadout.transverseProjectorXYValue = -12 / 25 := physical_fns_incompressible_velocity_and_noise_use_the_transverse_projector example : PhysicalFNSDynamicShellAndGalileanVertexAt physicalFNSDynamicShellReadout ∧ physicalFNSDynamicShellReadout.renormalizedAdvectiveVertexValue = 1 ∧ physicalFNSDynamicShellReadout.renormalizedViscosityValue = 3 := physical_fns_galilean_vertex_protection_does_not_freeze_viscosity example : PhysicalFNSDimensionDependentCriticalClassesAt physicalFNSCriticalDimensionReadout ∧ physicalFNSCriticalDimensionReadout.thermalCriticalDimensionValue = 2 ∧ physicalFNSCriticalDimensionReadout.constantNoiseCriticalDimensionValue = 4 := physical_fns_critical_dimensions_distinguish_thermal_constant_and_burgers_cases example : ¬PhysicalFNSThermalEquilibriumNoiseRelationAt physicalFNSConstantNoiseDeclaredThermalControl := by obtain ⟨_, _, _, _, notThermal, _, _, _, _, _, _⟩ := fns_transversality_noise_class_shell_vertex_and_critical_dimensions_are_independent exact notThermal example : ¬PhysicalKadanoffNeighborFreeEnergyAndCorrelationAt physicalKadanoffMissingNeighborClosureControl := by obtain ⟨_, _, _, _, _, missingClosure, _, _, _⟩ := kadanoff_scale_separation_cell_values_neighbor_closure_and_log_scope_are_independent exact missingClosure example : ¬physicalScalingEstablishment.establishedAt .blockTransformationIsAnExactMicroscopicSemigroup .kadanoff1966 := scaling_source_overclaims_remain_epistemically_unestablished.1 end OriginalCoarseGrainingSourcePhysicalReuseAudit namespace OriginalBrokenSymmetrySpectrumSourcePhysicalReuseAudit open PhysicsLogic.Framework.BrokenSymmetrySpectrumContext.Countermodels example : PhysicalGoldstoneStationaryBrokenDirectionAt physicalGoldstoneStationaryVacuumReadout ∧ physicalGoldstoneStationaryVacuumReadout.brokenFirstComponentValue ^ 2 + physicalGoldstoneStationaryVacuumReadout.brokenSecondComponentValue ^ 2 = 25 := physical_goldstone_stationary_vacuum_retains_a_nonzero_hessian_null_direction example : PhysicalGoldstoneCurrentSpectralPoleAt physicalGoldstoneCurrentPoleReadout ∧ ¬PhysicalGoldstoneCurrentSpectralPoleAt physicalGoldstoneMissingParticleBridgeControl := ⟨physical_goldstone_current_pole_requires_a_separate_particle_bridge.1, physical_goldstone_current_pole_requires_a_separate_particle_bridge.2.1⟩ example : PhysicalNambuGappedQuasiparticleAndNeutralPairAt physicalNambuNeutralPairReadout ∧ physicalNambuNeutralPairReadout.neutralCollectivePairEnergyValue ≠ physicalNambuNeutralPairReadout.twoQuasiparticleThresholdValue := physical_nambu_neutral_pair_differs_from_the_gapped_quasiparticle_threshold example : PhysicalAndersonCoulombPlasmaPoleAt physicalAndersonCoulombPlasmaReadout ∧ 0 < physicalAndersonCoulombPlasmaReadout.plasmaFrequencySquaredValue := physical_anderson_coulomb_plasma_has_a_positive_four_pi_pole example : PhysicalHigherFormMaxwellPhotonAt physicalHigherFormMaxwellPhotonReadout ∧ physicalHigherFormMaxwellPhotonContext.EstablishedGoldstoneSpectrumAt physicalHigherFormMaxwellPhotonModel := physical_higher_form_maxwell_photon_is_an_extended_global_goldstone example : ¬PhysicalHigherFormElectricMagneticMixedAnomalyAt physicalHigherFormSimultaneousGaugingControl := physical_higher_form_electric_and_magnetic_charges_obstruct_joint_gauging.2 example : ¬physicalBrokenSpectrumEstablishment.establishedAt .localGaugeRedundancyIsSpontaneouslyBroken .generalizedGlobalSymmetries2014 := by obtain ⟨_, _, _, _, notBroken, _⟩ := broken_symmetry_source_overclaims_remain_epistemically_unestablished exact notBroken end OriginalBrokenSymmetrySpectrumSourcePhysicalReuseAudit namespace OriginalGaugeMassSourcePhysicalReuseAudit open PhysicsLogic.Framework.GaugeFieldMassGenerationContext.Countermodels example : PhysicalEnglertBroutTransverseAbelianPolarizationAt physicalEnglertBroutAbelianPolarizationReadout ∧ physicalEnglertBroutAbelianPolarizationReadout.transversePolarizationValue = 36 := physical_englert_brout_abelian_polarization_is_transverse_and_normalized example : PhysicalEnglertBroutAbelianVacuumMassAt physicalEnglertBroutAbelianMassReadout ∧ PhysicalEnglertBroutCompactBrokenResidualMassesAt physicalEnglertBroutCompactMassReadout ∧ physicalEnglertBroutAbelianMassReadout.abelianMassSquaredValue = 36 ∧ physicalEnglertBroutCompactMassReadout.compactBrokenMassSquaredValue = 9 := physical_englert_brout_abelian_and_compact_masses_are_distinct example : PhysicalEnglertBroutChiralPseudovectorWardPoleAt physicalEnglertBroutChiralWardReadout ∧ ¬PhysicalEnglertBroutChiralPseudovectorWardPoleAt physicalEnglertBroutMissingChiralWardResidueControl := physical_englert_brout_chiral_ward_pole_spares_the_ordinary_vector example : PhysicalHiggsGaugeInvariantVectorRedefinitionAt physicalHiggsVectorRedefinitionReadout ∧ physicalHiggsVectorRedefinitionReadout.transformedVectorComponentValue = 4 := physical_higgs_vector_redefinition_retains_the_inverse_ev_gradient example : PhysicalHiggsDistinctRadialAndVectorMassesAt physicalHiggsRadialVectorMassReadout ∧ physicalHiggsRadialVectorMassReadout.radialScalarMassValue = 12 ∧ physicalHiggsRadialVectorMassReadout.vectorMassValue = 6 := physical_higgs_radial_and_vector_masses_remain_independently_determined example : PhysicalHiggsZeroCouplingGoldstoneRecoveryAt physicalHiggsZeroCouplingReadout ∧ PhysicalHiggsSemisimpleBrokenResidualDirectionsAt physicalHiggsSemisimpleMassReadout := physical_higgs_zero_coupling_and_semisimple_residual_branches_are_distinct example : ¬PhysicalHiggsGaugeInvariantVectorRedefinitionAt physicalHiggsWrongGradientCoefficientControl := by obtain ⟨_, _, _, _, wrongGradient, _, _, _⟩ := gauge_mass_source_formula_mutations_are_independently_falsifiable exact wrongGradient example : ¬physicalGaugeMassEstablishment.establishedAt .higgsClassicalArgumentProvesTheFullQuantumTheory .higgs1964 := by obtain ⟨_, _, _, _, notQuantum, _⟩ := gauge_mass_source_overclaims_remain_epistemically_unestablished exact notQuantum end OriginalGaugeMassSourcePhysicalReuseAudit namespace OriginalCosmologicalDefectSourcePhysicalReuseAudit open PhysicsLogic.Framework.CausalDomainDefectFormationContext.Countermodels example : PhysicalKibbleFiniteTemperaturePotentialAt physicalKibbleThermalTransitionReadout ∧ physicalKibbleThermalTransitionReadout.criticalTemperatureValue = 3 := physical_kibble_thermal_potential_retains_scalar_and_gauge_couplings example : PhysicalKibbleDistinctCorrelationAndPenetrationLengthsAt physicalKibbleDistinctLengthReadout ∧ physicalKibbleDistinctLengthReadout.scalarCorrelationLengthValue = 5 / 8 ∧ physicalKibbleDistinctLengthReadout.gaugePenetrationLengthValue = 15 / 4 := physical_kibble_scalar_coherence_and_gauge_penetration_lengths_differ example : PhysicalKibbleVacuumDomainWallTopologyAt physicalKibbleDomainWallReadout ∧ PhysicalKibbleVacuumStringTopologyAndFractionAt physicalKibbleStringReadout ∧ PhysicalKibbleVacuumMonopoleTopologyAndFractionAt physicalKibbleMonopoleReadout := physical_kibble_homotopy_degrees_distinguish_walls_strings_and_monopoles example : physicalKibbleDomainWallReadout.model.defect = .protodomainWall ∧ physicalKibbleStringReadout.model.defect = .protodomainString ∧ physicalKibbleMonopoleReadout.model.defect = .protodomainMonopole := ⟨physical_kibble_wall_string_and_monopole_select_different_defect_models.1, physical_kibble_wall_string_and_monopole_select_different_defect_models.2.1, physical_kibble_wall_string_and_monopole_select_different_defect_models.2.2.1⟩ example : PhysicalGuthCausalBubbleJunctionMonopoleAt physicalGuthBubbleMonopoleReadout ∧ ¬PhysicalGuthCausalBubbleJunctionMonopoleAt physicalGuthBubbleWithoutJunctionObstructionControl := physical_guth_bubble_monopoles_require_a_causal_topological_junction example : PhysicalGuthConditionalEntropyDilutionAt physicalGuthEntropyDilutionReadout ∧ physicalGuthEntropyDilutionReadout.dilutedMonopoleAbundanceValue = 1 / 125 := physical_guth_monopole_dilution_depends_on_entropy_and_reheating example : PhysicalGuthFailedBubblePercolationAt physicalGuthFailedPercolationReadout ∧ 2 < physicalGuthFailedPercolationReadout.expandedBubbleSeparationValue := physical_guth_old_inflation_does_not_supply_a_graceful_exit example : ¬PhysicalKibbleVacuumStringTopologyAndFractionAt physicalKibbleStringWithoutRandomOrientationsControl := by obtain ⟨_, _, _, notRandom, _, _, _, _⟩ := causal_defect_source_formula_mutations_are_independently_falsifiable exact notRandom example : ¬physicalCausalDefectEstablishment.establishedAt .oldInflationAlreadyProvidesAHomogeneousGracefulExit .guth1981 := by obtain ⟨_, _, _, noGracefulExit, _⟩ := causal_defect_source_overclaims_remain_epistemically_unestablished exact noGracefulExit end OriginalCosmologicalDefectSourcePhysicalReuseAudit namespace OriginalExteriorGravitySourcePhysicalReuseAudit open PhysicsLogic.Framework.ExactExteriorVacuumMetricContext.Countermodels example : PhysicalSchwarzschildDeterminantGaugeAndRadialTransformationAt physicalSchwarzschildDeterminantGaugeReadout ∧ physicalSchwarzschildDeterminantGaugeReadout.transformedArealRadiusCubedValue = 9 := physical_schwarzschild_determinant_gauge_retains_the_original_radial_cube example : PhysicalSchwarzschildExteriorDiagonalLineElementAt physicalSchwarzschildExteriorLineElementReadout ∧ physicalSchwarzschildExteriorLineElementReadout.staticTimeCoefficientValue * physicalSchwarzschildExteriorLineElementReadout.staticRadialCoefficientValue = -1 := physical_schwarzschild_exterior_retains_all_four_line_element_coefficients example : PhysicalSchwarzschildExactEquatorialOrbitAt physicalSchwarzschildExactOrbitReadout ∧ physicalSchwarzschildExactOrbitReadout.relativisticCubicCorrectionValue = 1 / 32 := physical_schwarzschild_exact_orbit_retains_a_nonzero_relativistic_cubic example : PhysicalKerrTwistingStationaryRetardedLineElementAt physicalKerrStationaryTwistingReadout ∧ physicalKerrStationaryTwistingReadout.stationaryLineElementValue = 12 := physical_kerr_twisting_type_d_retains_its_retarded_line_element example : PhysicalKerrImplicitCartesianRadialQuarticAt physicalKerrImplicitQuarticReadout ∧ physicalKerrImplicitQuarticReadout.euclideanRadiusSquaredValue = 5 := physical_kerr_cartesian_radius_retains_the_exact_implicit_quartic example : PhysicalKerrSchildOneFormAndCartesianMetricAt physicalKerrSchildMetricReadout ∧ physicalKerrSchildMetricReadout.asymptoticLineElementValue = 43 / 25 := physical_kerr_schild_one_form_retains_the_cartesian_line_element example : PhysicalKerrMassSpinAndQualifiedMultipolesAt physicalKerrMassSpinReadout ∧ PhysicalKerrExcludedRingAndUnconstructedInteriorAt physicalKerrExcludedRingReadout ∧ PhysicalKerrZeroSpinSchwarzschildReductionAt physicalKerrZeroSpinReadout := physical_kerr_mass_spin_ring_scope_and_zero_spin_limit_are_distinct example : ¬PhysicalKerrSchildOneFormAndCartesianMetricAt physicalKerrWrongSchildOneFormControl := by obtain ⟨_, _, _, _, _, _, notOneForm, _, _, _, _⟩ := exterior_metric_source_formula_mutations_are_independently_falsifiable exact notOneForm example : ¬physicalExteriorMetricEstablishment.establishedAt .kerr1963ProvesAnExactInvariantMultipoleNoHairTheorem .kerr1963 := by obtain ⟨_, _, _, _, noExactMultipoles, _⟩ := exterior_metric_source_overclaims_remain_epistemically_unestablished exact noExactMultipoles end OriginalExteriorGravitySourcePhysicalReuseAudit namespace OriginalOperationalMeasurementSourcePhysicalReuseAudit open PhysicsLogic.Framework.OperationalErrorDisturbanceContext.Countermodels example : PhysicalHeisenbergApertureLimitedResolutionAt physicalHeisenbergApertureResolutionReadout ∧ physicalHeisenbergApertureResolutionReadout.opticalResolutionValue = 2 := physical_heisenberg_microscope_resolution_retains_wavelength_and_aperture example : PhysicalHeisenbergPreparedDispersionIsSeparateAt physicalHeisenbergPreparedGaussianReadout ∧ physicalHeisenbergPreparedGaussianReadout.preparedPositionDispersionValue = 4 := physical_heisenberg_prepared_gaussian_widths_differ_from_apparatus_error example : PhysicalOzawaNoiseDisturbanceCommutatorIdentityAt physicalOzawaCommutatorIdentityReadout ∧ ¬PhysicalOzawaNoiseDisturbanceCommutatorIdentityAt physicalOzawaMissingCommutatorTermControl := physical_ozawa_commutator_identity_retains_both_intervention_terms example : PhysicalOzawaGeneralizedDispersionCrossTermsAt physicalOzawaGeneralizedCrossTermReadout ∧ PhysicalOzawaGeneralizedDispersionCrossTermsAt physicalOzawaZeroDisturbanceCrossTermReadout := physical_ozawa_generalized_bound_covers_zero_noise_and_zero_disturbance example : ¬PhysicalOzawaSelectedInputNoiseScopeAt physicalOzawaSingleStatePrecisionOverclaimControl := physical_ozawa_single_state_zero_noise_does_not_establish_universal_precision.2 example : ¬physicalOperationalMeasurementEstablishment.establishedAt .everyMeasurementSatisfiesTheProductOnlyBound .ozawa2003 := by obtain ⟨_, _, _, notUniversalProduct, _, _⟩ := operational_measurement_source_overclaims_remain_epistemically_unestablished exact notUniversalProduct end OriginalOperationalMeasurementSourcePhysicalReuseAudit namespace OriginalRelativeMeasurementSourcePhysicalReuseAudit open PhysicsLogic.Framework.LinearRelativeStateMeasurementContext.Countermodels example : PhysicalEverettSquareAmplitudeMeasureAt physicalEverettSquareAmplitudeReadout ∧ physicalEverettSquareAmplitudeReadout.firstBranchWeightValue = 9 / 25 ∧ physicalEverettSquareAmplitudeReadout.secondBranchWeightValue = 16 / 25 := physical_everett_branch_measure_is_additive_and_square_amplitude example : PhysicalEverettRepeatedMemoryBranchProductsAt physicalEverettMemoryTreeReadout ∧ physicalEverettMemoryTreeReadout.repeatedUpDownWeightValue = 144 / 625 := physical_everett_repeated_measurements_retain_four_memory_weights example : PhysicalZehDiagonalReducedDensityWithoutCollapseAt physicalZehPhaseAveragedDensityReadout ∧ ¬PhysicalZehDiagonalReducedDensityWithoutCollapseAt physicalZehDiagonalDensityDeclaredCollapseControl := physical_zeh_diagonal_density_does_not_select_a_global_outcome example : PhysicalZehEnvironmentOverlapSuppressionAt physicalZehEnvironmentalOverlapReadout ∧ 0 < physicalZehEnvironmentalOverlapReadout.globalInterbranchCoherenceValue := physical_zeh_environment_suppresses_reduced_but_not_global_coherence example : PhysicalRovelliPhysicalRecordComparisonAt physicalRovelliComparedRecordsReadout ∧ ¬PhysicalRovelliPhysicalRecordComparisonAt physicalRovelliAgreementWithoutComparisonControl := physical_rovelli_record_agreement_requires_an_actual_comparison example : PhysicalRovelliInformationPostulatesWithoutReconstructionAt physicalRovelliInformationPostulateReadout ∧ physicalRovelliInformationPostulateReadout.nextQuestionIndexValue = 5 := physical_rovelli_information_postulates_do_not_complete_reconstruction end OriginalRelativeMeasurementSourcePhysicalReuseAudit namespace OriginalBubbleCompletionSourcePhysicalReuseAudit open PhysicsLogic.Framework.BubbleTransitionCompletionContext.Countermodels example : PhysicalGuthSubcriticalDisconnectedPercolationAt physicalGuthSubcriticalPercolationReadout ∧ ¬PhysicalGuthSubcriticalDisconnectedPercolationAt physicalGuthMissingPercolationThresholdControl := physical_guth_percolation_failure_requires_an_independent_threshold example : PhysicalGuthLargeConvertedFractionWithoutConnectivityAt physicalGuthLargeConvertedFractionReadout ∧ physicalGuthLargeConvertedFractionReadout.newPhaseVolumeFractionValue = 9 / 10 := physical_guth_large_new_phase_fraction_need_not_form_a_connected_cluster example : PhysicalAlbrechtSteinhardtRadiativePotentialAndExpansionAt physicalAlbrechtSteinhardtRadiativePotentialReadout ∧ physicalAlbrechtSteinhardtRadiativePotentialReadout.radiativePotentialValue = -19 + 16 * Real.log 4 := physical_albrecht_steinhardt_potential_retains_log_thermal_and_friedmann_terms example : PhysicalAlbrechtSteinhardtHomogeneousFluctuationReplacementAt physicalAlbrechtSteinhardtFluctuationReplacementReadout ∧ physicalAlbrechtSteinhardtFluctuationReplacementReadout.independentFluctuationPathCountValue = 3 := physical_albrecht_steinhardt_homogeneous_fluctuation_replaces_standard_bubbles example : ¬physicalBubbleCompletionEstablishment.establishedAt .oldInflationAlreadyProvidesAHomogeneousGracefulExit .guth1981 := by obtain ⟨_, _, _, noGracefulExit, _, _, _⟩ := bubble_completion_source_overclaims_remain_epistemically_unestablished exact noGracefulExit end OriginalBubbleCompletionSourcePhysicalReuseAudit namespace OriginalNuclearRenormalizationSourcePhysicalReuseAudit open PhysicsLogic.Framework.RenormalizedPredictionContext.Countermodels example : PhysicalKSWComplexPDSLoopAt physicalKSWReferencePDSLoopReadout ∧ physicalKSWReferencePDSLoopReadout.referencePDSLoopValue ≠ physicalKSWReferencePDSLoopReadout.shiftedPDSLoopValue := physical_ksw_pds_loop_retains_scale_and_imaginary_on_shell_momentum example : PhysicalKSWRunningContactsAndBetaFunctionsAt physicalKSWRunningContactsReadout ∧ physicalKSWRunningContactsReadout.referenceContactCZeroValue = 1 ∧ physicalKSWRunningContactsReadout.shiftedContactCZeroValue = 2 ∧ physicalKSWRunningContactsReadout.referenceCTwoBetaValue = 2 := physical_ksw_running_contacts_have_the_original_exact_beta_functions example : PhysicalKSWBoundStatePoleIsNotAFiniteAmplitudeAt physicalKSWBoundStatePoleReadout ∧ physicalKSWBoundStatePoleReadout.boundStateDenominatorValue = 0 := physical_ksw_bound_state_pole_is_not_misread_as_a_finite_amplitude example : PhysicalKSWBothContactOrdersRunAt physicalKSWRunningContactsReadout ∧ physicalKSWRunningContactsReadout.referenceContactCTwoValue = 1 ∧ physicalKSWRunningContactsReadout.shiftedContactCTwoValue = 4 ∧ ¬PhysicalKSWBothContactOrdersRunAt physicalKSWFrozenShiftedCTwoControl := physical_ksw_c_two_runs_independently_at_both_pds_subtraction_scales example : PhysicalBHvKLogPeriodicThreeBodyForceAt physicalBHvKLimitCycleForceReadout ∧ physicalBHvKLimitCycleForceReadout.scalingExponentValue = 629 / 625 ∧ physicalBHvKLimitCycleForceReadout.referenceDimensionfulThreeBodyForceValue = 4 := physical_bhvk_limit_cycle_retains_the_original_arctangent_shifted_sine_ratio example : PhysicalBHvKShiftedCutoffLimitCycleAt physicalBHvKLimitCycleForceReadout ∧ physicalBHvKLimitCycleForceReadout.referenceThreeBodyCutoffValue = 1 ∧ physicalBHvKLimitCycleForceReadout.shiftedThreeBodyCutoffValue = 2 ∧ ¬PhysicalBHvKShiftedCutoffLimitCycleAt physicalBHvKWrongShiftedSineRatioForceControl := physical_bhvk_limit_cycle_running_holds_at_both_distinct_cutoffs example : PhysicalBHvKOneDatumPhaseLockedPredictionAt physicalBHvKPhaseLockedPredictionReadout ∧ physicalBHvKPhaseLockedPredictionReadout.phaseLockedReferencePredictionValue = 7 / 5 := physical_bhvk_one_independent_datum_locks_only_the_low_energy_prediction example : ¬physicalNuclearRenormalizationEstablishment.establishedAt .twoBodyDataAloneDetermineTheThreeBodyLimitCyclePhase .bedaqueHammerVanKolck1998 := by obtain ⟨_, _, notTwoBodyDatum, _, _, _⟩ := nuclear_renormalization_source_overclaims_remain_epistemically_unestablished exact notTwoBodyDatum end OriginalNuclearRenormalizationSourcePhysicalReuseAudit namespace OriginalTopologicalHallSourcePhysicalReuseAudit open PhysicsLogic.Framework.TopologicalTransverseResponseContext.Countermodels example : PhysicalTKNNMagneticFluxAndBlochSubbandsAt physicalTKNNMagneticFluxReadout ∧ physicalTKNNMagneticFluxReadout.magneticSubbandCountValue = 3 ∧ physicalTKNNMagneticFluxReadout.naiveEqualSubbandResponseValue = 1 / 3 := physical_tknn_rational_flux_splits_one_landau_level_into_numerator_many_subbands example : PhysicalTKNNDiophantineGapAndWeakConductanceAt physicalTKNNDiophantineGapReadout ∧ physicalTKNNDiophantineGapReadout.selectedGapResponseValue = 1 ∧ physicalTKNNDiophantineGapReadout.gapSCoefficientValue = -1 := physical_tknn_diophantine_gap_selects_an_integer_not_the_naive_subband_fraction example : PhysicalNTWFiniteTwistWithoutThermodynamicIndependenceAt physicalNTWFiniteTwistReadout ∧ physicalNTWFiniteTwistReadout.fixedTwistResponseValue = 1 / 2 ∧ physicalNTWFiniteTwistReadout.twistAveragedResponseValue = 1 := physical_ntw_finite_volume_fixed_twist_need_not_equal_its_torus_average example : PhysicalNTWDegenerateFractionalGroundSpaceAt physicalNTWFractionalGroundSpaceReadout ∧ physicalNTWFractionalGroundSpaceReadout.groundSpaceDegeneracyValue = 3 ∧ physicalNTWFractionalGroundSpaceReadout.fractionalGroundSpaceResponseValue = 1 / 3 := physical_ntw_ground_space_chern_sum_divided_by_degeneracy_is_fractional example : PhysicalNTWDegenerateFractionalGroundSpaceAt physicalNTWFractionalGroundSpaceReadout ∧ ¬physicalNTWFractionalGroundSpaceReadout.model.NonvacuousTopologicalTransverseResponseAt := physical_ntw_fractional_ground_space_does_not_satisfy_the_integer_framework example : ¬physicalHallSourceEstablishment.establishedAt .sphericalGeometryThermodynamicDegeneracyWasProved .niuThoulessWu1985 := by obtain ⟨_, _, _, noSphericalProof, _, _, _⟩ := hall_response_source_overclaims_remain_epistemically_unestablished exact noSphericalProof end OriginalTopologicalHallSourcePhysicalReuseAudit namespace OriginalMagneticBandTopologicalSourceCompositionReuseAudit example {T : PhysicsLogic.Papers.TKNN1982.TKNNTypes} (framework : PhysicsLogic.Papers.TKNN1982.PaperFramework T) (magneticSetup : framework.MagneticBlochSetup) (magneticCalculation : framework.MagneticBlochCalculation) (kuboSetup : framework.KuboCurvatureSetup) (kuboCalculation : framework.KuboCurvatureCalculation) (integerSetup : framework.IntegerQuantizationSetup) (integerCalculation : framework.IntegerQuantizationCalculation) : framework.tknnGappedBandTopologicalHandoff |>.SourceSensitiveBandGroundedTopologicalHallResponseAt framework.toGappedBandHallContext framework.toTopologicalTransverseResponseContext .magneticBlochKuboCurvaturePhaseWindingAndNormalization framework.TKNNGappedBandHallDerivationObtains .magneticBlochKuboCurvaturePhaseWindingAndNormalization framework.TKNNGappedBandHallDerivationObtains .magneticMomentumCurvatureBoundaryWindingAndConductanceNormalization framework.tknnBandTopologicalHandoffDerivationObtains := framework |>.tknn_magnetic_bloch_curvature_and_phase_winding_realize_source_sensitive_hall_composition magneticSetup magneticCalculation kuboSetup kuboCalculation integerSetup integerCalculation example {T : PhysicsLogic.Papers.TKNN1982.TKNNTypes} (framework : PhysicsLogic.Papers.TKNN1982.PaperFramework T) (sourceSensitive : framework.tknnGappedBandTopologicalHandoff |>.SourceSensitiveBandGroundedTopologicalHallResponseAt framework.toGappedBandHallContext framework.toTopologicalTransverseResponseContext .magneticBlochKuboCurvaturePhaseWindingAndNormalization framework.TKNNGappedBandHallDerivationObtains .magneticBlochKuboCurvaturePhaseWindingAndNormalization framework.TKNNGappedBandHallDerivationObtains .magneticMomentumCurvatureBoundaryWindingAndConductanceNormalization framework.tknnBandTopologicalHandoffDerivationObtains) (weak : framework.WeakPotentialResult) : framework.StokesBoundaryPhaseCalculation framework.occupiedSubbands framework.magneticBrillouinZone framework.selectedCurvatureIntegral /\ framework.EquationNineDiophantineGapFormulaAt /\ framework.EquationElevenWeakHallConductanceFormulaAt := framework |>.source_sensitive_tknn_hall_composition_and_weak_branch_retain_equations_nine_and_eleven sourceSensitive weak end OriginalMagneticBandTopologicalSourceCompositionReuseAudit namespace OriginalCollisionlessKineticPlasmaSourceReuseAudit example {T : PhysicsLogic.Papers.LandauDamping1946.Landau1946Types} (framework : PhysicsLogic.Papers.LandauDamping1946.PlasmaFramework T) (normalMode : framework.NormalModeInsufficiencyResult) (calculation : framework.LaplaceInitialValueCalculation) : (framework.toCollisionlessKineticPoissonContext |>.SourceStructuralCollisionlessKineticPoissonExplanationAt .fixedIonLinearizedFourierKineticPoissonAndOneSidedInitialValue framework.landauCollisionlessKineticPoissonDerivationObtains) /\ framework.EquationsSixThroughTenFormulaAt := by have source := framework |>.landau_initial_value_inputs_realize_source_structural_kinetic_poisson_explanation normalMode calculation exact ⟨source, framework |>.source_structural_landau_plasma_explanation_retains_printed_operational_formulas source⟩ example {T : PhysicsLogic.Papers.LandauDamping1946.Landau1946Types} (framework : PhysicsLogic.Papers.LandauDamping1946.PlasmaFramework T) (setup : framework.CollisionlessInitialValueSetup) (damping : framework.CollectiveFieldDampingResult) (distribution : framework.VelocityResolvedDistributionResult) : (framework.toCollisionlessKineticPoissonContext |>.NonvacuousCollisionlessKineticPoissonAt) /\ framework.CollectiveFieldDistributionSeparation := framework |>.landau_collisionless_kinetic_poisson_preserves_collective_distribution_separation setup damping distribution example {T : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.BGK1957Types} (framework : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.PlasmaWaveFramework T) (setup : framework.CollisionlessWaveFrameSetup) : (framework.toCollisionlessKineticPoissonContext |>.SourceStructuralCollisionlessKineticPoissonExplanationAt .stationaryNonlinearElectronIonKineticPoissonAndPrintedEquations framework.bgkCollisionlessKineticPoissonDerivationObtains) /\ framework.PrintedStationaryEquationsOneThroughFour := by have source := framework |>.bgk_stationary_wave_inputs_realize_source_structural_kinetic_poisson_explanation setup exact ⟨source, framework |>.source_structural_bgk_plasma_explanation_retains_printed_stationary_equations source⟩ example {T : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.BGK1957Types} (framework : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.PlasmaWaveFramework T) (setup : framework.CollisionlessWaveFrameSetup) (assessment : framework.PhysicalExistenceAssessment) : (framework.toCollisionlessKineticPoissonContext |>.NonvacuousCollisionlessKineticPoissonAt) /\ ¬framework.actualWaveExistenceEstablishedAt framework.actualPlasmaWave framework.selectedPlasma framework.actualPlasmaRegime := by have openScope := framework |>.bgk_nonvacuous_kinetic_poisson_preserves_unresolved_actual_plasma_scope setup assessment exact ⟨openScope.1, openScope.2.2.1⟩ end OriginalCollisionlessKineticPlasmaSourceReuseAudit namespace OriginalControlledResonanceExperimentalSourceReuseAudit example {T : PhysicsLogic.Papers.PoundRebka1960.PoundRebkaTypes} (framework : PhysicsLogic.Papers.PoundRebka1960.PoundRebkaFramework T) (result : framework.EstablishedControlledRedshiftAgreement) : (framework.toControlledResonanceModelTestContext |>.SourceStructuralControlledResonanceModelTestExplanationAt .dopplerCalibratedReversalOffsetIsolationAndFiniteRedshiftAgreement framework.poundRebkaControlledResonanceDerivationObtains) /\ framework.OriginalTwoWayBaselineFormulaAt /\ framework.ReportedTwoWayShiftValuesAt := by have source := framework |>.pound_rebka_reversal_inputs_realize_source_structural_resonance_explanation result have retained := framework |>.source_structural_pound_rebka_resonance_retains_original_formulas source exact ⟨source, retained.2.1, retained.2.2⟩ example {T : PhysicsLogic.Papers.PoundRebka1960.PoundRebkaTypes} (framework : PhysicsLogic.Papers.PoundRebka1960.PoundRebkaFramework T) (result : framework.EstablishedQualifiedRedshiftResult) : (framework.toControlledResonanceModelTestContext |>.NonvacuousControlledResonanceModelTestAt) /\ framework.inherentOffsetUnchangedByInversionAt framework.cobaltIronSource framework.ironAbsorber framework.selectedRegime /\ framework.measuredAgreesWithPredictionAt framework.measuredTwoWayShift framework.predictedTwoWayShift framework.reportedUncertainty framework.selectedRegime := by have retained := framework |>.pound_rebka_resonance_test_preserves_inversion_stability_and_finite_agreement result exact ⟨retained.1, retained.2.1, retained.2.2.1⟩ example {T : PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.HughesTypes} (framework : PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.HughesFramework T) (assessment : framework.EstablishedScopedAnisotropyAssessment) : (framework.toControlledResonanceModelTestContext |>.SourceStructuralControlledResonanceModelTestExplanationAt .protectedSiderealLithiumLineP2FormulaAndFiniteNuclearBound framework.hughesControlledResonanceDerivationObtains) /\ framework.SelectedAtomicP2EnergyFormulaAt /\ framework.PrintedNuclearReadoutAt := by have source := framework |>.hughes_sidereal_p2_inputs_realize_source_structural_resonance_explanation assessment have retained := framework |>.source_structural_hughes_resonance_retains_p2_formula_and_readouts source exact ⟨source, retained.1, retained.2.2⟩ example {T : PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.HughesTypes} (framework : PhysicsLogic.Papers.HughesRobinsonBeltranLopez1960.HughesFramework T) (assessment : framework.EstablishedScopedAnisotropyAssessment) : (framework.toControlledResonanceModelTestContext |>.NonvacuousControlledResonanceModelTestAt) /\ ¬framework.exactMassIsotropyEstablishedAt framework.selectedRegime /\ ¬framework.otherAngularDependenceTestedAt framework.selectedRegime /\ ¬framework.arbitraryPreferredDirectionTestedAt framework.selectedRegime := by have scope := framework |>.hughes_resonance_test_preserves_finite_bound_and_open_anisotropy_scope assessment exact ⟨scope.1, scope.2.2.1, scope.2.2.2.1, scope.2.2.2.2⟩ end OriginalControlledResonanceExperimentalSourceReuseAudit namespace OriginalDensityPotentialGaugeSourceReuseAudit example {Potential GroundState Density Approximation : Type} (framework : PhysicsLogic.Papers.HohenbergKohn1964.DensityFunctionalFramework Potential GroundState Density Approximation) (selectedState : GroundState) (selectedPotential : Potential) (particleNumber : Nat) (source : framework.HohenbergKohnDensityPotentialDerivationObtains selectedState selectedPotential particleNumber .nondegenerateStrictVariationalReductioAndConstantPotentialShift) : ((framework.toDensityPotentialGaugeContext selectedState selectedPotential particleNumber) |>.SourceStructuralDensityPotentialGaugeExplanationAt .nondegenerateStrictVariationalReductioAndConstantPotentialShift (framework.HohenbergKohnDensityPotentialDerivationObtains selectedState selectedPotential particleNumber)) /\ framework.StrictNondegenerateVariationalPrinciple /\ framework.ConstantShiftPreservesGroundStates := by have explanation := framework |>.hohenberg_kohn_strict_variational_source_realizes_density_potential_explanation selectedState selectedPotential particleNumber source have retained := framework |>.shared_source_hohenberg_kohn_density_route_retains_original_derivation selectedState selectedPotential particleNumber explanation exact ⟨explanation, retained.2.2.2.2.2.1, retained.2.2.2.2.2.2.1⟩ example {T : PhysicsLogic.Papers.RungeGross1984.RungeGrossTypes} (framework : PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework T) (source : framework.RungeGrossDensityDeterminationDerivationObtains .timeAnalyticTaylorCurrentContinuityBoundaryReductio) : (framework.toDensityPotentialGaugeContext |>.SourceStructuralDensityPotentialGaugeExplanationAt .timeAnalyticTaylorCurrentContinuityBoundaryReductio framework.RungeGrossDensityDeterminationDerivationObtains) /\ framework.TaylorCurrentSeparationCalculation /\ framework.ContinuityDensitySeparationCalculation /\ framework.BoundaryReductioCalculation := by have explanation := framework |>.runge_gross_source_derivation_realizes_shared_source_density_potential_explanation source have retained := framework |>.shared_source_runge_gross_density_route_retains_full_original_derivation explanation exact ⟨explanation, retained.2.2.1, retained.2.2.2.1, retained.2.2.2.2.1⟩ example {T : PhysicsLogic.Papers.RungeGross1984.RungeGrossTypes} (framework : PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework T) (complete : framework.CompleteScopedResult) : (framework.toDensityPotentialGaugeContext |>.NonvacuousDensityPotentialGaugeDeterminationAt) /\ framework.AllPrintedEquationSixDensitySeparationFormulas /\ ¬framework.vRepresentableDomainCharacterizedAt framework.initialState framework.interaction framework.exactRegime /\ ¬framework.practicalApproximationProvidedForExactFunctionalsAt framework.exactRegime := framework |>.shared_runge_gross_density_route_retains_equation_six_and_open_scope complete end OriginalDensityPotentialGaugeSourceReuseAudit namespace OriginalLargeComponentSphericalSourceReuseAudit example {T : PhysicsLogic.Papers.Stanley1968.StanleyTypes} (framework : PhysicsLogic.Papers.Stanley1968.SphericalLimitFramework T) (definitions : framework.ModelDefinitions) (positiveDimension : framework.positiveSpinDimension framework.spinConstraintFormulaReadout.spinDimensionValue) (temperature : framework.SelectedNoncriticalTemperatureSetup) (established : framework.EstablishedSphericalFreeEnergyLimit) : (framework.toLargeComponentSphericalCorrespondenceContext |>.SourceStructuralLargeComponentSphericalExplanationAt .localConstraintSaddleRemainderAndNoncriticalFreeEnergy framework.stanleySphericalCorrespondenceDerivationObtains) /\ framework.OriginalLocalAndGlobalConstraintFormulaAt /\ framework.EquationTwentyFourLimitAccountingAt framework.selectedNoncriticalTemperature framework.selectedFiniteSiteCount := by have explanation := framework |>.noncritical_stanley_source_realizes_spherical_correspondence_explanation definitions positiveDimension temperature established have retained := framework |>.shared_source_stanley_spherical_route_retains_constraints_and_equation_twenty_four explanation exact ⟨explanation, retained.2.2.2.2.1, retained.2.2.2.2.2⟩ example {T : PhysicsLogic.Papers.Stanley1968.StanleyTypes} (framework : PhysicsLogic.Papers.Stanley1968.SphericalLimitFramework T) (definitions : framework.ModelDefinitions) (positiveDimension : framework.positiveSpinDimension framework.spinConstraintFormulaReadout.spinDimensionValue) (temperature : framework.SelectedNoncriticalTemperatureSetup) (limit : framework.EstablishedSphericalFreeEnergyLimit) (scope : framework.CriticalIndexScopeAssessment) : (framework.toLargeComponentSphericalCorrespondenceContext |>.NonvacuousLargeComponentSphericalCorrespondenceAt) /\ Not (framework.criticalLimitOrderCommutationEstablished framework.selectedLattice) /\ Not (framework.criticalIndexAgreementEstablished framework.selectedLattice) := by have retained := framework |>.stanley_spherical_correspondence_preserves_noncritical_and_open_critical_scope definitions positiveDimension temperature limit scope exact ⟨retained.1, retained.2.2.2.1, retained.2.2.2.2⟩ example {T : PhysicsLogic.Papers.WegnerHoughton1973.WegnerHoughtonTypes} (framework : PhysicsLogic.Papers.WegnerHoughton1973.Framework T) (route : framework.LargeNRouteApplicable) : (framework.toLargeComponentSphericalCorrespondenceContext |>.SourceStructuralLargeComponentSphericalExplanationAt .infiniteComponentDiagonalFixedPointAndExactCriticalIndices framework.wegnerSphericalCorrespondenceDerivationObtains) /\ framework.LargeNSphericalExponentFormulaAt := by have explanation := framework.exact_wegner_large_n_source_realizes_spherical_correspondence_explanation route have retained := framework.shared_source_wegner_spherical_route_retains_exact_critical_exponents explanation exact ⟨explanation, retained.2.2⟩ example {T : PhysicsLogic.Papers.WegnerHoughton1973.WegnerHoughtonTypes} (framework : PhysicsLogic.Papers.WegnerHoughton1973.Framework T) (result : framework.EstablishedLargeNResults) (scope : framework.FiniteStepScope) : (framework.toLargeComponentSphericalCorrespondenceContext |>.NonvacuousLargeComponentSphericalCorrespondenceAt) /\ framework.LargeNSphericalExponentFormulaAt /\ 2 < framework.largeNSphericalFormulaReadout.spatialDimensionValue /\ framework.largeNSphericalFormulaReadout.spatialDimensionValue < 4 /\ framework.largeNSphericalFormulaReadout.inverseComponentCountValue = 0 /\ Not (framework.finiteCutoffRecursionsDerivedInPaperAt framework.exactGeneratorEquation framework.baseRegime) := framework |>.wegner_spherical_correspondence_preserves_exact_critical_and_open_finite_step_scope result scope end OriginalLargeComponentSphericalSourceReuseAudit namespace OriginalIncompressibleVelocitySourceReuseAudit example {T : PhysicsLogic.Papers.KolmogorovLocalStructure1941.Kolmogorov1941Types} (framework : PhysicsLogic.Papers.KolmogorovLocalStructure1941.LocalTurbulenceFramework T) (isotropic : framework.IsotropicMomentResult) (setup : framework.IncompressibleMomentSetup) (calculation : framework.IncompressibilityMomentCalculation) : (framework.toIncompressibleVelocityCorrelationContext |>.SourceStructuralIncompressibleVelocityExplanationAt .localIsotropicIncrementTensorAndRadialEquationTwelve framework.kolmogorovIncompressibleCorrelationDerivationObtains) /\ framework.EquationTwelveIncompressibleMomentFormulaAt := by have explanation := framework |>.local_isotropic_incompressibility_inputs_realize_source_velocity_explanation isotropic setup calculation have retained := framework.shared_source_kolmogorov_incompressibility_retains_equation_twelve explanation exact ⟨explanation, retained.2.2.2⟩ example {T : PhysicsLogic.Papers.KolmogorovLocalStructure1941.Kolmogorov1941Types} (framework : PhysicsLogic.Papers.KolmogorovLocalStructure1941.LocalTurbulenceFramework T) (complete : framework.CompleteKolmogorov1941Result) : (framework.toIncompressibleVelocityCorrelationContext |>.NonvacuousIncompressibleVelocityCorrelationAt) /\ framework.EquationTwelveIncompressibleMomentFormulaAt /\ Not (framework.hypothesisRigorouslyEstablishedAt .secondSimilarity framework.inertialRangeRegime) /\ Not (framework.exactGlobalValidityClaimedAt framework.selectedFlow framework.wholeFlowRegime) := framework |>.incompressible_velocity_realization_preserves_kolmogorov_local_and_tentative_scope complete example {T : PhysicsLogic.Papers.ForsterNelsonStephen1977.FNSTypes} (framework : PhysicsLogic.Papers.ForsterNelsonStephen1977.DynamicRGFramework T) (model : T.Model) (setup : framework.InfraredRGSetup model) : ((framework.toIncompressibleVelocityCorrelationContext model) |>.SourceStructuralIncompressibleVelocityExplanationAt .finiteWaveVectorSolenoidalProjectorAndInfraredVelocityTensor (framework.fnsIncompressibleCorrelationDerivationObtains model)) /\ framework.EquationThreeFiveSolenoidalCorrelationFormulaAt model := by have explanation := framework.infrared_incompressible_setup_realizes_source_velocity_explanation model setup have retained := framework.shared_source_fns_incompressibility_retains_equation_three_five model explanation exact ⟨explanation, retained.2⟩ example {T : PhysicsLogic.Papers.ForsterNelsonStephen1977.FNSTypes} (framework : PhysicsLogic.Papers.ForsterNelsonStephen1977.DynamicRGFramework T) (setup : framework.InfraredRGSetup framework.modelA) (scope : framework.ScopeAssessment) : (framework.toIncompressibleVelocityCorrelationContext framework.modelA |>.NonvacuousIncompressibleVelocityCorrelationAt) /\ framework.EquationThreeFiveSolenoidalCorrelationFormulaAt framework.modelA /\ Not (framework.exponentResultValidToAllOrdersFor framework.modelB) /\ Not (framework.ultravioletTurbulenceControlledByInfraredExpansion framework.modelB) := framework.incompressible_velocity_realization_preserves_fns_infrared_scope setup scope end OriginalIncompressibleVelocitySourceReuseAudit namespace OriginalCanonicalGibbsSourceReuseAudit example {T : PhysicsLogic.Papers.Jaynes1957.Jaynes1957Types} (framework : PhysicsLogic.Papers.Jaynes1957.JaynesFramework T) (setup : framework.CanonicalInferenceSetup) (calculation : framework.CanonicalPartitionCalculation) (scope : framework.CanonicalInferenceScope) : (framework.toCanonicalGibbsContext |>.SourceStructuralCanonicalGibbsExplanationAt .shannonMaximumEntropyMeanEnergyAndDiscretePartition framework.jaynesCanonicalGibbsDerivationObtains) /\ framework.MaximumEntropyInferenceResult /\ framework.EquationsTwoFourSixAndSevenCanonicalFormulaAt := by have explanation := framework.maximum_entropy_canonical_inputs_realize_source_gibbs_explanation setup calculation scope have retained := framework.shared_source_jaynes_gibbs_route_retains_original_derivation explanation exact ⟨explanation, retained.1.1, retained.2.2.2⟩ example {T : PhysicsLogic.Papers.Jaynes1957.Jaynes1957Types} (framework : PhysicsLogic.Papers.Jaynes1957.JaynesFramework T) (maximumEntropy : framework.MaximumEntropyInferenceResult) (canonical : framework.CanonicalInferenceResult) : (framework.toCanonicalGibbsContext |>.NonvacuousCanonicalGibbsNormalizationAt) /\ framework.epistemicProbability framework.selectedInformation framework.selectedEvent framework.selectedNumber framework.equilibriumRegime /\ Not (framework.microscopicDynamicsDerivedFromInference framework.canonicalAssignment) := framework |>.canonical_gibbs_realization_preserves_jaynes_epistemic_and_nondynamical_scope maximumEntropy canonical example {T : PhysicsLogic.Papers.Metropolis1953.MetropolisTypes} (framework : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework T) (finiteSetup : framework.FinitePeriodicSetup) (finiteCalculation : framework.FinitePeriodicModelCalculation) (targetSetup : framework.CanonicalTargetSetup) (reduction : framework.CanonicalReductionCalculation) : (framework.toCanonicalGibbsContext |>.SourceStructuralCanonicalGibbsExplanationAt .finitePeriodicVelocitySeparationAndNormalizedEquationTwo framework.metropolisCanonicalGibbsDerivationObtains) /\ framework.EquationTwoCanonicalConfigurationFormulaAt /\ framework.velocityIndependentForces framework.finitePeriodicDescription := by have explanation := framework.periodic_canonical_inputs_realize_source_gibbs_explanation finiteSetup finiteCalculation targetSetup reduction have retained := framework.shared_source_metropolis_gibbs_route_retains_original_derivation explanation exact ⟨explanation, retained.2.2.2.2, retained.2.2.1.2⟩ example {T : PhysicsLogic.Papers.Metropolis1953.MetropolisTypes} (framework : PhysicsLogic.Papers.Metropolis1953.ModifiedMonteCarloFramework T) (finite : framework.FinitePeriodicModelResult) (setup : framework.CanonicalTargetSetup) (target : framework.CanonicalTargetResult) (scope : framework.ScopeAssessment) : (framework.toCanonicalGibbsContext |>.NonvacuousCanonicalGibbsNormalizationAt) /\ framework.EquationTwoCanonicalAverageFormulaAt /\ Not (framework.finiteConvergenceRateEstablished framework.selectedUpdate) /\ Not (framework.updateRepresentsPhysicalMolecularDynamics framework.selectedUpdate) := by have retained := framework |>.canonical_gibbs_realization_preserves_metropolis_average_and_open_scope finite setup target scope exact ⟨retained.1, retained.2.1, retained.2.2.1, retained.2.2.2.1⟩ end OriginalCanonicalGibbsSourceReuseAudit namespace OriginalBrownianSourceCompositionReuseAudit example {T : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianTypes} (framework : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework T) (formulas : framework.OriginalGaussianSpreadFormulaPackageAt) : framework.RootTimeScalingAt := framework.original_gaussian_formulas_derive_root_time_scaling formulas example {T : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianTypes} (framework : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework T) (formulas : framework.OriginalGaussianSpreadFormulaPackageAt) (time : Real) (positive : 0 < time) : framework.rmsDisplacementValueAt framework.rmsDisplacement (4 * time) = 2 * framework.rmsDisplacementValueAt framework.rmsDisplacement time ∧ framework.rmsDisplacementValueAt framework.rmsDisplacement (4 * time) ≠ 4 * framework.rmsDisplacementValueAt framework.rmsDisplacement time := framework.quadrupling_time_doubles_original_rms_displacement formulas time positive example {T : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianTypes} (framework : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework T) (coefficient : framework.equationEightDiffusionCoefficientValue) (formulas : framework.OriginalGaussianSpreadFormulaPackageAt) : framework.SectionFiveMolecularCountFormulaAt := framework |>.original_coefficient_and_gaussian_spread_derive_molecular_count_formula coefficient formulas open PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels in example : positiveOneTimeDiffusiveObservation ballisticDisplacementAfterDiffusiveOneTimeFit ∧ ¬twoTimeDiffusiveRootScaling ballisticDisplacementAfterDiffusiveOneTimeFit := one_time_diffusive_fit_does_not_exclude_ballistic_scaling open PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels in example : coefficientAndDisplacementEquationsAt zeroDurationMolecularCountObservation ∧ ¬molecularCountRecoveredFromDisplacementAt zeroDurationMolecularCountObservation := zero_duration_displacement_does_not_determine_molecular_count example {T : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianTypes} (framework : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework T) (thermodynamic : framework.equationOneThermodynamicBalanceValueAt) (driftDiffusion : framework.equationTwoDriftDiffusionBalanceValueAt) (gradient : framework.NontrivialEquilibriumDensityGradientAt) : framework.equationEightDiffusionCoefficientValue := framework |>.equilibrium_balances_and_nonzero_gradient_determine_diffusion_coefficient thermodynamic driftDiffusion gradient open PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework.Countermodels in example : originalEquilibriumBalancesAt uniformEquilibriumWithUnderdeterminedDiffusion ∧ ¬originalEquilibriumCoefficientDeterminedAt uniformEquilibriumWithUnderdeterminedDiffusion ∧ ¬nontrivialEquilibriumGradientAt uniformEquilibriumWithUnderdeterminedDiffusion := uniform_equilibrium_balances_do_not_determine_diffusion_coefficient example {T : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianTypes} (framework : PhysicsLogic.Papers.EinsteinBrownian1905.BrownianFramework T) (diluteSetup : framework.DiluteSuspensionSetup) (osmoticCalculation : framework.OsmoticPressureCalculation) (equilibriumConditions : framework.DynamicEquilibriumConditions) (driftCalculation : framework.DriftDiffusionCalculation) (mesoscopicSetup : framework.MesoscopicStepSetup) (transitionCalculation : framework.TransitionLawExpansionCalculation) (pointSource : framework.RelativePointSourceInputAt) (pointCalculation : framework.PointSourceSolutionCalculation) : (framework.toThermalDragBrownianTransportContext |>.SourceSensitiveDiffusionGroundedThermalDragBrownianTransportAt framework.thermalDragBrownianTransportModel .symmetricTransitionExpansionAndPointSourceSolution framework.einsteinRandomDisplacementDiffusionDerivationObtains .osmoticPressureStokesFluxBalanceAndGaussianSpread framework.einsteinThermalDragDerivationObtains .symmetricGaussianDiffusionOsmoticStokesFluxBalanceAndExactCoefficient framework.einsteinDiffusionThermalDragHandoffDerivationObtains) ∧ framework.equationEightDiffusionCoefficientValue := by have composition := framework |>.symmetric_diffusion_and_osmotic_stokes_balance_realize_source_sensitive_brownian_transport diluteSetup osmoticCalculation equilibriumConditions driftCalculation mesoscopicSetup transitionCalculation pointSource pointCalculation have joinedSource := framework.toThermalDragBrownianTransportContext |>.source_sensitive_brownian_handoff_exposes_handoff_derivation composition.2 exact ⟨composition, joinedSource.2.2.2.2.2.2⟩ example {T : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianTypes} (framework : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework T) (formulas : framework.OriginalPersistentFlightFormulaPackageAt) : framework.ControlledSmallAngleTruncationGapAt := framework |>.original_persistent_flight_derives_strict_small_angle_truncation_gap formulas example {T : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianTypes} (framework : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework T) (setup : framework.SmallParticleSetup) (gap : framework.ControlledSmallAngleTruncationGapAt) : framework.toSmallAngleApproximationContext.ControlledApproximation .quadraticDeflectionModel .exactCosineGeometry framework.smallParticleRegime (fun observable => observable = .cosineLoss) framework.directionDeflectionApproximationToleranceValue := framework.small_angle_gap_realizes_foundational_controlled_approximation setup gap example {T : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianTypes} (framework : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework T) (setup : framework.SmallParticleSetup) (gap : framework.ControlledSmallAngleTruncationGapAt) : framework.ControlledPersistentDeflectionApproximationAt ∧ ¬framework.toSmallAngleApproximationContext.ControlledApproximation .quadraticDeflectionModel .exactCosineGeometry framework.smallParticleRegime (fun observable => observable = .cosineLoss) 0 := framework.controlled_small_angle_approximation_does_not_imply_exact_prediction setup gap example {T : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianTypes} (framework : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework T) (formulas : framework.OriginalRelaxationStokesFormulaPackageAt) : framework.CorrectedRelaxationApproximationAt := framework |>.original_relaxation_formulas_derive_distinct_corrected_prediction formulas open PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework.Countermodels in example : controlledSmallAngleApproximationAt piThirdControlledSmallAngleApproximation ∧ ¬exactQuadraticCosineIdentityAt piThirdControlledSmallAngleApproximation := controlled_small_angle_approximation_does_not_make_truncation_exact open PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework.Countermodels in example : originalRelaxationApproximationStagesAt positiveDistinctRelaxationApproximationStages ∧ ¬approximationStagesCoincideAt positiveDistinctRelaxationApproximationStages := original_relaxation_stages_do_not_identify_preliminary_and_corrected_spread example {T : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianTypes} (framework : PhysicsLogic.Papers.SmoluchowskiBrownian1906.BrownianFramework T) (kineticSetup : framework.KineticImpulseSetup) (kineticCalculation : framework.RandomExcessAndEquipartitionCalculation) (smallConditions : framework.SmallParticleConditions) (validDiffusion : framework.validAt framework.kineticDescription framework.diffusionRegime) (persistentCalculation : framework.PersistentPathCalculation) (regimeBridge : framework.SpreadRegimeBridge) (analogy : framework.gasMoleculeAnalogyApplicable framework.selectedParticle framework.selectedMedium framework.diffusionRegime) (diffusionCalculation : framework.DiffusionCalculation) (largeConditions : framework.LargeParticleConditions) (relaxationCalculation : framework.RelaxationDragCalculation) : (framework.toThermalDragBrownianTransportContext |>.SourceSensitiveDiffusionGroundedThermalDragBrownianTransportAt framework.thermalDragBrownianTransportModel .persistentCollisionFlightAndGasKineticDiffusionAnalogy framework.smoluchowskiRandomDisplacementDiffusionDerivationObtains .collisionEquipartitionPersistentFlightRelaxationStokesAndGasDiffusion framework.smoluchowskiThermalDragDerivationObtains .finiteCollisionExcessPersistentSmallFlightLargeRelaxationAndApproximateCoefficient framework.smoluchowskiDiffusionThermalDragHandoffDerivationObtains) ∧ framework.nearAgreementWithEinsteinCoefficient framework.rmsDisplacement framework.largeParticleRegime ∧ ¬framework.exactCoefficientAgreementWithEinstein framework.rmsDisplacement framework.largeParticleRegime := by have composition := framework |>.persistent_small_particle_diffusion_and_large_particle_relaxation_realize_source_sensitive_brownian_transport kineticSetup kineticCalculation smallConditions validDiffusion persistentCalculation regimeBridge analogy diffusionCalculation largeConditions relaxationCalculation have joinedSource := framework.toThermalDragBrownianTransportContext |>.source_sensitive_brownian_handoff_exposes_handoff_derivation composition.2 rcases joinedSource with ⟨_diffusionSource, _dragSource, _smallSetup, _largeSetup, _persistentFormulas, _relaxationFormulas, approximateAgreement, _differentPrefactor, notExactAgreement⟩ exact ⟨composition, approximateAgreement, notExactAgreement⟩ open PhysicsLogic.Framework.ThermalDragBrownianTransportContext.Countermodels example : physicalBrownianModel.stochasticModel.regime ≠ physicalBrownianModel.dragRegime ∧ ¬(physicalBrownianContext |>.SourceSensitiveDiffusionGroundedThermalDragBrownianTransportAt physicalBrownianModel () (fun _ : Unit => True) () (fun _ : Unit => True) () (fun _ : Unit => False)) := by have witness := completed_brownian_transport_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition exact ⟨witness.1, witness.2.2.2.2⟩ end OriginalBrownianSourceCompositionReuseAudit namespace OriginalSuperconductingSourceCompositionReuseAudit example {T : PhysicsLogic.Papers.BardeenCooperSchrieffer1957.BCSTypes} (framework : PhysicsLogic.Papers.BardeenCooperSchrieffer1957.SuperconductivityFramework T) (source : framework.bcsTransverseMeissnerScreeningDerivationObtains .coherenceFactorParamagneticDiamagneticLongWavelengthResponse) : (framework.toTransverseMeissnerScreeningContext |>.SourceSensitiveGapSupportedTransverseMeissnerScreeningAt framework.toSelfConsistentSpectralGapContext framework.selfConsistentSpectralGapModel framework.meissnerScreeningModel .coherentVariationalPairingGapAndPairBreakingSpectrum framework.BCSSelfConsistentSpectralGapDerivationObtains .coherenceFactorParamagneticDiamagneticLongWavelengthResponse framework.bcsTransverseMeissnerScreeningDerivationObtains .variationalGapCaseIICoherenceLondonCurrentsAndFieldExpulsion framework.bcsGapMeissnerHandoffDerivationObtains) ∧ framework.EquationTwoFortyGapSolutionFormulaAt ∧ framework.SelectedLongWavelengthMeissnerFormulaPackageAt := by have composition := framework |>.variational_gap_and_coherence_london_response_realize_source_sensitive_meissner_screening source have joinedSource := framework.toTransverseMeissnerScreeningContext |>.source_sensitive_gap_screening_handoff_exposes_handoff_derivation framework.toSelfConsistentSpectralGapContext composition.2 rcases joinedSource with ⟨_gapSource, _screeningSource, gapFormula, _dispersion, _pairBreaking, _caseII, _coherence, currentFormulas, _pippard, _penetration, _fieldExclusion⟩ exact ⟨composition, gapFormula, currentFormulas⟩ example {T : PhysicsLogic.Papers.Nambu1960.NambuTypes} (framework : PhysicsLogic.Papers.Nambu1960.SuperconductivityFramework T) (source : framework.nambuTransverseMeissnerScreeningDerivationObtains .anomalousGapWardVertexCollectiveCancellationAndRetainedTransverseResponse) : (framework.toTransverseMeissnerScreeningContext |>.SourceSensitiveGapSupportedTransverseMeissnerScreeningAt framework.toSelfConsistentSpectralGapContext framework.selfConsistentSpectralGapModel framework.meissnerScreeningModel .anomalousSelfEnergyGapAndQuasiparticleBranches framework.NambuSelfConsistentSpectralGapDerivationObtains .anomalousGapWardVertexCollectiveCancellationAndRetainedTransverseResponse framework.nambuTransverseMeissnerScreeningDerivationObtains .nonperturbativeGapMatrixWardNeutralCollectiveCancellationAndTransverseScreening framework.nambuGapMeissnerHandoffDerivationObtains) ∧ framework.EquationFourFourChargeWardFormulaAt ∧ framework.EquationFiveNineLongitudinalCancellationFormulaAt ∧ (¬framework.CoulombPairAnnihilationCreationIncluded framework.coulombInteraction framework.ladderPairSpectrum framework.weakCouplingRegime) ∧ framework.transverseBCSResponseRetained framework.correctedMeissnerResponse framework.staticLinearRegime := by have composition := framework |>.anomalous_gap_ward_collective_cancellation_realize_source_sensitive_meissner_screening source have joinedSource := framework.toTransverseMeissnerScreeningContext |>.source_sensitive_gap_screening_handoff_exposes_handoff_derivation framework.toSelfConsistentSpectralGapContext composition.2 rcases joinedSource with ⟨_gapSource, _screeningSource, _gapFormula, _gaugePhase, wardFormula, cancellationFormula, _nonperturbative, _bareDefect, _neutralPair, noCoulomb, _collectiveIntermediate, _longitudinalCancellation, _pureGauge, transverseRetained, _gaugeInvariant, _fieldExclusion⟩ exact ⟨composition, wardFormula, cancellationFormula, noCoulomb, transverseRetained⟩ open PhysicsLogic.Framework.TransverseMeissnerScreeningContext.Countermodels open PhysicsLogic.Framework.SelfConsistentSpectralGapContext.Countermodels example : physicalBcsSpectralGapModel.gap ≠ nambuSpectralGapControlModel.gap ∧ physicalMeissnerScreeningModel.current ≠ PhysicsLogic.Framework.TransverseMeissnerScreeningContext.Countermodels.ScreeningCurrent.aligned ∧ ¬(physicalMeissnerScreeningContext |>.SourceSensitiveGapSupportedTransverseMeissnerScreeningAt physicalBcsSpectralGapContext physicalBcsSpectralGapModel physicalMeissnerScreeningModel () (fun _ : Unit => True) () (fun _ : Unit => True) () (fun _ : Unit => False)) := by have witness := completed_gap_screening_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition exact ⟨witness.1, witness.2.2.1, witness.2.2.2.2.2.2.2⟩ end OriginalSuperconductingSourceCompositionReuseAudit namespace OriginalEffectiveRangeBoundStateSourceCompositionReuseAudit example {T : PhysicsLogic.Papers.Bethe1949.BetheTypes} (framework : PhysicsLogic.Papers.Bethe1949.EffectiveRangeFramework T) (reductionSource : framework.BetheEffectiveRangeDerivationObtains .comparisonIdentityAndInteriorReplacement) (extractionSource : framework.BetheBoundStateExtractionDerivationObtains .comparisonIdentityGroundStateSpecializationAndEmpiricalExtraction) : framework.NeutralRadialFormulaReadoutAt ∧ framework.tripletBoundStateRelationControlledAt framework.neutronProtonTriplet framework.deuteronBoundEnergy framework.nuclearRegime ∧ framework.reportedTripletEffectiveRangeQualifiedAt framework.neutronProtonExperiment framework.neutronProtonTriplet framework.reportedTripletEffectiveRange framework.nuclearRegime := by have composition := framework |>.bethe_comparison_ground_state_and_triplet_evidence_realize_source_sensitive_bound_state_extraction reductionSource extractionSource have joinedSource := framework.toEffectiveRangeBoundStateExtractionContext |>.source_sensitive_bound_state_extraction_exposes_handoff_derivation composition rcases joinedSource with ⟨_reduction, _extraction, radialFormulas, _boundSetup, _empiricalSetup, _calculatedRelation, controlledRelation, _extracted, qualifiedEstimate⟩ exact ⟨radialFormulas, controlledRelation, qualifiedEstimate⟩ example {T : PhysicsLogic.Papers.BlattJackson1949.BlattJacksonTypes} (framework : PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework T) (reductionSource : framework.BlattJacksonEffectiveRangeDerivationObtains .stationaryFunctionalAndZeroEnergyTrial) (extractionSource : framework.BlattJacksonBoundStateExtractionDerivationObtains .scatteringMatrixContinuationAndEmpiricalExtraction) : framework.BoundStateFormulaPackageAt ∧ framework.scatteringMatrixAnalyticallyContinuedAt framework.neutronProtonTriplet framework.deuteronBoundEnergy framework.nuclearRegime ∧ framework.boundStatePoleConditionCalculatedAt framework.neutronProtonTriplet framework.deuteronBoundEnergy framework.nuclearRegime ∧ framework.reportedTripletEffectiveRangeQualifiedAt framework.neutronProtonExperiment framework.neutronProtonTriplet framework.reportedTripletEffectiveRange framework.nuclearRegime := by have composition := framework |>.blatt_jackson_stationary_functional_pole_and_triplet_evidence_realize_source_sensitive_bound_state_extraction reductionSource extractionSource have joinedSource := framework.toEffectiveRangeBoundStateExtractionContext |>.source_sensitive_bound_state_extraction_exposes_handoff_derivation composition rcases joinedSource with ⟨_reduction, _extraction, _law, boundRelation, printedFormulas, _tripletSetup, _extracted, qualifiedEstimate⟩ rcases boundRelation with ⟨_wave, _asymptotics, continuation, pole, _relation, _formulas, _controlled⟩ exact ⟨printedFormulas, continuation, pole, qualifiedEstimate⟩ open PhysicsLogic.Framework.EffectiveRangeBoundStateExtractionContext.Countermodels example : physicalBetheGroundStateExtractionModel.reduction.selectedEnergy ≠ physicalBetheGroundStateExtractionModel.boundStateEnergy ∧ physicalBetheGroundStateExtractionModel.experiment ≠ physicalBlattJacksonPoleExtractionModel.experiment ∧ ¬(physicalBetheGroundStateExtractionContext |>.SourceSensitiveEffectiveRangeBoundStateExtractionAt physicalBetheGroundStateExtractionModel () (fun _ : Unit => True) () (fun _ : Unit => True) () (fun _ : Unit => False)) := by have witness := completed_nuclear_extraction_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition exact ⟨witness.1, witness.2.1, witness.2.2.2.2.2⟩ end OriginalEffectiveRangeBoundStateSourceCompositionReuseAudit namespace OriginalFiniteGaugeLoopConfinementSourceCompositionReuseAudit example {T : PhysicsLogic.Papers.Wilson1974.WilsonTypes} (framework : PhysicsLogic.Papers.Wilson1974.LatticeGaugeFramework T) (source : framework.WilsonLoopAreaConfinementDerivationObtains .strongCouplingSpanningSurfaceExpansionAndBinding) : framework.EquationThreeEightPlaquettePhaseFormulaAt ∧ framework.SelectedSurfaceFormulaPackageAt ∧ framework.SelectedAreaLawFormulaPackageAt ∧ framework.VacuumLoopScopeAssessment ∧ framework.quarkAntiquarkPathsRemainBound framework.latticeDescription framework.selectedQuarkPath := by have composition := framework |>.compact_gauge_minimal_surface_and_quenched_binding_realize_source_sensitive_loop_confinement source have joinedSource := framework.toLoopAreaConfinementContext |>.source_sensitive_finite_gauge_loop_confinement_exposes_handoff_derivation framework.toFiniteRegulatorGaugeInvarianceContext composition rcases joinedSource with ⟨_gauge, _loop, _gaugeTransformation, _angularRescaling, plaquetteFormula, surfaceFormulas, areaFormulas, vacuumScope, _tiling, _surfaceSum, bound⟩ exact ⟨plaquetteFormula, surfaceFormulas, areaFormulas, vacuumScope, bound⟩ example {T : PhysicsLogic.Papers.Creutz1980.CreutzTypes} (framework : PhysicsLogic.Papers.Creutz1980.MonteCarloGaugeFramework T) (source : framework.CreutzLoopAreaConfinementDerivationObtains .finiteCutoffMonteCarloLoopFitAndStaticPotentialInterpretation) : framework.pureSU2GaugeFields framework.latticeDescription ∧ framework.ReliableLoopData ∧ framework.fitStatisticalUncertaintyAssessed framework.selectedFit ∧ framework.EquationFourThreeSquareLoopFitFormulaAt ∧ framework.EquationFourFourAreaCoefficientFormulaAt ∧ framework.RegulatedConfinementResult := by have composition := framework |>.pure_su2_equilibrium_and_finite_square_loop_fit_realize_source_sensitive_loop_confinement source have joinedSource := framework.toLoopAreaConfinementContext |>.source_sensitive_finite_gauge_loop_confinement_exposes_handoff_derivation framework.toFiniteRegulatorGaugeInvarianceContext composition rcases joinedSource with ⟨_gauge, _loop, pureSU2, _regulatedGauge, _equilibrium, _reliabilitySetup, reliability, statisticalUncertainty, _areaFormula, _latticeAreaFormula, squareLoopFit, areaCoefficient, confinement⟩ exact ⟨pureSU2, reliability, statisticalUncertainty, squareLoopFit, areaCoefficient, confinement⟩ open PhysicsLogic.Framework.LoopAreaConfinementContext.Countermodels example : physicalWilson1974StrongSurfaceReadout.gaugeGroup ≠ physicalCreutz1980FiniteLoopFitReadout.gaugeGroup ∧ physicalWilson1974StrongSurfaceReadout.derivation ≠ physicalCreutz1980FiniteLoopFitReadout.derivation ∧ ¬(completeFiniteGaugeSupportedLoopAreaRoute |>.SourceSensitiveFiniteGaugeSupportedLoopAreaConfinementAt PhysicsLogic.Framework.FiniteRegulatorGaugeInvarianceContext.Countermodels.invariantOnIncompleteOrbits PhysicsLogic.Framework.FiniteRegulatorGaugeInvarianceContext.Countermodels.invariantOnIncompleteOrbits.selectedCalculation completeFiniteGaugeSupportedLoopAreaModel () (fun _ : Unit => True) () (fun _ : Unit => True) () (fun _ : Unit => False)) := by have witness := completed_finite_gauge_confinement_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition exact ⟨witness.1, witness.2.1, witness.2.2.2.2.2.2.2⟩ end OriginalFiniteGaugeLoopConfinementSourceCompositionReuseAudit namespace OriginalFiniteVelocityClusteringSourceCompositionReuseAudit example (framework : PhysicsLogic.Papers.HastingsKoma2006.PaperFramework) (coneSource : framework.HastingsKomaFiniteVelocityDerivationObtains .exponentialTimeSlicingAndSupportChainIteration) (clusteringSource : framework.hastingsKomaCorrectedClusteringDerivationObtains .gaussianFilteringRealTimePropagationAndFiniteVolumeTransfer) : framework.EstablishedExponentialPropagation ∧ framework.EstablishedFiniteVolumeThermodynamicClustering ∧ framework.groundSectorContributionSeparated framework.selectedGroundState framework.selectedGroundProjection framework.sourceObservable framework.targetObservable ∧ framework.CorrectedBosonicClusteringFormulaAt := by have composition := framework |>.exponential_support_chain_gaussian_filter_and_finite_volume_transfer_realize_source_sensitive_clustering coneSource clusteringSource have joinedSource := framework.toSpectralGapClusteringContext |>.source_sensitive_finite_velocity_clustering_exposes_handoff_derivation framework.toFiniteVelocityCommutatorConeContext composition rcases joinedSource with ⟨_cone, _clustering, exponential, _finiteVolume, thermodynamic, _filterLemma, groundCorrection, _bosonicCorrection, printedFormula, _result⟩ exact ⟨exponential, thermodynamic, groundCorrection, printedFormula⟩ example (framework : PhysicsLogic.Papers.NachtergaeleSims2006.PaperFramework) (coneSource : framework.NachtergaeleSimsFiniteVelocityDerivationObtains .commutatorIdentityMatrixUnitsAndPositiveIteration) (clusteringSource : framework.nachtergaeleSimsCorrectedClusteringDerivationObtains .analyticContourSpectralGaussianCommutatorOptimization) : framework.FiniteVelocityFormulaPackageAt ∧ framework.targetGroundSectorComponentsVanish framework.selectedGroundProjection framework.targetObservable framework.selectedGroundState ∧ framework.ExponentialClusteringFormulaAt := by have composition := framework |>.weighted_metric_cone_analytic_contour_and_imaginary_time_window_realize_source_sensitive_clustering coneSource clusteringSource have joinedSource := framework.toSpectralGapClusteringContext |>.source_sensitive_finite_velocity_clustering_exposes_handoff_derivation framework.toFiniteVelocityCommutatorConeContext composition rcases joinedSource with ⟨_cone, _clustering, _establishedCone, printedCone, _gapped, projection, _contour, _gaussian, _optimized, printedClustering, _result⟩ exact ⟨printedCone, projection, printedClustering⟩ open PhysicsLogic.Framework.SpectralGapClusteringContext.Countermodels example : physicalHastingsKoma2006BosonicExponentialReadout.geometry ≠ physicalNachtergaeleSims2006MetricImaginaryTimeReadout.geometry ∧ physicalHastingsKoma2006BosonicExponentialReadout.groundSectorCorrectionValue = 4 ∧ physicalNachtergaeleSims2006MetricImaginaryTimeReadout.selectedImaginaryTime = 1 ∧ ¬(testContext |>.SourceSensitiveFiniteVelocitySupportedSpectralGapClusteringAt completeConeContext completeConeModel testModel () (fun _ : Unit => True) () (fun _ : Unit => True) () (fun _ : Unit => False)) := by have witness := completed_finite_velocity_clustering_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition exact ⟨witness.1, witness.2.2.1, witness.2.2.2.2.1, witness.2.2.2.2.2.2.2.2⟩ end OriginalFiniteVelocityClusteringSourceCompositionReuseAudit namespace OriginalTrappedSurfaceSourceCompositionReuseAudit example {T : PhysicsLogic.Papers.Penrose1965.PenroseTypes} (framework : PhysicsLogic.Papers.Penrose1965.CollapseFramework T) (stabilitySource : framework.penroseTrappedSurfaceStabilityDerivationObtains .controlledNonsphericalCollapseConstruction) (reductioSource : framework.PenroseDirectionalNullIncompletenessDerivationObtains .futureTrappedSurfaceOpticalScalarCompactBoundaryDegreeObstruction) : framework.PrintedOpticalScalarFocusingFormulaAt framework.nullCrossSectionalArea framework.orthogonalFutureNullCongruence framework.futureDirection framework.initialAffineParameter framework.selectedSpacetime framework.selectedRegime ∧ framework.trappedSurfaceConstructionCalculated framework.nonsphericalCollapse framework.selectedTrappedSurface framework.selectedSpacetime framework.selectedRegime ∧ framework.deviationsFromSphericalSymmetryControlled framework.nonsphericalCollapse framework.sphericalReference framework.selectedRegime ∧ ¬framework.nullCompleteToward framework.selectedSpacetime framework.futureDirection framework.selectedRegime := by have composition := framework |>.controlled_future_collapse_and_optical_degree_reductio_realize_source_sensitive_incompleteness stabilitySource reductioSource have joinedSource := framework.toTrappedSurfaceIncompletenessContext |>.source_sensitive_stability_supported_incompleteness_exposes_handoff_derivation framework.toTrappedSurfaceStabilityContext composition have physical := framework.toTrappedSurfaceIncompletenessContext |>.source_sensitive_stability_supported_incompleteness_retains_trapping_topology_and_direction framework.toTrappedSurfaceStabilityContext composition rcases joinedSource with ⟨_stability, _reductio, printed, constructed, controlled, _global, _focusing, _boundary, _topology⟩ rcases physical with ⟨_referenceClosed, _referenceTrapped, _controlled, _targetClosed, _targetTrapped, _noncompact, _energy, _focusingLaw, _compactLaw, _topologyLaw, incomplete⟩ exact ⟨printed, constructed, controlled, incomplete⟩ example {T : PhysicsLogic.Papers.HawkingOpenUniverses1965.HawkingTypes} (framework : PhysicsLogic.Papers.HawkingOpenUniverses1965.OpenUniverseFramework T) (stabilitySource : framework.hawkingTrappedSurfaceStabilityDerivationObtains .finiteConvergenceMarginPerturbation) (reductioSource : framework.hawkingPerturbedDirectionalNullIncompletenessDerivationObtains .finiteMarginPerturbedPastTrappedSphereWithImportedPenroseReductio) : framework.PrintedLargeSphereConvergenceFormulaAt ∧ framework.convergencePositiveByMargin framework.inwardPastFamily framework.homogeneousLargeSphere framework.pastDirection framework.homogeneousSpacetime framework.positiveConvergenceMargin framework.selectedRegime ∧ framework.convergencePositiveByMargin framework.outwardPastFamily framework.homogeneousLargeSphere framework.pastDirection framework.homogeneousSpacetime framework.positiveConvergenceMargin framework.selectedRegime ∧ framework.convergenceMarginPreserved framework.selectedPerturbation framework.inwardPastFamily framework.perturbedLargeSphere framework.pastDirection framework.perturbedSpacetime framework.positiveConvergenceMargin framework.selectedRegime ∧ framework.convergenceMarginPreserved framework.selectedPerturbation framework.outwardPastFamily framework.perturbedLargeSphere framework.pastDirection framework.perturbedSpacetime framework.positiveConvergenceMargin framework.selectedRegime ∧ (¬framework.exactLocalSymmetry framework.perturbedSpacetime framework.selectedRegime) ∧ ¬framework.nullCompleteToward framework.perturbedSpacetime framework.pastDirection framework.selectedRegime := by have composition := framework |>.large_sphere_finite_margin_and_perturbed_penrose_reductio_realize_source_sensitive_incompleteness stabilitySource reductioSource have joinedSource := framework.toPerturbedTrappedSurfaceIncompletenessContext |>.source_sensitive_stability_supported_incompleteness_exposes_handoff_derivation framework.toTrappedSurfaceStabilityContext composition have physical := framework.toPerturbedTrappedSurfaceIncompletenessContext |>.source_sensitive_stability_supported_incompleteness_retains_trapping_topology_and_direction framework.toTrappedSurfaceStabilityContext composition rcases joinedSource with ⟨_stability, _reductio, printed, inward, outward, preservedInward, preservedOutward, _finite, _small, noExactSymmetry, _global, _focusing, _boundary, _topology⟩ rcases physical with ⟨_referenceClosed, _referenceTrapped, _controlled, _targetClosed, _targetTrapped, _noncompact, _energy, _focusingLaw, _compactLaw, _topologyLaw, incomplete⟩ exact ⟨printed, inward, outward, preservedInward, preservedOutward, noExactSymmetry, incomplete⟩ open PhysicsLogic.Framework.TrappedSurfaceIncompletenessContext.Countermodels example : physicalPenrose1965FutureCollapseReadout.direction ≠ physicalHawking1965OpenCoveringReadout.direction ∧ physicalHawking1965OpenCoveringReadout.inwardNullConvergence = 3 / 2 ∧ physicalHawking1965OpenCoveringReadout.outwardNullConvergence = 13 / 2 ∧ ¬(establishedWithoutTheoryModelGrounding |>.SourceSensitiveStabilitySupportedDirectionalNullIncompletenessAt completeStabilityWithRejectedUpperTrapping completeStabilityWithRejectedUpperTrapping.calculationModel establishedWithoutTheoryModelGrounding.calculationModel () (fun _ : Unit => True) () (fun _ : Unit => True) () (fun _ : Unit => False)) := by have witness := completed_directional_incompleteness_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition rcases witness with ⟨_geometry, direction, _globalRoute, _optical, inward, outward, _perturbation, _margin, _supported, _stabilitySource, _reductioSource, noComposition⟩ exact ⟨direction, inward, outward, noComposition⟩ end OriginalTrappedSurfaceSourceCompositionReuseAudit namespace OriginalFractionalHallChargeSourceCompositionReuseAudit example {T : PhysicsLogic.Papers.Laughlin1983.LaughlinTypes} (framework : PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework T) (source : framework.laughlinFractionalHallChargeDerivationObtains .plasmaPhantomChargeScreening) : framework.EquationsSevenAndEightPlasmaFormulaAt ∧ framework.phantomChargeInsertedAt framework.selectedPlasma framework.insertionPoint ∧ framework.plasmaCompletelyScreensAt framework.selectedPlasma framework.insertionPoint ∧ (framework.exponent : Real) * framework.chargeMagnitudeAt framework.quasihole framework.bulkRegime = framework.electronChargeMagnitude := by have composition := framework |>.plasma_phantom_screening_realizes_source_sensitive_fractional_hall_charge source have joinedSource := framework.toFractionalHallQuasiholeChargeContext |>.source_sensitive_fractional_hall_charge_exposes_handoff_derivation composition rcases joinedSource with ⟨_source, formulas, _boltzmann, _plasma, _coupling, phantom, screening, _orbitals, chargeEquation⟩ exact ⟨formulas, phantom, screening, chargeEquation⟩ example {T : PhysicsLogic.Papers.Laughlin1983.LaughlinTypes} (framework : PhysicsLogic.Papers.Laughlin1983.QuantumHallFramework T) (source : framework.laughlinFractionalHallChargeDerivationObtains .fluxInsertionOrbitalStateCounting) : framework.fluxQuantumInsertedAdiabatically framework.electronGas framework.insertedFlux framework.insertionPoint ∧ framework.fluxInsertionMovesFluidByOneOrbital framework.insertedFlux framework.selectedGroundState ∧ (framework.exponent : Real) * framework.chargeMagnitudeAt framework.quasihole framework.bulkRegime = framework.electronChargeMagnitude := by have composition := framework |>.single_orbital_flux_counting_realizes_source_sensitive_fractional_hall_charge source have joinedSource := framework.toFractionalHallQuasiholeChargeContext |>.source_sensitive_fractional_hall_charge_exposes_handoff_derivation composition rcases joinedSource with ⟨_source, inserted, _allOrbitals, oneOrbital, _filling, chargeEquation⟩ exact ⟨inserted, oneOrbital, chargeEquation⟩ example {T : PhysicsLogic.Papers.ArovasSchriefferWilczek1984.ArovasTypes} (framework : PhysicsLogic.Papers.ArovasSchriefferWilczek1984.FractionalStatisticsFramework T) (source : framework.arovasFractionalHallChargeDerivationObtains .normalizedDensityBerryPhaseAndElectromagneticLoopMatching) : framework.stateNormalized framework.oneQuasiholeState ∧ framework.adiabaticPhaseSeparateFromDynamicalPhase framework.quasiholeHamiltonian framework.oneQuasiholeState framework.transportLoop framework.singleQuasiholeLoopPhase ∧ framework.phaseDerivativeEqualsStateInnerProduct framework.oneQuasiholeState framework.movingQuasiholePosition framework.transportLoop ∧ framework.DensityPhaseCalculation ∧ framework.electromagneticLoopPhase (framework.chargeMagnitudeAt framework.quasihole framework.bulkRegime) framework.loopFlux framework.singleQuasiholeLoopPhase ∧ framework.chargeMagnitudeAt framework.quasihole framework.bulkRegime = framework.filling * framework.electronChargeMagnitude := by have composition := framework |>.normalized_berry_density_and_electromagnetic_matching_realize_source_sensitive_fractional_hall_charge source have joinedSource := framework.toFractionalHallQuasiholeChargeContext |>.source_sensitive_fractional_hall_charge_exposes_handoff_derivation composition rcases joinedSource with ⟨_source, normalized, nondynamical, _duration, innerProduct, density, electromagnetic, _matching, _identification, chargeEquation⟩ exact ⟨normalized, nondynamical, innerProduct, density, electromagnetic, chargeEquation⟩ open PhysicsLogic.Framework.FractionalHallQuasiholeChargeContext.Countermodels example : physicalLaughlinPlasmaScreeningReadout.branch ≠ physicalLaughlinFluxInsertionReadout.branch ∧ physicalLaughlinPlasmaScreeningReadout.quasiholeChargeMagnitudeValue = (1 : Real) / 3 ∧ physicalArovasAdiabaticChargeReadout.quasiholeChargeMagnitudeValue = (1 : Real) / 5 ∧ ¬(physicalLaughlinFractionalChargeContext |>.SourceSensitiveFunctionalChargeSupportedFractionalHallChargeAt physicalLaughlinFractionalChargeModel.selectedChargeModel physicalLaughlinFractionalChargeModel () (fun _ : Unit => True) () (fun _ : Unit => True) () (fun _ : Unit => False)) := by have witness := completed_fractional_hall_charge_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition rcases witness with ⟨_source, _derivation, independentBranches, _phantom, _temperature, laughlinCharge, arovasCharge, _quasiparticle, _correction, _variational, _fractional, _chargeSource, _hallSource, noComposition⟩ exact ⟨independentBranches, laughlinCharge, arovasCharge, noComposition⟩ end OriginalFractionalHallChargeSourceCompositionReuseAudit namespace OriginalGoldstoneModeConversionSourceCompositionReuseAudit example {T : PhysicsLogic.Papers.Anderson1963.AndersonTypes} (framework : PhysicsLogic.Papers.Anderson1963.PlasmonFramework T) (plasmaSetup : framework.GaugeInvariantPlasmaSetup) (plasmaCalculation : framework.PlasmaResponseCalculation) (undressedSetup : framework.UndressedMatterResponseSetup) (undressedCriterion : framework.UndressedResponseSpectrumCriterion) (matterSetup : framework.UncoupledMatterBrokenSetup) (goldstoneCalculation : framework.UncoupledGoldstoneCalculation) (conversionLocalSetup : framework.SuperconductingConversionLocalSetup) (conversionCalculation : framework.CoulombModeConversionCalculation) (analogyScope : framework.RelativisticAnalogyLocalSetup) (programScope : framework.ProgramScope) : framework.EquationsFourSevenAndEightPlasmaPoleFormulaAt framework.electronPlasma ∧ framework.nonrelativisticExample framework.electronPlasma ∧ framework.longitudinalTransverseDispersionsInequivalent framework.electronPlasma ∧ (¬framework.nonAbelianGaugeGroupsAnalyzedFor framework.electromagneticField) ∧ framework.longRangeCoulombAbsentIn framework.uncoupledMatterSpectrum framework.superconductor ∧ framework.longRangeCoulombPresentIn framework.interactingPhysicalSpectrum framework.superconductor ∧ ¬framework.modeInSpectrum framework.uncoupledGoldstoneMode framework.interactingPhysicalSpectrum := by have composition := framework |>.conductor_pole_undressed_goldstone_and_gauge_mixing_realize_source_sensitive_mode_conversion plasmaSetup plasmaCalculation undressedSetup undressedCriterion matterSetup goldstoneCalculation conversionLocalSetup conversionCalculation analogyScope programScope have joinedSource := framework.toBrokenSymmetryModeConversionContext |>.source_sensitive_mode_conversion_exposes_handoff_derivation composition rcases joinedSource with ⟨_goldstone, _conversion, printed, _conducting, _induced, _gauge, _charge, _longitudinal, _transverse, _longWavelength, analogy, scope, absent, present, _mixed, _converted, _finite, _inSpectrum, removed⟩ exact ⟨printed, analogy.1, analogy.2, scope.2.1, absent, present, removed⟩ example {T : PhysicsLogic.Papers.Nambu1960.NambuTypes} (framework : PhysicsLogic.Papers.Nambu1960.SuperconductivityFramework T) (zeroModeSetup : framework.ZeroMomentumCollectiveSetup) (zeroModeCalculation : framework.ZeroMomentumPairCalculation) (wardCalculation : framework.WardIdentityCalculation) (dispersionLocalSetup : framework.FiniteMomentumCollectiveLocalSetup) (dispersionCalculation : framework.FiniteMomentumDispersionCalculation) (coulombLocalSetup : framework.CoulombPlasmaLocalSetup) (plasmaCalculation : framework.CoulombPlasmaCalculation) (scope : framework.ApproximationScope) : framework.EquationFourFourChargeWardFormulaAt ∧ framework.EquationSixElevenCollectiveDispersionFormulaAt ∧ framework.EquationEightElevenPlasmaFrequencyFormulaAt ∧ (¬framework.CoulombPairAnnihilationCreationIncluded framework.coulombInteraction framework.ladderPairSpectrum framework.weakCouplingRegime) ∧ framework.CoulombPairAnnihilationCreationIncluded framework.coulombInteraction framework.coulombCoupledSpectrum framework.weakCouplingRegime ∧ framework.polarizationBubblesResummed framework.coulombInteraction framework.correctedMeissnerResponse framework.weakCouplingRegime ∧ (¬framework.modeInSpectrum framework.zeroMomentumPairMode framework.coulombCoupledSpectrum) ∧ (¬framework.higherExternalFieldOrdersAnalyzedIn framework.staticLinearRegime) ∧ ¬framework.neglectedTermsControlledIn framework.weakCouplingRegime := by have composition := framework |>.matrix_ward_neutral_pair_and_coulomb_resummation_realize_source_sensitive_mode_conversion zeroModeSetup zeroModeCalculation wardCalculation dispersionLocalSetup dispersionCalculation coulombLocalSetup plasmaCalculation scope have joinedSource := framework.toBrokenSymmetryModeConversionContext |>.source_sensitive_mode_conversion_exposes_handoff_derivation composition rcases joinedSource with ⟨_goldstone, _conversion, ward, neutralDispersion, plasmaFrequency, _scope, _zeroMode, _matchedKernel, _continuity, coulombAbsent, coulombPresent, polarization, _shifted, _finite, _plasmaInSpectrum, originalRemoved, _poleRemoved, _linear, _weak, higherOpen, uncontrolled, _hartreeFockOpen⟩ exact ⟨ward, neutralDispersion, plasmaFrequency, coulombAbsent, coulombPresent, polarization, originalRemoved, higherOpen, uncontrolled⟩ open PhysicsLogic.Framework.BrokenSymmetryModeConversionContext.Countermodels example : physicalNambu1960NeutralCollectivePairReadout.chargeWardVertexZeroOne = -4 ∧ physicalNambu1960CoulombConvertedPlasmaReadout.nambuPlasmaFrequencySquared = 4 ∧ physicalAnderson1963NormalConductorPlasmaReadout.andersonPlasmaFrequencySquaredOverPi = 16 ∧ physicalAnderson1963NormalConductorPlasmaReadout.brokenMatterSymmetryEstablished = false ∧ physicalAnderson1963SuperconductingGoldstoneReadout.brokenMatterSymmetryEstablished = true ∧ ¬(selectedModeConversionDataContext |>.SourceSensitiveGoldstoneSupportedModeConversionAt selectedModeConversionDataCalculation () (fun _ : Unit => True) () (fun _ : Unit => True) () (fun _ : Unit => False)) := by have witness := completed_goldstone_mode_conversion_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition rcases witness with ⟨_source, _nambuSystems, _andersonSystems, _branches, ward, _neutralFrequency, _zeroFrequency, nambuPlasma, _response, andersonPlasma, _longitudinal, _transverse, normalUnbroken, superconductorBroken, _originated, _goldstoneSource, _conversionSource, noComposition⟩ exact ⟨ward, nambuPlasma, andersonPlasma, normalUnbroken, superconductorBroken, noComposition⟩ end OriginalGoldstoneModeConversionSourceCompositionReuseAudit namespace OriginalValidatedNumericalNearReturnSourceCompositionReuseAudit example {T : PhysicsLogic.Papers.FermiPastaUlam1955.FPUTypes} (framework : PhysicsLogic.Papers.FermiPastaUlam1955.NonlinearChainFramework T) (source : framework.fpuValidatedNearReturnDerivationObtains .finiteDifferenceModalEnergyValidationAndOnePercentReturn) (modalCalculation : framework.ModalEnergyFormulaCalculation) (figureOne : framework.FigureOneQuadraticRunParameters) (modalApproximation : framework.nonlinearPotentialContributionNeglectedInModalEnergy framework.quadraticPureModeRun) (equipartitionInterpretation : framework.EquipartitionInterpretationBridge) (scope : framework.ScopeAssessment) : framework.EquationOneQuadraticChainFormulaAt framework.quadraticPureModeRun ∧ framework.SelectedModalEnergyFormulaAt ∧ framework.FigureOneQuadraticRunParameters ∧ framework.modalEnergyFormulaReadout.FirstModeOnePercentReturnAt ∧ framework.fixedEndpoints framework.selectedSystem ∧ framework.nonlinearPotentialContributionNeglectedInModalEnergy framework.quadraticPureModeRun ∧ framework.HamiltonianEnergyValidationAt framework.quadraticPureModeRun ∧ framework.availableLinearControlsPass framework.quadraticPureModeRun ∧ framework.firstModeEnergyReturnsWithin framework.quadraticPureModeRun ((1 : Real) / 100) ∧ (¬framework.equipartitionObservedOverRun framework.quadraticPureModeRun) ∧ (¬framework.exactFullStateRecurrenceEstablished framework.quadraticPureModeRun) ∧ ¬framework.eventualEquipartitionExcluded framework.quadraticPureModeRun := by have composition := framework |>.fixed_endpoint_quadratic_chain_and_approximate_modal_energy_realize_source_sensitive_near_return source modalCalculation figureOne modalApproximation equipartitionInterpretation scope have joinedSource := framework.toValidatedNumericalNearReturnContext |>.source_sensitive_numerical_near_return_exposes_handoff_derivation composition rcases joinedSource with ⟨_source, equation, modalFormula, selectedFigure, numericReturn, _chain, fixed, _neighbor, _positiveCount, _countBound, _ratio, _rest, _initialMode, _difference, _coordinates, _fourier, _modal, neglectedPotential, energyCheck, linearCheck, _reliable, _fewModes, _higherModes, firstReturn, _nearProfile, noEquipartition, sourceScope⟩ exact ⟨equation, modalFormula, selectedFigure, numericReturn, fixed, neglectedPotential, energyCheck, linearCheck, firstReturn, noEquipartition, sourceScope.2.1, sourceScope.2.2.1⟩ example : PhysicsLogic.Papers.FermiPastaUlam1955.argument.ClaimIndependentOfStep .numericFirstModeReturn .supplyEquipartitionBridge := PhysicsLogic.Papers.FermiPastaUlam1955.numeric_first_mode_return_is_independent_of_equipartition_interpretation example : PhysicsLogic.Papers.FermiPastaUlam1955.Fidelity.countermodelMatrix .approximateModalSumNotConservedTotalEnergy := by obtain ⟨accounting, control, notExact⟩ := PhysicsLogic.Papers.FermiPastaUlam1955.NonlinearChainFramework.Countermodels.omitted_nonlinear_potential_separates_modal_sum_from_exact_total_energy exact ⟨⟨accounting, control⟩, notExact⟩ example {T : PhysicsLogic.Papers.ZabuskyKruskal1965.ZKTypes} (framework : PhysicsLogic.Papers.ZabuskyKruskal1965.KdVFramework T) (source : framework.zabuskyKruskalValidatedNearReturnDerivationObtains .periodicKdVValidationSolitonFocusingAndWaveformReconstruction) (scope : framework.ScopeAssessment) : framework.EquationOneKdVEvolutionFormulaAt ∧ framework.EquationFivePeriodicCosineInitialConditionAt ∧ framework.EquationTwoThroughFourIsolatedPulseFormulaBundleAt ∧ framework.momentumExactlyConserved framework.selectedRun ∧ framework.energyConservedToFiveSignificantFigures framework.selectedRun ∧ (¬framework.closeInteractionMechanismResolved framework.selectedRun) ∧ framework.profileNear (framework.stateAt framework.selectedRun framework.initialTime) (framework.stateAt framework.selectedRun framework.recurrenceTime) ∧ framework.secondProfileLessNearThanFirst framework.selectedRun ∧ (¬framework.exactFullStateRecurrenceEstablished framework.selectedRun) ∧ ¬framework.asymptoticThermalizationExcluded framework.selectedRun := by have composition := framework |>.periodic_kdv_soliton_focusing_and_imperfect_waveform_realize_source_sensitive_near_return source scope have joinedSource := framework.toValidatedNumericalNearReturnContext (framework.stateAt framework.selectedRun framework.recurrenceTime) |>.source_sensitive_numerical_near_return_exposes_handoff_derivation composition rcases joinedSource with ⟨_source, equation, cosine, pulseFormulas, _domain, _period, _dispersion, _scheme, _stable, momentum, approximateEnergy, _reliable, _steepening, _dispersionPreventsShock, _train, _widths, _velocities, _identified, _overlap, unresolved, _focus, _phase, _explains, _evolution, nearProfile, sourceScope⟩ exact ⟨equation, cosine, pulseFormulas, momentum, approximateEnergy, unresolved, nearProfile, sourceScope.2.2.2.1, sourceScope.2.2.1, sourceScope.2.2.2.2.1⟩ open PhysicsLogic.Framework.ValidatedNumericalNearReturnContext.Countermodels example : physicalFPU1955FixedEndpointReadout.observable ≠ physicalZK1965PeriodicKdVReadout.observable ∧ physicalFPU1955FixedEndpointReadout.firstModeReturnEnergy = 99 ∧ physicalZK1965PeriodicKdVReadout.dispersionParameter = (11 : Real) / 500 ∧ physicalZK1965PeriodicKdVReadout.numericalEnergyDrift = (1 : Real) / 100000 ∧ physicalZK1965PeriodicKdVReadout.firstProfileResidual < physicalZK1965PeriodicKdVReadout.secondProfileResidual ∧ ¬(selectedSemanticsContext |>.SourceSensitivePredictionSupportedNumericalNearReturnAt (selectedSemanticsContext |>.validatedNumericalPredictionModelAt selectedSemanticsModel) selectedSemanticsModel () (fun _ : Unit => True) () (fun _ : Unit => True) () (fun _ : Unit => False)) := by have witness := completed_validated_numerical_near_return_and_child_sources_without_handoff_derivation_have_no_source_sensitive_composition rcases witness with ⟨_source, _system, observable, _validation, _count, _cycles, _forceRatio, _initialEnergy, returnEnergy, _modalTolerance, _controlError, _fullStateResidual, dispersion, _period, _pulseCount, _momentumDrift, energyDrift, _returnTime, worsening, _collision, _thermalization, _nearReturn, _predictionSource, _returnSource, noComposition⟩ exact ⟨observable, returnEnergy, dispersion, energyDrift, worsening, noComposition⟩ end OriginalValidatedNumericalNearReturnSourceCompositionReuseAudit namespace OriginalAnomalySelectedVacuumSourceCompositionReuseAudit example {Volume Regime Energy : Type} (system : PhysicsLogic.Papers.WittenIndex1982.IndexSystem Volume Regime Energy) (colors : Nat) (scope : system.PureN1SUTheory colors) (anomaly : system.ChiralAnomalyCalculation colors) (condensate : system.GluinoCondensateBreakingHypothesis colors) (counting : system.DiscreteChiralBreakingVacuumCountRule) : system.gluinoBilinearCondensateIsNonzero colors ∧ system.unbrokenDiscreteChiralGroupOrder colors = 2 ∧ system.anomalyFreeDiscreteChiralGroupOrder colors = 2 * colors ∧ system.infiniteVolumeChiralVacuumCount colors = colors ∧ (system.ChiralLargeVolumeIdentificationContract → (∀ volume, system.HasAtLeastZeroEnergyStates (system.periodic volume) colors) → system.finiteVolumeGroundStatesApproachChiralVacua colors) := by have composition := system |>.conditional_gluino_condensate_and_identity_parent_realize_source_sensitive_anomaly_selected_vacua colors scope anomaly condensate counting have handoff := (system.toAnomalySelectedVacuumContext colors) |>.source_sensitive_anomaly_selected_vacua_exposes_handoff_derivation composition rcases handoff with ⟨_anomaly, _vacuum, _composite, gluino, residual, parent, count, identification⟩ exact ⟨gluino, residual, parent, count, identification⟩ example {T : PhysicsLogic.Papers.SeibergWitten1994.SWTypes} [CommRing T.Value] [IsDomain T.Value] (framework : PhysicsLogic.Papers.SeibergWitten1994.Framework T) (patch : PhysicsLogic.Papers.SeibergWitten1994.MonopolePatch T.Value) (vacuum : PhysicsLogic.Papers.SeibergWitten1994.MonopoleVacuum T.Value) (anomaly : framework.N2ChiralAnomalyCalculation) (restriction : framework.N1MassDeformationSelectsZ4) (spontaneous : framework.SpontaneousChiralZ4ToZ2) (counting : framework.ChiralBreakingVacuumCountRule) (minimal : framework.MinimalSingularityHypothesis) (singularities : framework.MasslessMonopoleDyonHypothesis) (matter : patch.HasMonopoleMatterContent) (superpotential : patch.HasExactSuperpotential) (nontrivial : vacuum.HasNontrivialDeformationData patch) (vacuumCalculation : PhysicsLogic.Papers.SeibergWitten1994.Framework.MonopoleVacuumEquationCalculation patch vacuum) (dyonCondenses : framework.DyonCondensation) (twoVacua : framework.TwoSingularitiesGiveTwoVacua patch vacuum) : framework.monopolePoint ≠ framework.dyonPoint ∧ patch.HasExactSuperpotential ∧ vacuum.SatisfiesVacuumEquations patch ∧ vacuum.LiesAtMonopolePoint ∧ vacuum.RealizesChargeCondensate patch PhysicsLogic.Papers.SeibergWitten1994.ElectromagneticCharge.monopole ∧ (¬PhysicsLogic.Papers.SeibergWitten1994.ElectromagneticCharge.MutuallyLocal PhysicsLogic.Papers.SeibergWitten1994.ElectromagneticCharge.monopole PhysicsLogic.Papers.SeibergWitten1994.ElectromagneticCharge.dyon) ∧ framework.DyonCondensation ∧ framework.HasTwoIsolatedVacua ∧ framework.ChiralBreakingMatchesTwoVacua := by have composition := framework |>.mass_deformed_monopole_and_dyon_singularities_realize_source_sensitive_anomaly_selected_vacua patch vacuum anomaly restriction spontaneous counting minimal singularities matter superpotential nontrivial vacuumCalculation dyonCondenses twoVacua have handoff := framework.toAnomalySelectedVacuumContext |>.source_sensitive_anomaly_selected_vacua_exposes_handoff_derivation composition rcases handoff with ⟨_anomaly, _vacuum, _composite, _restriction, distinct, _singularities, _matter, exactPotential, _nontrivial, equations, atMonopole, monopoleCondenses, _monopoleMagnetic, _monopoleElectric, _dyonMagnetic, _dyonElectric, nonlocalCharges, dyon, isolated, agreement⟩ exact ⟨distinct, exactPotential, equations, atMonopole, monopoleCondenses, nonlocalCharges, dyon, isolated, agreement⟩ open PhysicsLogic.Framework.AnomalySelectedVacuumContext.Countermodels example : physicalWitten1982ConditionalSUNReadout.finiteVolumeIndexEstablished = true ∧ physicalWitten1982ConditionalSUNReadout.gluinoCondensateEstablished = false ∧ physicalWitten1982ConditionalSUNReadout.finiteVolumeVacuumIdentificationEstablished = false ∧ physicalSeibergWitten1994MonopoleDyonReadout.anomalySurvivingOrder = 8 ∧ physicalSeibergWitten1994MonopoleDyonReadout.vacuumParentOrder = 4 ∧ physicalSeibergWitten1994MonopoleDyonReadout.residualVacuumOrder = 2 ∧ physicalSeibergWitten1994MonopoleDyonReadout.dyonElectricCharge = -1 ∧ physicalSeibergWitten1994MonopoleDyonReadout.monopoleSingularity ≠ physicalSeibergWitten1994MonopoleDyonReadout.dyonSingularity ∧ ¬(validCompositeVacuumRoute |>.SourceSensitiveAnomalySupportedVacuumSelectionAt (validCompositeVacuumRoute.anomalyModelAt () () 4) (validCompositeVacuumRoute.vacuumModelAt () () 4 2 2) validCompositeCalculation () (fun _ : Unit => True) () (fun _ : Unit => True) () (fun _ : Unit => True) () (fun _ : Unit => False)) := by have witness := completed_anomaly_selected_vacua_and_all_sources_without_handoff_derivation_have_no_source_sensitive_composition rcases witness with ⟨_sources, _anomalyTheories, _distinctEndpoints, _wittenAnomaly, _wittenParent, _wittenCount, finiteIndex, unprovedCondensate, unprovedIdentification, microscopic, deformedParent, residual, _count, _monopoleMagnetic, dyonElectric, distinctSingularities, _independentMonodromy, _inhabitedRoute, _anomalySource, _vacuumSource, _compositeSource, noComposition⟩ exact ⟨finiteIndex, unprovedCondensate, unprovedIdentification, microscopic, deformedParent, residual, dyonElectric, distinctSingularities, noComposition⟩ end OriginalAnomalySelectedVacuumSourceCompositionReuseAudit namespace OriginalSourceSpecificFrameworkCountermodelReuseAudit section CurrentAnomaly open PhysicsLogic.Framework.QuantumCurrentAnomalyContext.Countermodels example : PhysicalAdlerAxialTriangleAnomalyAt physicalAdler1969AxialTriangleReadout ∧ PhysicalWittenAdjointCurrentAnomalyAt physicalWitten1982AdjointCurrentReadout ∧ PhysicalSeibergWittenAdjointCurrentAnomalyAt physicalSeibergWitten1994AdjointCurrentReadout := ⟨quantum_current_anomalies_distinguish_adler_triangles_witten_gluinos_and_seiberg_witten_r_currents.1, quantum_current_anomalies_distinguish_adler_triangles_witten_gluinos_and_seiberg_witten_r_currents.2.1, quantum_current_anomalies_distinguish_adler_triangles_witten_gluinos_and_seiberg_witten_r_currents.2.2.1⟩ end CurrentAnomaly section AnomalyRestriction open PhysicsLogic.Framework.AnomalyRestrictedSymmetryContext.Countermodels example : PhysicalWittenAnomalyRestrictionAt physicalWitten1982AnomalyRestrictionReadout ∧ PhysicalSeibergWittenAnomalyRestrictionAt physicalSeibergWitten1994AnomalyRestrictionReadout := ⟨anomaly_restrictions_distinguish_witten_chiral_subgroups_and_seiberg_witten_r_symmetry.1, anomaly_restrictions_distinguish_witten_chiral_subgroups_and_seiberg_witten_r_symmetry.2.1⟩ end AnomalyRestriction section DiscreteVacua open PhysicsLogic.Framework.DiscreteSymmetryVacuumContext.Countermodels example : PhysicalWittenConditionalDiscreteVacuaAt physicalWitten1982ConditionalDiscreteVacua ∧ PhysicalSeibergWittenChargedDiscreteVacuaAt physicalSeibergWitten1994ChargedDiscreteVacua := ⟨discrete_vacua_distinguish_conditional_gluino_cosets_and_mass_deformed_monopole_vacua.1, discrete_vacua_distinguish_conditional_gluino_cosets_and_mass_deformed_monopole_vacua.2.1⟩ end DiscreteVacua section SupersymmetricGroundStates open PhysicsLogic.Framework.SupersymmetricVacuumEnergyContext.Countermodels example : PhysicalWittenInstantonLiftedVacuumAt physicalWitten1981InstantonLiftedVacuumReadout ∧ PhysicalWittenFiniteVolumeZeroEnergyVacuumAt physicalWitten1982FiniteVolumeVacuumReadout := ⟨supersymmetric_vacua_distinguish_perturbative_zero_modes_exact_normalizability_and_protected_indices.1, supersymmetric_vacua_distinguish_perturbative_zero_modes_exact_normalizability_and_protected_indices.2.1⟩ end SupersymmetricGroundStates section ProtectedIndices open PhysicsLogic.Framework.ProtectedIndexTransportContext.Countermodels example : PhysicalWittenFiniteVolumeIndexTransportAt physicalWitten1982FiniteVolumeIndexTransport ∧ PhysicalStromingerVafaEllipticGenusTransportAt physicalStromingerVafa1996EllipticGenusTransport := ⟨protected_indices_distinguish_finite_volume_supertraces_elliptic_genera_and_large_charge_scope.1, protected_indices_distinguish_finite_volume_supertraces_elliptic_genera_and_large_charge_scope.2.1⟩ end ProtectedIndices section SelectedVacua open PhysicsLogic.Framework.AnomalySelectedVacuumContext.Countermodels example : ¬PhysicalWittenConditionalChiralVacuumRouteAt physicalWittenCondensatePresentedAsEstablished ∧ ¬PhysicalSeibergWittenMonopoleDyonVacuumRouteAt physicalSeibergWittenWrongDyonElectricCharge := ⟨physical_witten_selected_vacua_reject_index_and_epistemic_overclaims.2.1, physical_seiberg_witten_selected_vacua_reject_wrong_anomalies_charges_and_monodromy.2.2.1⟩ end SelectedVacua section BulkScalar open PhysicsLogic.Framework.BulkBoundaryScalarContext.Countermodels example : PhysicalGKPMasslessBulkBoundaryScalarAt physicalGKPMasslessBulkBoundaryScalar ∧ PhysicalWittenMassiveBulkBoundaryScalarAt physicalWittenMassiveBulkBoundaryScalar := ⟨bulk_boundary_scalars_distinguish_gkp_massless_modes_witten_massive_roots_and_bf_scope.1, bulk_boundary_scalars_distinguish_gkp_massless_modes_witten_massive_roots_and_bf_scope.2.1⟩ end BulkScalar section ScalarCorrelator open PhysicsLogic.Framework.ScalarCorrelatorCompositionContext.Countermodels example : PhysicalGKPMasslessScalarCorrelatorCompositionAt physicalGKPMasslessScalarCorrelatorComposition ∧ PhysicalWittenMassiveScalarCorrelatorCompositionAt physicalWittenMassiveScalarCorrelatorComposition := ⟨scalar_correlators_distinguish_massless_flux_logs_massive_branches_and_source_handoffs.1, scalar_correlators_distinguish_massless_flux_logs_massive_branches_and_source_handoffs.2.1⟩ end ScalarCorrelator section BraneDecoupling open PhysicsLogic.Framework.BraneWorldvolumeDecouplingContext.Countermodels example : PhysicalMaldacenaD3DecouplingAt physicalMaldacena1997D3Decoupling ∧ PhysicalItzhakiD2DecouplingAt physicalItzhakiEtAl1998D2Decoupling ∧ PhysicalItzhakiD6NonDecouplingAt physicalItzhakiEtAl1998D6NonDecoupling := ⟨brane_limits_distinguish_d3_fixed_coupling_d2_running_and_d6_bulk_escape.1, brane_limits_distinguish_d3_fixed_coupling_d2_running_and_d6_bulk_escape.2.1, brane_limits_distinguish_d3_fixed_coupling_d2_running_and_d6_bulk_escape.2.2.1⟩ end BraneDecoupling section RRFlux open PhysicsLogic.Framework.QuantizedBraneRRFluxChargeContext.Countermodels example : PhysicalPolchinskiD1RRFluxChargeAt physicalPolchinski1995TypeIIBD1Flux ∧ PhysicalMaldacenaD3RRFluxChargeAt physicalMaldacena1997TypeIIBD3FiveFormFlux := ⟨rr_flux_charges_distinguish_d1_linking_forms_d3_self_duality_and_integer_stacks.1, rr_flux_charges_distinguish_d1_linking_forms_d3_self_duality_and_integer_stacks.2.1⟩ end RRFlux section MagneticMassFlow open PhysicsLogic.Framework.MagneticMassDeformationContext.Countermodels example : PhysicalSeibergFundamentalMagneticMassFlowAt physicalSeiberg1994FundamentalMagneticMassFlow ∧ PhysicalKutasovAdjointMagneticMassFlowAt physicalKutasovSchwimmer1995AdjointMagneticMassFlow := ⟨magnetic_mass_flows_distinguish_fundamental_rank_one_and_adjoint_rank_k_higgsing.1, magnetic_mass_flows_distinguish_fundamental_rank_one_and_adjoint_rank_k_higgsing.2.1⟩ end MagneticMassFlow section LargeRankDiagrams open PhysicsLogic.Framework.PlanarDiagramExpansionContext.Countermodels example : PhysicalTHooftPurePlanarDiagramAt physicalTHooft1974PurePlanarDiagram ∧ PhysicalTHooftQuarkBoundaryDiagramAt physicalTHooft1974OneQuarkBoundaryDiagram ∧ PhysicalWittenLargeRankBaryonAt physicalWitten1979LargeRankBaryon := ⟨planar_diagrams_distinguish_euler_handles_quark_boundaries_and_large_rank_baryons.1, planar_diagrams_distinguish_euler_handles_quark_boundaries_and_large_rank_baryons.2.1, planar_diagrams_distinguish_euler_handles_quark_boundaries_and_large_rank_baryons.2.2.1⟩ end LargeRankDiagrams section KaluzaKleinTowers open PhysicsLogic.Framework.KaluzaKleinTowerContext.Countermodels example : PhysicalTownsendMagneticSixbraneTowerAt physicalTownsend1995MagneticSixbraneTower ∧ PhysicalWittenStrongCouplingRRTowerAt physicalWitten1995StrongCouplingRRTower := ⟨kk_towers_distinguish_regular_sixbrane_fibers_inverse_coupling_masses_and_scope.1, kk_towers_distinguish_regular_sixbrane_fibers_inverse_coupling_masses_and_scope.2.1⟩ end KaluzaKleinTowers section ForwardAmplitude open PhysicsLogic.Framework.ForwardAmplitudePositivityContext.Countermodels example : PhysicalAdamsRegulatedForwardPositivityAt physicalAdamsEtAl2006RegulatedForwardAmplitude ∧ PhysicalAdamsWrongSignSuperluminalityAt physicalAdamsEtAl2006WrongSignSuperluminalBackground ∧ PhysicalKomargodskiSchwimmerDilatonPositivityAt physicalKomargodskiSchwimmer2011DilatonAmplitude := ⟨forward_amplitudes_distinguish_regulated_crossing_wrong_sign_backgrounds_and_dilaton_anomalies.1, forward_amplitudes_distinguish_regulated_crossing_wrong_sign_backgrounds_and_dilaton_anomalies.2.1, forward_amplitudes_distinguish_regulated_crossing_wrong_sign_backgrounds_and_dilaton_anomalies.2.2.1⟩ end ForwardAmplitude section BlackHoleMicrostates open PhysicsLogic.Framework.BlackHoleMicrostateEntropyContext.Countermodels example : PhysicalStromingerVafaWeightedMicrostateEntropyAt physicalStromingerVafa1996WeightedMicrostateEntropy ∧ PhysicalCallanMaldacenaExtremalMicrostateEntropyAt physicalCallanMaldacena1996ExtremalMicrostateEntropy ∧ PhysicalHorowitzStromingerChiralBlackStringEntropyAt physicalHorowitzStrominger1996ChiralBlackStringEntropy := ⟨microstate_entropies_distinguish_weighted_indices_d1d5p_counts_and_chiral_black_strings.1, microstate_entropies_distinguish_weighted_indices_d1d5p_counts_and_chiral_black_strings.2.1, microstate_entropies_distinguish_weighted_indices_d1d5p_counts_and_chiral_black_strings.2.2.1⟩ end BlackHoleMicrostates section HolographicRegions open PhysicsLogic.Framework.HolographicRegionEntropyContext.Countermodels example : PhysicalRyuTakayanagiAdS3IntervalAt physicalRyuTakayanagi2006AdS3Interval ∧ PhysicalLewkowyczMaldacenaReplicaSurfaceAt physicalLewkowyczMaldacena2013ReplicaSurface := ⟨holographic_entropies_distinguish_ads3_interval_logs_replica_assumptions_and_einstein_surfaces.1, holographic_entropies_distinguish_ads3_interval_logs_replica_assumptions_and_einstein_surfaces.2.1⟩ end HolographicRegions section ParentAction open PhysicsLogic.Framework.ParentActionDualReductionCountermodels example : PhysicalRocekVerlindeAbelianParentAt physicalRocekVerlinde1991AbelianParent ∧ PhysicalDeLaOssaQuevedoNonabelianParentAt physicalDeLaOssaQuevedo1992NonabelianParent := ⟨parent_actions_distinguish_abelian_radius_inversion_nonabelian_curvature_and_global_scope.1, parent_actions_distinguish_abelian_radius_inversion_nonabelian_curvature_and_global_scope.2.1⟩ end ParentAction section LinearResponseNoise open PhysicsLogic.Framework.LinearResponseNoiseDissipationHandoff.Countermodels open PhysicsLogic.Framework.EquilibriumNoiseDissipationContext.Countermodels example : (physicalResponseNoiseHandoff |>.NonvacuousResponseGroundedNoiseDissipationAt) /\ PhysicalKuboSymmetrizedConductivityNoiseAt physicalKubo1957SymmetrizedConductivityReadout /\ ¬(wrongDissipationCoefficientHandoff |>.SelectedResponseNoiseDissipationCompatibilityAt) /\ ¬(wrongResponseFrequencyHandoff |>.SelectedResponseNoiseDissipationCompatibilityAt) /\ ¬(wrongNoiseConventionHandoff |>.SelectedResponseNoiseDissipationCompatibilityAt) := by obtain ⟨composition, symmetrized, _system, coefficient, frequency, convention, _missingDerivation⟩ := response_noise_handoffs_distinguish_systems_coefficients_frequency_conventions_and_source_derivations exact ⟨composition, symmetrized, coefficient, frequency, convention⟩ end LinearResponseNoise section CosmologicalCollapseAbundance open PhysicsLogic.Framework.CollapseCriterionAbundanceHandoff.Countermodels open PhysicsLogic.Framework.CosmologicalPerturbationCollapseContext.Countermodels open PhysicsLogic.Framework.ThresholdedCollapseAbundanceContext.Countermodels example : (physicalPressSchechterCollapseAbundanceHandoff |>.NonvacuousCollapseGroundedThresholdAbundanceAt) /\ PhysicalPressSchechterMaximumVarianceCountingAt physicalPressSchechter1974CondensationReadout /\ PhysicalPressSchechterGaussianAccretionAbundanceAt physicalPressSchechter1974AbundanceReadout /\ ¬(pressSchechterMissingDistributionHandoff |>.SelectedCollapseCriterionAbundanceCompatibilityAt) /\ ¬(pressSchechterErasedMassCorrectionHandoff |>.SelectedCollapseCriterionAbundanceCompatibilityAt) := by obtain ⟨composed, spherical, abundance, _subcritical, _wrongField, noDistribution, noCorrection, _noSource⟩ := collapse_abundance_handoffs_distinguish_spherical_thresholds_gaussian_populations_and_independent_accretion exact ⟨composed, spherical, abundance, noDistribution, noCorrection⟩ end CosmologicalCollapseAbundance section MagneticHallComposition open PhysicsLogic.Framework.GappedBandTopologicalResponseHandoff.Countermodels open PhysicsLogic.Framework.GappedBandHallContext.Countermodels open PhysicsLogic.Framework.TopologicalTransverseResponseContext.Countermodels example : (physicalTKNNGappedBandTopologicalHandoff |>.NonvacuousBandGroundedTopologicalHallResponseAt) /\ PhysicalTKNNMagneticBlochBoundaryWindingAt physicalTKNN1982MagneticBlochReadout /\ PhysicalTKNNBerryCurvatureChernNormalizationAt physicalTKNNBerryChernReadout /\ PhysicalHaldaneZeroFluxHoneycombMassFormulaAt physicalHaldane1988ZeroFluxReadout /\ PhysicalNTWBoundaryTwistTorusAndIntegerResponseAt physicalNTWNondegenerateTwistReadout /\ ¬(twistBoundarySubstitutedForMagneticMomentumHandoff |>.SelectedGappedBandTopologicalResponseCompatibilityAt) /\ ¬(wrongMagneticChernIntegerHandoff |>.SelectedGappedBandTopologicalResponseCompatibilityAt) /\ ¬(wrongMagneticHallConductanceUnitHandoff |>.SelectedGappedBandTopologicalResponseCompatibilityAt) /\ ¬(missingMagneticBoundaryPhaseWindingHandoff |>.SelectedGappedBandTopologicalResponseCompatibilityAt) /\ ¬(missingMagneticBulkCurvatureIdentityHandoff |>.SelectedGappedBandTopologicalResponseCompatibilityAt) /\ ¬(interactingTwistScopeSubstitutedForMagneticBandHandoff |>.SelectedGappedBandTopologicalResponseCompatibilityAt) := by obtain ⟨composed, tknn, berry, haldane, ntw, noTwist, noInteger, noUnit, noWinding, noCurvature, noScope, _noSource⟩ := hall_handoffs_distinguish_magnetic_bands_twist_tori_chern_integers_units_and_boundary_winding exact ⟨composed, tknn, berry, haldane, ntw, noTwist, noInteger, noUnit, noWinding, noCurvature, noScope⟩ end MagneticHallComposition section CollisionlessKineticPlasma open PhysicsLogic.Framework.CollisionlessKineticPoissonContext.Countermodels example : PhysicalLandauLinearInitialValueAndCollectiveDampingAt physicalLandau1946CollisionlessReadout /\ PhysicalBGKStationaryNonlinearTrappedWaveAt physicalBGK1957StationaryReadout /\ ¬PhysicalLandauLinearInitialValueAndCollectiveDampingAt landauWrongFourierPoissonSignReadout /\ ¬PhysicalBGKStationaryNonlinearTrappedWaveAt bgkWrongTwoSpeciesPoissonSignReadout /\ ¬physicalCollisionlessPlasmaEstablishment.establishedAt .actualCollisionalPlasmaWaveExistence .bernsteinGreeneKruskal1957 := by obtain ⟨landau, bgk, _landauNonvacuous, _bgkNonvacuous, wrongLandauSign, _falseDistribution, wrongBGKSign, _negativeTrapped, noActual, _noRefutation⟩ := collisionless_plasma_sources_distinguish_linear_landau_damping_stationary_bgk_trapping_poisson_signs_and_open_scope exact ⟨landau, bgk, wrongLandauSign, wrongBGKSign, noActual⟩ end CollisionlessKineticPlasma section ControlledResonanceExperimentalGravitation open PhysicsLogic.Framework.ControlledResonanceModelTestContext.Countermodels example : PhysicalPoundRebkaControlledRedshiftAgreementAt physicalPoundRebka1960ResonanceReadout /\ PhysicalHughesControlledP2NuclearBoundAt physicalHughes1960NuclearResonanceReadout /\ ¬PhysicalPoundRebkaControlledRedshiftAgreementAt poundDriftingInherentOffsetReadout /\ ¬PhysicalHughesControlledP2NuclearBoundAt hughesWithoutNuclearConversionReadout /\ ¬physicalControlledResonanceEstablishment.establishedAt .exactMassIsotropy .hughesRobinsonBeltranLopez1960 := by obtain ⟨pound, hughes, _poundNonvacuous, _hughesNonvacuous, _finiteAgreement, drifting, _unprotected, noConversion, noIsotropy, _noGravityTheory⟩ := resonance_sources_distinguish_gamma_reversal_p2_null_bounds_references_and_open_scope exact ⟨pound, hughes, drifting, noConversion, noIsotropy⟩ end ControlledResonanceExperimentalGravitation section DensityPotentialGaugeDetermination open PhysicsLogic.Framework.DensityPotentialGaugeContext.Countermodels example : PhysicalHohenbergKohnStaticConstantGaugeAt physicalHohenbergKohn1964DensityPotentialReadout /\ PhysicalRungeGrossDynamicTimeGaugeAt physicalRungeGross1984DensityPotentialReadout /\ ¬PhysicalHohenbergKohnStaticConstantGaugeAt hohenbergKohnDegenerateGroundReadout /\ ¬PhysicalRungeGrossDynamicTimeGaugeAt rungeGrossWrongEquationSixSignReadout /\ ¬physicalDensityPotentialEstablishment.establishedAt .timeDependentActionMinimumEstablished .rungeGross1984 := by obtain ⟨hohenberg, runge, _hohenbergNonvacuous, _rungeNonvacuous, degenerate, _nonconstant, _unfixed, wrongEquation, _literalEquality, noMinimum⟩ := density_potential_sources_distinguish_static_constants_time_gauges_equation_six_and_open_scope exact ⟨hohenberg, runge, degenerate, wrongEquation, noMinimum⟩ end DensityPotentialGaugeDetermination section LargeComponentSphericalCorrespondence open PhysicsLogic.Framework.LargeComponentSphericalCorrespondenceContext.Countermodels example : PhysicalStanleyNoncriticalSphericalFreeEnergyAt physicalStanley1968SphericalReadout /\ PhysicalWegnerInfiniteComponentSphericalCriticalExponentsAt physicalWegner1973SphericalReadout /\ Not (PhysicalStanleyNoncriticalSphericalFreeEnergyAt stanleyWrongLocalConstraintReadout) /\ Not (PhysicalStanleyNoncriticalSphericalFreeEnergyAt stanleyFalseCriticalExponentConvergenceReadout) /\ Not (PhysicalWegnerInfiniteComponentSphericalCriticalExponentsAt wegnerWrongSusceptibilityExponentReadout) /\ Not (physicalLargeComponentSphericalEstablishment.establishedAt .componentAndCriticalLimitsCommute .stanley1968) := by obtain ⟨stanley, wegner, _stanleyNonvacuous, _wegnerNonvacuous, wrongConstraint, falseConvergence, wrongSusceptibility, _finiteComponent, noInterchange, _noWegnerCritical⟩ := spherical_sources_distinguish_stanley_noncritical_constraints_wegner_critical_indices_and_open_limit_orders exact ⟨stanley, wegner, wrongConstraint, falseConvergence, wrongSusceptibility, noInterchange⟩ end LargeComponentSphericalCorrespondence section IncompressibleVelocityCorrelation open PhysicsLogic.Framework.IncompressibleVelocityCorrelationContext.Countermodels example : PhysicalKolmogorovRadialIncrementIncompressibilityAt physicalKolmogorov1941IncompressibleReadout /\ PhysicalFNSFourierProjectorIncompressibilityAt physicalFNS1977IncompressibleReadout /\ Not (PhysicalKolmogorovRadialIncrementIncompressibilityAt kolmogorovWrongRadialDerivativeReadout) /\ Not (PhysicalFNSFourierProjectorIncompressibilityAt fnsWrongProjectorCrossTermReadout) /\ Not (PhysicalFNSFourierProjectorIncompressibilityAt fnsHighReTurbulenceOverclaimReadout) /\ Not (physicalIncompressibleVelocityEstablishment.establishedAt .infraredControlsUltravioletTurbulence .forsterNelsonStephen1977) := by obtain ⟨kolmogorov, forster, _kolmogorovNonvacuous, _forsterNonvacuous, wrongDerivative, wrongProjector, wrongReynolds, _noCascade, noUltraviolet⟩ := incompressible_velocity_sources_distinguish_kolmogorov_radial_derivatives_fns_fourier_projectors_and_open_regimes exact ⟨kolmogorov, forster, wrongDerivative, wrongProjector, wrongReynolds, noUltraviolet⟩ end IncompressibleVelocityCorrelation section CanonicalGibbsNormalization open PhysicsLogic.Framework.CanonicalGibbsContext.Countermodels example : PhysicalJaynesDiscreteMaximumEntropyCanonicalAt physicalJaynes1957CanonicalReadout /\ PhysicalMetropolisConfigurationIntegralCanonicalAt physicalMetropolis1953CanonicalReadout /\ physicalJaynesCanonicalGibbsContext.selectedInverseTemperature < 0 /\ 1 < physicalMetropolisCanonicalGibbsContext.probabilityValueAt .metropolisConfigurationDensity .groundConfiguration /\ Not (PhysicalJaynesDiscreteMaximumEntropyCanonicalAt jaynesWrongDiscreteProbabilityReadout) /\ Not (PhysicalMetropolisConfigurationIntegralCanonicalAt metropolisWrongWeightedObservableReadout) /\ Not (physicalCanonicalGibbsEstablishment.establishedAt .targetAloneEstablishesDetailedBalance .metropolis1953) := by obtain ⟨jaynes, metropolis, _jaynesNonvacuous, _metropolisNonvacuous, negativeMultiplier, highDensity, wrongJaynesWeight, _missingMomentum, wrongMetropolisObservable, noBalance, _noFiniteRate⟩ := canonical_gibbs_sources_distinguish_jaynes_discrete_inference_metropolis_configuration_integrals_and_open_scope exact ⟨jaynes, metropolis, negativeMultiplier, highDensity, wrongJaynesWeight, wrongMetropolisObservable, noBalance⟩ end CanonicalGibbsNormalization end OriginalSourceSpecificFrameworkCountermodelReuseAudit section HawkingOriginalCosmologicalFormulaReuse open PhysicsLogic.Papers.HawkingOpenUniverses1965 open PhysicsLogic.Papers.HawkingOpenUniverses1965.OpenUniverseFramework.Countermodels example (readout : HyperbolicCoveringReadout) (deckOrbit : readout.NontrivialDeckTranslationOrbitAt) : readout.NoncompactCoverAt := readout.deck_translation_orbit_makes_the_cover_noncompact deckOrbit example (readout : HyperbolicCoveringReadout) (descent : readout.CoveringPastIncompletenessDescendsAt) : 0 < readout.quotientPastAffineLength ∧ 0 < readout.quotientPastExtensionObstruction := readout.covering_incompleteness_descends_to_compact_quotient descent example (readout : FiniteConvergenceMarginReadout) (margin : readout.SourceMarginAt) : 0 < readout.perturbedInwardConvergence ∧ 0 < readout.perturbedOutwardConvergence := readout.finite_margin_preserves_both_past_null_convergences margin example {T : HawkingTypes} (framework : OpenUniverseFramework T) (setup : framework.OpenUniverseModelSetup) : framework.SelectedRobertsonWalkerMetricFormulaAt ∧ framework.SelectedHyperbolicCoveringAlternativeAt := framework.cosmological_setup_retains_metric_and_hyperbolic_cover_alternative setup example {T : HawkingTypes} (framework : OpenUniverseFramework T) (conditions : framework.WeakerGlobalConditions) : framework.SelectedMatterFlowGlobalConditionsAt := framework.weaker_conditions_retain_original_matter_flow_bounds conditions example : PhysicsLogic.Papers.HawkingOpenUniverses1965.Fidelity.countermodelMatrix .negativeCurvatureNotNoncompactQuotient := negative_curvature_does_not_make_the_original_quotient_noncompact example : PhysicsLogic.Papers.HawkingOpenUniverses1965.Fidelity.countermodelMatrix .closedMetricFormulaNotFlatOrOpenProvedScope := closed_metric_is_defined_but_outside_hawkings_proved_curvature_scope example : PhysicsLogic.Papers.HawkingOpenUniverses1965.Fidelity.countermodelMatrix .nonstrictMarginNotStrictPerturbedTrapping := by have witness := PhysicsLogic.Papers.HawkingOpenUniverses1965.RobertsonWalkerGeometry.PerturbationGeometry.a_nonstrict_half_unit_budget_allows_a_marginal_target exact ⟨⟨witness.1, witness.2.1⟩, witness.2.2⟩ example : PhysicsLogic.Papers.HawkingOpenUniverses1965.Fidelity.countermodelMatrix .noncompactMatterSurfaceNotCauchyComplete := noncompact_matter_surface_does_not_imply_cauchy_completeness end HawkingOriginalCosmologicalFormulaReuse section WittenOriginalDynamicalBreakingFormulaReuse open PhysicsLogic.Papers.WittenDynamicalBreaking1981 open PhysicsLogic.Papers.WittenDynamicalBreaking1981.DynamicalBreakingFramework.SourceFormulaCountermodels example (readout : GoldstinoCurrentFormulaReadout) (source : readout.SourceFormulaAt) : readout.nonderivativeCouplingValue ≠ 0 := readout.positive_vacuum_energy_requires_nonderivative_goldstino_coupling source example (readout : GoldstinoCurrentFormulaReadout) (derivative : readout.DerivativeCurrentMatrixElementAt) : readout.derivativeMatrixElementAt 0 = 0 := readout.derivative_current_coupling_vanishes_at_zero_momentum derivative example (readout : SupersymmetricQuantumMechanicsFormulaReadout) (source : readout.SourceFormulaAt) (component : QuantumMechanicsSpinComponent) : ¬readout.SpinComponentNormalizableAt component := readout.two_zero_model_has_no_normalizable_zero_energy_spin_component source component example (readout : MonopoleInstantonFormulaReadout) (potential : readout.EquationThirtyTwoScalarPotentialAt) : readout.scalarPotentialValueAtMagnitude 0 = 0 ∧ readout.scalarPotentialValueAtMagnitude readout.higgsVacuumMagnitude = 0 := readout.scalar_potential_has_both_symmetric_and_higgsed_zero_energy_vacua potential example (readout : MonopoleInstantonFormulaReadout) (zeroModes : readout.TwoZeroModeSaturationAt) : readout.vacuumInsertionCapacity < readout.instantonFermionZeroModeCount ∧ readout.orderParameterInsertionCapacity = readout.instantonFermionZeroModeCount := readout.two_zero_modes_forbid_vacuum_but_saturate_bilinear zeroModes example (readout : MonopoleInstantonFormulaReadout) (source : readout.SourceZeroModeFormulaAt) : readout.undeformedSignedOrderParameter = 0 := readout.opposite_monopoles_cancel_before_discrete_deformation source.2.1 source.2.2.2 example (readout : MonopoleInstantonFormulaReadout) (source : readout.SourceDeformationFormulaAt) : readout.deformedSignedOrderParameter ≠ 0 := readout.unequal_deformed_weights_generate_nonzero_order_parameter source.1.2.1 source.2 example : PhysicsLogic.Papers.WittenDynamicalBreaking1981.Fidelity.countermodelMatrix .twoClassicalZerosNotNormalizableExactState := two_classical_zeros_do_not_supply_a_normalizable_exact_state example : PhysicsLogic.Papers.WittenDynamicalBreaking1981.Fidelity.countermodelMatrix .superchargeScalarCoefficientsNotPauliAxisAssignment := scalar_supercharge_coefficients_do_not_determine_pauli_axis_assignment example : PhysicsLogic.Papers.WittenDynamicalBreaking1981.Fidelity.countermodelMatrix .symmetricSuTwoVacuumNotHiggsedMonopoleBranch := symmetric_su_two_vacuum_is_not_the_analyzed_higgsed_monopole_branch example : PhysicsLogic.Papers.WittenDynamicalBreaking1981.Fidelity.countermodelMatrix .oppositeMonopoleFluxesNotBianchiIdentity := opposite_nonzero_monopole_fluxes_do_not_supply_bianchi_identity example : PhysicsLogic.Papers.WittenDynamicalBreaking1981.Fidelity.countermodelMatrix .twoTotalZeroModesNotMixedSpeciesSaturation := two_total_zero_modes_do_not_supply_mixed_fermion_bilinear_saturation example : PhysicsLogic.Papers.WittenDynamicalBreaking1981.Fidelity.countermodelMatrix .totalDerivativeThetaNotGenuineDiscreteBreaking := total_derivative_theta_term_does_not_supply_genuine_discrete_breaking example : PhysicsLogic.Papers.WittenDynamicalBreaking1981.Fidelity.countermodelMatrix .threeDimensionalDeformationNotFourDimensionalBreaking := three_dimensional_deformed_weights_do_not_establish_four_dimensional_breaking end WittenOriginalDynamicalBreakingFormulaReuse section SeibergWittenOriginalMonopoleFormulaReuse open PhysicsLogic.Papers.SeibergWitten1994 open PhysicsLogic.Papers.SeibergWitten1994.Framework.Countermodels open PhysicsLogic.Papers.SeibergWitten1994.MonopoleVacuum.Countermodels example (readout : MonopolePointPeriodReadout) (normalization : readout.HasPrintedSourceNormalization) : readout.electricPeriodValue ≠ 0 := readout.printed_electric_period_is_nonzero normalization example {T : SWTypes} [AddCommGroup T.Period] [Norm T.Period] (framework : Framework T) (calculation : framework.EllipticCurvePeriodCalculation) : framework.monopolePointPeriodReadout.modulusCoordinateValue = 1 ∧ framework.monopolePointPeriodReadout.magneticPeriodValue = 0 ∧ framework.monopolePointPeriodReadout.electricPeriodValue = Complex.ofReal (4 / Real.pi) := framework.period_calculation_retains_printed_monopole_point_values calculation example {T : SWTypes} [AddCommGroup T.Period] [Norm T.Period] (framework : Framework T) (solution : framework.ExactLowEnergySolution) : framework.monopolePointPeriodReadout.electricPeriodValue = Complex.ofReal (4 / Real.pi) := (framework.exact_solution_retains_printed_monopole_point_values solution).2.2 example : PhysicsLogic.Papers.SeibergWitten1994.Fidelity.countermodelMatrix .fTermStationarityNotDFlatVacuum := f_term_stationarity_does_not_supply_d_flatness example : PhysicsLogic.Papers.SeibergWitten1994.Fidelity.countermodelMatrix .unitWindingNotQuantizedMagneticFlux := by rcases unit_higgs_winding_does_not_supply_quantized_magnetic_flux with ⟨charge, winding, noFlux⟩ exact ⟨⟨charge, winding⟩, noFlux⟩ example : PhysicsLogic.Papers.SeibergWitten1994.Fidelity.countermodelMatrix .magneticHiggsMassNotElectricConfinement := by rcases magnetic_higgs_mass_does_not_supply_electric_confinement with ⟨condensate, higgs, noConfinement⟩ exact ⟨⟨condensate, higgs⟩, noConfinement⟩ end SeibergWittenOriginalMonopoleFormulaReuse section FixsenOriginalInstrumentAndMeasurementReuse open PhysicsLogic.Papers.FixsenEtAl1996 open PhysicsLogic.Papers.FixsenEtAl1996.FirasTemperatureMethodReadout open PhysicsLogic.Papers.FixsenEtAl1996.FirasDipoleMeasurementReadout open PhysicsLogic.Papers.FixsenEtAl1996.FirasDistortionEpochReadout example {T : FixsenTypes} (framework : FirasFramework T) (setup : framework.FirasObservationSetup) : framework.instrumentCalibrationReadout.HasSourceAbsoluteCalibrationScope := framework.observation_setup_retains_printed_absolute_calibration_bounds setup example {T : FixsenTypes} (framework : FirasFramework T) (calculation : framework.TemperatureScaleCalculation) : framework.temperatureMethodReadout.HasSourceTemperatureAssessment := framework.temperature_calculation_retains_three_printed_correlated_methods calculation example {T : FixsenTypes} (framework : FirasFramework T) (calculation : framework.DipoleAnalysisCalculation) : framework.dipoleMeasurementReadout.HasSourceDipoleMeasurement ∧ framework.dipoleMeasurementReadout.HasImportedDopplerSpeedInference := framework.dipole_analysis_retains_printed_beam_correction_and_speed_bridge calculation example {T : FixsenTypes} (framework : FirasFramework T) (limits : framework.EstablishedDistortionLimits) : framework.distortionEpochReadout.HasSourceDistortionMeasurement := framework.distortion_result_retains_separate_source_redshift_epochs limits example : printedSourceTemperatureAssessment.thermometryKelvin = (2730 / 1000 : Real) ∧ printedSourceTemperatureAssessment.colorKelvin = (27255 / 10000 : Real) ∧ printedSourceTemperatureAssessment.dipoleKelvin = (2717 / 1000 : Real) := by norm_num [printedSourceTemperatureAssessment] example : PhysicsLogic.Papers.FixsenEtAl1996.Fidelity.countermodelMatrix .confidenceLabelNotIndependentStatisticalCoverage := by rcases source_confidence_label_does_not_license_independent_statistics with ⟨numbers, sharedScale, notStatistical⟩ exact ⟨⟨numbers, sharedScale⟩, notStatistical⟩ example : PhysicsLogic.Papers.FixsenEtAl1996.Fidelity.countermodelMatrix .measuredDipoleNotDopplerVelocity := measured_dipole_does_not_supply_doppler_speed_without_conversion example : PhysicsLogic.Papers.FixsenEtAl1996.Fidelity.countermodelMatrix .sourceEpochsNotImportedEnergyConversions := source_epoch_assignment_does_not_supply_imported_energy_conversion end FixsenOriginalInstrumentAndMeasurementReuse section BohmOriginalGuidanceAndApparatusReuse open PhysicsLogic.Papers.Bohm1952 open PhysicsLogic.Papers.Bohm1952.BohmStationaryStateReadout open PhysicsLogic.Papers.Bohm1952.BohmApparatusPacketReadout open PhysicsLogic.Papers.Bohm1952.BohmContextualMeasurementReadout open PhysicsLogic.Papers.Bohm1952.BohmEPRCorrelationReadout example {T : BohmTypes} (framework : HiddenVariableFramework T) (postulates : framework.ConfigurationGuidancePostulates) : framework.guidanceKinematicsReadout.HasSourceGuidanceKinematics := framework.guidance_postulates_retain_printed_phase_gradient_and_mass postulates example {T : BohmTypes} (framework : HiddenVariableFramework T) (setup : framework.StationaryStateSetup) (calculation : framework.StationaryStateCalculation) : framework.stationaryStateReadout.particleVelocity = 0 ∧ framework.stationaryStateReadout.particleKineticEnergy = 0 := framework.stationary_example_retains_zero_velocity_and_kinetic_energy setup calculation example {T : BohmTypes} (framework : HiddenVariableFramework T) (setup : framework.StationaryStateSetup) (calculation : framework.StationaryStateCalculation) (positiveEnergy : 0 < framework.stationaryStateReadout.totalEnergy) : framework.stationaryStateReadout.particleVelocity = 0 ∧ framework.stationaryStateReadout.particleKineticEnergy = 0 ∧ 0 < framework.stationaryStateReadout.totalEnergy := framework.stationary_example_retains_zero_kinetic_positive_total_energy setup calculation positiveEnergy example {T : BohmTypes} (framework : HiddenVariableFramework T) (setup : framework.MeasurementPacketSetup) (calculation : framework.MeasurementPacketCalculation) : framework.apparatusPacketReadout.actualOccupiedOutcomeCount = 1 ∧ framework.apparatusPacketReadout.leftBornWeight = framework.apparatusPacketReadout.leftWaveCoefficientMagnitude ^ 2 ∧ framework.apparatusPacketReadout.rightBornWeight = framework.apparatusPacketReadout.rightWaveCoefficientMagnitude ^ 2 := framework.apparatus_measurement_retains_one_actual_outcome_and_born_weights setup calculation example {T : BohmTypes} (framework : HiddenVariableFramework T) (run : framework.StandardMomentumMeasurementRun) (calculation : framework.ContextualObservableCalculation) : framework.contextualMeasurementReadout.positionContextDistributionWeight ≠ framework.contextualMeasurementReadout.momentumContextDistributionWeight := (framework.contextual_measurement_retains_distinct_apparatus_distributions run calculation).2 example {T : BohmTypes} (framework : HiddenVariableFramework T) (setup : framework.EPRSetup) (calculation : framework.EPRCalculation) : framework.eprCorrelationReadout.remoteMomentumBeforeChoice ≠ framework.eprCorrelationReadout.remoteMomentumAfterChoice ∧ framework.eprCorrelationReadout.remoteOperationalMarginalAfter = framework.eprCorrelationReadout.remoteOperationalMarginalBefore := framework.epr_analysis_retains_equal_marginal_and_remote_motion_change setup calculation example : PhysicsLogic.Papers.Bohm1952.Fidelity.countermodelMatrix .positiveStationaryEnergyNotParticleMotion := by have witness := StationaryGaussianOscillator.actual_positive_energy_does_not_require_particle_motion exact ⟨⟨StationaryGaussianOscillator.the_same_stationary_wave_and_flow_realize_the_dynamics, witness.1, witness.2.1⟩, witness.2.2⟩ example : PhysicsLogic.Papers.Bohm1952.Fidelity.countermodelMatrix .separatedPacketsNotActualOccupiedBranch := by rcases separated_packets_do_not_select_an_actual_occupied_branch with ⟨separated, bornWeights, unoccupied⟩ exact ⟨⟨separated, bornWeights⟩, unoccupied⟩ example : PhysicsLogic.Papers.Bohm1952.Fidelity.countermodelMatrix .apparatusContextNotOneHiddenVariableDistribution := contextual_apparatuses_do_not_share_one_hidden_variable_distribution example : PhysicsLogic.Papers.Bohm1952.Fidelity.countermodelMatrix .unchangedMarginalNotLocalParticleMotion := unchanged_operational_marginal_does_not_imply_local_particle_motion end BohmOriginalGuidanceAndApparatusReuse section EpelbaumOriginalNuclearForceFormulaReuse open PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998 open PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ChiralVertexPowerCountingReadout open PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.LeadingPionExchangePotentialReadout open PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ProjectionDiagramReadout open PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.ThreeNucleonCancellationReadout example {Sector Projector Operator Energy Potential Term Order Regime : Type} (framework : ChiralNuclearFramework Sector Projector Operator Energy Potential Term Order Regime) (setup : framework.ProjectionSetup) : framework.projectionSectorReadout.HasPrintedSourceProjectors := framework.projection_setup_retains_printed_orthogonal_complementary_projectors setup example {Sector Projector Operator Energy Potential Term Order Regime : Type} (framework : ChiralNuclearFramework Sector Projector Operator Energy Potential Term Order Regime) (setup : framework.LowEnergyChiralSetup) : framework.chiralVertexReadout.HasPrintedVertexDimension ∧ framework.chiralVertexReadout.HasPrintedProjectedIndex ∧ 0 < framework.chiralVertexReadout.doubledVertexDimension ∧ framework.chiralVertexReadout.HasEqualVerticesAndDenominators := framework.low_energy_setup_retains_printed_chiral_vertex_and_projection_indices setup example {Sector Projector Operator Energy Potential Term Order Regime : Type} (framework : ChiralNuclearFramework Sector Projector Operator Energy Potential Term Order Regime) (result : framework.TwoNucleonPotentialResult) : framework.leadingPionExchangeReadout.HasPrintedLeadingTwoNucleonPotential := framework |>.two_nucleon_result_retains_spin_isospin_exchange_and_independent_contacts result example {Sector Projector Operator Energy Potential Term Order Regime : Type} (framework : ChiralNuclearFramework Sector Projector Operator Energy Potential Term Order Regime) (result : framework.TwoNucleonPotentialResult) : framework.projectionDiagramReadout.projectionNucleonOnlyDenominatorCount = 0 := (framework |>.second_order_projection_graphs_exclude_nucleon_only_iteration_denominators result).1 example {Sector Projector Operator Energy Potential Term Order Regime : Type} (framework : ChiralNuclearFramework Sector Projector Operator Energy Potential Term Order Regime) (result : framework.ThreeNucleonCancellationResult) : framework.threeNucleonCancellationReadout.totalProjectionThreeBodyContribution = 0 := (framework |>.three_body_cancellation_retains_opposite_printed_pion_energy_factors result).2.2 example : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.Fidelity.countermodelMatrix .printedVertexIdentityNotPositiveChiralIndex := by rcases printed_vertex_dimension_does_not_supply_positive_chiral_counting with ⟨vertexIdentity, projectedIdentity, nonpositive⟩ exact ⟨⟨vertexIdentity, projectedIdentity⟩, nonpositive⟩ example : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.Fidelity.countermodelMatrix .scalarContactNotIndependentTensorContact := by rcases scalar_contact_does_not_supply_independent_tensor_contact with ⟨exchangeFormula, scalarOnly, missingTensor⟩ exact ⟨⟨exchangeFormula, scalarOnly⟩, missingTensor⟩ example : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.Fidelity.countermodelMatrix .projectionReducibleGraphNotTrueIteration := projection_reducible_label_does_not_mean_lippmann_schwinger_iteration example : PhysicsLogic.Papers.EpelbaumGloeckleMeissner1998.Fidelity.countermodelMatrix .projectionCancellationNotOldFashionedIteration := projection_cancellation_does_not_supply_old_fashioned_recoil_iteration end EpelbaumOriginalNuclearForceFormulaReuse section JaynesOriginalThermometerSpinAndCorrelationReuse open PhysicsLogic.Papers.Jaynes1957 open PhysicsLogic.Papers.Jaynes1957.JaynesFramework example {T : Jaynes1957Types} (framework : JaynesFramework T) (result : framework.ThermodynamicDictionaryResult) : framework.selectedThermometerReadout.energyConjugateValue * framework.selectedThermometerReadout.temperatureValue = 1 := framework.calibrated_energy_conjugate_is_reciprocal_temperature result example {T : Jaynes1957Types} (framework : JaynesFramework T) (result : framework.RotationalPolarizationResult) : framework.selectedRotationalReadout.spinPolarizationValue = -(1 / 2 : Real) * Real.tanh (framework.selectedRotationalReadout.angularMomentumConjugateValue / 2) := framework.rotational_polarization_gives_original_spin_half_tanh_formula result example {T : Jaynes1957Types} (framework : JaynesFramework T) (result : framework.RotationalPolarizationResult) : framework.selectedRotationalReadout.spinEnergyTransferValue = 0 ∧ framework.selectedRotationalReadout.appliedMagneticFieldValue = 0 := (framework |>.rotational_polarization_requires_no_spin_energy_exchange_or_magnetic_field result).2 example {T : Jaynes1957Types} (framework : JaynesFramework T) (result : framework.DensityMatrixRepresentationResult) : framework.selectedDensityArrayReadout.firstDensityDiagonalValue + framework.selectedDensityArrayReadout.secondDensityDiagonalValue = 1 := framework.density_representation_has_unit_trace_from_normalized_array result example {T : Jaynes1957Types} (framework : JaynesFramework T) (result : framework.InformationLossResult) : framework.selectedCorrelationRemovalReadout.originalAdditiveMeanValue = framework.selectedCorrelationRemovalReadout.productAdditiveMeanValue ∧ framework.selectedCorrelationRemovalReadout.productCorrelationValue = 0 := framework |>.correlation_removal_preserves_additive_means_but_erases_correlations result example : PhysicsLogic.Papers.Jaynes1957.Fidelity.countermodelMatrix .factorizedPartitionNotCalibratedTemperature := by rcases Countermodels.partition_factorization_does_not_identify_an_uncalibrated_conjugate_with_temperature with ⟨partition, probabilities, _wrongConjugate, notCalibrated⟩ exact ⟨⟨partition, probabilities⟩, notCalibrated⟩ example : PhysicsLogic.Papers.Jaynes1957.Fidelity.countermodelMatrix .larmorComparisonNotMagneticMechanism := by rcases Countermodels.larmor_equivalent_rotational_polarization_does_not_supply_a_magnetic_mechanism with ⟨polarization, _larmorFrequency, noMagneticField⟩ exact ⟨polarization, noMagneticField⟩ example : PhysicsLogic.Papers.Jaynes1957.Fidelity.countermodelMatrix .densityMatrixNotUniquePureStateArray := Countermodels.the_density_matrix_does_not_determine_its_pure_state_array example : PhysicsLogic.Papers.Jaynes1957.Fidelity.countermodelMatrix .preservedSubsystemMeansNotPreservedCorrelations := by rcases Countermodels.preserving_subsystem_predictions_does_not_preserve_joint_correlations with ⟨marginals, means, changedCorrelations⟩ exact ⟨⟨marginals, means⟩, changedCorrelations⟩ end JaynesOriginalThermometerSpinAndCorrelationReuse section RungeGrossOriginalBoundaryActionAndOrbitalReuse open PhysicsLogic.Papers.RungeGross1984 open PhysicsLogic.Papers.RungeGross1984.TimeDependentDensityFramework example {T : RungeGrossTypes} (framework : TimeDependentDensityFramework T) (result : framework.EstablishedDensityHydrodynamics) : HydrodynamicReadout.RetainsEquationsSevenAndEight framework.selectedHydrodynamicReadout := framework |>.hydrodynamic_result_retains_continuity_force_and_both_initial_conditions result example {T : RungeGrossTypes} (framework : TimeDependentDensityFramework T) (result : framework.EstablishedStationaryDensityAction) : framework.selectedDensityActionReadout.totalDensityActionValue = framework.selectedDensityActionReadout.universalActionValue - framework.selectedDensityActionReadout.densityValue * framework.selectedDensityActionReadout.externalPotentialValue * framework.selectedDensityActionReadout.spacetimeMeasureValue := framework.stationary_action_keeps_external_density_potential_coupling result example {T : RungeGrossTypes} (framework : TimeDependentDensityFramework T) (result : framework.EstablishedTimeDependentKohnSham) : EffectiveOrbitalReadout.RetainsEquationsFifteenSixteenAndSeventeen framework.selectedEffectiveOrbitalReadout := framework |>.effective_orbitals_retain_printed_density_orbital_and_potential_equations result example : BoundaryIntegrationReadout.RetainsPrintedBoundaryReductio Countermodels.physicalBoundaryIntegrationReadout Countermodels.physicalTaylorCurrentReadout ∧ BoundaryIntegrationReadout.weightedDivergenceIntegralValue Countermodels.physicalBoundaryIntegrationReadout = -18 := by rcases Countermodels.occupied_support_and_zero_surface_realize_the_original_boundary_reductio with ⟨source, _bulk, divergence⟩ exact ⟨source, divergence⟩ example : PhysicsLogic.Papers.RungeGross1984.Fidelity.countermodelMatrix .occupiedGradientNotDivergenceWithoutFalloff := by rcases Countermodels.occupied_nonconstant_gradient_does_not_force_nonzero_divergence_without_boundary_falloff with ⟨identity, occupied, _surface, zeroDivergence⟩ exact ⟨⟨identity, occupied⟩, zeroDivergence⟩ example : PhysicsLogic.Papers.RungeGross1984.Fidelity.countermodelMatrix .boundaryFalloffNotOccupiedSupport := by rcases Countermodels.vanishing_surface_and_nonconstant_potential_do_not_supply_occupied_density_support with ⟨identity, surface, _gradient, unoccupied⟩ exact ⟨⟨identity, surface⟩, unoccupied⟩ example : PhysicsLogic.Papers.RungeGross1984.Fidelity.countermodelMatrix .flatTaylorJetNotTimeAnalytic := by rcases Countermodels.identical_taylor_jets_do_not_determine_nonanalytic_future_potentials with ⟨zeroJets, _laterDifference, nonanalytic⟩ exact ⟨zeroJets, nonanalytic⟩ example : PhysicsLogic.Papers.RungeGross1984.Fidelity.countermodelMatrix .fixedInitialDensityNotFixedInitialCurrent := by rcases Countermodels.a_fixed_initial_density_does_not_fix_the_initial_current with ⟨fixedDensity, _otherCurrent, currentNotFixed⟩ exact ⟨fixedDensity, currentNotFixed⟩ example : PhysicsLogic.Papers.RungeGross1984.Fidelity.countermodelMatrix .externalHartreeNotExchangeCorrelation := by rcases Countermodels.external_and_hartree_terms_do_not_supply_the_exchange_correlation_action_derivative with ⟨density, hartreeOnly, _nonzeroXC, missingXC⟩ exact ⟨⟨density, hartreeOnly⟩, missingXC⟩ example : PhysicsLogic.Papers.RungeGross1984.Fidelity.countermodelMatrix .selectedRepresentabilityNotUniversalRepresentability := by rcases Countermodels.selected_density_representability_does_not_characterize_all_candidate_densities with ⟨selected, _otherDensity, notUniversal⟩ exact ⟨selected, notUniversal⟩ end RungeGrossOriginalBoundaryActionAndOrbitalReuse section BellOriginalThresholdProjectorAndContextualityReuse open PhysicsLogic.Papers.Bell1966 open PhysicsLogic.Papers.Bell1966.HiddenVariableFramework example : SpinHalfFormulaReadout.alignedPhysicalSpinHalfReadout |>.EquationsOneThroughThreeAndUniformAverageFormulaAt := SpinHalfFormulaReadout.aligned_spin_half_readout_realizes_all_printed_formulas example : SpinHalfFormulaReadout.alignedPhysicalSpinHalfReadout.hiddenOutcomeValueAt (-(1 / 2 : Real)) = 1 ∧ SpinHalfFormulaReadout.alignedPhysicalSpinHalfReadout.spectralValueAt (SpinHalfFormulaReadout.alignedPhysicalSpinHalfReadout |>.hiddenOutcomeValueAt (-(1 / 2 : Real))) := SpinHalfFormulaReadout.aligned_spin_half_threshold_is_still_a_spectral_outcome example {State Observable Context HiddenState : Type} (framework : HiddenVariableFramework State Observable Context HiddenState) (calculation : framework.SpinHalfConstructionCalculation) : framework.incompatibleSpinAdditionReadout.summedOutcomeValue ≠ framework.incompatibleSpinAdditionReadout.firstOutcomeValue + framework.incompatibleSpinAdditionReadout.secondOutcomeValue := (framework |>.spin_half_construction_retains_incompatible_stern_gerlach_spectrum calculation).2 example {State Observable Context HiddenState : Type} (framework : HiddenVariableFramework State Observable Context HiddenState) (calculation : framework.JauchPironNoGoCalculation) : ¬framework.noncommutingProjectorMeetReadout.ProjectorsCommute ∧ framework.noncommutingProjectorMeetReadout.meetAssignedValue = 0 := by have source := framework.jauch_piron_route_retains_noncommuting_zero_meet_projectors calculation exact ⟨source.2.1, source.2.2.2.2⟩ example {State Observable Context HiddenState : Type} (framework : HiddenVariableFramework State Observable Context HiddenState) (state : State) (construction : framework.ContextualProjectorDecompositionAt state) : framework.contextualProjectorPartitionReadout.SameSelectedBornProbability ∧ ¬(framework.contextualProjectorPartitionReadout |>.ContextIndependentSelectedValue) := (framework |>.contextual_escape_retains_printed_cumulative_born_intervals state construction).2 example : PhysicsLogic.Papers.Bell1966.Fidelity.countermodelMatrix .ternaryThresholdNotBinarySpinEigenvalue := Countermodels.ordinary_ternary_sign_at_threshold_is_not_a_spin_eigenvalue example : PhysicsLogic.Papers.Bell1966.Fidelity.countermodelMatrix .twoDimensionalSpinNotGleasonDomain := Countermodels.physical_spin_half_dimension_does_not_satisfy_gleason_hypothesis example : PhysicsLogic.Papers.Bell1966.Fidelity.countermodelMatrix .equalBornProbabilityNotContextIndependent := Countermodels.equal_quantum_projector_probabilities_do_not_fix_hidden_values example : PhysicsLogic.Papers.Bell1966.Fidelity.countermodelMatrix .normalizedBornWeightsNotCumulativeAssignment := Countermodels.normalized_born_weights_do_not_supply_cumulative_interval_assignment example : PhysicsLogic.Papers.Bell1966.Fidelity.countermodelMatrix .selectedNonlocalMechanismNotUniversalNonlocality := Countermodels.one_nonlocal_two_spin_mechanism_does_not_prove_universal_nonlocality end BellOriginalThresholdProjectorAndContextualityReuse section KnoxOriginalSamplingLikelihoodAndCosmologicalInferenceReuse open PhysicsLogic.Papers.Knox1995 open PhysicsLogic.Papers.Knox1995.CMBInferenceFramework example : EstimatorVarianceFormulaReadout.physicalQuadrupoleVarianceReadout |>.SourceFormulaAt := (EstimatorVarianceFormulaReadout.physical_quadrupole_realizes_printed_variance_and_cosmic_variance).1 example {T : CMBTypes} (framework : CMBInferenceFramework T) (experiment : T.Experiment) (setup : framework.SingleSkyEstimatorSetup experiment) : framework.harmonicSamplingReadout.independentModeCountValue = 2 * framework.estimatorVarianceFormulaReadout.multipoleIndexValue + 1 := (framework |>.single_sky_setup_retains_printed_harmonic_covariance_and_mode_count experiment setup).2.1 example {T : CMBTypes} (framework : CMBInferenceFramework T) (experiment : T.Experiment) (setup : framework.SingleSkyEstimatorSetup experiment) (calculation : framework.EstimatorVarianceCalculation experiment) : framework.chiSquareLikelihoodReadout.degreeOfFreedomValue = 2 * framework.estimatorVarianceFormulaReadout.multipoleIndexValue + 1 := (framework |>.estimator_route_retains_printed_chi_square_likelihood_variable experiment setup calculation).2 example {T : CMBTypes} (framework : CMBInferenceFramework T) (narrow wide : T.Experiment) (foreground : T.ForegroundModel) (setup : framework.AngularResolutionSetup narrow wide) (calculation : framework.AngularResolutionCalculation narrow wide foreground) : (framework.beamComparisonReadout |>.RetainsPrintedFixedWeightBeamComparison) ∧ (framework.foregroundCorrelationReadout |>.RetainsPrintedForegroundScope) := (framework |>.beam_route_retains_detector_weight_beam_and_foreground_equations narrow wide foreground setup calculation).2 example {T : CMBTypes} (framework : CMBInferenceFramework T) (experiment : T.Experiment) (likelihood : T.Likelihood) (setup : framework.LikelihoodForecastSetup experiment likelihood) (calculation : framework.LikelihoodRecoveryCalculation experiment likelihood) : framework.chiSquareLikelihoodReadout |>.RetainsPrintedEstimatorLikelihood := (framework |>.likelihood_recovery_retains_original_chi_square_density experiment likelihood setup calculation).2 example {T : CMBTypes} (framework : CMBInferenceFramework T) (setup : framework.CosmologicalConstantCorrectionSetup) (calculation : framework.CosmologicalConstantCorrectionCalculation) : framework.quadrupoleCorrectionReadout |>.RetainsPrintedScalarTensorCorrections := (framework |>.cosmological_correction_retains_both_squared_quadrupole_ratios setup calculation).2 example {T : CMBTypes} (framework : CMBInferenceFramework T) (experiment : T.Experiment) (nuisance : T.CosmologicalParameter) (setup : framework.CosmicConfusionSetup experiment nuisance) (calculation : framework.CosmicConfusionCalculation experiment nuisance) : framework.baryonNuisanceResponseReadout |>.RetainsPrintedBaryonResponseEquations := framework |>.cosmic_confusion_retains_printed_baryon_response_coefficients experiment nuisance setup calculation example : PhysicsLogic.Papers.Knox1995.Fidelity.countermodelMatrix .fiveQuadrupoleModesNotZeroCosmicVariance := Countermodels.five_exactly_measured_full_sky_modes_do_not_remove_cosmic_variance example : PhysicsLogic.Papers.Knox1995.Fidelity.countermodelMatrix .fullSkyCoverageNotPrintedModeCount := Countermodels.full_sky_coverage_does_not_supply_the_printed_mode_count example : PhysicsLogic.Papers.Knox1995.Fidelity.countermodelMatrix .fixedDetectorWeightNotEqualPixelNoise := Countermodels.fixed_detector_weight_does_not_mean_equal_pixel_noise example : PhysicsLogic.Papers.Knox1995.Fidelity.countermodelMatrix .correctModeCountNotPrintedLikelihoodVariable := Countermodels.a_correct_mode_count_does_not_supply_the_printed_likelihood_variable example : PhysicsLogic.Papers.Knox1995.Fidelity.countermodelMatrix .scalarForegroundDegradationNotUncorrelatedPixels := Countermodels.scalar_foreground_degradation_does_not_remove_pixel_correlations example : PhysicsLogic.Papers.Knox1995.Fidelity.countermodelMatrix .correctedConsistencyNotUncorrectedConsistency := Countermodels.corrected_inflationary_relation_does_not_imply_uncorrected_relation example : PhysicsLogic.Papers.Knox1995.Fidelity.countermodelMatrix .printedBaryonResponseNotSubdominantNuisance := by have response := Countermodels.original_baryon_response_exceeds_both_conditional_forecast_errors exact ⟨response.1, response.2.2.2⟩ end KnoxOriginalSamplingLikelihoodAndCosmologicalInferenceReuse section ItzhakiOriginalRegimeThroatAndD6RadiationReuse open PhysicsLogic.Papers.ItzhakiMaldacenaSonnenscheinYankielowicz1998 open PhysicsLogic.Papers.ItzhakiMaldacenaSonnenscheinYankielowicz1998.PaperFramework example : DpSupergravityValidityReadout.physicalD6SupergravityWindow |>.RetainsPrintedEffectiveCouplingCurvatureAndDilaton := DpSupergravityValidityReadout.d6_witness_realizes_printed_coupling_curvature_dilaton_and_window.1 example : forall bound : Real, exists step, bound < D5InfiniteThroatReadout.physicalInfiniteThroat.properThroatLengthAt step := D5InfiniteThroatReadout.physical_ns5_throat_retains_curvature_escape_and_unbounded_length.2 example (framework : PaperFramework) (validity : framework.GeneralValidityWindowCalculation) : framework.supergravityValidityReadout |>.RetainsPrintedEffectiveCouplingCurvatureAndDilaton := (framework.general_validity_retains_printed_upper_bound_curvature_and_dilaton validity).1 example (framework : PaperFramework) (calculation : framework.D2RegimeCalculation) : exists energy, framework.d2IIALowerThreshold < energy /\ energy < framework.d2IIAUpperThreshold := (framework.d2_regime_retains_large_rank_and_inhabited_supergravity_interval calculation).2.2.2 example (framework : PaperFramework) (calculation : framework.D5RegimeCalculation) : framework.d5InfiniteThroatReadout.RetainsPrintedInfiniteThroat := (framework.d5_regime_retains_infinite_ns5_throat_and_escaping_mode calculation).1 example (framework : PaperFramework) (escape : framework.D6MassiveEscapeCalculation) : framework.d6AsymptoticGeometryReadout |>.RetainsPrintedALEGeometryAndMassiveEscape := (framework.d6_escape_retains_flat_ale_orbifold_and_massive_geodesic escape).1 example (framework : PaperFramework) (escape : framework.D6MassiveEscapeCalculation) (radiation : framework.D6ThermalRadiationCalculation) : framework.d6ThermalEmissionReadout |>.RetainsPrintedNonextremalHorizonTemperatureAndFlux := (framework.d6_radiation_retains_nonextremal_horizon_temperature_and_bulk_flux escape radiation).1 example : PhysicsLogic.Papers.ItzhakiMaldacenaSonnenscheinYankielowicz1998.Fidelity.countermodelMatrix .rankOneThresholdsNotInhabitedD2Window := Countermodels.printed_d2_thresholds_at_rank_one_do_not_supply_a_supergravity_region example : PhysicsLogic.Papers.ItzhakiMaldacenaSonnenscheinYankielowicz1998.Fidelity.countermodelMatrix .positiveTemperatureNotHawkingFlux := Countermodels.a_positive_hawking_temperature_does_not_supply_bulk_emission example : PhysicsLogic.Papers.ItzhakiMaldacenaSonnenscheinYankielowicz1998.Fidelity.countermodelMatrix .hawkingFluxNotWorldvolumeEnergyBridge := Countermodels.asymptotic_emission_requires_a_separate_worldvolume_energy_bridge end ItzhakiOriginalRegimeThroatAndD6RadiationReuse section NakanoOriginalFrequencyMemoryAndConductivityReuse open PhysicsLogic.Papers.Nakano1956 open PhysicsLogic.Papers.Nakano1956.ConductivityFramework example : FrequencyResolvedInteractionReadout.physicalFrequencyResolvedInteraction |>.RetainsPrintedFrequencySectorsAndMemoryKernel := FrequencyResolvedInteractionReadout.original_frequency_sectors_realize_the_double_commutator_memory_kernel example : RelaxationMemoryConductivityReadout.physicalMemoryRelaxationConductivity |>.RetainsEquationsSevenEightAndNine := RelaxationMemoryConductivityReadout.original_memory_relaxation_and_zero_frequency_conductivity_are_inhabited example : RelaxationMemoryConductivityReadout.physicalMemoryRelaxationConductivity.perturbativeKernelValueAt 0 = 2 := RelaxationMemoryConductivityReadout.printed_memory_kernel_starts_at_the_equilibrium_current_variance example {T : NakanoTypes} (framework : ConductivityFramework T) (static : framework.EstablishedStaticConductivity) (setup : framework.RelaxationApproximationSetup) (calculation : framework.RelaxationApproximationCalculation) : framework.relaxationMemoryConductivityReadout |>.RetainsPrintedZeroFrequencyConductivity := (framework.relaxation_route_retains_original_frequency_memory_and_conductivity_formulas static setup calculation).2.2.2 example : 2 * Countermodels.strongerZeroFrequencyScattering.approximateConductivityValue = RelaxationMemoryConductivityReadout.physicalMemoryRelaxationConductivity.approximateConductivityValue := Countermodels.doubled_zero_frequency_scattering_halves_the_printed_conductivity.2.2 example : PhysicsLogic.Papers.Nakano1956.Fidelity.countermodelMatrix .finiteFrequencyNotZeroFrequencyCollision := Countermodels.a_finite_frequency_interaction_does_not_supply_zero_frequency_conductivity example : PhysicsLogic.Papers.Nakano1956.Fidelity.countermodelMatrix .secondOrderMemoryNotExponentialRelaxation := Countermodels.second_order_memory_does_not_imply_a_single_relaxation_time_tail example : PhysicsLogic.Papers.Nakano1956.Fidelity.countermodelMatrix .positiveConductivityNotPrintedEquationNine := Countermodels.a_positive_named_conductivity_does_not_supply_equation_nine example : PhysicsLogic.Papers.Nakano1956.Fidelity.countermodelMatrix .fixedCollisionWeightNotTemperatureIndependentConductivity := Countermodels.fixed_collision_weight_does_not_make_conductivity_temperature_independent end NakanoOriginalFrequencyMemoryAndConductivityReuse section EvansColemanOriginalSimilarityFitAndRedshiftReuse open PhysicsLogic.Papers.EvansColeman1994 open PhysicsLogic.Papers.EvansColeman1994.RadiationFluidCollapseContext example : Regularity.RegularCenter Regularity.centerRegularProfile := Regularity.the_cartesian_center_regularity_contract_is_inhabited example : Regularity.RegularSimilarityBoundary Regularity.boundaryRegularProfile := Regularity.the_similarity_boundary_regularity_contract_is_inhabited example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .centerRegularityNotBoundaryRegularity := Regularity.a_regular_center_does_not_impose_similarity_boundary_regularity example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .centerRegularityNotFieldEquations := Regularity.a_regular_center_does_not_solve_the_radiation_field_equations example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (setup : context.SelfSimilarConstructionSetup) (calculation : context.SelfSimilarConstructionCalculation) (nonflat : ¬ context.asymptoticallyFlatAt context.selfSimilarSolution context.asymptoticExteriorRegime) (scope : context.LocalSimilarityScopeBridge) : context.SelectedRegularityAt := (context.regularity_calculation_constructs_local_self_similar_solution setup calculation nonflat scope).2.2.2 example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (conclusions : PaperConclusions context) : Regularity.RegularCenter context.selectedSimilarityProfile /\ Regularity.RegularSimilarityBoundary context.selectedSimilarityProfile /\ Regularity.RegularSonicPoint context.selectedSimilarityProfile := conclusions.actualRegularity example : Construction.CalibratedGeometry Construction.calibratedKinematicProfile Construction.sampleEstimates := Construction.nonexact_geometry_calibration_is_inhabited example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .calibratedGeometryNotDynamicalSolution := Construction.calibrated_geometry_does_not_solve_the_field_equations example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (setup : context.SelfSimilarConstructionSetup) (calculation : context.SelfSimilarConstructionCalculation) : Construction.AdmissibleIndex (context.similarityIndexValueAt context.similarityIndex) := context.the_constructed_similarity_index_is_admissible setup calculation example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (setup : context.SelfSimilarConstructionSetup) (calculation : context.SelfSimilarConstructionCalculation) : context.selectedSimilarityProfile.RadiationFieldEquations := (context.the_construction_calculation_solves_the_selected_boundary_value_problem setup calculation).2.2.2.1 example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (setup : context.SelfSimilarConstructionSetup) (calculation : context.SelfSimilarConstructionCalculation) : ¬ context.asymptoticallyFlatAt context.selfSimilarSolution context.asymptoticExteriorRegime := context.calibrated_similarity_construction_excludes_asymptotic_flatness setup calculation example : RadiationThresholdNumericalFitReadout.reportedRadiationThresholdFit |>.RetainsPrintedCriticalSearchAndMassFit := RadiationThresholdNumericalFitReadout.original_threshold_fit_retains_critical_strength_mass_law_and_independent_grids example : Intermediate.WitnessedTransientComparison Intermediate.finitelyMatchingEvolution Similarity.Profile.homotheticNonSolution Intermediate.comparisonWindow Intermediate.comparisonSamples (fun _ => 0) := Intermediate.finite_comparisons_and_later_departure_exhibit_a_transient_window _ _ _ _ _ Intermediate.two_exact_sample_comparisons_have_a_later_physical_field_discrepancy.1 Intermediate.two_exact_sample_comparisons_have_a_later_physical_field_discrepancy.2 example (readout : CriticalSingularityRedshiftReadout) {profile : Similarity.Profile} (calibration : readout.IndexCalibration) (calculation : readout.LapseCalculation profile) : readout.limits.LapseVanishes := readout.source_redshift_lapse_tends_to_zero_at_the_central_singularity calibration calculation example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (setup : context.ThresholdSetup) (calculation : context.MassScalingCalculation) : context.SelectedPrintedThresholdNumericalFitAt := context.threshold_fit_retains_original_critical_strength_coefficient_and_grids setup calculation example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (setup : context.SelfSimilarConstructionSetup) (calculation : context.SelfSimilarConstructionCalculation) : context.SelectedCalibratedSimilarityGeometryAt := context.self_similar_route_retains_sonic_regular_eigenvalue_and_nonflat_asymptotics setup calculation example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (calculation : context.IntermediateAsymptoticCalculation) : context.SelectedTransientSimilarityAt := context.finite_numerical_comparison_establishes_transient_similarity calculation example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .finiteMetricComparisonNotDynamicalSolution := Intermediate.finite_metric_agreement_does_not_solve_the_compared_profile example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (calculation : context.IntermediateAsymptoticCalculation) : context.parameterValueAt context.supercriticalParameter ≠ context.parameterValueAt context.criticalParameter := context.near_critical_calculation_has_a_nonzero_actual_parameter_gap calculation example : Intermediate.PointwisePersistence Intermediate.returningEvolution Similarity.Profile.homotheticNonSolution := Intermediate.finite_later_departure_does_not_exclude_eventual_profile_convergence.2.2 example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .metricSamplesNotFullFluidAgreement := Intermediate.metric_samples_do_not_determine_fluid_samples example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .exteriorMismatchNotInnerDeparture := by have witness := Intermediate.exterior_discrepancy_is_not_evidence_of_departure_at_the_inner_scale exact ⟨⟨witness.1, witness.2.1⟩, witness.2.2⟩ example : Intermediate.ObservedTransientComparison Intermediate.returningEvolution Similarity.Profile.homotheticNonSolution Intermediate.comparisonWindow Intermediate.comparisonSamples {.radialMetric} (fun _ => 0) := Intermediate.observed_comparisons_and_departure_exhibit_a_transient_window _ _ _ _ _ _ Intermediate.finite_observed_departure_allows_return_to_the_profile.1 Intermediate.finite_observed_departure_allows_return_to_the_profile.2.1 example : Intermediate.isolatedCriticalFamily.OnlyCriticalPersists := Intermediate.the_only_critical_continuation_expectation_is_inhabited example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (calculation : context.IntermediateAsymptoticCalculation) (continuation : context.PreciselyCriticalContinuationExpectation) : context.SelectedNoncriticalNonpersistenceAt := context.noncritical_nonpersistence_requires_the_separate_continuation_expectation calculation continuation example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (curvature : context.SelectedCurvatureCalculationAt) (calibration : context.SelectedIndexCalibrationAt) (lapse : context.SelectedLapseCalculationAt) : context.SelectedSingularityLimitsAt := context.calculated_curvature_and_lapse_asymptotics_establish_the_selected_limits curvature calibration lapse example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (curvature : context.SelectedCurvatureCalculationAt) (calibration : context.SelectedIndexCalibrationAt) (lapse : context.SelectedLapseCalculationAt) : context.SelectedSingularityFieldBindingsAt := (context.calculated_curvature_and_lapse_asymptotics_establish_the_selected_limits curvature calibration lapse).2.2.2 example : SingularityFields.quadraticReadout.DensityScaling (1 / (4 * Real.pi)) := SingularityFields.the_bound_density_has_proper_time_inverse_square_scaling example : SingularityFields.quadraticTrace.MatchesProfile SingularityFields.quadraticProfile /\ SingularityFields.quadraticClock.IsProperFor SingularityFields.quadraticTrace /\ ¬ SingularityFields.identityClock.IsProperFor SingularityFields.quadraticTrace := SingularityFields.coordinate_time_cannot_replace_proper_time_for_the_same_lapse example : Singularity.nonunitLapseNormalization.LapseAsymptotics 2 /\ ¬ SingularityFields.BoundaryLapseMatches Singularity.nonunitLapseNormalization Regularity.boundaryRegularProfile := SingularityFields.lapse_asymptotics_alone_do_not_bind_the_boundary_field example : SingularityFields.doubledDensityReadout.DensityScaling (2 * (1 / (4 * Real.pi))) /\ ¬ SingularityFields.DensityMatches SingularityFields.doubledDensityReadout SingularityFields.quadraticTrace SingularityFields.quadraticClock := SingularityFields.a_proper_time_density_limit_does_not_supply_the_same_field_binding example {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (calibration : context.SelectedIndexCalibrationAt) (lapse : context.SelectedLapseCalculationAt) (transport : context.SelectedObserverTransportAt) : context.SelectedObservedRedshiftAt := context.lapse_and_observer_transport_establish_infinite_redshift calibration lapse transport example : let point : Singularity.RadialPhotonEndpoint := ⟨2, 3, 5⟩ point.metricPair point.coordinateStationaryObserver point.coordinateStationaryObserver = -1 /\ point.metricPair point.outwardPhoton point.outwardPhoton = 0 /\ point.observedFrequency = 5 / 2 := by dsimp only have physical : (⟨2, 3, 5⟩ : Singularity.RadialPhotonEndpoint).Admissible := by norm_num [Singularity.RadialPhotonEndpoint.Admissible] have kinematics := Singularity.RadialPhotonEndpoint.endpoint_kinematics_are_timelike_and_null _ physical exact ⟨kinematics.1, kinematics.2.1, Singularity.RadialPhotonEndpoint.measured_frequency_is_energy_over_lapse _ (by norm_num)⟩ example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .vanishingLapseNotObservedRedshift := by have witness := Singularity.a_vanishing_lapse_does_not_alone_give_observed_redshift exact ⟨⟨witness.1, witness.2.1, witness.2.2.1⟩, witness.2.2.2⟩ example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .sonicRegularityNotSimilarityEigenvalue := Countermodels.regular_acoustic_sonic_point_does_not_select_the_nonlinear_eigenvalue example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .similarityEigenvalueNotMassScalingExponent := Countermodels.calibrated_similarity_index_is_not_the_black_hole_mass_exponent example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .finiteObservedMassRangeNotAllScaleLaw := Countermodels.two_observed_mass_decades_do_not_supply_an_all_scale_mass_law example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .criticalPointNullityDependsOnMeasure := Singularity.a_single_critical_parameter_is_not_null_for_every_measure example : PhysicsLogic.Papers.EvansColeman1994.Fidelity.countermodelMatrix .massExponentNotRedshiftExponent := Countermodels.the_black_hole_mass_exponent_does_not_determine_the_lapse_redshift end EvansColemanOriginalSimilarityFitAndRedshiftReuse section HannayOriginalNonzeroHolonomyAndIndependentExamplesReuse open PhysicsLogic.Papers.Hannay1985 open PhysicsLogic.Papers.Hannay1985.AngleHolonomyFramework example : HannayAngleFormulaReadout.OriginalEquationsOneThroughNineFormulaPackageAt HannayAngleFormulaReadout.physicalNonzeroAngleHolonomyReadout := HannayAngleFormulaReadout.original_nonzero_holonomy_realizes_printed_equations_one_through_nine.1 example : RotatingHoopIllustrationReadout.physicalNoncircularRotatingHoop.extraAngleValue = -(Real.pi ^ 2 / 2) := by rfl example : RotatingHoopIllustrationReadout.physicalCircularRotatingHoop.extraAngleValue = -(2 * Real.pi) := RotatingHoopIllustrationReadout.original_circular_hoop_realizes_one_negative_full_relative_turn.2.2 example : DrivenTopIllustrationReadout.physicalNonspinningDrivenTop.axialAngularVelocityValue = 0 /\ 0 < DrivenTopIllustrationReadout.physicalNonspinningDrivenTop.extraGeometricAngleValue := DrivenTopIllustrationReadout.a_nonspinning_driven_top_still_has_nonzero_parallel_transport_holonomy.2 example {T : HannayTypes} (framework : AngleHolonomyFramework T) (holonomy : framework.AngleHolonomyResult) (setup : framework.QuadraticOscillatorSetup) (calculation : framework.QuadraticOscillatorCalculation) : framework.SelectedPrintedQuadraticOscillatorIllustrationAt := framework.oscillator_route_retains_elliptic_hamiltonian_connection_and_curvature holonomy setup calculation example {T : HannayTypes} (framework : AngleHolonomyFramework T) (holonomy : framework.AngleHolonomyResult) (setup : framework.RotatingHoopSetup) (calculation : framework.RotatingHoopCalculation) : framework.SelectedPrintedRotatingHoopIllustrationsAt := framework.rotating_hoop_route_retains_distinct_noncircular_and_circular_systems holonomy setup calculation example {T : HannayTypes} (framework : AngleHolonomyFramework T) (result : framework.RotatingHoopResult) : framework.rotatingHoop ≠ framework.circularHoop := by rcases result with ⟨_, _, _, ⟨_, _, _, _, _, _, _, _, _, _, distinctSystems⟩⟩ exact distinctSystems example {T : HannayTypes} (framework : AngleHolonomyFramework T) (holonomy : framework.AngleHolonomyResult) (setup : framework.DrivenTopSetup) (calculation : framework.DrivenTopCalculation) : framework.SelectedPrintedDrivenTopIllustrationAt := framework.driven_top_route_retains_independent_spin_and_solid_angle_rotations holonomy setup calculation example : PhysicsLogic.Papers.Hannay1985.Fidelity.countermodelMatrix .positiveQuadraticDiagonalsNotEllipticContours := Countermodels.positive_quadratic_diagonal_terms_do_not_exclude_hyperbolic_contours example : PhysicsLogic.Papers.Hannay1985.Fidelity.countermodelMatrix .oneHoopRotationNotCircularAngle := Countermodels.one_complete_rotation_of_a_noncircular_hoop_is_not_one_negative_turn example : PhysicsLogic.Papers.Hannay1985.Fidelity.countermodelMatrix .sourceGeneralHoopNotCircularSpecialization := Countermodels.the_general_hoop_source_witness_is_genuinely_noncircular example : PhysicsLogic.Papers.Hannay1985.Fidelity.countermodelMatrix .geometricTopAngleNotPositiveAxialSpin := Countermodels.zero_axial_spin_does_not_remove_geometric_top_rotation end HannayOriginalNonzeroHolonomyAndIndependentExamplesReuse section HalperinNelsonOriginalNeutralHexaticAndSubstrateReuse open PhysicsLogic.Papers.HalperinNelson1978 open PhysicsLogic.Papers.HalperinNelson1978.MeltingFramework example : HalperinNelsonFormulaReadout.physicalCriticalTriangularCrystal.CriticalCouplingAndEquationSevenAt := HalperinNelsonFormulaReadout.original_critical_crystal_realizes_the_source_elastic_jump.2 example : HalperinNelsonFormulaReadout.physicalQuadraticFugacityCrystal.EquationsThreeThroughFiveAndCouplingAt := HalperinNelsonFormulaReadout.original_nonzero_fugacity_realizes_all_three_bessel_recursions.1 example : 0 < HalperinNelsonFormulaReadout.importedModifiedBesselI0 1 := HalperinNelsonFormulaReadout.imported_modified_bessel_i0_is_positive 1 example : 0 < HalperinNelsonFormulaReadout.importedModifiedBesselI1 2 := HalperinNelsonFormulaReadout.imported_modified_bessel_i1_is_positive (by norm_num) example : HalperinNelsonFormulaReadout.physicalQuadraticFugacityCrystal.reducedShearInverseScaleDerivative < HalperinNelsonFormulaReadout.physicalQuadraticFugacityCrystal.reducedShearPlusLambdaInverseScaleDerivative := HalperinNelsonFormulaReadout.original_nonzero_fugacity_has_distinct_shear_and_bulk_recursions example : TriangularBurgersNeutralityReadout.physicalNeutralTriangularDislocations.RetainsPrintedTriangularChargeNeutralityAndCoreFugacity := TriangularBurgersNeutralityReadout.original_triangular_burgers_vectors_are_neutral_with_nonzero_core_fugacity example : SixfoldOrientationalCorrelationReadout.physicalIntermediateHexatic.RetainsPrintedHexaticSixfoldCorrelationAndSeparatedDefects := SixfoldOrientationalCorrelationReadout.original_hexatic_has_sixfold_order_but_no_free_disclinations.1 example : EssentialMeltingCorrelationLengthReadout.physicalEssentialMeltingLength.exponentEstimateValue ≠ (1 : Real) / 2 := EssentialMeltingCorrelationLengthReadout.original_melting_length_retains_the_bounded_non_kt_exponent.2 example : PeriodicSubstrateRelevanceReadout.physicalFloatingTriangularSolid.RetainsPrintedSubstrateEigenvalueAndFloatingStability := PeriodicSubstrateRelevanceReadout.original_floating_solid_satisfies_both_independent_stability_tests example {T : MeltingTypes} (framework : MeltingFramework T) (setup : framework.SmoothSolidSetup) (calculation : framework.SmoothSolidCalculation) : framework.SelectedPrintedTriangularBurgersNeutralityAt := framework.smooth_solid_route_retains_triangular_burgers_neutrality_and_core_fugacity setup calculation example {T : MeltingTypes} (framework : MeltingFramework T) (smooth : framework.SmoothSolidResult) (rg : framework.DislocationRGCalculation) (calculation : framework.CriticalBehaviorCalculation) : framework.SelectedPrintedEssentialMeltingLengthAt := framework.critical_route_retains_essential_singularity_and_non_kt_exponent smooth rg calculation example {T : MeltingTypes} (framework : MeltingFramework T) (dislocations : framework.DislocationUnbindingResult) (calculation : framework.OrientationalCalculation) : framework.SelectedPrintedHexaticSixfoldCorrelationsAt := framework.orientational_route_retains_sixfold_correlations_and_distinct_defects dislocations calculation example {T : MeltingTypes} (framework : MeltingFramework T) (substrate : framework.substrateSelected framework.periodicSubstrate) (dislocations : framework.DislocationUnbindingResult) (calculation : framework.PeriodicSubstrateCalculation) : framework.SelectedPrintedFloatingAndPinnedSubstrateTestsAt := framework.substrate_route_retains_both_independent_floating_stability_tests substrate dislocations calculation example : PhysicsLogic.Papers.HalperinNelson1978.Fidelity.countermodelMatrix .integerBurgersVectorsNotChargeNeutrality := Countermodels.integer_triangular_burgers_vectors_do_not_supply_charge_neutrality example : PhysicsLogic.Papers.HalperinNelson1978.Fidelity.countermodelMatrix .exponentialTranslationNotExponentialHexaticOrder := Countermodels.exponential_translational_decay_does_not_impose_exponential_hexatic_decay example : PhysicsLogic.Papers.HalperinNelson1978.Fidelity.countermodelMatrix .boundedApproximationNotExactTruncatedDecimal := Countermodels.the_printed_approximation_does_not_fix_its_truncated_decimal_exactly example : PhysicsLogic.Papers.HalperinNelson1978.Fidelity.countermodelMatrix .substrateIrrelevanceNotDislocationStability := Countermodels.substrate_irrelevance_does_not_supply_dislocation_stability example : PhysicsLogic.Papers.HalperinNelson1978.Fidelity.countermodelMatrix .dislocationStabilityNotSubstrateIrrelevance := Countermodels.dislocation_stability_does_not_make_a_commensurate_substrate_irrelevant end HalperinNelsonOriginalNeutralHexaticAndSubstrateReuse section KomargodskiSchwimmerOriginalDilatonAnomalyAndScalarLoopReuse open PhysicsLogic.Papers.KomargodskiSchwimmer2011 open PhysicsLogic.Papers.KomargodskiSchwimmer2011.RGFlowFramework example : WeylCompensatorAnomalyReadout.physicalEulerAnomalyDilatonScattering.RetainsPrintedWeylCompensationAndEulerForwardAmplitude := WeylCompensatorAnomalyReadout.original_weyl_compensation_selects_only_the_euler_anomaly.1 example : CompensatorSpectatorHierarchyReadout.physicalWeakSpectatorCompensatorHierarchy.RetainsPrintedCompensatorRecoveryHierarchyAndVacuumTuning := CompensatorSpectatorHierarchyReadout.original_compensator_has_independent_spectator_and_vacuum_hierarchies example : DilatonSpectralDispersionReadout.physicalStrictPositiveDilatonDispersion.RetainsPrintedOpticalTheoremAndStrictDispersionSumRule := DilatonSpectralDispersionReadout.original_positive_dilaton_channel_realizes_optical_and_dispersion_equations example : DilatonRunningTailReadout.physicalNestedDilatonRunningTails.RetainsPrintedNestedRunningTailsAndOpenInterpretations := DilatonRunningTailReadout.original_nested_spectral_tails_give_a_strictly_decreasing_running_anomaly.1 example : FreeScalarOneLoopData.physicalCompensatedMassiveScalarLoop.HasPaperOneLoopCoefficients := FreeScalarOneLoopData.original_compensated_scalar_realizes_all_appendix_b_loop_equations.2.2.2.1 example : FreeScalarOneLoopData.physicalCompensatedMassiveScalarLoop.HasPairProductionThreshold := FreeScalarOneLoopData.original_compensated_scalar_realizes_all_appendix_b_loop_equations.2.2.2.2 example {Scale : Type} {Mass : Type} (flow : RGFlowFramework Scale Mass) (lowEnergy : flow.DilatonLowEnergyTheorem) : flow.SelectedPrintedWeylCompensationAndEulerAmplitudeAt := flow.low_energy_route_retains_weyl_compensation_and_euler_only_scattering lowEnergy example {Scale : Type} {Mass : Type} (flow : RGFlowFramework Scale Mass) (construction : flow.GeneralCompensatorConstruction) : flow.SelectedPrintedCompensatorSpectatorHierarchyAt := flow.compensator_route_retains_spectator_hierarchy_and_vacuum_tuning construction example {Scale : Type} {Mass : Type} (flow : RGFlowFramework Scale Mass) (assumptions : flow.HasDispersiveAssumptions) : flow.SelectedPrintedStrictDilatonDispersionAt := flow.dispersive_route_retains_optical_channel_and_exact_sum_rule_prefactor assumptions example {Scale : Type} {Mass : Type} (flow : RGFlowFramework Scale Mass) (monotone : flow.MonotoneAFunction) : flow.SelectedPrintedNestedDilatonRunningTailsAt := flow.monotone_running_route_retains_source_spectral_tail_values monotone example {Scale : Type} {Mass : Type} (flow : RGFlowFramework Scale Mass) (data : FreeScalarOneLoopData) (conclusion : flow.FreeScalarATheoremConclusion data) : data.HasPaperOneLoopCoefficients := flow.appendix_b_conclusion_retains_the_actual_one_loop_coefficients data conclusion example : PhysicsLogic.Papers.KomargodskiSchwimmer2011.Fidelity.countermodelMatrix .changingWeylAnomalyNotFlatAmplitudeContribution := ATheoremCountermodels.a_changing_weyl_anomaly_does_not_contribute_to_flat_dilaton_scattering example : PhysicsLogic.Papers.KomargodskiSchwimmer2011.Fidelity.countermodelMatrix .unitCompensatorVevNotWeakSpectatorHierarchy := ATheoremCountermodels.a_unit_compensator_vev_does_not_supply_a_weak_spectator_hierarchy example : PhysicsLogic.Papers.KomargodskiSchwimmer2011.Fidelity.countermodelMatrix .nonnegativeCrossSectionNotStrictPositiveChannel := ATheoremCountermodels.nonnegative_cross_section_does_not_supply_a_nontrivial_scattering_channel example : PhysicsLogic.Papers.KomargodskiSchwimmer2011.Fidelity.countermodelMatrix .monotoneRunningAnomalyNotGradientFlowProof := ATheoremCountermodels.a_monotone_running_anomaly_does_not_prove_gradient_flow example : PhysicsLogic.Papers.KomargodskiSchwimmer2011.Fidelity.countermodelMatrix .firstLoopIntegralNotSourceLaplacianNormalization := ATheoremCountermodels.the_first_loop_integral_does_not_fix_the_source_laplacian_normalization example : PhysicsLogic.Papers.KomargodskiSchwimmer2011.Fidelity.countermodelMatrix .exactLoopCoefficientsNotPairProductionThreshold := ATheoremCountermodels.exact_appendix_b_loop_coefficients_do_not_supply_the_pair_threshold end KomargodskiSchwimmerOriginalDilatonAnomalyAndScalarLoopReuse section GurneyCondonOriginalAlphaTurningAndSourceScopeReuse open PhysicsLogic.Papers.GurneyCondon1928 open PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework example : NatureAlphaIllustrationReadout.physicalNatureAlphaAndUraniumScattering |>.RetainsPrintedAlphaEnergyRangeAndExteriorTurningPoint := NatureAlphaIllustrationReadout.original_four_million_volt_alpha_turns_outside_the_uranium_nuclear_region example {T : GurneyCondonTypes} (framework : RadioactiveDisintegrationFramework T) (result : framework.EstablishedScatteringConsistency) : framework.SelectedPrintedNatureAlphaScatteringAt := framework.scattering_route_retains_original_alpha_energy_range_and_exterior_turning result example : PhysicsLogic.Papers.GurneyCondon1928.Fidelity.countermodelMatrix .printedAlphaEnergyNotUraniumRange := Countermodels.the_four_million_volt_alpha_example_does_not_fix_the_uranium_range example : PhysicsLogic.Papers.GurneyCondon1928.Fidelity.countermodelMatrix .exteriorWaveTailNotIndependentInteriorTail := Countermodels.the_exterior_wave_tail_does_not_supply_the_independent_interior_tail example : PhysicsLogic.Papers.GurneyCondon1928.Fidelity.countermodelMatrix .finiteOverlapNotPrintedQuantitativeTunnelingFormula := Countermodels.a_finite_overlap_does_not_print_a_quantitative_tunneling_formula example : PhysicsLogic.Papers.GurneyCondon1928.Fidelity.countermodelMatrix .higherAlphaEnergyNotComparableNuclearPotentials := Countermodels.higher_alpha_energy_does_not_supply_similar_nuclear_potentials example : PhysicsLogic.Papers.GurneyCondon1928.Fidelity.countermodelMatrix .subBarrierIncomingAlphaNotNuclearPenetration := Countermodels.a_sub_barrier_incoming_alpha_does_not_reach_the_nuclear_region example : PhysicsLogic.Papers.GurneyCondon1928.Fidelity.countermodelMatrix .atomicFieldReversalNotNuclearEscapeDerivation := Countermodels.atomic_field_reversal_does_not_derive_nuclear_alpha_escape end GurneyCondonOriginalAlphaTurningAndSourceScopeReuse section VishveshwaraOriginalMasterBranchDipoleAndResonanceReuse open PhysicsLogic.Papers.Vishveshwara1970 open PhysicsLogic.Papers.Vishveshwara1970.Framework example : Countermodels.realizesPrintedOddMasterEquations Countermodels.physicalPrintedOddMasterEquationModel := Countermodels.original_nonzero_odd_master_mode_realizes_all_printed_source_equations example : Countermodels.RetainsPrintedOuterDecayAndKruskalHorizonDivergence Countermodels.physicalOddHorizonGrowingBranch := Countermodels.original_outer_decaying_odd_mode_grows_beyond_the_kruskal_bound example : Countermodels.RetainsPrintedDipoleGaugeAndStationaryRotation Countermodels.physicalOddDipoleGaugeAndStationaryRotation := Countermodels.original_nonstationary_dipole_is_gauge_but_stationary_rotation_remains example : Countermodels.RetainsPrintedDampingResonanceAndOpenEvenCompleteness Countermodels.physicalDampedResonanceWithOpenEvenCompleteness := Countermodels.original_damped_resonance_retains_the_missing_even_completeness_theorem example {T : VishveshwaraTypes} (framework : Framework T) (result : framework.EstablishedOddHigherModeStability) (radius : Real) (exterior : 2 * framework.schwarzschildMassValueAt framework.schwarzschildMetric < radius) : Countermodels.realizesPrintedOddMasterEquations (Countermodels.selectedPrintedOddMasterEquationModelAt framework radius) := framework.odd_higher_stability_retains_complete_selected_printed_master_equations result radius exterior example : PhysicsLogic.Papers.Vishveshwara1970.Fidelity.countermodelMatrix .printedTortoiseNotOddMasterNormalization := Countermodels.the_tortoise_formula_does_not_normalize_the_odd_master_variable example : PhysicsLogic.Papers.Vishveshwara1970.Fidelity.countermodelMatrix .outerDecayingBranchNotKruskalRegularity := Countermodels.the_outer_decaying_odd_branch_does_not_remain_kruskal_regular example : PhysicsLogic.Papers.Vishveshwara1970.Fidelity.countermodelMatrix .dipoleGaugeRemovalNotErasedStationaryRotation := Countermodels.nonstationary_gauge_removal_does_not_erase_stationary_source_rotation example : PhysicsLogic.Papers.Vishveshwara1970.Fidelity.countermodelMatrix .outgoingInfinityNotIncomingHorizon := Countermodels.outgoing_infinity_radiation_does_not_impose_the_ingoing_horizon_branch example : PhysicsLogic.Papers.Vishveshwara1970.Fidelity.countermodelMatrix .oddSturmLiouvilleCompletenessNotEvenCompleteness := Countermodels.odd_sturm_liouville_completeness_does_not_supply_even_completeness example : PhysicsLogic.Papers.Vishveshwara1970.Fidelity.countermodelMatrix .dampedResonanceNotNonlinearStability := Countermodels.a_damped_lower_half_plane_resonance_does_not_prove_nonlinear_stability end VishveshwaraOriginalMasterBranchDipoleAndResonanceReuse section LindeLindeMezhlumianOriginalNormalizedVolumeAndHubbleReuse open PhysicsLogic.Papers.LindeLindeMezhlumian1993 open PhysicsLogic.Papers.LindeLindeMezhlumian1993.StochasticInflationFramework example : HubbleFluctuationSourceReadout.physicalFlatHubbleFluctuations |>.RetainsPrintedHubbleStepsVarianceAndLightMassLimit := HubbleFluctuationSourceReadout.original_hubble_kicks_variance_diffusion_and_massive_limit_are_inhabited example : SelfReproducingHubbleDomainReadout.physicalSelfReproducingHubbleDomains |>.RetainsPrintedIndependentHubbleDomainReproduction := SelfReproducingHubbleDomainReadout.original_e_cubed_daughters_have_a_growing_half_and_positive_reproduction example : SpectralVolumeStationarityReadout.physicalNormalizedStationaryVolumeMode |>.RetainsPrintedGrowingVolumeNormalizedStationarityAndSpectralGap := SpectralVolumeStationarityReadout.original_leading_eigenmode_grows_while_its_normalized_density_is_stationary example {T : LLMTypes} (framework : StochasticInflationFramework T) (setup : framework.OrdinaryDiffusionSetup) (calculation : framework.BrownianStepSpreadCalculation) : framework.SelectedPrintedHubbleFluctuationSourceAt := framework.brownian_route_retains_original_hubble_step_variance_and_light_mass_limit setup calculation example {T : LLMTypes} (framework : StochasticInflationFramework T) (setup : framework.SelfReproductionSetup) (calculation : framework.MicrostationarityCalculation) : framework.SelectedPrintedSelfReproducingHubbleDomainsAt := framework.reproduction_route_retains_original_e_cubed_growing_hubble_domains setup calculation example {T : LLMTypes} (framework : StochasticInflationFramework T) (model : T.Model) (forward : framework.ForwardBranchingEquation) (boundary : framework.BoundaryControlledSetup model) (solved : framework.SolvedStationaryModelClass model) (calculation : framework.StationaryVolumeCalculation model) : framework.SelectedPrintedNormalizedSpectralStationarityAt model := framework.stationary_route_retains_growing_raw_volume_and_normalized_spectral_mode model forward boundary solved calculation example : PhysicsLogic.Papers.LindeLindeMezhlumian1993.Fidelity.countermodelMatrix .brownianVarianceNotDiffusionCoefficient := Countermodels.brownian_variance_does_not_fix_the_factor_of_two_diffusion_coefficient example : PhysicsLogic.Papers.LindeLindeMezhlumian1993.Fidelity.countermodelMatrix .eCubedDomainsNotGrowingHalf := Countermodels.e_cubed_total_domains_do_not_select_the_rising_field_half example : PhysicsLogic.Papers.LindeLindeMezhlumian1993.Fidelity.countermodelMatrix .normalizedStationarityNotRawVolumeStationarity := Countermodels.stationary_normalized_volume_does_not_make_raw_volume_stationary example : PhysicsLogic.Papers.LindeLindeMezhlumian1993.Fidelity.countermodelMatrix .rawVolumeGrowthNotNormalizedStationarity := Countermodels.growing_raw_volume_does_not_supply_normalized_stationarity example : PhysicsLogic.Papers.LindeLindeMezhlumian1993.Fidelity.countermodelMatrix .spectralGapNotInitialPlanckBoundary := Countermodels.a_spectral_gap_does_not_supply_the_initial_planck_boundary example : PhysicsLogic.Papers.LindeLindeMezhlumian1993.Fidelity.countermodelMatrix .initialPlanckBoundaryNotFinalPlanckBoundary := Countermodels.the_initial_planck_boundary_does_not_supply_the_final_planck_boundary example : PhysicsLogic.Papers.LindeLindeMezhlumian1993.Fidelity.countermodelMatrix .normalizedStationarityNotClockIndependentDensity := Countermodels.a_normalized_stationary_volume_mode_does_not_identify_clock_densities end LindeLindeMezhlumianOriginalNormalizedVolumeAndHubbleReuse section CasimirOriginalGaussianRegressionKnudsenAndHallReuse open PhysicsLogic.Papers.Casimir1945 open PhysicsLogic.Papers.Casimir1945.MicroscopicReversibilityAnalysis example : (EquilibriumGaussianRegressionReadout.physicalDiagonalGaussianAndIntermediateRegression |>.RetainsPrintedQuadraticEntropyGaussianAndConjugateCovariance) ∧ (EquilibriumGaussianRegressionReadout.physicalDiagonalGaussianAndIntermediateRegression |>.RetainsPrintedIntermediateLagAndIndependentFiniteTimeRegression) := EquilibriumGaussianRegressionReadout.original_gaussian_fluctuations_and_independent_lagged_regression_are_inhabited example : ThermomolecularOrificeReadout.physicalSmallAndLargeOrificeKnudsenTransport |>.RetainsPrintedSmallHoleLargeHoleAndKnudsenPressureLaws := ThermomolecularOrificeReadout.original_small_and_large_orifices_have_distinct_knudsen_transport_energies example : FieldReversedResistanceHallReadout.physicalFieldReversedOddHallTensor |>.RetainsPrintedFieldReversedResistanceAndOddHallTensor := FieldReversedResistanceHallReadout.original_field_reversal_preserves_even_resistance_and_reverses_nonzero_hall_response example {T : CasimirTypes} (analysis : MicroscopicReversibilityAnalysis T) (statistics : analysis.FluctuationStatisticsResult) : analysis.gaussianRegressionReadout |>.RetainsPrintedQuadraticEntropyGaussianAndConjugateCovariance := analysis.fluctuation_statistics_route_retains_original_gaussian_entropy_and_covariance statistics example {T : CasimirTypes} (analysis : MicroscopicReversibilityAnalysis T) (regression : analysis.RegressionScopeResult) : analysis.gaussianRegressionReadout |>.RetainsPrintedIntermediateLagAndIndependentFiniteTimeRegression := analysis.regression_route_retains_original_intermediate_lag_and_zero_instantaneous_derivative regression example {T : CasimirTypes} (analysis : MicroscopicReversibilityAnalysis T) (transport : analysis.ThermomolecularResult) : analysis.thermomolecularOrificeReadout |>.RetainsPrintedSmallHoleLargeHoleAndKnudsenPressureLaws := analysis.thermomolecular_route_retains_original_small_and_large_orifice_energies transport example {T : CasimirTypes} (analysis : MicroscopicReversibilityAnalysis T) (electrical : analysis.ElectricalFourPoleResult) : analysis.fieldReversedResistanceHallReadout |>.RetainsPrintedFieldReversedResistanceAndOddHallTensor := analysis.electrical_route_retains_original_field_reversed_odd_hall_tensor electrical example : PhysicsLogic.Papers.Casimir1945.Fidelity.countermodelMatrix .gaussianFluctuationsNotRegressionWindow := Countermodels.complete_gaussian_fluctuation_data_do_not_supply_the_regression_window example : PhysicsLogic.Papers.Casimir1945.Fidelity.countermodelMatrix .regressionWindowNotPositiveEntropyCurvature := Countermodels.an_intermediate_regression_window_does_not_supply_positive_entropy_curvature example : PhysicsLogic.Papers.Casimir1945.Fidelity.countermodelMatrix .zeroInstantaneousDerivativeNotLaggedRegression := by obtain ⟨slope, lag, zero⟩ := GaussianOscillator.the_same_zero_initial_slope_can_have_zero_finite_lag_change exact ⟨⟨GaussianOscillator.the_reversible_gaussian_oscillator_has_two_nonvacuous_fluctuating_observables, slope, lag⟩, fun nonzero => nonzero zero⟩ example : PhysicsLogic.Papers.Casimir1945.Fidelity.countermodelMatrix .smallOrificeEnergyNotLargeOrificeEnergy := Countermodels.small_orifice_energy_does_not_fix_large_orifice_energy example : PhysicsLogic.Papers.Casimir1945.Fidelity.countermodelMatrix .virtualEntropyStationarityNotZeroEntropyProduction := Countermodels.stationary_virtual_entropy_does_not_eliminate_irreversible_entropy_production example : PhysicsLogic.Papers.Casimir1945.Fidelity.countermodelMatrix .fieldReversedResistanceNotSameFieldSymmetry := Countermodels.field_reversed_cross_resistance_does_not_imply_same_field_symmetry example : PhysicsLogic.Papers.Casimir1945.Fidelity.countermodelMatrix .oneSelectedFourPoleNotRequiredFamily := Countermodels.one_selected_four_pole_does_not_cover_the_required_four_pole_family end CasimirOriginalGaussianRegressionKnudsenAndHallReuse section HaldaneOriginalMonopoleQuasiparticleAndHierarchyReuse open PhysicsLogic.Papers.Haldane1983 open PhysicsLogic.Papers.Haldane1983.HierarchyFramework example : SphericalMonopoleFormulaReadout.physicalThreeElectronMonopoleSphere |>.SphericalMonopoleAndLowestLandauFormulaAt := SphericalMonopoleFormulaReadout.original_six_flux_monopole_realizes_radial_field_and_seven_landau_states example : (ParentFluxFillingFormulaReadout.physicalOddThreeParentFilling |>.ParentFluxAndFillingFormulaAt) ∧ (ParentFluxFillingFormulaReadout.physicalOddThreeParentFilling |>.selectedFiniteFillingValue) ≠ (ParentFluxFillingFormulaReadout.physicalOddThreeParentFilling |>.thermodynamicFillingValue) := ParentFluxFillingFormulaReadout.original_finite_sphere_filling_differs_from_its_one_third_limit example : FixedFieldChargeFormulaReadout.physicalOddThreeFixedFieldQuasiparticles |>.ExcitationAndFixedFieldChargeFormulaAt := FixedFieldChargeFormulaReadout.original_fixed_field_electron_creates_three_fractionally_charged_holes example : ContinuedFractionHierarchyReadout.physicalOneThirdParentTwoSeventhsAndTwoFifthsChildren |>.RetainsPrintedOddParentEvenBosonicChildrenAndHallSequence := ContinuedFractionHierarchyReadout.original_one_third_parent_has_distinct_two_sevenths_and_two_fifths_children.1 example {T : HaldaneTypes} (framework : HierarchyFramework T) (observables : framework.EstablishedHierarchyObservables) : framework.continuedFractionHierarchyReadout |>.RetainsPrintedOddParentEvenBosonicChildrenAndHallSequence := framework.hierarchy_observable_route_retains_original_continued_fraction_branches observables example : PhysicsLogic.Papers.Haldane1983.Fidelity.countermodelMatrix .diracTotalFluxNotLowestLandauDegeneracy := Countermodels.dirac_flux_count_does_not_fix_the_lowest_landau_degeneracy example : PhysicsLogic.Papers.Haldane1983.Fidelity.countermodelMatrix .finiteSphereFillingNotThermodynamicParentLimit := Countermodels.finite_sphere_filling_does_not_equal_the_thermodynamic_parent_limit example : PhysicsLogic.Papers.Haldane1983.Fidelity.countermodelMatrix .fixedFieldHoleCountNotHoleDegeneracy := Countermodels.fixed_field_hole_count_does_not_fix_the_one_hole_degeneracy example : PhysicsLogic.Papers.Haldane1983.Fidelity.countermodelMatrix .twoSeventhsBranchNotTwoFifthsBranch := Countermodels.the_two_sevenths_branch_does_not_determine_the_two_fifths_branch example : PhysicsLogic.Papers.Haldane1983.Fidelity.countermodelMatrix .positiveFiniteChildFillingNotTwoSevenths := Countermodels.positive_child_finite_filling_does_not_equal_two_sevenths example : PhysicsLogic.Papers.Haldane1983.Fidelity.countermodelMatrix .positiveHierarchyChargeNotHallResistance := Countermodels.positive_hierarchy_charge_does_not_fix_its_hall_resistance example : PhysicsLogic.Papers.Haldane1983.Fidelity.countermodelMatrix .weakExcitationInteractionNotPositiveChildGap := Countermodels.weak_excitation_interactions_do_not_supply_a_positive_child_gap example : PhysicsLogic.Papers.Haldane1983.Fidelity.countermodelMatrix .completePrintedHierarchyNotPositiveChildGap := Countermodels.complete_printed_hierarchy_data_do_not_supply_a_positive_child_gap end HaldaneOriginalMonopoleQuasiparticleAndHierarchyReuse section ForsterNelsonStephenOriginalForcingProjectorAndCriticalDimensionReuse open PhysicsLogic.Papers.ForsterNelsonStephen1977 open PhysicsLogic.Papers.ForsterNelsonStephen1977.DynamicRGFramework example : EquationThreeFiveProjectorReadout.physicalThreeFourZeroTransverseProjector |>.SourceFormulaAt := EquationThreeFiveProjectorReadout.original_nonzero_wave_vector_realizes_the_printed_transverse_projector example : (EquationThreeFiveProjectorReadout.physicalThreeFourZeroTransverseProjector |>.projectorValueAt 0 1) = -(12 / 25 : Real) ∧ (EquationThreeFiveProjectorReadout.physicalThreeFourZeroTransverseProjector |>.correlationValueAt 0 1) = -12 := EquationThreeFiveProjectorReadout.original_transverse_projector_has_nonzero_negative_cross_correlation example : EquationThreeNineFormulaReadout.physicalNontrivialShellVelocityScaling |>.SourceFormulaAt := EquationThreeNineFormulaReadout.original_shell_formula_has_distinct_wave_frequency_and_coupling_rescalings.1 example : FiniteBandForcingReadout.physicalQuadraticConstantAndFiniteBandForcing |>.RetainsPrintedQuadraticConstantAndFiniteBandForcing := FiniteBandForcingReadout.original_models_a_b_and_c_have_distinct_zero_mode_and_finite_band_forcing example : InfraredCriticalDimensionReadout.physicalThreeDimensionalDistinctCriticalRegimes |>.RetainsPrintedTwoAndFourDimensionalInfraredThresholds := InfraredCriticalDimensionReadout.original_three_dimensional_models_have_distinct_critical_dimensions_and_exponents example {T : FNSTypes} (framework : DynamicRGFramework T) (forcing : framework.forcingNonzeroAtZeroWaveNumber framework.modelB) (calculation : framework.ModelBCalculation) : (framework.finiteBandForcingReadout |>.RetainsPrintedQuadraticConstantAndFiniteBandForcing) ∧ (framework.infraredCriticalDimensionReadout |>.RetainsPrintedTwoAndFourDimensionalInfraredThresholds) := framework.model_b_route_retains_original_zero_mode_forcing_and_four_dimensional_threshold forcing calculation example {T : FNSTypes} (framework : DynamicRGFramework T) (setup : framework.ModelCUniversalitySetup) (calculation : framework.ModelCFlowCalculation) (bridge : framework.InfraredClassBridge) : framework.finiteBandForcingReadout |>.RetainsPrintedQuadraticConstantAndFiniteBandForcing := framework.model_c_route_retains_original_finite_pulse_and_generated_quadratic_force setup calculation bridge example : PhysicsLogic.Papers.ForsterNelsonStephen1977.Fidelity.countermodelMatrix .nonzeroWaveVectorNotCrossProjector := Countermodels.a_nonzero_wave_vector_does_not_fix_the_transverse_cross_projector example : PhysicsLogic.Papers.ForsterNelsonStephen1977.Fidelity.countermodelMatrix .distinctRescalingsNotRunningCoupling := Countermodels.distinct_wave_and_frequency_scales_do_not_force_coupling_running example : PhysicsLogic.Papers.ForsterNelsonStephen1977.Fidelity.countermodelMatrix .vanishingModelAZeroModeNotQuadraticForcing := Countermodels.vanishing_model_a_zero_mode_does_not_fix_quadratic_infrared_forcing example : PhysicsLogic.Papers.ForsterNelsonStephen1977.Fidelity.countermodelMatrix .finitePulseNotGeneratedQuadraticForce := Countermodels.a_finite_band_pulse_does_not_derive_generated_quadratic_infrared_force example : PhysicsLogic.Papers.ForsterNelsonStephen1977.Fidelity.countermodelMatrix .modelACriticalDimensionNotModelBThreshold := Countermodels.the_model_a_critical_dimension_does_not_fix_the_model_b_critical_dimension example : PhysicsLogic.Papers.ForsterNelsonStephen1977.Fidelity.countermodelMatrix .modelAAllOrderProtectionNotModelBAllOrderProof := Countermodels.model_a_higher_order_protection_does_not_prove_model_b_to_all_orders example : PhysicsLogic.Papers.ForsterNelsonStephen1977.Fidelity.countermodelMatrix .infraredScaleSeparationNotUltravioletControl := Countermodels.separated_infrared_and_ultraviolet_scales_do_not_control_ultraviolet_turbulence example : PhysicsLogic.Papers.ForsterNelsonStephen1977.Fidelity.countermodelMatrix .selectedModelCFlowNotAllFiniteBandUniversality := Countermodels.the_selected_model_c_flow_does_not_prove_universality_for_every_finite_band_force end ForsterNelsonStephenOriginalForcingProjectorAndCriticalDimensionReuse section MinkowskiOriginalHyperboloidProperTimeFourForceAndRetardationReuse open PhysicsLogic.Papers.Minkowski1908 open PhysicsLogic.Papers.Minkowski1908.SpacetimeContext example : OriginalFiniteBoostReadout.physicalThreeFifthsHyperbolicBoost |>.RetainsOriginalBoostHyperboloidLightRayAndReciprocalContraction := OriginalFiniteBoostReadout.original_nonidentity_boost_preserves_upper_hyperboloid_light_and_reciprocal_rods example : OriginalConeProperTimeReadout.physicalThreeFourFiveProperTimeAndReversedSpacelikeOrder |>.RetainsOriginalConeProperTimeAccelerationAndOrder := OriginalConeProperTimeReadout.original_cones_have_real_proper_time_orthogonal_acceleration_and_reversible_spacelike_order example : OriginalFourForceRetardedChargeReadout.physicalFourForceAndThreeFourFiveRetardedCharge |>.RetainsOriginalFourForceRetardedChargeAndQualifiedAnalogy := OriginalFourForceRetardedChargeReadout.original_four_force_retarded_potential_and_qualified_gravity_analogy_are_inhabited example {T : MinkowskiTypes} (context : SpacetimeContext T) (invariance : context.EstablishedFiniteCInvariance) : context.originalFiniteBoostReadout |>.RetainsOriginalBoostHyperboloidLightRayAndReciprocalContraction := context.finite_c_route_retains_original_nontrivial_boost_light_and_reciprocal_rods invariance example {T : MinkowskiTypes} (context : SpacetimeContext T) (geometry : context.EstablishedCausalProperTimeGeometry) : context.originalConeProperTimeReadout |>.RetainsOriginalConeProperTimeAccelerationAndOrder := context.causal_route_retains_original_cones_proper_time_and_spacelike_order geometry example {T : MinkowskiTypes} (context : SpacetimeContext T) (dynamics : context.EstablishedFourDimensionalDynamics) : context.originalFourForceRetardedChargeReadout |>.HasOriginalFourMomentumForceAndEnergy := context.dynamics_route_retains_original_four_momentum_force_and_energy dynamics example {T : MinkowskiTypes} (context : SpacetimeContext T) (electromagnetism : context.EstablishedPointChargeElectrodynamics) : context.originalFourForceRetardedChargeReadout |>.HasOriginalUniqueRetardedNullChargePotential := context.point_charge_route_retains_original_unique_retarded_null_potential electromagnetism example {T : MinkowskiTypes} (context : SpacetimeContext T) (gravity : context.EstablishedQualifiedGravityProposal) : context.originalFourForceRetardedChargeReadout |>.HasOriginalQualifiedChargeMassReplacement := context.gravity_route_retains_original_charge_mass_replacement_and_open_scope gravity example : PhysicsLogic.Papers.Minkowski1908.Fidelity.countermodelMatrix .preservedQuadraticValueNotUpperSheet := Countermodels.preserving_the_quadratic_value_does_not_select_the_upper_hyperboloid_sheet example : PhysicsLogic.Papers.Minkowski1908.Fidelity.countermodelMatrix .oneRodContractionNotReciprocalContraction := Countermodels.one_contracted_rod_does_not_supply_the_reciprocal_contraction example : PhysicsLogic.Papers.Minkowski1908.Fidelity.countermodelMatrix .timelikeIntervalNotProperTimeValue := Countermodels.a_timelike_interval_does_not_fix_an_unrelated_proper_time_value example : PhysicsLogic.Papers.Minkowski1908.Fidelity.countermodelMatrix .normalizedFourVelocityNotOrthogonalAcceleration := Countermodels.normalized_four_velocity_does_not_make_an_independent_acceleration_orthogonal example : PhysicsLogic.Papers.Minkowski1908.Fidelity.countermodelMatrix .nullRetardationNotUniqueIntersection := Countermodels.a_null_retarded_separation_does_not_prove_unique_source_intersection example : PhysicsLogic.Papers.Minkowski1908.Fidelity.countermodelMatrix .electricChargeProductNotGravityMassProduct := Countermodels.the_electric_charge_product_is_not_the_proposed_mass_product end MinkowskiOriginalHyperboloidProperTimeFourForceAndRetardationReuse section EverettOriginalRelativeStateMemoryProductMeasureAndInterventionReuse open PhysicsLogic.Papers.Everett1957 open PhysicsLogic.Papers.Everett1957.RelativeStateFramework example : OriginalRelativeSubsystemReadout.physicalThreeFifthsFourFifthsRelativeStates |>.RetainsOriginalCompositeExpansionAndRelativeSubsystemStates := OriginalRelativeSubsystemReadout.original_unequal_composite_coefficients_have_distinct_normalized_relative_states example : OriginalCalibratedBranchMemoryReadout.physicalUnequalTwoBranchObserverMemory |>.RetainsOriginalLinearBranchMemoryAndUnequalWeights := OriginalCalibratedBranchMemoryReadout.original_two_retained_observer_branches_have_unequal_phase_insensitive_weights example : SquareAmplitudeFormulaReadout.physicalUnequalNormalizedSquareAmplitudeMeasure |>.SourceFormulaAt := SquareAmplitudeFormulaReadout.original_normalized_square_amplitude_measure_has_unequal_branch_weights.1 example : OriginalFiniteProductMeasureReadout.physicalUnequalTwoObservationProductMeasure |>.RetainsOriginalProductMeasureFiniteExceptionsAndTypicalityLimit := OriginalFiniteProductMeasureReadout.original_product_measure_differs_from_branch_counting_and_retains_finite_exceptions example (length : Nat) : 0 < OriginalFiniteProductMeasureReadout.physicalUnequalTwoObservationProductMeasure.exceptionalTailValueAt length := OriginalFiniteProductMeasureReadout.original_selected_exceptional_tail_is_positive_at_every_finite_depth length example : OriginalObserverInterventionReadout.physicalCommunicatingLocalAndRemoteObservers |>.RetainsOriginalAgreementLocalDisturbanceAndRemoteNoninteraction := OriginalObserverInterventionReadout.original_noncommuting_observation_disturbs_local_repeat_but_remote_observation_does_not example {T : EverettTypes} (framework : RelativeStateFramework T) (setup : framework.RelativeStateSetup) (calculation : framework.RelativeStateCalculation) : framework.originalRelativeSubsystemReadout |>.RetainsOriginalCompositeExpansionAndRelativeSubsystemStates := framework.relative_state_route_retains_original_normalized_conditional_coefficients setup calculation example {T : EverettTypes} (framework : RelativeStateFramework T) (setup : framework.GoodMeasurementSetup) (calculation : framework.LinearMeasurementCalculation) : framework.originalCalibratedBranchMemoryReadout |>.RetainsOriginalLinearBranchMemoryAndUnequalWeights := framework.measurement_route_retains_original_unequal_relative_memory_branches setup calculation example {T : EverettTypes} (framework : RelativeStateFramework T) (setup : framework.StatisticalTypicalitySetup) (calculation : framework.StatisticalTypicalityCalculation) : framework.originalFiniteProductMeasureReadout |>.RetainsOriginalProductMeasureFiniteExceptionsAndTypicalityLimit := framework.statistical_route_retains_original_product_measure_and_finite_exceptions setup calculation example {T : EverettTypes} (framework : RelativeStateFramework T) (setup : framework.StatisticalTypicalitySetup) (calculation : framework.StatisticalTypicalityCalculation) : Filter.Tendsto framework.originalFiniteProductMeasureReadout.exceptionalTailValueAt Filter.atTop (nhds 0) ∧ framework.exceptionalSetHasMeasureTendingToZero framework.exceptionalSequences framework.selectedMeasure framework.asymptoticRegime := framework.statistical_route_distinguishes_selected_history_from_exceptional_set_limit setup calculation example {T : EverettTypes} (framework : RelativeStateFramework T) (setup : framework.MultipleObserverSetup) (calculation : framework.MultipleObserverCalculation) : framework.originalObserverInterventionReadout |>.RetainsOriginalAgreementLocalDisturbanceAndRemoteNoninteraction := framework.observer_route_retains_original_local_disturbance_and_remote_stability setup calculation example : PhysicsLogic.Papers.Everett1957.Fidelity.countermodelMatrix .distinctRelativeStatesNotAbsoluteSubsystemState := Countermodels.distinct_relative_states_do_not_supply_an_independent_absolute_state example : PhysicsLogic.Papers.Everett1957.Fidelity.countermodelMatrix .equalBranchCountNotEqualBranchWeight := Countermodels.two_equally_counted_branches_do_not_have_equal_square_amplitude_weight example : PhysicsLogic.Papers.Everett1957.Fidelity.countermodelMatrix .phaseInsensitiveWeightNotSignedAmplitude := Countermodels.an_amplitude_phase_reversal_does_not_preserve_the_signed_amplitude example : PhysicsLogic.Papers.Everett1957.Fidelity.countermodelMatrix .mixedBranchCountNotMixedBranchMeasure := Countermodels.counting_mixed_branches_does_not_give_their_square_amplitude_measure example : PhysicsLogic.Papers.Everett1957.Fidelity.countermodelMatrix .zeroAsymptoticExceptionNotZeroFiniteException := Countermodels.a_vanishing_asymptotic_exceptional_measure_does_not_vanish_at_finite_length example : PhysicsLogic.Papers.Everett1957.Fidelity.countermodelMatrix .selectedHistoryDecayNotWholeExceptionalSetDecay := Countermodels.a_vanishing_selected_product_history_does_not_make_the_whole_exceptional_set_vanish example : PhysicsLogic.Papers.Everett1957.Fidelity.countermodelMatrix .localNoncommutingDisturbanceNotRemoteDisturbance := Countermodels.a_noncommuting_local_disturbance_does_not_disturb_a_remote_repeat end EverettOriginalRelativeStateMemoryProductMeasureAndInterventionReuse section LandauOriginalRotonCounterflowAndSecondSoundReuse open PhysicsLogic.Papers.LandauSuperfluidity1941 open PhysicsLogic.Papers.LandauSuperfluidity1941.HeliumTwoMotionFramework example : OriginalPhononRotonThermodynamicReadout.physicalOriginalQuadraticRotonAndLowTemperatureGas |>.RetainsOriginalSpectrumThresholdAndThermodynamics := OriginalPhononRotonThermodynamicReadout.original_zero_momentum_roton_threshold_and_dilute_heat_laws_are_inhabited example : OriginalTwoMotionCounterflowReadout.physicalTwoMotionCounterflow |>.HasZeroMassCounterflowWithNonzeroHeat := OriginalTwoMotionCounterflowReadout.the_physical_control_realizes_heat_counterflow example : OriginalTwoMotionCounterflowReadout.physicalTwoMotionCounterflow |>.RetainsRotatingMassFractions := OriginalTwoMotionCounterflowReadout.the_physical_control_retains_rotating_mass_fractions example : OriginalTwoSoundHydrodynamicReadout.physicalOriginalTwoSoundModes |>.RetainsOriginalSoundSpeedsLimitsAndViscosityQualification := OriginalTwoSoundHydrodynamicReadout.original_second_sound_and_temperature_limits_do_not_erase_normal_viscosity example {T : LandauTypes} (framework : HeliumTwoMotionFramework T) (thermodynamics : framework.EstablishedExcitationThermodynamics) : framework.originalPhononRotonThermodynamicReadout |>.RetainsOriginalSpectrumAndLowTemperatureThermodynamics := framework.thermodynamic_route_retains_original_roton_dispersion_and_heat_capacity thermodynamics example {T : LandauTypes} (framework : HeliumTwoMotionFramework T) (flow : framework.EstablishedSubthresholdFlow) : framework.originalPhononRotonThermodynamicReadout |>.HasOriginalAttainedPhononAndRotonWallThresholds := framework.wall_flow_route_retains_original_phonon_and_roton_thresholds flow example {T : LandauTypes} (framework : HeliumTwoMotionFramework T) (statistics : framework.EstablishedTwoMotionStatistics) : framework.originalTwoMotionCounterflowReadout |>.RetainsRotatingMassFractions := framework.rotating_vessel_route_retains_carried_inertia_and_temperature_fractions statistics example {T : LandauTypes} (framework : HeliumTwoMotionFramework T) (hydrodynamics : framework.EstablishedHydrodynamics) : framework.originalTwoSoundHydrodynamicReadout |>.RetainsOriginalSoundSpeedsLimitsAndViscosityQualification := framework.hydrodynamic_route_retains_original_second_sound_and_viscosity_scope hydrodynamics example : PhysicsLogic.Papers.LandauSuperfluidity1941.Fidelity.countermodelMatrix .rotonEnergyMinimumNotVelocityMinimum := Countermodels.the_original_roton_energy_minimum_is_not_the_velocity_minimizing_momentum example : PhysicsLogic.Papers.LandauSuperfluidity1941.Fidelity.countermodelMatrix .phononSubthresholdNotRotonSubthreshold := Countermodels.the_phonon_speed_bound_does_not_exclude_roton_creation example : PhysicsLogic.Papers.LandauSuperfluidity1941.Fidelity.countermodelMatrix .completeThermodynamicsNotIndependentWallThreshold := Countermodels.complete_low_temperature_thermodynamics_does_not_supply_the_independent_wall_threshold example : PhysicsLogic.Papers.LandauSuperfluidity1941.Fidelity.countermodelMatrix .twoMotionDensityNotParticleSpeciesPartition := Countermodels.a_two_motion_density_decomposition_is_not_a_partition_of_helium_atoms example : PhysicsLogic.Papers.LandauSuperfluidity1941.Fidelity.countermodelMatrix .zeroMassCurrentNotZeroEntropyTransport := Countermodels.zero_total_mass_current_does_not_mean_zero_entropy_transport example : PhysicsLogic.Papers.LandauSuperfluidity1941.Fidelity.countermodelMatrix .firstSoundSpeedNotSecondSoundSpeed := Countermodels.the_first_sound_speed_does_not_equal_the_second_sound_speed example : PhysicsLogic.Papers.LandauSuperfluidity1941.Fidelity.countermodelMatrix .omittedNormalViscosityNotVanishingPhysicalViscosity := Countermodels.omitting_normal_viscosity_from_the_displayed_equations_does_not_make_it_zero end LandauOriginalRotonCounterflowAndSecondSoundReuse section CallenWeltonOriginalQuantumNoiseBrownianRadiationAcousticAndCoherenceReuse open PhysicsLogic.Papers.CallenWelton1951 open PhysicsLogic.Papers.CallenWelton1951.GeneralizedNoiseFramework example : GeneralizedNyquistFormulaReadout.physicalOriginalQuantumNyquistReadout |>.SourceFormulaAt := GeneralizedNyquistFormulaReadout.original_quantum_nyquist_formula_and_inhabited_classical_regime_are_consistent.1 example : OriginalSpectralBalanceReadout.physicalOriginalBoltzmannQuantumNoiseAndClassicalLimit |>.RetainsOriginalDetailedBalanceQuantumNoiseAndClassicalScope := OriginalSpectralBalanceReadout.original_boltzmann_difference_sum_zero_point_and_classical_scope_are_inhabited example : OriginalBrownianRadiationAcousticReadout.physicalOriginalBrownianVacuumAndWallNoise |>.RetainsOriginalBrownianVacuumAndAcousticApplications := OriginalBrownianRadiationAcousticReadout.original_brownian_nonwhite_dipole_planck_and_double_wall_noise_are_inhabited example : OriginalTwoSystemCoherenceReadout.physicalOriginalDistinctSourceWithOpenIrreversibilityProgram |>.RetainsOriginalTwoSystemBackreactionAndOpenProgram := OriginalTwoSystemCoherenceReadout.original_distinct_source_loses_coherence_without_completing_irreversibility_theory example {T : CallenWeltonTypes} (framework : GeneralizedNoiseFramework T) (generalized : framework.EstablishedGeneralizedNoiseDissipation) (classicalRegime : framework.originalSpectralBalanceReadout.reducedPlanckConstantValue * framework.originalSpectralBalanceReadout.frequencyValue < framework.originalSpectralBalanceReadout.classicalThermalEnergyValue) : framework.originalSpectralBalanceReadout |>.RetainsOriginalDetailedBalanceQuantumNoiseAndClassicalScope := by fail_if_success exact framework.generalized_relation_retains_original_detailed_balance_and_quantum_noise generalized exact framework.generalized_relation_retains_original_detailed_balance_and_quantum_noise generalized classicalRegime example {T : CallenWeltonTypes} (framework : GeneralizedNoiseFramework T) (setup : framework.BrownianApplicationSetup) (calculation : framework.BrownianApplicationCalculation) (frequency : T.Frequency) (inSector : framework.selectedFrequencySector frequency) : framework.originalApplicationReadout.HasOriginalNonwhiteBrownianNoise := framework.brownian_route_retains_original_friction_and_nonwhite_quantum_spectrum setup calculation frequency inSector example {T : CallenWeltonTypes} (framework : GeneralizedNoiseFramework T) (setup : framework.VacuumRadiationSetup) (calculation : framework.VacuumRadiationCalculation) (frequency : T.Frequency) (inSector : framework.selectedFrequencySector frequency) : framework.originalApplicationReadout |>.HasOriginalDipoleResistanceAndSplitVacuumDensity := framework.vacuum_route_retains_original_dipole_resistance_and_split_energy_density setup calculation frequency inSector example {T : CallenWeltonTypes} (framework : GeneralizedNoiseFramework T) (setup : framework.AcousticApplicationSetup) (calculation : framework.AcousticApplicationCalculation) (frequency : T.Frequency) (inSector : framework.selectedFrequencySector frequency) : framework.originalApplicationReadout |>.HasOriginalVolumeWallNodeAndAdiabaticSound := framework.acoustic_route_retains_original_double_wall_noise_and_adiabatic_sound setup calculation frequency inSector example {T : CallenWeltonTypes} (framework : GeneralizedNoiseFramework T) (result : framework.CompleteInterpretationAndScope) : framework.originalTwoSystemCoherenceReadout |>.RetainsOriginalTwoSystemBackreactionAndOpenProgram := framework.coherence_route_retains_original_two_system_transfer_and_open_program result example : PhysicsLogic.Papers.CallenWelton1951.Fidelity.countermodelMatrix .detailedBalanceNotDissipativeDifference := Countermodels.detailed_balance_does_not_supply_the_dissipative_transition_difference example : PhysicsLogic.Papers.CallenWelton1951.Fidelity.countermodelMatrix .dissipativeDifferenceNotFluctuationSum := Countermodels.dissipative_transition_difference_does_not_supply_fluctuation_sum example : PhysicsLogic.Papers.CallenWelton1951.Fidelity.countermodelMatrix .zeroPointEnergyNotPlanckOccupation := Countermodels.positive_zero_point_energy_does_not_supply_the_planck_occupation_term example : PhysicsLogic.Papers.CallenWelton1951.Fidelity.countermodelMatrix .classicalRegimeNotExactEquipartition := by obtain ⟨formula, regime, notEquality⟩ := QuantumResponse.Countermodels.the_computed_oscillator_energy_is_not_exactly_classical_in_the_comparison_regime exact ⟨⟨formula, regime⟩, notEquality⟩ example : PhysicsLogic.Papers.CallenWelton1951.Fidelity.countermodelMatrix .lowFrequencyBrownianNoiseNotHighFrequencyWhiteNoise := Countermodels.correct_low_frequency_brownian_noise_does_not_make_high_frequency_noise_white example : PhysicsLogic.Papers.CallenWelton1951.Fidelity.countermodelMatrix .volumePressureNotDoubleWallPressure := Countermodels.positive_volume_pressure_does_not_supply_the_doubled_wall_pressure example : PhysicsLogic.Papers.CallenWelton1951.Fidelity.countermodelMatrix .doubleWallPressureNotVelocityNode := Countermodels.doubled_wall_pressure_does_not_supply_the_rigid_wall_velocity_node example : PhysicsLogic.Papers.CallenWelton1951.Fidelity.countermodelMatrix .coherenceLossNotEnergyTransferBalance := Countermodels.loss_of_source_coherence_does_not_supply_the_energy_transfer_balance example : PhysicsLogic.Papers.CallenWelton1951.Fidelity.countermodelMatrix .proposedIrreversibilityNotCompletedTheory := Countermodels.a_proposed_future_irreversibility_program_is_not_a_completed_theory end CallenWeltonOriginalQuantumNoiseBrownianRadiationAcousticAndCoherenceReuse section AndersonOriginalConductorSpectrumScreeningGravityAndScopeReuse open PhysicsLogic.Papers.Anderson1963 open PhysicsLogic.Papers.Anderson1963.PlasmonFramework open PhysicsLogic.Framework.BrokenSymmetryModeConversionContext.Countermodels example : PlasmaPoleFormulaReadout.physicalOriginalConductorPoleReadout |>.EquationsFourSevenAndEightAt := PlasmaPoleFormulaReadout.original_positive_density_charge_mass_and_plasma_pole_are_inhabited.1 example : OriginalConductorResponseReadout.physicalOriginalNormalConductorReadout |>.RetainsOriginalConductorPoleScreeningAndGaplessMatter := OriginalConductorResponseReadout.original_self_consistent_plasma_has_gapped_gauge_modes_but_gapless_normal_matter.1 example : OriginalSpectralRenormalizationReadout.physicalOriginalConductorSpectralSaturationReadout |>.RetainsOriginalSaturationRenormalizationAndSurfaceCharge := OriginalSpectralRenormalizationReadout.original_saturated_spectral_weight_prevents_renormalization_without_losing_particles.1 example : OriginalSpectrumComparisonReadout.physicalOriginalNormalSuperconductingAndGravitationalReadout |>.RetainsOriginalDistinctSpectraGravityAndUnfinishedScope := OriginalSpectrumComparisonReadout.original_normal_plasma_superconducting_gap_and_gravitational_instability_are_distinct.1 example : physicalAnderson1963NormalConductorPlasmaReadout.physicalSpectrumContainsZeroMassMode = true ∧ physicalAnderson1963NormalConductorPlasmaReadout.physicalGaugeSectorContainsZeroMassMode = false ∧ physicalAnderson1963SuperconductingGoldstoneReadout.physicalSpectrumContainsZeroMassMode = false := by obtain ⟨_, gaplessNormal, gappedGauge, _, gappedSuperconductor, _⟩ := anderson_normal_plasma_retains_gapless_matter_while_gauge_and_superconductor_are_gapped exact ⟨gaplessNormal, gappedGauge, gappedSuperconductor⟩ example {T : AndersonTypes} (framework : PlasmonFramework T) (setup : framework.PlasmaResponseSetup) (calculation : framework.PlasmaResponseCalculation) : framework.originalConductorResponseReadout |>.HasOriginalGappedGaugeSectorAndGaplessNormalMatter := framework.normal_plasma_response_retains_gapless_matter_beneath_gapped_gauge_modes setup calculation example {T : AndersonTypes} (framework : PlasmonFramework T) (setup : framework.PlasmaResponseSetup) (calculation : framework.PlasmaResponseCalculation) : framework.originalSpectralRenormalizationReadout |>.RetainsOriginalSaturationRenormalizationAndSurfaceCharge := framework.normal_plasma_response_retains_saturated_spectral_weight_and_surface_charge setup calculation example {T : AndersonTypes} (framework : PlasmonFramework T) (setup : framework.PlasmaResponseSetup) (calculation : framework.PlasmaResponseCalculation) : framework.SelectedOriginalConductorResponseAt := framework.normal_plasma_response_retains_original_screened_conductor_values setup calculation example {T : AndersonTypes} (framework : PlasmonFramework T) (setup : framework.SuperconductingConversionSetup) (calculation : framework.CoulombModeConversionCalculation) : framework.originalSpectrumComparisonReadout |>.HasOriginalGaplessNormalMatterAndGappedSuperconductor := framework |>.superconducting_conversion_retains_gapless_normal_matter_and_fully_gapped_superconductor setup calculation example {T : AndersonTypes} (framework : PlasmonFramework T) (setup : framework.SuperconductingConversionSetup) (calculation : framework.CoulombModeConversionCalculation) : (framework.originalSpectrumComparisonReadout |>.HasOriginalGravitationalInstabilityInsteadOfPositiveMass) ∧ (framework.originalSpectrumComparisonReadout |>.HasOriginalUnfinishedNonAbelianAndBaryonProgram) := framework |>.superconducting_comparison_retains_gravitational_instability_and_open_gauge_program setup calculation example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .conductorResponseNotSelfConsistentInducedField := PlasmonFramework.Countermodels.conductor_response_does_not_supply_self_consistent_induced_field example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .gappedGaugeSectorNotFullyGappedNormalSpectrum := PlasmonFramework.Countermodels.a_gapped_gauge_sector_does_not_fully_gap_the_normal_plasma example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .longWavelengthDegeneracyNotFiniteMomentumDegeneracy := PlasmonFramework.Countermodels.equal_long_wavelength_plasma_frequencies_do_not_imply_equal_finite_momentum_dispersion example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .conductorSaturationNotPositiveRenormalization := PlasmonFramework.Countermodels.saturated_conductor_spectral_weight_does_not_leave_positive_renormalization example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .spectralSaturationNotSurfaceChargeRelocation := PlasmonFramework.Countermodels.spectral_saturation_does_not_supply_surface_charge_relocation example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .localizedChargeLossNotParticleLoss := PlasmonFramework.Countermodels.loss_of_localized_charge_does_not_destroy_particle_conservation example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .zeroMomentumPoleNotAllMomentumSingularity := PlasmonFramework.Countermodels.the_zero_momentum_polarizability_pole_is_not_a_pole_at_every_momentum example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .normalGaplessMatterNotGaplessSuperconductor := PlasmonFramework.Countermodels.gapless_normal_matter_does_not_imply_gapless_superconducting_matter example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .representationNoninvarianceNotObservableGaugeViolation := PlasmonFramework.Countermodels.noninvariant_matter_representation_does_not_break_observable_gauge_symmetry example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .positiveGravitationalScaleNotStableMass := PlasmonFramework.Countermodels.a_positive_gravitational_frequency_scale_does_not_make_the_mode_stable example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .phononGravitonCountNotStableTensorMode := PlasmonFramework.Countermodels.five_phonon_graviton_components_do_not_supply_a_stable_massive_tensor example : PhysicsLogic.Papers.Anderson1963.Fidelity.countermodelMatrix .unfinishedNonAbelianProgramNotBaryonGaugeModel := PlasmonFramework.Countermodels.an_unfinished_nonabelian_program_does_not_construct_baryon_gauge_conservation end AndersonOriginalConductorSpectrumScreeningGravityAndScopeReuse section ReggeWheelerOriginalWormholeConstraintAndStaticSourceReuse open PhysicsLogic.Papers.ReggeWheeler1957 open PhysicsLogic.Papers.ReggeWheeler1957.Framework open PhysicsLogic.Papers.ReggeWheeler1957.Framework.Countermodels example : OriginalOddExteriorReadout.originalExteriorAtThreeSchwarzschildMasses |>.HasOriginalOddMasterReduction := OriginalOddExteriorReadout.original_schwarzschild_exterior_realizes_the_tortoise_potential_and_odd_wave_equation example : OriginalWormholeBoundaryReadout.originalTwoEndedWormholeBoundary |>.HasSourceOddGrowingModeExclusion := OriginalWormholeBoundaryReadout.original_wormhole_has_double_ended_decay_but_no_smooth_growing_mode example : OriginalEvenStaticScopeReadout.originalConstrainedEvenStaticAndLinearScope |>.HasSeparateTwoEndedStaticAnalysis := OriginalEvenStaticScopeReadout.original_even_constraint_static_sources_and_first_order_scope_are_inhabited.2.2.1 example {T : ReggeWheelerTypes} (framework : Framework T) (parity : framework.EstablishedParityDecomposition) (setup : framework.OddMasterSetup) (calculation : framework.OddMasterCalculation) : framework.OriginalOddExteriorValuesAt := framework |>.odd_master_route_retains_original_exterior_tortoise_and_potential_values parity setup calculation example {T : ReggeWheelerTypes} (framework : Framework T) (master : framework.EstablishedOddMasterReduction) (setup : framework.OddStabilitySetup) (calculation : framework.OddStabilityCalculation) : framework.originalWormholeBoundaryReadout |>.HasSourceOddGrowingModeExclusion := framework |>.odd_stability_route_retains_double_decay_and_failed_wormhole_continuation master setup calculation example : PhysicsLogic.Papers.ReggeWheeler1957.Fidelity.countermodelMatrix .schwarzschildLapseNotTortoiseDerivative := schwarzschild_lapse_does_not_supply_the_printed_tortoise_derivative example : PhysicsLogic.Papers.ReggeWheeler1957.Fidelity.countermodelMatrix .throatMismatchNotInnerDivergence := the_original_failed_continuation_retains_decay_at_the_inner_boundary example : PhysicsLogic.Papers.ReggeWheeler1957.Fidelity.countermodelMatrix .wormholeExclusionNotUnequalWaveAmplitudes := the_wormhole_boundary_argument_does_not_require_unequal_wave_amplitudes example : PhysicsLogic.Papers.ReggeWheeler1957.Fidelity.countermodelMatrix .threeEvenEquationsNotIndependentConstraint := three_first_order_even_equations_do_not_supply_constraint_twenty_eight example : PhysicsLogic.Papers.ReggeWheeler1957.Fidelity.countermodelMatrix .firstIntegralTransportNotZeroConstraint := constraint_first_integral_transport_does_not_set_the_constraint_to_zero example : PhysicsLogic.Papers.ReggeWheeler1957.Fidelity.countermodelMatrix .evenConstraintNotLaterZerilliPotential := the_original_even_constraint_does_not_construct_a_later_zerilli_potential example : PhysicsLogic.Papers.ReggeWheeler1957.Fidelity.countermodelMatrix .oneSidedSourcesNotTwoEndedStaticAnalysis := a_static_source_on_one_side_does_not_supply_sources_on_both_wormhole_sides end ReggeWheelerOriginalWormholeConstraintAndStaticSourceReuse section ZehOriginalSpinEnvironmentChiralEprAndSymmetryReuse open PhysicsLogic.Papers.Zeh1970 open PhysicsLogic.Papers.Zeh1970.MeasurementFramework open PhysicsLogic.Papers.Zeh1970.MeasurementFramework.Countermodels example : OriginalSpinEnsembleReadout.originalOrthogonalSpinPreparations |>.HasOriginalDensityMatrixUnderdetermination := OriginalSpinEnsembleReadout.original_x_and_y_spin_ensembles_share_i_over_two_without_selecting_an_outcome example : OriginalEnvironmentalScaleReadout.originalDipoleAndRotatorScaleComparison |>.HasOriginalEnvironmentalScaleArgument := OriginalEnvironmentalScaleReadout.original_rotator_spacing_and_dipole_coupling_realize_a_distance_dependent_ratio example : OriginalBranchDynamicsReadout.originalRetainedChiralBranchDynamics |>.HasOriginalBranchDynamics := OriginalBranchDynamicsReadout.retained_chiral_branches_are_globally_pure_operationally_decoupled_and_uncollapsed example : OriginalEprSymmetryReadout.originalEprBranchesAndInvariantUniversalState |>.HasSymmetryInvariantGlobalSum := OriginalEprSymmetryReadout.epr_anticorrelation_and_asymmetric_branches_preserve_global_symmetry.2.2 example {T : ZehTypes} (framework : MeasurementFramework T) (setup : framework.DensityMatrixSetup) (calculation : framework.DensityMatrixCalculation) : framework.originalSpinEnsembleReadout |>.HasOriginalDensityMatrixUnderdetermination := framework.density_matrix_route_retains_original_x_and_y_spin_projectors setup calculation example {T : ZehTypes} (framework : MeasurementFramework T) (setup : framework.DensityMatrixSetup) (calculation : framework.DensityMatrixCalculation) : framework.originalSpinEnsembleReadout |>.HasAssumedRatherThanDerivedMeasurementPostulate := framework |>.density_matrix_trace_rule_presupposes_rather_than_derives_measurement_postulate setup calculation example {T : ZehTypes} (framework : MeasurementFramework T) (setup : framework.UniversalCorrelationSetup) (calculation : framework.UniversalCorrelationCalculation) : framework.originalEnvironmentalScaleReadout |>.HasOriginalEnvironmentalScaleArgument := framework |>.environmental_correlation_route_retains_original_rotator_and_dipole_scales setup calculation example {T : ZehTypes} (framework : MeasurementFramework T) (setup : framework.EnvironmentalSuperselectionSetup) (calculation : framework.EnvironmentalSuperselectionCalculation) : framework.originalEprSymmetryReadout.HasSymmetryBreakingObservedComponent ∧ framework.originalEprSymmetryReadout.HasSymmetryInvariantGlobalSum := framework |>.superselection_route_distinguishes_asymmetric_branch_from_invariant_world setup calculation example : PhysicsLogic.Papers.Zeh1970.Fidelity.countermodelMatrix .sameMixedDensityNotSamePreparation := the_same_maximally_mixed_density_does_not_identify_x_and_y_preparations example : PhysicsLogic.Papers.Zeh1970.Fidelity.countermodelMatrix .bornTraceNotDerivedMeasurementPostulate := the_born_trace_formula_does_not_independently_derive_the_measurement_postulate example : PhysicsLogic.Papers.Zeh1970.Fidelity.countermodelMatrix .strongNearCouplingNotAllDistanceStrong := strong_nearby_environmental_coupling_does_not_extend_to_every_distance example : PhysicsLogic.Papers.Zeh1970.Fidelity.countermodelMatrix .operationalDecouplingNotZeroCoupling := operational_branch_decoupling_does_not_require_zero_interbranch_coupling example : PhysicsLogic.Papers.Zeh1970.Fidelity.countermodelMatrix .slowSugarNotSlowAmmonia := slow_sugar_tunneling_does_not_imply_slow_ammonia_tunneling example : PhysicsLogic.Papers.Zeh1970.Fidelity.countermodelMatrix .observedAsymmetryNotGlobalAsymmetry := an_asymmetric_observed_component_does_not_break_global_symmetry example (state : EnvironmentalBranchState) (normalized : state.Normalized) : PhysicsLogic.Framework.FiniteQuantumState.IsPure state.globalDensity ∧ PhysicsLogic.Framework.FiniteQuantumState.IsDensity state.reducedDensity ∧ EnvironmentalBranchState.purity state.reducedDensity = state.weight 0 ^ 2 + state.weight 1 ^ 2 + 2 * state.weight 0 * state.weight 1 * Complex.normSq state.recordOverlap := ⟨state.actual_global_state_is_pure normalized, state.actual_reduced_state_is_density normalized, state.actual_reduced_purity normalized⟩ example (state : EnvironmentalBranchState) (normalized : state.Normalized) (populated : 0 < state.weight 0 ∧ 0 < state.weight 1) (distinguished : Complex.normSq state.recordOverlap < 1) : ¬PhysicsLogic.Framework.FiniteQuantumState.IsPure state.reducedDensity := (state.distinguished_records_give_actual_mixedness normalized populated distinguished).2.2 example : PhysicsLogic.Papers.Zeh1970.Fidelity.countermodelMatrix .reducedMixednessNotExactDiagonality := a_mixed_reduced_state_need_not_be_exactly_diagonal example : PhysicsLogic.Papers.Zeh1970.Fidelity.countermodelMatrix .twoPopulatedBranchesNotMixedness := two_populated_branches_do_not_supply_environmental_distinguishability example : PhysicsLogic.Papers.Zeh1970.Fidelity.countermodelMatrix .unequalRecordVectorsNotMixedness := unequal_environment_vectors_do_not_supply_environmental_distinguishability end ZehOriginalSpinEnvironmentChiralEprAndSymmetryReuse section YoungOriginalTriangularFusionBesselExponentAndBulkReuse open PhysicsLogic.Papers.Young1979 open PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework open PhysicsLogic.Papers.Young1979.VectorCoulombGasFramework.DependencyCountermodels example : YoungRecursionFormulaReadout.physicalNonzeroFugacityTriangularRecursion |>.EquationsTwentyFiveAt := YoungRecursionFormulaReadout.original_nonzero_fugacity_realizes_both_genuine_bessel_recursions.1 example : OriginalTriangularDefectReadout.originalTriangularAndSquareBurgersGeometries |>.HasOriginalTriangularDefectMechanism := OriginalTriangularDefectReadout.triangular_unit_burgers_fusion_has_a_genuine_quadratic_channel_absent_on_the_square_lattice example : OriginalMeltingCriticalScopeReadout.originalEqualForceCriticalResponseAndOpenScope |>.HasVacancyCorrectedTrueBulkModulus := OriginalMeltingCriticalScopeReadout.original_exponent_shear_loss_bulk_correction_and_unanalyzed_transition_are_inhabited.2.2.1 example {T : MeltingTypes} (framework : VectorCoulombGasFramework T) (setup : framework.RecursionSetup) (calculation : framework.RecursionCalculation) : framework.OriginalTriangularRecursionValuesAt := framework |>.recursion_route_retains_triangular_burgers_fusion_and_genuine_quadratic_channel setup calculation example {T : MeltingTypes} (framework : VectorCoulombGasFramework T) (assessment : framework.ScopeAssessment) : framework.originalMeltingCriticalScopeReadout.HasRestrictedSolidSideScope := framework |>.scope_route_leaves_disclination_and_first_order_alternatives_unproved assessment example : PhysicsLogic.Papers.Young1979.Fidelity.countermodelMatrix .triangularFusionNotSquareUnitFusion := triangular_unit_burgers_fusion_does_not_exist_for_two_square_unit_vectors example : PhysicsLogic.Papers.Young1979.Fidelity.countermodelMatrix .unitFusionNotPhysicalAngularForce := unit_burgers_fusion_does_not_supply_the_physical_angular_force example : PhysicsLogic.Papers.Young1979.Fidelity.countermodelMatrix .triangularQuadraticChannelNotSquareChannel := the_triangular_quadratic_fugacity_channel_is_absent_on_the_square_lattice example : PhysicsLogic.Papers.Young1979.Fidelity.countermodelMatrix .physicalExponentNotScalarCoulombGasExponent := the_physical_equal_force_exponent_is_not_the_scalar_coulomb_gas_exponent example : PhysicsLogic.Papers.Young1979.Fidelity.countermodelMatrix .dislocationBulkNotTrueBulkModulus := the_dislocation_only_bulk_response_is_not_the_true_bulk_modulus example : PhysicsLogic.Papers.Young1979.Fidelity.countermodelMatrix .bulkCorrectionNotPositiveSign := a_distinct_vacancy_corrected_bulk_modulus_does_not_determine_the_correction_sign example : PhysicsLogic.Papers.Young1979.Fidelity.countermodelMatrix .solidSideScopeNotDisclinationCalculation := a_solid_side_recursion_does_not_analyze_the_later_disclination_transition end YoungOriginalTriangularFusionBesselExponentAndBulkReuse section SourceFaithfulPhysicalCommitmentReuse open PhysicsLogic.Philosophy open PhysicsLogic.Philosophy.Observer open PhysicsLogic.Philosophy.Probability example : Observer.Countermodels.consciouslyCompletedPhysicalObserverRoles.toPhysicalObserverRoleContext = Observer.Countermodels.physicalObserverRoles.toPhysicalObserverRoleContext := Observer.Countermodels.identical_physical_observer_roles_allow_opposite_consciousness.1 example : Probability.Countermodels.twoOutcomePossibility.OnticallyBranchesAt .branching ∧ ¬(Probability.Countermodels.incompleteClassificationOfSameTwoOutcomes |>.OnticallyBranchesAt .branching) := ⟨Probability.Countermodels.identical_outcome_support_allows_opposite_ontic_classifications.2.1, Probability.Countermodels.identical_outcome_support_allows_opposite_ontic_classifications.2.2.1⟩ example {State Outcome : Type} (possibility : PhysicalPossibilityContext State Outcome) (stateComplete : State → Prop) (state : State) (complete : stateComplete state) (branching : possibility.PhysicallyBranchesAt state) : (possibility.withCompleteness stateComplete).OnticallyBranchesAt state := physical_branching_is_ontic_only_with_explicit_completeness possibility stateComplete state complete branching end SourceFaithfulPhysicalCommitmentReuse section OrderedPhysicalLimitCrossFamilyReuse open PhysicsLogic.Paradigm.OrderedLimits example : Countermodels.forwardOrderOnly.EstablishedOrderedValueAt .magnetic .infiniteVolume .zeroField .magnetization 1 .selected ∧ ¬∃ value, Countermodels.forwardOrderOnly.EstablishedOrderedValueAt .magnetic .zeroField .infiniteVolume .magnetization value .selected := Countermodels.established_order_does_not_supply_reversed_order example : (Countermodels.ambiguouslyCertifiedOrders |>.NoncommutingEstablishedOrdersAt .magnetic .infiniteVolume .zeroField .magnetization .selected) ∧ ¬(Countermodels.ambiguouslyCertifiedOrders |>.NonvacuousOrderCommutationAt .magnetic .infiniteVolume .zeroField .magnetization .selected) := ⟨(Countermodels.equal_multivalued_witnesses_do_not_establish_order_commutation).2.2.1, (Countermodels.equal_multivalued_witnesses_do_not_establish_order_commutation).2.2.2⟩ example {T : PhysicsLogic.Papers.Stanley1968.StanleyTypes} (framework : PhysicsLogic.Papers.Stanley1968.SphericalLimitFramework T) (scope : framework.CriticalIndexScopeAssessment) : ¬(framework.toCriticalOrderedLimitContext |>.NonvacuousOrderCommutationAt framework.selectedLattice .infiniteSpinDimension .criticalTemperature framework.criticalTemperature framework.criticalTemperature) ∧ ¬(framework.toCriticalOrderedLimitContext |>.NoncommutingEstablishedOrdersAt framework.selectedLattice .infiniteSpinDimension .criticalTemperature framework.criticalTemperature framework.criticalTemperature) := (framework.critical_scope_realizes_shared_unproved_order_comparison scope).2 example (model : PhysicsLogic.Papers.MerminWagner1966.Model) (channel : PhysicsLogic.Papers.MerminWagner1966.OrderChannel) (scope : PhysicsLogic.Papers.MerminWagner1966.SourceScope model) (spontaneous : PhysicsLogic.Papers.MerminWagner1966.SpontaneousLimitAt model channel) : (PhysicsLogic.Papers.MerminWagner1966.toOrderedMagnetizationLimitContext model).EstablishedOrderedValueAt channel .infiniteVolume .zeroField channel (model.spontaneousMagnetization channel) .sourceScope := PhysicsLogic.Papers.MerminWagner1966.spontaneous_limit_realizes_selected_volume_then_field_order model channel scope spontaneous example (framework : PhysicsLogic.Papers.WittenStringDynamics1995.PaperFramework) (limit : framework.ConditionalHeteroticSevenLimit) : (framework.toHeteroticSevenOrderedLimitContext |>.EstablishedOrderedValueAt .sevenDimensionalHeterotic .circleDecompactification .strongCoupling .elevenDimensionalSupergravityOnK3 .elevenDimensionalSupergravityOnK3 .sevenDimensions) ∧ ¬∃ target, framework.toHeteroticSevenOrderedLimitContext |>.EstablishedOrderedValueAt .sevenDimensionalHeterotic .strongCoupling .circleDecompactification .elevenDimensionalSupergravityOnK3 target .sevenDimensions := framework.source_certified_heterotic_order_does_not_supply_reversed_order limit example : PhysicsLogic.Papers.MerminWagner1966.Fidelity.countermodelMatrix .sourceSelectedOrderNotReversedOrder := PhysicsLogic.Papers.MerminWagner1966.source_selected_magnetic_limit_does_not_certify_reversed_order example : PhysicsLogic.Papers.Wetterich1993.Fidelity.countermodelMatrix .namedShapeNotOriginalExponentialProfile := by have witness := PhysicsLogic.Papers.Wetterich1993.RegulatorShapeFormulaReadout.Countermodels.named_regulator_shape_does_not_supply_original_exponential_profile exact ⟨⟨witness.1, witness.2.1⟩, witness.2.2⟩ example : PhysicsLogic.Papers.Wetterich1993.Fidelity.countermodelMatrix .preferredShapeCommutationNotSharperShapeCommutation := PhysicsLogic.Papers.Wetterich1993.RegulatorShapeFormulaReadout.Countermodels.preferred_shape_commutation_does_not_transfer_to_sharper_shape example : PhysicsLogic.Papers.Wetterich1993.Fidelity.countermodelMatrix .sharperShapeNoncommutationNotPreferredShapeNoncommutation := PhysicsLogic.Papers.Wetterich1993.RegulatorShapeFormulaReadout.Countermodels.sharper_shape_noncommutation_does_not_transfer_to_preferred_shape example {T : PhysicsLogic.Papers.Wetterich1993.WetterichTypes} (framework : PhysicsLogic.Papers.Wetterich1993.Framework T) (scope : framework.ScopeQualifications) : (framework.regulatorShapeFormulaReadout.toOrderedRegulatorLimitContext |>.NonvacuousOrderCommutationAt .preferred .zeroMomentum .infraredScale () ()) ∧ (framework.regulatorShapeFormulaReadout.toOrderedRegulatorLimitContext |>.NoncommutingEstablishedOrdersAt .sharper .zeroMomentum .infraredScale () ()) := framework.scope_qualifications_distinguish_commuting_preferred_and_noncommuting_sharper_shapes scope end OrderedPhysicalLimitCrossFamilyReuse section GenuinePartitionZeroAccumulationReuse open PhysicsLogic.Paradigm.ThermodynamicLimit example : ¬(Countermodels.improperEventualityWithoutSelectedZeros |>.ExceptionalPointsAccumulateAt .critical) ∧ (Countermodels.improperEventualityWithoutSelectedZeros |>.ExceptionalPointsAccumulateAt .offCritical) := ⟨(Countermodels.improper_eventuality_cannot_create_nonexistent_partition_zeros).2.2.1, (Countermodels.improper_eventuality_cannot_create_nonexistent_partition_zeros).2.2.2⟩ example {Size Description Control Region : Type} (model : PhysicsLogic.Framework.PartitionZeroPhaseBoundaryCalculationModel Size Description Control Region) (accumulation : model.ZerosAccumulateAtSelectedControl) : ∃ size zero, model.finitePartitionZeroAt size zero := PhysicsLogic.Framework.PartitionZeroPhaseBoundaryCalculationModel.zero_accumulation_exhibits_actual_finite_partition_zero accumulation example {T : PhysicsLogic.Papers.LeeYang1952.LeeYangTypes} (framework : PhysicsLogic.Papers.LeeYang1952.LeeYangFramework T) (description : T.Description) (region : T.Region) (activity : T.Activity) (accumulation : (framework.partitionZeroPhaseBoundaryModelAt description region activity).ZerosAccumulateAtSelectedControl) : ∃ size zero, framework.finitePartitionZeroAt (framework.finiteDescription description size) size zero := framework.selected_accumulation_exhibits_actual_finite_volume_partition_zero description region activity accumulation example {T : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroTypes} (framework : PhysicsLogic.Papers.BiskupBorgsChayesKotecky2004.PartitionZeroFramework T) (accumulation : framework.partitionZeroPhaseBoundaryModel |>.ZerosAccumulateAtSelectedControl) : ∃ size zero, framework.finitePartitionZeroAt framework.selectedModel size zero := framework.selected_accumulation_exhibits_actual_finite_torus_partition_zero accumulation end GenuinePartitionZeroAccumulationReuse namespace ComputedSpacetimeGeometryReuse open PhysicsLogic.Framework.FiniteMetricGeometry example (jet : MetricTwoJet) (mu nu : CoordinateIndex) : jet.sourceRicci mu nu = -jet.modernRicci mu nu := jet.source_contraction_is_negative_modern_ricci mu nu example (mu nu : CoordinateIndex) : ¬(euclideanFlatJet.toDirectContext mu nu).NonvacuousMetricConnectionCurvatureAt := euclidean_metric_cannot_realize_spacetime_contract mu nu example : vacuumTidalJet.ContractedVacuum ∧ vacuumTidalJet.riemann 0 3 0 3 = -2 ∧ vacuumTidalJet.riemann 1 3 1 3 = 1 ∧ vacuumTidalJet.riemann 2 3 2 3 = 1 ∧ ¬vacuumTidalJet.FullCurvatureZero := contracted_vacuum_does_not_erase_tidal_curvature example (mu nu : CoordinateIndex) : (vacuumTidalJet.toDirectContext mu nu).NonvacuousMetricConnectionCurvatureAt ∧ vacuumTidalJet.ContractedVacuum ∧ ¬vacuumTidalJet.FullCurvatureZero := PhysicsLogic.Papers.Einstein1916.PointwiseGeometry.pointwise_vacuum_realizes_original_shared_geometry mu nu example {T : PhysicsLogic.Papers.Einstein1916.RelativityTypes} (context : PhysicsLogic.Papers.Einstein1916.SemanticContext T) (certificate : PhysicsLogic.Papers.Einstein1916.routedArgument.RouteSemanticallyCertified context.interpretAtom .vacuumFieldEquation) : context.framework.EquationFortyFourIndexContractionAt context.metric ∧ context.framework.EquationFortySevenContractedVacuumFormulaAt context.metric context.fieldCoordinates := by have formulas := context.certified_vacuum_route_retains_original_curvature_formulas certificate exact ⟨formulas.2.1, formulas.2.2.2.2.2⟩ example : (PhysicsLogic.Papers.Rosen1940.PointwiseGeometry.partIReadout sphericalFlatJet sphericalFlatJet).SourceFormulaAt ∧ sphericalFlatJet.FullCurvatureZero ∧ ¬sphericalFlatJet.ConnectionZero := PhysicsLogic.Papers.Rosen1940.PointwiseGeometry.original_spherical_auxiliary_has_nonzero_coordinate_connection example (mu nu : CoordinateIndex) : (PhysicsLogic.Papers.Rosen1940.PointwiseGeometry.partIReadout nonvacuumCurvatureJet cartesianFlatJet).SourceFormulaAt ∧ ((nonvacuumCurvatureJet.toFormalFlatContext cartesianFlatJet mu nu) |>.NonvacuousMetricConnectionCurvatureAt) ∧ ¬nonvacuumCurvatureJet.ContractedVacuum := PhysicsLogic.Papers.Rosen1940.PointwiseGeometry.original_formal_flat_rewrite_does_not_impose_vacuum mu nu example : PhysicsLogic.Papers.Einstein1916.Fidelity.countermodelMatrix .invertibleEuclideanGeometryNotLorentzianSpacetime := PhysicsLogic.Papers.Einstein1916.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.Rosen1940.Fidelity.countermodelMatrix .connectionDifferenceNotFlatAuxiliary := PhysicsLogic.Papers.Rosen1940.Fidelity.countermodel_matrix_is_realized _ end ComputedSpacetimeGeometryReuse namespace ComputedMemoryCovarianceReuse open PhysicsLogic.Framework.FiniteResidualCovariance example {dimension outputs : Nat} (model : LinearResidualModel dimension outputs) (physical : model.initialCovariance.PosSemidef) (time : Real) : (model.twoTimeCovariance time time).PosSemidef := model.equal_time_covariance_is_positive_semidefinite physical time example (first second : Real) : PhysicsLogic.Papers.Mori1965.FiniteProjection.selectedResidualModel.NonvacuousStationaryResidualCovariance ∧ PhysicsLogic.Papers.Mori1965.FiniteProjection.oscillator.projectedCorrelation first second = PhysicsLogic.Papers.Mori1965.FiniteProjection.selectedResidualModel.twoTimeCovariance first second := PhysicsLogic.Papers.Mori1965.FiniteProjection.projected_force_realizes_shared_residual_covariance first second example (component : Fin 2) (time : Real) : deriv (fun t => PhysicsLogic.Papers.Mori1965.FiniteProjection.oscillatorVariable t component) time + intervalIntegral (fun t => PhysicsLogic.Papers.Mori1965.FiniteProjection.oscillatorVariable t component) 0 time MeasureTheory.volume = PhysicsLogic.Papers.Mori1965.FiniteProjection.oscillator.projectedForce time component 0 := (PhysicsLogic.Papers.Mori1965.FiniteProjection.oscillator_satisfies_the_exact_memory_equation component time).2.2 example {dimension : Nat} (model : PhysicsLogic.Papers.Zwanzig1973.FiniteQuadraticBath.Model dimension) (thermalEnergy first second : Real) (domain : model.AlgebraicBathDomain) (group : model.PropagatorGroup) (identity : model.EquationEighteen) : (model.residualModel thermalEnergy).twoTimeCovariance first second = thermalEnergy • model.memoryKernel (first - second) := model.canonical_initial_covariance_and_equation_eighteen_give_noise_memory_relation thermalEnergy first second domain group identity example : (PhysicsLogic.Papers.Zwanzig1973.FiniteQuadraticBath.unitOscillatorBath.residualModel 1) |>.NonvacuousStationaryResidualCovariance := PhysicsLogic.Papers.Zwanzig1973.FiniteQuadraticBath.finite_bath_realizes_shared_residual_covariance example {T : PhysicsLogic.Papers.Mori1965.MoriTypes} (context : PhysicsLogic.Papers.Mori1965.SemanticContext T) (certificate : PhysicsLogic.Papers.Mori1965.routedArgument.RouteSemanticallyCertified context.interpretAtom .forceCorrelation) : context.framework.SelectedCovarianceInverseAt := (context.certified_force_correlation_retains_the_covariance_inverse certificate).2 example {T : PhysicsLogic.Papers.Zwanzig1973.ZwanzigTypes} (context : PhysicsLogic.Papers.Zwanzig1973.SemanticContext T) (certificate : PhysicsLogic.Papers.Zwanzig1973.routedArgument.RouteSemanticallyCertified context.interpretAtom .canonicalNoiseRelation) : context.framework.PositiveCanonicalBathQuadraticFormAt ∧ context.framework.EquationEighteenBathCovarianceIdentityAt := by have formulas := context.certified_noise_route_retains_the_matrix_covariance_derivation certificate exact ⟨formulas.1, formulas.2.2.2.2.2⟩ example {T : PhysicsLogic.Papers.Zwanzig1973.ZwanzigTypes} (context : PhysicsLogic.Papers.Zwanzig1973.SemanticContext T) (certificate : PhysicsLogic.Papers.Zwanzig1973.routedArgument.RouteSemanticallyCertified context.interpretAtom .markovApproximation) : context.framework.EquationTwentyThreeDiscreteFrictionFormulaAt ∧ context.framework.EquationsTwentyNineAndThirtyFormulaAt := by have formulas := context.certified_markov_route_retains_the_particle_and_spectral_formulas certificate exact ⟨formulas.2.1, formulas.2.2.2.2⟩ example : PhysicsLogic.Papers.Mori1965.Fidelity.countermodelMatrix .exactProjectionNotDecayingMemory := PhysicsLogic.Papers.Mori1965.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.Zwanzig1973.Fidelity.countermodelMatrix .nonsingularBathNotPhysicalCanonicalCovariance := PhysicsLogic.Papers.Zwanzig1973.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.Zwanzig1973.Fidelity.countermodelMatrix .printedDebyeDensityNotNormalizedFrequencyLaw := PhysicsLogic.Papers.Zwanzig1973.Fidelity.countermodel_matrix_is_realized _ end ComputedMemoryCovarianceReuse namespace ComputedDensityEntanglementReuse open PhysicsLogic.Framework.FiniteQuantumState open scoped ComplexOrder Kronecker example {A B : Type} [Fintype A] [Fintype B] (rho : Operator (A × B)) (separable : Separable rho) : IsDensity rho ∧ (partialTransposeRight rho).PosSemidef := ⟨separable_isDensity separable, separable_positive_partial_transpose separable⟩ example {A B : Type} [Fintype A] [Fintype B] (rho : Operator (A × B)) (testVector : A × B -> Complex) (separable : Separable rho) : pptWitness testVector rho <= 0 := separable_pptWitness_nonpositive testVector separable example : IsPure singlet ∧ ¬Separable singlet ∧ vonNeumannEntropy singlet = 0 ∧ vonNeumannEntropy (partialTraceRight singlet) = Real.log 2 := by refine ⟨singlet_isPure, singlet_not_separable, pure_vonNeumannEntropy_zero _ singlet_isPure, ?_⟩ rw [singlet_marginals.1, maximallyMixedQubit_entropy] example : IsDensity classicalPair ∧ Separable classicalPair ∧ 0 < vonNeumannEntropy (partialTraceRight classicalPair) ∧ ¬IsPure classicalPair := positive_marginal_entropy_does_not_imply_entanglement example : PhysicsLogic.Papers.CoffmanKunduWootters2000.FiniteThreeQubit.SpectralRootsAt singlet PhysicsLogic.Papers.CoffmanKunduWootters2000.FiniteThreeQubit.singletRoots ∧ (PhysicsLogic.Papers.CoffmanKunduWootters2000.FiniteThreeQubit.tangleDefinitionReadout singlet PhysicsLogic.Papers.CoffmanKunduWootters2000.FiniteThreeQubit.singletRoots).SourceFormulaAt := PhysicsLogic.Papers.CoffmanKunduWootters2000.FiniteThreeQubit.actual_spectrum_realizes_original_tangle_definition _ _ PhysicsLogic.Papers.CoffmanKunduWootters2000.FiniteThreeQubit.singlet_spectrum example {T : PhysicsLogic.Papers.CoffmanKunduWootters2000.CKWTypes} (context : PhysicsLogic.Papers.CoffmanKunduWootters2000.SemanticContext T) (certificate : PhysicsLogic.Papers.CoffmanKunduWootters2000.routedArgument.RouteSemanticallyCertified context.interpretAtom .pureMonogamy) : context.framework.tangleDefinitionFormulaABReadout.MatrixDomainAt ∧ context.framework.tangleDefinitionFormulaACReadout.MatrixDomainAt := context.certified_tangle_definition_retains_both_actual_matrix_spectra certificate example : (pptContext PhysicsLogic.Papers.CoffmanKunduWootters2000.FiniteThreeQubit.singletWithThirdQubit bellPairVector).EstablishedBipartiteEntanglementAt (pptContext PhysicsLogic.Papers.CoffmanKunduWootters2000.FiniteThreeQubit.singletWithThirdQubit bellPairVector).selectedModel := PhysicsLogic.Papers.CoffmanKunduWootters2000.FiniteThreeQubit.singlet_with_third_qubit_realizes_shared_entanglement.2.2 example : (pptContext (bellPair ⊗ₖ zeroState) singletVector).EstablishedBipartiteEntanglementAt (pptContext (bellPair ⊗ₖ zeroState) singletVector).selectedModel := PhysicsLogic.Papers.AMPS2012.FiniteEntropy.smooth_pair_realizes_shared_matrix_entanglement example (left right : PhysicsLogic.Papers.Bell1964.SpinDirection) : PhysicsLogic.Papers.Bell1964.FiniteSpinPair.correlation PhysicsLogic.Papers.Bell1964.FiniteSpinPair.isotropicState left right = -(left.dot right) / 3 := PhysicsLogic.Papers.Bell1964.FiniteSpinPair.actual_isotropic_product_mixture_is_equation_eleven left right example : PhysicsLogic.Papers.CoffmanKunduWootters2000.Fidelity.countermodelMatrix .namedRootsNotActualMatrixSpectrum := PhysicsLogic.Papers.CoffmanKunduWootters2000.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.CoffmanKunduWootters2000.Fidelity.countermodelMatrix .ghzMixedMarginalNotPairEntanglement := PhysicsLogic.Papers.CoffmanKunduWootters2000.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.AMPS2012.Fidelity.countermodelMatrix .actualSmoothPairNotPageDecrease := PhysicsLogic.Papers.AMPS2012.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.AMPS2012.Fidelity.countermodelMatrix .actualPageDecreaseNotPureInteriorPair := PhysicsLogic.Papers.AMPS2012.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.Bell1964.Fidelity.countermodelMatrix .equalMarginalsNotEqualSpinCorrelations := PhysicsLogic.Papers.Bell1964.Fidelity.countermodel_matrix_is_realized _ end ComputedDensityEntanglementReuse namespace ActualCollisionlessPlasmaReuse open PhysicsLogic.Framework.ElectrostaticKinetic example (particle : Species) (equilibrium : Real -> Real) (perturbation : Distribution) (potential : Potential) (time position velocity : Real) (equilibriumRegular : DifferentiableAt Real equilibrium velocity) (regular : RegularAt perturbation potential time position velocity) : HasDerivAt (fun amplitude => residual particle (perturbedDensity amplitude equilibrium perturbation) (perturbedPotential amplitude potential) time position velocity) (linearizedResidual particle equilibrium perturbation potential time position velocity) 0 := linearized_residual_is_first_variation particle equilibrium perturbation potential time position velocity equilibriumRegular regular example (particle : Species) (density : Real -> Real -> Real) (potential profile : Real -> Real) (position velocity : Real) (physical : particle.Physical) (potentialRegular : DifferentiableAt Real potential position) (profileRegular : DifferentiableAt Real profile (energy particle potential position velocity)) (branch : LocalEnergyBranchAt particle density potential profile position velocity) : StationaryAt particle density potential position velocity := local_energy_branch_is_stationary particle density potential profile position velocity physical potentialRegular profileRegular branch example : PhysicsLogic.Papers.LandauDamping1946.KineticFormulaReadout.normalizedEquilibriumControl.SourceFormulaAt := PhysicsLogic.Papers.LandauDamping1946.KineticFormulaReadout.normalized_maxwellian_control_realizes_source_equations example {T : PhysicsLogic.Papers.LandauDamping1946.Landau1946Types} (context : PhysicsLogic.Papers.LandauDamping1946.SemanticContext T) (certificate : PhysicsLogic.Papers.LandauDamping1946.routedArgument.RouteSemanticallyCertified context.interpretAtom .normalModeInsufficiency) : context.framework.kineticFormulaReadout.EquationFourAt /\ context.framework.kineticFormulaReadout.EquationFiveAt := by have source := context.certified_initial_setup_retains_actual_kinetic_and_poisson_equations certificate exact source.2.2.2 example {T : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.BGK1957Types} (context : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.SemanticContext T) (certificate : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.routedArgument.RouteSemanticallyCertified context.interpretAtom .orbitAndTrapping) (time position : Real) : context.framework.actualPlasma.PoissonAt time position := (context.certified_orbit_route_retains_actual_kinetic_and_poisson_equations certificate time position).2 example : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.ActualStationaryControls.neutralControl |>.NonvacuousVlasovPoissonAt 0 0 := PhysicsLogic.Papers.BernsteinGreeneKruskal1957.ActualStationaryControls.neutral_two_species_control_is_genuine_vlasov_poisson example : PhysicsLogic.Papers.LandauDamping1946.Fidelity.countermodelMatrix .linearizedSolutionNotExactNonlinearSolution := PhysicsLogic.Papers.LandauDamping1946.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.Fidelity.countermodelMatrix .actualStationaryPopulationsNotPoissonClosure := PhysicsLogic.Papers.BernsteinGreeneKruskal1957.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.BernsteinGreeneKruskal1957.Fidelity.countermodelMatrix .namedDerivativeNotActualKineticEquation := PhysicsLogic.Papers.BernsteinGreeneKruskal1957.Fidelity.countermodel_matrix_is_realized _ end ActualCollisionlessPlasmaReuse namespace ActualElasticNuclearScatteringReuse open PhysicsLogic.Framework.ElasticSWave example (effectiveRangeFunction momentum : Real) (positive : 0 < momentum) : OpticalTheoremAt momentum (reducedAmplitude effectiveRangeFunction momentum) := (nonvacuous_elastic_s_wave_has_unitarity_and_optical_theorem _ _ _ (real_effective_range_function_realizes_elastic_scattering _ _ positive)).2 example (Channel Scale Momentum Coupling Term Order Regime Fit : Type) (context : PhysicsLogic.Papers.KaplanSavageWise1998.SemanticContext Channel Scale Momentum Coupling Term Order Regime Fit) (certificate : PhysicsLogic.Papers.KaplanSavageWise1998.routedArgument.RouteSemanticallyCertified context.interpretAtom .pionlessPDS) : Complex.normSq (context.framework.pionlessPDSFormulaReadout.elasticSMatrixValueAt context.lowChannel context.lowScale context.lowMomentum) = 1 := context.certified_pionless_route_retains_actual_elastic_unitarity certificate example : (PhysicsLogic.Papers.KaplanSavageWise1998.PionlessPDSFormulaReadout.Countermodels.elasticPDSControl 1) |>.EquationsFiveAndSixAt := PhysicsLogic.Papers.KaplanSavageWise1998.PionlessPDSFormulaReadout.Countermodels.real_contact_bubble_sum_realizes_source_unitarity.1 example {T : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.ThreeBodyTypes} (context : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.SemanticContext T) (certificate : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.routedArgument.RouteSemanticallyCertified context.interpretAtom .largeLengthResummation) : NonvacuousElasticSWaveAt (-context.framework.twoBodyFormulaReadout.scatteringLength⁻¹) (context.framework.twoBodyFormulaReadout.momentumValueAt context.framework.onShellMomentum) (context.framework.twoBodyFormulaReadout.reducedAmplitudeValueAt context.framework.onShellMomentum) := (context.certified_resummation_retains_actual_elastic_two_body_amplitude certificate).2 example : (PhysicsLogic.Papers.BedaqueHammerVanKolck1998.DressedTwoBodyFormulaReadout.Countermodels.dressedControl 2) |>.LeadingEffectiveRangeRecoveredAt () := (PhysicsLogic.Papers.BedaqueHammerVanKolck1998.DressedTwoBodyFormulaReadout.Countermodels.dressed_control_realizes_leading_elastic_scattering 2 (by norm_num)).2 example : PhysicsLogic.Papers.KaplanSavageWise1998.Fidelity.countermodelMatrix .complexContactNotElasticUnitarity := PhysicsLogic.Papers.KaplanSavageWise1998.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.Fidelity.countermodelMatrix .principalSquareRootNotRetardedBoundary := PhysicsLogic.Papers.BedaqueHammerVanKolck1998.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.BedaqueHammerVanKolck1998.Fidelity.countermodelMatrix .auxiliaryPropagatorNotPhysicalAmplitude := PhysicsLogic.Papers.BedaqueHammerVanKolck1998.Fidelity.countermodel_matrix_is_realized _ end ActualElasticNuclearScatteringReuse namespace ActualCoherentSpectroscopyReuse open PhysicsLogic.Framework.CoherentTwoState example (s c frequency : Real) (normalized : s ^ 2 + c ^ 2 = 1) (initial : StateVector) : SolvesSchrodingerAt (rotatingGenerator s c frequency) (fun time => (pulseMatrix s c (frequency * time / 2)).mulVec initial) := pulse_solves_schrodinger s c frequency normalized initial example {T : PhysicsLogic.Papers.Rabi1939.RabiTypes} (context : PhysicsLogic.Papers.Rabi1939.SemanticContext T) (certificate : PhysicsLogic.Papers.Rabi1939.routedArgument.RouteSemanticallyCertified context.interpretAtom .resonantTransitionDynamics) : 0 ≤ context.framework.selectedResonanceFormulaReadout.transitionProbability ∧ context.framework.selectedResonanceFormulaReadout.transitionProbability ≤ 1 := nonvacuous_transition_probability_is_admissible _ _ _ _ (context.certified_resonance_route_retains_actual_coherent_pulse certificate) example {T : PhysicsLogic.Papers.Ramsey1950.RamseyTypes} (context : PhysicsLogic.Papers.Ramsey1950.SemanticContext T) (certificate : PhysicsLogic.Papers.Ramsey1950.routedArgument.RouteSemanticallyCertified context.interpretAtom .separatedTransition) : NormalizedAt context.framework.coherentPulseSequenceReadout.afterSecondPulse := nonvacuous_transition_has_normalized_final_state _ _ _ _ (context.certified_separated_transition_retains_actual_coherent_dynamics certificate) example : PhysicsLogic.Papers.Rabi1939.Fidelity.countermodelMatrix .coherentWeakFieldProbabilityNotExactLaboratoryEvolution := PhysicsLogic.Papers.Rabi1939.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.Rabi1939.Fidelity.countermodelMatrix .weakDetuningNotReciprocalRatioAgreement := PhysicsLogic.Papers.Rabi1939.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.Ramsey1950.Fidelity.countermodelMatrix .coherentReinforcementNotIncoherentProbabilitySum := PhysicsLogic.Papers.Ramsey1950.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.Ramsey1950.Fidelity.countermodelMatrix .driveFreeIntervalNotAbsentRelativePhase := PhysicsLogic.Papers.Ramsey1950.Fidelity.countermodel_matrix_is_realized _ end ActualCoherentSpectroscopyReuse namespace ActualRamseyProfileReuse open PhysicsLogic.Papers.Ramsey1950 open SourceFreeRegionProfile example (drive speed s c x : Real) (positive : 0 < speed) : Countermodels.uniformComparisonProfile.coherentTransitionProbability drive speed s c x = Countermodels.commonEnergyShiftProfile.coherentTransitionProbability drive speed s c x := Countermodels.common_energy_shift_preserves_the_entire_fringe drive speed s c x positive example (profile : SourceFreeRegionProfile) (source : profile.CommonEnergyBasisAt) (segment : FreeRegionSegment) (member : segment ∈ profile.segments) (initial : PhysicsLogic.Framework.CoherentTwoState.StateVector) : PhysicsLogic.Framework.CoherentTwoState.SolvesSchrodingerAt segment.angularGenerator (fun time => (diagonalEvolution (fun i => segment.energy i / profile.reducedPlanckConstant) time).mulVec initial) := profile.every_source_segment_solves_its_actual_schrodinger_equation source segment member initial example (first second : SourceFreeRegionProfile) (same : first.SameSpaceMeansAt second) (readout : MolecularBeamVelocityAverageReadout) (weight : readout.NormalizedMolecularBeamWeightAt) (positive : 0 < readout.mostProbableSpeedValue) (drive : Real) (s c x : Real -> Real) : readout.weightedIntegralValueAt (fun y => MolecularBeamVelocityAverageReadout.molecularBeamWeightAt y * first.coherentTransitionProbability drive (readout.mostProbableSpeedValue * y) (s y) (c y) (x y)) = readout.weightedIntegralValueAt (fun y => MolecularBeamVelocityAverageReadout.molecularBeamWeightAt y * second.coherentTransitionProbability drive (readout.mostProbableSpeedValue * y) (s y) (c y) (x y)) := first.same_space_means_preserve_positive_velocity_average second same readout weight positive drive s c x example : Countermodels.unequalSpeedProfile.SameSpaceMeansAt Countermodels.uniformComparisonProfile ∧ Countermodels.unequalSpeedProfile.segments ≠ Countermodels.uniformComparisonProfile.segments := Countermodels.genuinely_different_longitudinal_profiles_have_the_same_space_means example {T : RamseyTypes} (context : SemanticContext T) (certificate : routedArgument.RouteSemanticallyCertified context.interpretAtom .separatedTransition) : context.framework.sourceFreeRegionProfile.LongitudinalSpaceMeanReductionAt := (context.certified_separated_transition_retains_source_space_mean_reduction certificate).2 example {T : RamseyTypes} (context : SemanticContext T) (certificate : routedArgument.RouteSemanticallyCertified context.interpretAtom .fieldAveragingAndPrecision) : context.framework.sourceFreeRegionProfile.LongitudinalResponseInvariantAt ∧ ¬TransverseMeanDeterminesCoherentResponse := context.certified_precision_route_retains_actual_longitudinal_invariance certificate example : Fidelity.countermodelMatrix .variableSpeedNotSpaceMeanWeighting := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .sameMeanSplittingNotSameIndividualMeans := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .staticHamiltonianNotCommonEnergyBasis := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .transverseMeanNotMeanFringe := Fidelity.countermodel_matrix_is_realized _ end ActualRamseyProfileReuse namespace ActualRamseyBeamReuse open PhysicsLogic.Papers.Ramsey1950 open MolecularBeamVelocityAverageReadout example : sourceIntegral molecularBeamWeightAt = 1 := beam_weight_has_integral_one example (parameters : SourceBeamParameters) (physical : parameters.PhysicalParametersAt) : 0 ≤ parameters.exactAverage ∧ parameters.exactAverage ≤ 1 := parameters.exact_average_is_admissible physical.1 example (parameters : SourceBeamParameters) (physical : parameters.PhysicalParametersAt) (resonant : parameters.interactionDetuning = 0) : parameters.ControlledNearAverageAt 0 0 := parameters.pointwise_near_reduction_derives_controlled_source_average 0 0 (fun _ => 0) (parameters.exact_pulse_resonance_has_zero_near_reduction_error physical resonant) example (parameters : SourceBeamParameters) (threshold cosineError sineError : Real) (estimates : parameters.FarPhaseMixingInputAt threshold cosineError sineError) : |parameters.exactAverage - parameters.derivedFarHarmonicValue| ≤ cosineError + sineError := (parameters.weighted_phase_mixing_estimates_derive_controlled_far_average threshold cosineError sineError estimates).2.2.2.2.2 example (parameters : SourceBeamParameters) (physical : parameters.PhysicalParametersAt) : parameters.nearAverage = parameters.equationSeventeenValue ∧ parameters.farAverage = parameters.derivedFarHarmonicValue := ⟨parameters.near_average_is_printed_equation_seventeen physical.2.2.2, parameters.far_average_has_the_derived_double_harmonic physical⟩ example {T : RamseyTypes} (framework : SeparatedFieldFramework T) (physical : framework.SourceBeamPhysicalCalibrationAt) (drive y : Real) (positive : 0 < y) : (framework.sourceBeamParametersAt drive).exactProbabilityAt y = framework.sourceFreeRegionProfile.coherentTransitionProbability drive ((framework.sourceBeamParametersAt drive).mostProbableSpeed * y) (framework.sourceBeamParametersAt drive).mixingSine (framework.sourceBeamParametersAt drive).mixingCosine ((framework.sourceBeamParametersAt drive).pulsePhaseAt y) := framework.source_beam_response_is_actual_profile_coherent_transition physical drive y positive example {T : RamseyTypes} (framework : SeparatedFieldFramework T) (physical : framework.SourceBeamPhysicalCalibrationAt) (drive : Real) (initial : PhysicsLogic.Framework.CoherentTwoState.StateVector) : PhysicsLogic.Framework.CoherentTwoState.SolvesSchrodingerAt ({ framework.sourceHamiltonianReadout with oscillatorAngularFrequency := drive } : SourceHamiltonianReadout).rotatingAngularGenerator (fun time => (PhysicsLogic.Framework.CoherentTwoState.pulseMatrix (framework.sourceBeamParametersAt drive).mixingSine (framework.sourceBeamParametersAt drive).mixingCosine ((framework.sourceBeamParametersAt drive).generalizedFrequency * time / 2)).mulVec initial) := framework.source_beam_pulse_solves_actual_schrodinger_equation physical drive initial example {T : RamseyTypes} (framework : SeparatedFieldFramework T) (transition : framework.EstablishedSeparatedFieldTransition) (input : framework.SourceBeamIntegralInputAt) : framework.probabilityValue framework.averagedTransitionProbability = framework.selectedSourceBeamParameters.exactAverage := (framework.source_beam_integral_input_derives_exact_average transition input).2.2.2.2 example {T : RamseyTypes} (context : SemanticContext T) (certificate : routedArgument.RouteSemanticallyCertified context.interpretAtom .velocityDistributionAverage) : context.framework.ExactSourceBeamAverageAt ∧ context.framework.SourceNearFrequencySweepAt ∧ context.framework.SourceFarFrequencySweepAt ∧ context.framework.SourceBeamHarmonicReductionAt := context.certified_velocity_average_retains_actual_coherent_integral_and_frequency_sweeps certificate example : Fidelity.countermodelMatrix .positiveSpeedOrderNotTotalizedIntegralOrder := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .selectedVelocityNotAllSpeedCoherentResponse := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .unweightedPhaseCancellationNotWeightedDephasing := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .coherentResonanceNotFarDephasedAverage := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .equationFifteenNotPrintedEquationEighteen := Fidelity.countermodel_matrix_is_realized _ end ActualRamseyBeamReuse namespace ActualCrooksPathMeasureReuse open PhysicsLogic.Papers.Crooks1999 example {State Path Protocol Ensemble : Type} (framework : FluctuationFramework State Path Protocol Ensemble) (protocol : Protocol) (setup : framework.GeneralFluctuationSetup protocol) (microscopic : framework.MicroscopicReversibilityEquation protocol) (entropy : framework.EntropyProductionEquation protocol) (oddness : framework.EntropyProductionOddUnderReversal protocol) (average : framework.EntropyExponentialAverageDefinition protocol) : framework.MeasureFluctuationRelation protocol ∧ framework.EntropyExponentialIdentity protocol ∧ framework.NormalizedExponentialEntropyEnsembleAt protocol := (framework.source_raw_equations_derive_measure_fluctuation_and_entropy_identity protocol setup microscopic entropy oddness average).2 example {State Path Protocol Ensemble : Type} (context : SemanticContext State Path Protocol Ensemble) (setup : context.framework.GeneralFluctuationSetup context.protocol) (microscopic : context.framework.MicroscopicReversibilityEquation context.protocol) (entropy : context.framework.EntropyProductionEquation context.protocol) (oddness : context.framework.EntropyProductionOddUnderReversal context.protocol) (presentation : context.framework.EntropyDistributionPresentationAt context.protocol) (versions : context.framework.ContinuousEntropyDensityVersionsAt context.protocol) : routedArgument.RouteSemanticallyCertified context.interpretAtom .generalFluctuation := context.actual_path_measure_realizes_general_fluctuation_route setup microscopic entropy oddness presentation versions example {State Path Protocol Ensemble : Type} (framework : FluctuationFramework State Path Protocol Ensemble) (protocol : Protocol) (domain : framework.PathIntegralRealizationAt protocol) (endpoints : framework.ForwardReverseEndpointMatch protocol) (relation : framework.MeasureFluctuationRelation protocol) (forwardWork : framework.EntropyWorkEquation protocol) (reverseWork : framework.EntropyWorkEquation (framework.reverseProtocol protocol)) : framework.WorkMeasureFluctuationRelation protocol := framework.source_entropy_measure_and_work_equations_derive_work_measure_relation protocol domain endpoints relation forwardWork reverseWork example {State Path Protocol Ensemble : Type} (framework : FluctuationFramework State Path Protocol Ensemble) (protocol : Protocol) (betaPositive : 0 < framework.beta) (setup : framework.GeneralFluctuationSetup protocol) (equilibrium : framework.EquilibriumEndpointSetup protocol) (canonical : framework.CanonicalEndpointDensityEquations protocol) (microscopic : framework.MicroscopicReversibilityEquation protocol) (forwardEntropy : framework.EntropyProductionEquation protocol) (reverseEntropy : framework.EntropyProductionEquation (framework.reverseProtocol protocol)) (energy : framework.EnergyChangeEndpointEquation protocol) (firstLaw : framework.FirstLawEquation protocol) (forwardBath : framework.SingleHeatBathEntropyEquation protocol) (reverseBath : framework.SingleHeatBathEntropyEquation (framework.reverseProtocol protocol)) (heatOdd : framework.HeatIsOddUnderReversal protocol) (entropyAverage : framework.EntropyExponentialAverageDefinition protocol) (workAverage : framework.WorkExponentialAverageDefinition protocol) (meanDefinition : framework.MeanWorkDefinition protocol) : framework.sharedNonequilibriumWorkModel protocol |>.NonvacuousNonequilibriumWorkIdentityAndBoundAt := (framework.raw_path_measure_route_exhibits_shared_work_identity_and_bound protocol betaPositive setup equilibrium canonical microscopic forwardEntropy reverseEntropy energy firstLaw forwardBath reverseBath heatOdd entropyAverage workAverage meanDefinition).2 example : Fidelity.countermodelMatrix .actualMeasureNotArbitraryPointwiseDensity := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .actualPathMeasureNotPathwiseSecondLaw := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .printedWorkCoefficientNotDerivedWorkCoefficient := Fidelity.countermodel_matrix_is_realized _ end ActualCrooksPathMeasureReuse namespace ActualQuantumCommutatorReuse open PhysicsLogic.Framework open scoped ComplexOrder example {Operator : Type*} [CStarAlgebra Operator] (hamiltonian : selfAdjoint Operator) (time : Real) (observable : Operator) : QuantumCommutator.evolve (QuantumCommutator.hamiltonianTimeEvolution hamiltonian) time observable = QuantumCommutator.finiteVolumeHeisenberg (hamiltonian : Operator) time observable := QuantumCommutator.selfadjoint_hamiltonian_gives_actual_heisenberg_evolution _ _ _ example {Operator : Type*} [CStarAlgebra Operator] (dynamics : QuantumCommutator.TimeEvolution (Operator := Operator)) (first second : Operator) (time bound : Real) (estimate : QuantumCommutator.commutatorNorm dynamics first second time ≤ bound) (functional : Operator →L[Complex] Complex) (contracting : ‖functional‖ ≤ 1) : ‖functional (QuantumCommutator.evolvedCommutator dynamics first second time)‖ ≤ bound := QuantumCommutator.norm_bound_controls_every_contracting_functional dynamics first second time bound estimate functional contracting open QuantumCommutator.TwoSpin in example (time : Real) : spinModel.commutatorNorm () firstObservable secondObservable time = 2 * |Real.sin (2 * time)| := actual_ising_commutator_norm time open QuantumCommutator.TwoSpin in example (velocity : Real) : ActualSeparatedLocalHamiltonian ∧ ¬ ExactSeparatedCone 1 velocity := actual_local_hamiltonian_does_not_imply_exact_cone velocity namespace LiebRobinson open PhysicsLogic.Papers.LiebRobinson1972 example (context : SemanticContext) (setup : context.framework.FiniteRangeDynamicsSetup) (regularity : context.framework.FiniteRangeDynamicsSetup → context.framework.operatorModel.StronglyContinuousAt context.framework.finiteRangeInteraction ∧ context.framework.localTimeEvolutionIsAnalytic context.framework.finiteRangeInteraction ∧ context.framework.operatorModel.FiniteVolumeConvergenceAt context.framework.finiteRangeInteraction) : routedArgument.RouteSemanticallyCertified context.interpretAtom .quasiLocalDynamics := context.actual_operator_regularity_realizes_source_dynamics_route setup regularity example (framework : PaperFramework) (cone : framework.EstablishedStrictLocalCommutatorCone) : framework.operatorModel.AdmissibleDynamicsAt framework.finiteRangeInteraction ∧ framework.OriginalStrictLocalCommutatorConeFormulaAt := ⟨framework.strict_local_cone_retains_actual_operator_dynamics cone, framework.strict_local_cone_retains_original_asymptotic_formula cone⟩ example : Fidelity.countermodelMatrix .actualFiniteSpinHamiltonianNotExactCone := Fidelity.countermodel_matrix_is_realized _ end LiebRobinson namespace NachtergaeleSims open PhysicsLogic.Papers.NachtergaeleSims2006 example (lambda : Real) : TwoSpinMetricControl.geometry.weightedInteractionNormValue () lambda = 32 * Real.exp lambda := TwoSpinMetricControl.actual_weighted_interaction_norm_value lambda example (lambda : Real) (positive : 0 < lambda) : TwoSpinMetricControl.ActualFiniteMetricSpinSystemAt lambda ∧ ¬ TwoSpinMetricControl.ExactSourceConeAt lambda := TwoSpinMetricControl.actual_source_bound_does_not_imply_exact_cone lambda positive example (context : SemanticContext) (setup : context.framework.InteractionAndDynamicsSetup) (convergence : context.framework.InteractionAndDynamicsSetup → context.framework.operatorModel.FiniteVolumeConvergenceAt context.framework.selectedInteraction) (continuity : context.framework.InteractionAndDynamicsSetup → context.framework.operatorModel.StronglyContinuousAt context.framework.selectedInteraction) : routedArgument.RouteSemanticallyCertified context.interpretAtom .quantumDynamics := context.actual_operator_limits_realize_source_dynamics_route setup convergence continuity example (framework : PaperFramework) (cone : framework.EstablishedFiniteVelocityCone) : framework.operatorModel.AdmissibleDynamicsAt framework.selectedInteraction ∧ framework.metricGeometry.FiniteWeightedInteractionNorm framework.selectedInteraction framework.selectedCommutatorBound.decayParameter ∧ framework.commutatorNormValueAt framework.sourceObservable framework.targetObservable 0 = 0 ∧ framework.FiniteVelocityFormulaPackageAt := framework.finite_velocity_cone_retains_actual_operator_and_metric_semantics cone example : Fidelity.countermodelMatrix .actualWeightedMetricBoundNotExactCone := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .actualInteractionNotAssignedZeroNorm := Fidelity.countermodel_matrix_is_realized _ end NachtergaeleSims end ActualQuantumCommutatorReuse namespace ActualCubicLatticeReuse open PhysicsLogic.Framework open PhysicsLogic.Papers.LiebRobinson1972 example {Operator : Type*} [CStarAlgebra Operator] (automorphism : Operator ≃⋆ₐ[Complex] Operator) (hamiltonian observable : Operator) (time : Real) : automorphism (QuantumCommutator.finiteVolumeHeisenberg hamiltonian time observable) = QuantumCommutator.finiteVolumeHeisenberg (automorphism hamiltonian) time (automorphism observable) := QuantumCommutator.automorphism_maps_finite_volume_heisenberg _ _ _ _ example : QuantumCommutator.TwoSpin.spinModel.DisjointTensorGeneration := QuantumCommutator.TwoSpin.spin_model_disjoint_tensor_generation example {Interaction Observable Region : Type} (model : QuantumSpinOperatorModel Interaction Observable Region) (geometry : CubicLatticeOperatorGeometry model) (interaction : Interaction) (invariant : geometry.TranslationInvariantAt interaction) (finiteRange : geometry.FiniteRangeAt interaction) (center : model.Site) : (model.interactionNeighborhood interaction center).Finite := geometry.finite_origin_range_is_uniform interaction invariant finiteRange center example {Interaction Observable Region : Type} (model : QuantumSpinOperatorModel Interaction Observable Region) (geometry : CubicLatticeOperatorGeometry model) (interaction : Interaction) (invariant : geometry.TranslationInvariantAt interaction) (convergence : model.FiniteVolumeConvergenceAt interaction) : geometry.DynamicsTranslationCovariantAt interaction := geometry.actual_finite_volume_limit_derives_spacetime_covariance interaction invariant convergence example (context : SemanticContext) (operators : context.framework.OperatorKinematicsAt) (lattice : context.framework.CubicLatticeKinematicsAt) (interaction : context.framework.operatorModel.LocalSelfAdjointInteractionAt context.framework.finiteRangeInteraction) (invariant : context.framework.latticeGeometry.TranslationInvariantAt context.framework.finiteRangeInteraction) (finiteRange : context.framework.latticeGeometry.FiniteRangeAt context.framework.finiteRangeInteraction) (regularity : context.framework.FiniteRangeDynamicsSetup → context.framework.operatorModel.StronglyContinuousAt context.framework.finiteRangeInteraction ∧ context.framework.localTimeEvolutionIsAnalytic context.framework.finiteRangeInteraction ∧ context.framework.operatorModel.FiniteVolumeConvergenceAt context.framework.finiteRangeInteraction) : routedArgument.RouteSemanticallyCertified context.interpretAtom .quasiLocalDynamics := context.actual_cubic_lattice_inputs_realize_source_dynamics_route operators lattice interaction invariant finiteRange regularity example (context : SemanticContext) (certificate : routedArgument.RouteSemanticallyCertified context.interpretAtom .quasiLocalDynamics) : context.framework.CubicLatticeKinematicsAt ∧ context.framework.latticeGeometry.TranslationInvariantAt context.framework.finiteRangeInteraction ∧ context.framework.latticeGeometry.FiniteRangeAt context.framework.finiteRangeInteraction ∧ context.framework.latticeGeometry.DynamicsTranslationCovariantAt context.framework.finiteRangeInteraction := context.certified_dynamics_retains_actual_cubic_lattice certificate example (framework : PaperFramework) (positive : 0 < framework.latticeGeometry.dimension) (velocity : Real) : framework.ExteriorConeDomainNonvacuousAt velocity := framework.actual_cubic_lattice_has_unbounded_exterior_domain positive velocity example : Fidelity.countermodelMatrix .exponentialPairDecayNotFiniteRange := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .finiteOriginRangeNotUniformWithoutTranslations := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .lawfulSpatialActionNotFaithful := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .zeroDimensionalCoordinatesNotExteriorDomain := Fidelity.countermodel_matrix_is_realized _ example (framework : PhysicsLogic.Papers.NachtergaeleSims2006.PaperFramework) (setup : framework.DiscreteMetricSystemSetup) : framework.operatorModel.DisjointTensorGeneration := framework.discrete_setup_retains_actual_tensor_generation setup end ActualCubicLatticeReuse namespace ActualGroundSectorReuse open PhysicsLogic.Framework open QuantumGroundSector example {StateSpace : Type*} [NormedAddCommGroup StateSpace] [InnerProductSpace Complex StateSpace] [CompleteSpace StateSpace] (hamiltonian : selfAdjoint (Operator StateSpace)) (positive : (hamiltonian : Operator StateSpace).IsPositive) : (HamiltonianDynamics.ofBounded hamiltonian).RealTimeGeneratorAt ∧ (HamiltonianDynamics.ofBounded hamiltonian).ImaginaryTimeGeneratorAt ∧ NonnegativeHamiltonianAt (HamiltonianDynamics.ofBounded hamiltonian).hamiltonian := ⟨bounded_real_time_generator hamiltonian, bounded_imaginary_time_generator hamiltonian positive, bounded_positive_hamiltonian_is_nonnegative _ positive⟩ example {Basis : Type} [Fintype Basis] [DecidableEq Basis] (energy : Basis → Real) (ground : Finset Basis) (gap : Real) (positive : 0 < gap) (lower : ∀ index, energy index = 0 ∨ gap ≤ energy index) (attained : ∃ index, energy index = gap) : ExactSpectralGapAt ((FiniteSpectralSector.diagonal energy ground).hamiltonianOperator.toPMap ⊤) gap := FiniteSpectralSector.diagonal_exact_spectral_gap energy ground gap positive lower attained example : TwoLevelGroundSectorControls.ActualGappedTwoLevelSystem := TwoLevelGroundSectorControls.actual_gapped_two_level_system example (time : NNReal) : TwoLevelGroundSectorControls.dynamics.correlation (TwoLevelGroundSectorControls.basisVector 0) (TwoLevelGroundSectorControls.matrixUnit 0 1) (TwoLevelGroundSectorControls.matrixUnit 1 0) time = (Real.exp (-(time : Real)) : Complex) := TwoLevelGroundSectorControls.actual_nonzero_imaginary_time_correlation time example {StateSpace : Type*} [NormedAddCommGroup StateSpace] [InnerProductSpace Complex StateSpace] [CompleteSpace StateSpace] (vector : StateSpace) (normalized : ‖vector‖ = 1) (observable : Operator StateSpace) : TargetGroundComponentsVanish vector (InnerProductSpace.rankOne Complex vector vector) (centeredObservable vector observable) := rank_one_ground_projection_annihilates_centered_target vector normalized observable example {StateSpace : Type*} [NormedAddCommGroup StateSpace] [InnerProductSpace Complex StateSpace] [CompleteSpace StateSpace] (hamiltonian : Operator StateSpace) : realSpectrum (hamiltonian.toPMap ⊤) = {energy : Real | (energy : Complex) ∈ spectrum Complex hamiltonian} := bounded_real_spectrum_agrees_with_mathlib hamiltonian example (sector : FiniteSpectralSector) (ground : sector.GroundBandAt) : sector.averageFunctional.NormalizedPositiveAt ∧ sector.averageFunctional.GroundAverageIdentityAt := ⟨sector.average_functional_normalized_positive ground, sector.average_functional_ground_average_identity⟩ example {Algebra : Type*} [CStarAlgebra Algebra] (state : ProjectedCorrelationFunctional Algebra) (finite : Nat → ProjectedCorrelationFunctional Algebra) (converges : state.WeakStarLimitAt finite Filter.atTop) (first second : Algebra) (bound : Real) (estimate : ∀ᶠ index in Filter.atTop, ‖(finite index).bosonic first second‖ ≤ bound) : ‖state.bosonic first second‖ ≤ bound := state.weak_star_limit_preserves_bosonic_bound finite Filter.atTop converges first second bound estimate example : IsingGroundSectorControls.ActualUniformlyGappedIsingFamily ∧ ¬ IsingGroundSectorControls.UnsubtractedExponentialDecay := IsingGroundSectorControls.actual_corrected_ising_clustering_does_not_imply_raw_decay namespace NachtergaeleSims open PhysicsLogic.Papers.NachtergaeleSims2006 example : OneSiteGroundControl.operatorModel.AdmissibleDynamicsAt OneSiteGroundControl.selectedInteraction ∧ OneSiteGroundControl.groundData.AdmissibleAt OneSiteGroundControl.selectedInteraction OneSiteGroundControl.selectedState OneSiteGroundControl.selectedProjection OneSiteGroundControl.exactGap := OneSiteGroundControl.actual_ground_representation_is_admissible example (time : Real) (nonnegative : 0 ≤ time) : OneSiteGroundControl.groundData.imaginaryTimeCorrelation OneSiteGroundControl.selectedState OneSiteGroundControl.loweringObservable OneSiteGroundControl.raisingObservable time = (Real.exp (-time) : Complex) := OneSiteGroundControl.actual_source_imaginary_time_correlation time nonnegative example (framework : PaperFramework) (setup : framework.NondegenerateGroundStateSetup) : framework.ConnectedEqualTimeClusteringFormulaAt := framework.nondegenerate_setup_derives_connected_equal_time_formula setup example (context : SemanticContext) (dependencies : routedArgument.RoutesSemanticallyCertified context.interpretAtom (routedArgument.routeDependencyIds clusteringRoute)) (cone : context.framework.EstablishedFiniteVelocityCone) (ground : context.framework.groundRepresentation.AdmissibleAt context.framework.selectedInteraction context.framework.selectedGroundState context.framework.selectedGroundProjection context.framework.selectedGap) (target : context.framework.groundRepresentation.TargetComponentsVanishAt context.framework.selectedGroundProjection context.framework.targetObservable context.framework.selectedGroundState) (analytic : context.framework.AnalyticApproximationCalculation) (contour : context.framework.CauchyContourCalculation) (spectral : context.framework.SpectralGapEstimateCalculation) (gaussian : context.framework.GaussianSplitCalculation) (commutator : context.framework.CommutatorIntegralCalculation) (optimization : context.framework.CorrelationOptimizationCalculation) : routedArgument.RouteSemanticallyCertified context.interpretAtom .exponentialClustering := context.actual_ground_representation_inputs_realize_source_clustering_route dependencies cone ground target analytic contour spectral gaussian commutator optimization example : Fidelity.countermodelMatrix .actualTargetProjectionNotAdjointCondition := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .degenerateZeroExpectationNotProjectionCondition := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .actualGapLowerBoundNotExactGap := Fidelity.countermodel_matrix_is_realized _ end NachtergaeleSims namespace HastingsKoma open PhysicsLogic.Papers.HastingsKoma2006 example (context : SemanticContext) (clustering : context.interpretAtom .bosonicClustering) (volume : Lattice) (admissible : context.framework.groundData.admissibleVolume volume) : ∃ dynamics : HamiltonianDynamics (context.framework.groundData.finiteSector volume).StateSpace, dynamics.hamiltonian = (context.framework.groundData.finiteSector volume).hamiltonianOperator.toPMap ⊤ ∧ dynamics.RealTimeGeneratorAt := by obtain ⟨dynamics, hamiltonian, generated, _⟩ := context.bosonic_clustering_atom_realizes_finite_hamiltonian_dynamics clustering volume admissible exact ⟨dynamics, hamiltonian, generated⟩ example (framework : PaperFramework) : framework.FiniteVolumeThermodynamicTransfer := framework.actual_weak_star_semantics_supply_finite_volume_transfer example (framework : PaperFramework) : framework.FermionicProjectionIdentityBridge := framework.actual_ground_average_correction_derives_fermionic_bridge example (context : SemanticContext) (dependencies : routedArgument.RoutesSemanticallyCertified context.interpretAtom (routedArgument.routeDependencyIds fermionicRoute)) (setup : context.framework.FermionicClusteringSetup) (integral : context.framework.finitePropagationControlsRealTimeIntegral context.framework.selectedPropagationBound context.framework.selectedGap context.framework.selectedCorrelationBound) (optimization : context.framework.decayParameterOptimized context.framework.selectedGap context.framework.selectedPropagationBound context.framework.selectedCorrelationBound) (calculation : context.framework.FermionicClusteringCalculation) (average : context.framework.groundData.AverageWeakStarLimitAt) : routedArgument.RouteSemanticallyCertified context.interpretAtom .fermionicCorrelationDecay := context.actual_ground_average_inputs_realize_source_fermionic_route dependencies setup integral optimization calculation average example : Fidelity.countermodelMatrix .actualGroundVectorNotTraceIdentity := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .uniformQuasiDegeneracyNotExactFiniteDegeneracy := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .actualFiniteBandNotExactKernel := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .uniformlyGappedIsingCorrectionNotRawDecay := Fidelity.countermodel_matrix_is_realized _ end HastingsKoma end ActualGroundSectorReuse namespace ActualReciprocalCorrelationReuse example {T : PhysicsLogic.Papers.Onsager1931.OnsagerTypes} (framework : PhysicsLogic.Papers.Onsager1931.ReciprocalRelationsFramework T) (inputs : framework.ActualEquilibriumCorrelationInputs) : framework.equilibriumDynamics.NonvacuousReversalCorrelationAt framework.zeroField framework.selectedForce₁ framework.selectedForce₂ := framework.the_actual_heat_ensemble_realizes_nonvacuous_correlation_reversal inputs example {T : PhysicsLogic.Papers.Casimir1945.CasimirTypes} (analysis : PhysicsLogic.Papers.Casimir1945.MicroscopicReversibilityAnalysis T) (field : T.Field) (inputs : analysis.ActualEquilibriumCorrelationInputsAt field) : analysis.equilibriumDynamics.NonvacuousReversalCorrelationAt field analysis.selectedEvenVariable₁ analysis.selectedOddVariable := analysis.the_actual_fluctuation_ensemble_realizes_nonvacuous_correlation_reversal field inputs example {T : PhysicsLogic.Papers.Casimir1945.CasimirTypes} (analysis : PhysicsLogic.Papers.Casimir1945.MicroscopicReversibilityAnalysis T) (field : T.Field) (calculation : analysis.SharedCoefficientCalculationAt field) : (analysis.toReciprocalTransportContext field).RegressionCoefficientCalculationAt (analysis.reciprocalTransportModelAt field) := analysis.finite_lag_correlation_moments_derive_the_shared_coefficient_calculation field calculation example (boundary : PhysicsLogic.Papers.Casimir1945.Fidelity.BoundaryId) : PhysicsLogic.Papers.Casimir1945.Fidelity.countermodelMatrix boundary := PhysicsLogic.Papers.Casimir1945.Fidelity.countermodel_matrix_is_realized boundary end ActualReciprocalCorrelationReuse namespace ActualDecayTimeRecordReuse open PhysicsLogic.Framework example {T : PhysicsLogic.Papers.Gamow1928.GamowTypes} (framework : PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework T) (damped : framework.EstablishedDampedOutgoingDecay) (age elapsed : Real) (nonnegative : 0 ≤ age) (positive : 0 < elapsed) : framework.selectedDecayTimeRecordLaw.NonvacuousExponentialDecayTimeAt (framework.decayRateValueAt framework.lowerDecayRate) age elapsed := (framework.damped_result_exhibits_actual_decay_time_records damped age elapsed nonnegative positive).1 example {T : PhysicsLogic.Papers.Gamow1928.GamowTypes} {Record : Type*} [MeasurableSpace Record] (framework : PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework T) (records : DecayTimeRecordLaw Record) (damped : framework.EstablishedDampedOutgoingDecay) (normalized : records.NormalizedPositiveTimeLawAt) (represents : records.RepresentsSurvivalAt (fun time => PhysicsLogic.Papers.Gamow1928.WaveMechanics.interiorProbability framework.simpleBarrierFormulaReadout.planckConstantValue framework.simpleBarrierFormulaReadout.complexEnergyValue framework.simpleBarrierFormulaReadout.outgoingSpatialAmplitudeValueAt framework.simpleBarrierFormulaReadout.outgoingBoundaryRadius time / framework.simpleBarrierFormulaReadout.initialInteriorProbability)) : (records.toSurvivalReadout (framework.decayRateValueAt framework.lowerDecayRate)).ExponentialMemorylessDecayAt := framework.interior_probability_representation_derives_actual_record_conditioning records damped normalized represents example {T : PhysicsLogic.Papers.GurneyCondon1928.GurneyCondonTypes} (framework : PhysicsLogic.Papers.GurneyCondon1928.RadioactiveDisintegrationFramework T) (selection : BarrierDecaySelection T.System T.State T.Channel T.Barrier T.Energy (PhysicsLogic.Papers.GurneyCondon1928.BarrierWavePair T) T.Rate T.Regime) (potential : T.Potential) (result : framework.EstablishedWaveOverlapDecayAt selection potential) (age elapsed : Real) (nonnegative : 0 ≤ age) (positive : 0 < elapsed) : ∃ records rate, framework.SelectedRecordLawAt selection records rate ∧ (records.jointSurvivalEvent age elapsed).Nonempty ∧ (records.escapeEvent age elapsed).Nonempty := by obtain ⟨records, rate, selected, actual⟩ := framework.established_overlap_decay_has_actual_decay_time_records selection potential result age elapsed nonnegative positive exact ⟨records, rate, selected, records.nonvacuous_decay_has_actual_surviving_and_decaying_records rate age elapsed actual⟩ example : PhysicsLogic.Papers.Gamow1928.Fidelity.countermodelMatrix .coherentOccupationRevivalNotDecayTimeLaw := PhysicsLogic.Papers.Gamow1928.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.GurneyCondon1928.Fidelity.countermodelMatrix .normalizedLifetimeRecordsNotExponentialLaw := PhysicsLogic.Papers.GurneyCondon1928.Fidelity.countermodel_matrix_is_realized _ example : PhysicsLogic.Papers.GurneyCondon1928.Fidelity.countermodelMatrix .zeroSurvivorMassNotConditionalLaw := PhysicsLogic.Papers.GurneyCondon1928.Fidelity.countermodel_matrix_is_realized _ end ActualDecayTimeRecordReuse namespace ActualPolytropicMassReuse open PhysicsLogic.Papers.Chandrasekhar1931 example {T : ChandrasekharTypes} (framework : Framework T) (eos : framework.EstablishedRelativisticEquationOfState) (setup : framework.PolytropicMassLocalSetup) (calculation : framework.PolytropicMassCalculation) : (framework.polytropicMassModelAt framework.polytropicMassFormula).PrintedMassRelationAt (framework.massInGramsValueAt framework.selectedMass) := framework.unique_polytropic_mass_retains_printed_relation (framework.index_three_polytrope_establishes_unique_model_mass eos setup calculation) example {T : ChandrasekharTypes} (framework : Framework T) (eos : framework.EstablishedRelativisticEquationOfState) (setup : framework.PolytropicMassLocalSetup) (calculation : framework.PolytropicMassCalculation) (candidate : Real) (equation : (framework.polytropicMassModelAt framework.polytropicMassFormula).PrintedMassRelationAt candidate) : candidate = framework.massInGramsValueAt framework.selectedMass := framework.unique_polytropic_mass_excludes_distinct_positive_solution (framework.index_three_polytrope_establishes_unique_model_mass eos setup calculation) candidate equation example {T : ChandrasekharTypes} (framework : Framework T) (eos : framework.EstablishedRelativisticEquationOfState) (setup : framework.PolytropicMassLocalSetup) (calculation : framework.PolytropicMassCalculation) (coordinate : Real) (positive : 0 < coordinate) (interior : coordinate < (framework.polytropicMassModelAt framework.polytropicMassFormula).profile.surface) : HasDerivAt (framework.polytropicMassModelAt framework.polytropicMassFormula).profile.enclosedMass (coordinate ^ 2 * (framework.polytropicMassModelAt framework.polytropicMassFormula).profile.potential coordinate ^ 3) coordinate := framework.unique_polytropic_mass_retains_local_density_balance (framework.index_three_polytrope_establishes_unique_model_mass eos setup calculation) coordinate positive interior example (model : PolytropicMass.IndexThreeModel) (physical : model.PhysicalPolytropeAt) (centralPotential : Real) (positive : 0 < centralPotential) : model.surfaceMass centralPotential = model.surfaceMass (2 * centralPotential) ∧ model.radius centralPotential ≠ model.radius (2 * centralPotential) := by refine ⟨?_, Ne.symm (model.doubled_central_potential_changes_radius physical.1 physical.2.1 centralPotential positive)⟩ rw [model.actual_surface_mass_is_independent_of_central_potential physical.1 physical.2.1 centralPotential positive, model.actual_surface_mass_is_independent_of_central_potential physical.1 physical.2.1 (2 * centralPotential) (by positivity)] example {T : ChandrasekharTypes} (framework : Framework T) (complete : framework.CompleteScopedResult) : framework.reportedMassInGramsAt framework.selectedMass = Framework.reportedIdealWhiteDwarfMassGrams ∧ ∀ candidate, framework.idealEquilibriumMassSupportedAt candidate framework.selectedStar .extremeRelativisticDegeneracy -> framework.massInGramsValueAt candidate ≤ (framework.polytropicMassModelAt framework.polytropicMassFormula).uniqueMass := ⟨(framework.equation_five_retains_gram_and_solar_mass_reports complete.2.2.1).1, (framework.complete_result_retains_inhabited_universal_gram_bound complete).2⟩ example (boundary : Fidelity.BoundaryId) : Fidelity.countermodelMatrix boundary := Fidelity.countermodel_matrix_is_realized boundary end ActualPolytropicMassReuse namespace ActualStellarRadialReuse namespace WhiteDwarfs open PhysicsLogic.Papers.Chandrasekhar1931 example (model : PolytropicMass.IndexThreeModel) (physical : model.PhysicalPolytropeAt) (centralPotential : Real) (positive : 0 < centralPotential) (radius : Real) (radialPositive : 0 < radius) (interior : radius < model.radius centralPotential) : HasDerivAt (model.massAtRadius centralPotential) (4 * Real.pi * radius ^ 2 * model.massDensity centralPotential radius) radius ∧ HasDerivAt (model.pressure centralPotential) (-(model.gravitationalConstant * model.massAtRadius centralPotential radius * model.massDensity centralPotential radius) / radius ^ 2) radius := by have fields := model.physical_polytrope_establishes_radial_equilibrium physical centralPotential positive radius radialPositive interior exact ⟨fields.2.2.2.1.2.2.1, fields.2.2.2.2⟩ example {T : ChandrasekharTypes} (framework : Framework T) (eos : framework.EstablishedRelativisticEquationOfState) (setup : framework.PolytropicMassLocalSetup) (calculation : framework.PolytropicMassCalculation) : ((framework.polytropicMassModelAt framework.polytropicMassFormula).radialMassProfile 1).NonvacuousMassGrowthAt ((framework.polytropicMassModelAt framework.polytropicMassFormula).radius 1 / 2) := framework.unique_polytropic_mass_exhibits_nonvacuous_radial_mass_growth (framework.index_three_polytrope_establishes_unique_model_mass eos setup calculation) 1 (by norm_num) example : PolytropicMass.Countermodels.hasHydrostaticPolytropicFields PolytropicMass.Countermodels.unitProfileModel 1 ∧ ¬PolytropicMass.Countermodels.hasGravitationalDensityBalance PolytropicMass.Countermodels.unitProfileModel 1 := PolytropicMass.Countermodels.hydrostatic_polytropic_fields_do_not_supply_gravitational_density_balance end WhiteDwarfs namespace NeutronCores open PhysicsLogic.Papers.OppenheimerVolkoff1939 example (fields : RadialEquilibrium.Fields) (radius : Real) (positive : 0 < radius) (reduction : fields.EinsteinReductionAt radius) : fields.FirstOrderSystemAt radius := fields.einstein_reduction_establishes_actual_first_order_system radius positive reduction example {T : OppenheimerVolkoffTypes} (framework : Framework T) (setup : framework.RelativisticEquilibriumSetup) (calculation : framework.RelativisticEquilibriumCalculation) (radius : Real) (interior : framework.RegularInteriorPointAt framework.relativisticEquilibriumEquations framework.coldNeutronEquationOfState radius) : (framework.radialFieldsAt framework.relativisticEquilibriumEquations framework.coldNeutronEquationOfState).FirstOrderSystemAt radius := framework.established_equilibrium_retains_actual_radial_system (framework.einstein_equations_establish_relativistic_hydrostatic_system setup calculation) radius interior example {T : OppenheimerVolkoffTypes} (framework : Framework T) (setup : framework.ColdNeutronIntegrationSetup) (calculation : framework.ColdNeutronIntegrationCalculation) : framework.selectedRadialMassProfile.NonvacuousMassGrowthAt framework.coldNeutronEquationOfStateFormulaReadout.geometrizedRadiusValue := framework.neutron_family_exhibits_nonvacuous_radial_mass_growth (framework.cold_neutron_equation_of_state_and_numerics_establish_equilibrium_family setup calculation) example {T : OppenheimerVolkoffTypes} (framework : Framework T) (family : framework.EstablishedColdNeutronEquilibriumFamily) : HasDerivAt (framework.fermiParameterValueAt framework.relativisticEquilibriumEquations framework.coldNeutronEquationOfState) framework.coldNeutronEquationOfStateFormulaReadout.geometrizedFermiParameterRadialDerivativeValue framework.coldNeutronEquationOfStateFormulaReadout.geometrizedRadiusValue ∧ HasDerivAt (framework.dimensionlessFermiParameterValueAt framework.relativisticEquilibriumEquations framework.coldNeutronEquationOfState) framework.coldNeutronEquationOfStateFormulaReadout.dimensionlessFermiParameterRadialDerivativeValue framework.coldNeutronEquationOfStateFormulaReadout.dimensionlessRadiusValue := by have rates := framework.neutron_family_retains_actual_fermi_and_rescaled_mass_derivatives family exact ⟨rates.2.1, rates.2.2.2⟩ example {T : OppenheimerVolkoffTypes} (framework : Framework T) (family : framework.EstablishedColdNeutronEquilibriumFamily) : HasDerivAt (framework.dimensionlessEnclosedMassValueAt framework.relativisticEquilibriumEquations) 0 framework.coldNeutronSurfaceFormulaReadout.dimensionlessRadiusValue := framework.neutron_vacuum_surface_has_actual_zero_dimensionless_mass_derivative family example : RadialEquilibrium.Countermodels.hasPositiveRelativisticRadialSystemAt RadialEquilibrium.Countermodels.relativisticPointFields 1 ∧ ¬RadialEquilibrium.Countermodels.hasNewtonianPressureGradientAt RadialEquilibrium.Countermodels.relativisticPointFields 1 := RadialEquilibrium.Countermodels.relativistic_radial_balance_does_not_supply_newtonian_pressure_law end NeutronCores end ActualStellarRadialReuse namespace OVUnitRescalingReuse open PhysicsLogic.Papers.OppenheimerVolkoff1939 noncomputable section def independentUnits : ColdNeutronUnits where particleMass := 2 lightSpeed := 3 planckConstant := 5 gravitationalConstant := 4 theorem independent_units_are_positive : independentUnits.PositiveConstants := by norm_num [ColdNeutronUnits.PositiveConstants, independentUnits] example : independentUnits.lengthUnit ^ 2 = 125 / (192 * Real.pi ^ 2) := by have formula := (independentUnits.original_length_and_mass_scales independent_units_are_positive).2.2.1 rw [formula] change (5 : Real) ^ 3 / (Real.pi ^ 2 * 2 ^ 4 * 3 * 4) = 125 / (192 * Real.pi ^ 2) ring example : independentUnits.lightSpeed * independentUnits.timeUnit / independentUnits.lengthUnit = 1 ∧ independentUnits.gravitationalConstant * independentUnits.massUnit * independentUnits.timeUnit ^ 2 / independentUnits.lengthUnit ^ 3 = 1 := independentUnits.restored_units_make_light_speed_and_gravity_one independent_units_are_positive example : independentUnits.geometrizedMomentum 7 / independentUnits.geometrizedParticleMass = 7 / 6 := by rw [independentUnits.geometrized_momentum_preserves_the_fermi_parameter_ratio independent_units_are_positive 7] change (7 : Real) / (2 * 3) = 7 / 6 norm_num example : independentUnits.physicalEOSScale / independentUnits.energyDensityUnit = 1 / (4 * Real.pi) := independentUnits.energy_density_units_give_the_normalized_eos independent_units_are_positive example (factor : Real) (positive : 0 < factor) : (independentUnits.withParticleMassFactor factor).lengthUnit = independentUnits.lengthUnit / factor ^ 2 ∧ (independentUnits.withParticleMassFactor factor).massUnit = independentUnits.massUnit / factor ^ 2 := independentUnits.particle_mass_changes_both_units_by_inverse_square factor independent_units_are_positive positive /-- An independent smooth coordinate-change test, not an asserted cold-neutron solution. -/ def independentRadialProfile : ColdNeutronRescaling.RadialProfile where enclosedMass := fun radius => radius ^ 3 / 4 fermiParameter := fun radius => radius ^ 2 + 1 theorem independent_mass_has_derivative (radius : Real) : HasDerivAt independentRadialProfile.enclosedMass ((3 / 4) * radius ^ 2) radius := by convert ((hasDerivAt_id radius).pow 3).div_const 4 using 1 simp only [id_eq] ring theorem independent_parameter_has_derivative (radius : Real) : HasDerivAt independentRadialProfile.fermiParameter (2 * radius) radius := by convert ((hasDerivAt_id radius).pow 2).add_const 1 using 1 simp only [id_eq] ring example : independentRadialProfile.dimensionlessMass 3 (1 / 2) = 9 / 32 ∧ independentRadialProfile.dimensionlessParameter 3 (1 / 2) = 13 / 4 := by norm_num [independentRadialProfile, ColdNeutronRescaling.RadialProfile.dimensionlessMass, ColdNeutronRescaling.RadialProfile.dimensionlessParameter] example : HasDerivAt (independentRadialProfile.dimensionlessMass 3) (27 / 16) (1 / 2) := by convert independentRadialProfile.rescaled_mass_has_actual_derivative 3 (1 / 2) _ (by norm_num) (independent_mass_has_derivative (3 * (1 / 2))) using 1 norm_num example : HasDerivAt (independentRadialProfile.dimensionlessParameter 3) 9 (1 / 2) := by convert independentRadialProfile.rescaled_parameter_has_actual_derivative 3 (1 / 2) _ (independent_parameter_has_derivative (3 * (1 / 2))) using 1 norm_num example {T : OppenheimerVolkoffTypes} (framework : Framework T) (setup : framework.ColdNeutronIntegrationSetup) (calculation : framework.ColdNeutronIntegrationCalculation) : framework.SelectedColdNeutronEquationOfStateFormulaPackageAt := framework.cold_neutron_calculation_derives_rescaled_formula_package setup calculation example {T : OppenheimerVolkoffTypes} (framework : Framework T) (family : framework.EstablishedColdNeutronEquilibriumFamily) : framework.coldNeutronEquationOfStateFormulaReadout.RescalingIdentitiesAt framework.coldNeutronUnits.lengthUnit := (framework.neutron_family_retains_actual_unit_map family).1 example {T : OppenheimerVolkoffTypes} (framework : Framework T) (family : framework.EstablishedColdNeutronEquilibriumFamily) : framework.coldNeutronEquationOfStateFormulaReadout.DerivedDimensionlessSystemAt ∧ ¬framework.coldNeutronEquationOfStateFormulaReadout.EquationsEighteenAndNineteenDimensionlessSystemAt := framework.neutron_family_exposes_printed_equation_nineteen_discrepancy family example {T : OppenheimerVolkoffTypes} (framework : Framework T) (family : framework.EstablishedColdNeutronEquilibriumFamily) : framework.coldNeutronUnits.physicalGravitationalMass framework.coldNeutronEquationOfStateFormulaReadout.geometrizedEnclosedMassValue = framework.coldNeutronUnits.massUnit * framework.coldNeutronEquationOfStateFormulaReadout.dimensionlessEnclosedMassValue := (framework.neutron_family_restores_dimensional_radius_and_gravitational_mass family).2 example (boundary : Fidelity.BoundaryId) : Fidelity.countermodelMatrix boundary := Fidelity.countermodel_matrix_is_realized boundary end end OVUnitRescalingReuse namespace PhysicalRecordAccessReuse open PhysicsLogic.Framework.CoherentRecordMeasurement section Everett open PhysicsLogic.Papers.Everett1957 example {T : EverettTypes} (framework : RelativeStateFramework T) (setup : framework.MultipleObserverSetup) (calculation : framework.MultipleObserverCalculation) : framework.recordProcess.NonvacuousRecordComparisonAt := framework.observer_comparison_derives_actual_record_access setup calculation example {T : EverettTypes} (framework : RelativeStateFramework T) (setup : framework.MultipleObserverSetup) (calculation : framework.MultipleObserverCalculation) : framework.originalObserverInterventionReadout.firstCommunicatedRecordValue = framework.originalObserverInterventionReadout.secondCommunicatedRecordValue := (framework.observer_route_retains_original_local_disturbance_and_remote_stability setup calculation).1.1 example {T : EverettTypes} (framework : RelativeStateFramework T) (setup : framework.MultipleObserverSetup) (calculation : framework.MultipleObserverCalculation) (outcome : Bool) : framework.recordProcess.comparedState (outcome, outcome, outcome) = framework.objectAmplitudesAt framework.initialObjectState outcome := by have inputs := (calculation setup).1 exact (framework.recordProcess.calibration_preserves_the_same_input_coefficients inputs.1.2 inputs.2 outcome).2 example {T : EverettTypes} (context : SemanticContext T) (result : context.interpretAtom .observerConsistency) : context.framework.recordProcess.NonvacuousRecordComparisonAt := context.observer_consistency_atom_retains_actual_record_access result example (boundary : Fidelity.BoundaryId) : Fidelity.countermodelMatrix boundary := Fidelity.countermodel_matrix_is_realized boundary end Everett section Rovelli open PhysicsLogic.Papers.Rovelli1996 example {T : RovelliTypes} (framework : RelationalFramework T) (setup : framework.CorrelationInformationSetup) (calculation : framework.CorrelationInformationCalculation) : HasCertainCorrelation framework.recordProcess.measuredState ∧ ∀ outcome, ¬HasCertainOutcome framework.recordProcess.measuredState outcome := framework.correlation_information_retains_actual_projector_and_outcome_obstruction setup calculation example {T : RovelliTypes} (framework : RelationalFramework T) (setup : framework.ObserverComparisonSetup) (calculation : framework.ObserverComparisonCalculation) : (framework.recordProcessAt framework.comparisonReference).NonvacuousRecordComparisonAt := framework.physical_comparison_derives_actual_record_access setup calculation example {T : RovelliTypes} (framework : RelationalFramework T) (setup : framework.CorrelationInformationSetup) (calculation : framework.CorrelationInformationCalculation) (outcome : Bool) : let state := outcomeComponent (framework.measuredRecordCoefficientsAt framework.secondObserver framework.correlatedState) outcome (∃ pair, state pair ≠ 0) ∧ ∀ pair, state pair ≠ 0 -> pair.1 = outcome ∧ pair.2 = outcome := framework.conditioned_comparison_in_p_account_has_a_matching_pointer setup calculation outcome example {T : RovelliTypes} (context : SemanticContext T) (result : context.interpretAtom .measurementKnownWithoutOutcome) : HasCertainCorrelation (context.framework.measuredRecordCoefficientsAt context.framework.secondObserver context.framework.correlatedState) := context.information_atom_retains_actual_correlation_projector result example {T : RovelliTypes} (context : SemanticContext T) (result : context.interpretAtom .consistentComparisonRecords) : context.framework.comparisonProcess.NonvacuousRecordComparisonAt := context.comparison_atom_retains_actual_record_access result example (boundary : Fidelity.BoundaryId) : Fidelity.countermodelMatrix boundary := Fidelity.countermodel_matrix_is_realized boundary end Rovelli end PhysicalRecordAccessReuse namespace WeightedReversalReuse section Evans open PhysicsLogic.Papers.EvansCohenMorriss1993 example {measure : FiniteSegmentMeasureFormulaReadout} {window : StressWindowRatioFormulaReadout} (data : StressWindowWeights measure window) (inputs : data.PhysicalInputsAt) : data.model.ratio data.selectedEvent = data.model.reverseWeightedAverage data.selectedEvent (fun index => Real.exp (data.pairExponent index)) := data.segment_weights_derive_the_reverse_weighted_exponential_average inputs example {measure : FiniteSegmentMeasureFormulaReadout} {window : StressWindowRatioFormulaReadout} (data : StressWindowWeights measure window) (inputs : data.PhysicalInputsAt) (printed : data.PrintedStarredAggregationClaimAt) : window.SourceFormulaAt := data.the_additional_printed_aggregation_claim_gives_equation_eight inputs printed example {T : EvansCohenMorrissTypes} (framework : Framework T) (setup : framework.ProbabilityRatioSetup) (inputs : framework.SelectedWeightAggregationInputsAt) : framework.stressWindowWeights.model.NonvacuousWeightedAggregationAt framework.stressWindowWeights.selectedEvent := framework.selected_weight_inputs_derive_shared_nonvacuous_aggregation setup inputs example {T : EvansCohenMorrissTypes} (framework : Framework T) (setup : framework.ProbabilityRatioSetup) (calculation : framework.ProbabilityRatioCalculation) : framework.EstablishedProbabilityRatio := framework.proposed_measure_and_reversal_pair_yield_probability_ratio setup calculation example {T : EvansCohenMorrissTypes} (context : SemanticContext T) (result : context.interpretAtom .establishedProbabilityRatio) : context.framework.stressWindowWeights.model.NonvacuousWeightedAggregationAt context.framework.stressWindowWeights.selectedEvent := context.ratio_result_atom_retains_actual_weighted_aggregation result example : StressWindowWeights.Countermodels.constantContractionWeights.PrintedStarredAggregationClaimAt := StressWindowWeights.Countermodels.constant_contraction_supports_the_printed_starred_average.2.1 example (boundary : Fidelity.BoundaryId) : Fidelity.countermodelMatrix boundary := Fidelity.countermodel_matrix_is_realized boundary end Evans section Gallavotti open PhysicsLogic.Papers.GallavottiCohen1995 example {partition : SRBFinitePartitionFormulaReadout} (data : FiniteFluctuationWeights partition) (fluctuation scale : Real) (inputs : data.PhysicalInputsAt fluctuation scale) : data.NonvacuousProbabilityRatioAt fluctuation scale := data.physical_weights_derive_the_computed_correction fluctuation scale inputs example {partition : SRBFinitePartitionFormulaReadout} (data : FiniteFluctuationWeights partition) (fluctuation scale : Real) (inputs : data.PhysicalInputsAt fluctuation scale) (preferred : 1 < data.correctionLower * Real.exp (fluctuation * scale)) : data.srbModel.toWeightedReversalContext.StatisticalAsymmetryAt data.selectedEvent := (data.the_corrected_lower_bound_gives_actual_statistical_preference fluctuation scale inputs preferred).2 example {T : GallavottiCohenTypes} (framework : Framework T) (srb : framework.EstablishedSRBConstruction) (inputs : framework.SelectedWeightCalculationInputsAt) : framework.fluctuationWeights.srbModel.NonvacuousWeightedAggregationAt framework.fluctuationWeights.selectedEvent := framework.selected_weight_inputs_derive_shared_nonvacuous_aggregation srb inputs example {T : GallavottiCohenTypes} (framework : Framework T) (srb : framework.EstablishedSRBConstruction) (setup : framework.FluctuationSymmetrySetup) (calculation : framework.TimeReversalRatioCalculation) (bridge : framework.FiniteAndLongTimeBridge) : framework.EstablishedFluctuationSymmetry := framework.time_reversal_and_srb_weights_yield_long_time_symmetry srb setup calculation bridge example {T : GallavottiCohenTypes} (context : SemanticContext T) (result : context.interpretAtom .establishedFluctuationSymmetry) : context.framework.fluctuationWeights.srbModel.NonvacuousWeightedAggregationAt context.framework.fluctuationWeights.selectedEvent := context.fluctuation_result_atom_retains_actual_weighted_aggregation result example : (∑ index, FiniteFluctuationWeights.Countermodels.measureCorrectionWeights.srbTileProbabilityValueAt index * FiniteFluctuationWeights.Countermodels.measureCorrectionWeights.normalizedFluctuationValueAt index) = 1 := FiniteFluctuationWeights.Countermodels.both_controls_retain_unit_mean_normalized_production.1 example (boundary : Fidelity.BoundaryId) : Fidelity.countermodelMatrix boundary := Fidelity.countermodel_matrix_is_realized boundary end Gallavotti end WeightedReversalReuse namespace FiniteResolutionReuse section Evans open PhysicsLogic.Papers.EvansCohenMorriss1993 example {measure : FiniteSegmentMeasureFormulaReadout} {window : StressWindowRatioFormulaReadout} (data : StressWindowWeights measure window) (inputs : data.PhysicalInputsAt) : data.model.NonvacuousLogarithmicEstimateAt data.selectedEvent data.printedMeanExponent data.contractionExponentSpread := data.physical_inputs_derive_the_actual_contraction_spread_estimate inputs example {measure : FiniteSegmentMeasureFormulaReadout} {window : StressWindowRatioFormulaReadout} (data : StressWindowWeights measure window) (inputs : data.PhysicalInputsAt) : (data.model.logRatioApproximationContext data.printedMeanExponent).ControlledApproximation false true () (fun event => event = data.selectedEvent) data.contractionExponentSpread := data.physical_inputs_realize_a_controlled_log_ratio_approximation inputs example {measure : FiniteSegmentMeasureFormulaReadout} {window : StressWindowRatioFormulaReadout} (data : StressWindowWeights measure window) (inputs : data.PhysicalInputsAt) (zero : data.contractionExponentSpread = 0) : data.PrintedStarredAggregationClaimAt := data.zero_contraction_spread_supplies_the_printed_aggregation_claim inputs zero example {T : EvansCohenMorrissTypes} (framework : Framework T) (setup : framework.ProbabilityRatioSetup) (inputs : framework.SelectedWeightAggregationInputsAt) : framework.stressWindowWeights.model.NonvacuousLogarithmicEstimateAt framework.stressWindowWeights.selectedEvent framework.stressWindowWeights.printedMeanExponent framework.stressWindowWeights.contractionExponentSpread := framework.selected_weight_inputs_derive_shared_logarithmic_estimate setup inputs example {T : EvansCohenMorrissTypes} (context : SemanticContext T) (result : context.interpretAtom .establishedProbabilityRatio) : context.framework.stressWindowWeights.model.NonvacuousLogarithmicEstimateAt context.framework.stressWindowWeights.selectedEvent context.framework.stressWindowWeights.printedMeanExponent context.framework.stressWindowWeights.contractionExponentSpread := context.ratio_result_atom_retains_the_actual_logarithmic_error_bound result example : Fidelity.countermodelMatrix .controlledSpreadNotExactCentralExponential := Fidelity.countermodel_matrix_is_realized _ end Evans section Gallavotti open PhysicsLogic.Papers.GallavottiCohen1995 example {partition : SRBFinitePartitionFormulaReadout} (data : FiniteFluctuationWeights partition) (reference lower upper scale : Real) (inputs : data.FiniteBinPhysicalInputsAt lower upper scale) : data.srbModel.NonvacuousLogarithmicEstimateAt data.selectedEvent (reference * scale) (data.finiteResolutionError reference lower upper scale) := data.finite_bin_weights_derive_the_actual_logarithmic_error reference lower upper scale inputs example {partition : SRBFinitePartitionFormulaReadout} (data : FiniteFluctuationWeights partition) (reference lower upper scale : Real) (inputs : data.FiniteBinPhysicalInputsAt lower upper scale) : (data.srbModel.logRatioApproximationContext (reference * scale)).ControlledApproximation false true () (fun event => event = data.selectedEvent) (data.finiteResolutionError reference lower upper scale) := data.finite_bin_inputs_realize_a_controlled_log_ratio_approximation reference lower upper scale inputs example {T : GallavottiCohenTypes} (framework : Framework T) (srb : framework.EstablishedSRBConstruction) (lower upper : Real) (inputs : framework.SelectedFiniteBinCalculationInputsAt lower upper) : framework.FiniteResolutionFluctuationEstimateAt lower upper := framework.selected_finite_bin_inputs_derive_the_physical_resolution_estimate srb lower upper inputs example : FiniteFluctuationWeights.Countermodels.finiteBinWeights.srbModel.ratio FiniteFluctuationWeights.Countermodels.finiteBinWeights.selectedEvent = 7 / 3 := FiniteFluctuationWeights.Countermodels.using_the_literal_negative_bin_changes_the_actual_ratio.1 example : Fidelity.countermodelMatrix .literalNegativeIntervalNotReflectedEvent := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .finiteBinNotPositiveExactCenterProbability := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .fixedResolutionNotReferenceRate := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .leadingRateNotUniformMultiplicativeCorrection := Fidelity.countermodel_matrix_is_realized _ end Gallavotti end FiniteResolutionReuse namespace WetterichInverseReuse open PhysicsLogic.Papers.Wetterich1993 abbrev Configuration := Fin 2 -> Real abbrev Kernel := Matrix (Fin 2) (Fin 2) Real def massAt (index : Fin 2) : Real := if index = 0 then 1 else 3 noncomputable def cutoff (time : Real) : Real := Real.exp (2 * time) noncomputable def matrixFlow : ExactFlowFormulaReadout Configuration Kernel where referenceConfiguration := fun _ => 0 scaleValueAt := Real.exp averageActionValueAt := fun time field => field 0 ^ 2 / 2 + 3 * field 1 ^ 2 / 2 + (Real.log (1 + cutoff time) + Real.log (3 + cutoff time)) / 2 hessianKernelAt := fun _ _ => Matrix.diagonal massAt regulatorKernelAt := fun time => Matrix.diagonal (fun _ => cutoff time) regulatorScaleDerivativeKernelAt := fun time => Matrix.diagonal (fun _ => 2 * cutoff time) connectedPropagatorKernelAt := fun time _ => Matrix.diagonal (fun index => (massAt index + cutoff time)⁻¹) traceValue := Matrix.trace regulatorDerivativeRepresentsAt := fun time => HasDerivAt cutoff (2 * cutoff time) time traceDefinedAt := fun _ => True theorem matrix_action_has_the_actual_coordinate_gradient (time : Real) (field : Configuration) (index : Fin 2) : HasDerivAt (fun value => matrixFlow.averageActionValueAt time (Function.update field index value)) (massAt index * field index) (field index) := by fin_cases index · have calculation := (((hasDerivAt_id (field 0)).pow 2).div_const 2) |>.add_const (3 * field 1 ^ 2 / 2) |>.add_const ((Real.log (1 + cutoff time) + Real.log (3 + cutoff time)) / 2) convert calculation using 1 norm_num [massAt] · have calculation := ((((hasDerivAt_id (field 1)).pow 2).const_mul 3).div_const 2) |>.const_add (field 0 ^ 2 / 2) |>.add_const ((Real.log (1 + cutoff time) + Real.log (3 + cutoff time)) / 2) convert calculation using 1 norm_num [massAt] ring theorem matrix_hessian_is_computed_from_the_same_action (time : Real) (field : Configuration) (row column : Fin 2) : matrixFlow.hessianKernelAt time field row column = deriv (fun value => deriv (fun next => matrixFlow.averageActionValueAt time (Function.update (Function.update field column value) row next)) ((Function.update field column value) row)) (field column) := by have gradient : (fun value => deriv (fun next => matrixFlow.averageActionValueAt time (Function.update (Function.update field column value) row next)) ((Function.update field column value) row)) = fun value => massAt row * (Function.update field column value) row := funext fun value => (matrix_action_has_the_actual_coordinate_gradient time (Function.update field column value) row).deriv rw [gradient] by_cases same : row = column · subst column simpa [matrixFlow] using ((hasDerivAt_id (field row)).const_mul (massAt row)).deriv.symm · simp [matrixFlow, Function.update_of_ne same, Matrix.diagonal_apply_ne _ same] theorem cutoff_has_the_actual_scale_derivative (time : Real) : HasDerivAt cutoff (2 * cutoff time) time := by have calculation := (Real.hasDerivAt_exp (2 * time)).comp time ((hasDerivAt_id time).const_mul 2) simpa [cutoff, mul_comm] using calculation theorem matrix_flow_has_actual_connected_inverse_identities (time : Real) (field : Configuration) : matrixFlow.ConnectedInverseIdentityAt time field := by have cutoffPositive : 0 < cutoff time := Real.exp_pos _ have first : 1 + cutoff time ≠ 0 := ne_of_gt (by positivity) have second : 3 + cutoff time ≠ 0 := ne_of_gt (by positivity) constructor <;> ext row column <;> fin_cases row <;> fin_cases column <;> simp [matrixFlow, massAt, ExactFlowFormulaReadout.regulatedHessianKernelAt, Matrix.mul_apply, Fin.sum_univ_two, first, second] theorem matrix_flow_has_the_actual_scale_derivative (time : Real) (field : Configuration) : HasDerivAt (fun next => matrixFlow.averageActionValueAt next field) ((1 / 2 : Real) * Matrix.trace (matrixFlow.connectedPropagatorKernelAt time field * matrixFlow.regulatorScaleDerivativeKernelAt time)) time := by have cutoffPositive : 0 < cutoff time := Real.exp_pos _ have first : 1 + cutoff time ≠ 0 := ne_of_gt (by positivity) have second : 3 + cutoff time ≠ 0 := ne_of_gt (by positivity) have firstLog := ((cutoff_has_the_actual_scale_derivative time).const_add 1).log first have secondLog := ((cutoff_has_the_actual_scale_derivative time).const_add 3).log second have calculation := ((firstLog.add secondLog).div_const 2) |>.const_add (field 0 ^ 2 / 2 + 3 * field 1 ^ 2 / 2) convert calculation using 1 simp [matrixFlow, massAt, Matrix.diagonal_mul_diagonal, Matrix.trace_diagonal, Fin.sum_univ_two] ring theorem matrix_flow_has_the_connected_trace_calculation : matrixFlow.ConnectedTraceEvolutionEquationAt := by refine ⟨?_, ?_, ?_⟩ · intro time exact ⟨Real.exp_pos time, (Real.hasDerivAt_exp time).differentiableAt, (Real.hasDerivAt_exp time).deriv⟩ · intro time field exact (matrix_flow_has_the_actual_scale_derivative time field).differentiableAt · intro time field exact ⟨cutoff_has_the_actual_scale_derivative time, trivial, (matrix_flow_has_the_actual_scale_derivative time field).deriv⟩ theorem matrix_control_uses_the_public_inverse_substitution : matrixFlow.ExactEvolutionEquationAt := matrixFlow.connected_trace_calculation_and_inverse_identities_give_exact_flow matrix_flow_has_the_connected_trace_calculation matrix_flow_has_actual_connected_inverse_identities theorem matrix_control_retains_the_actual_inverse (time : Real) (field : Configuration) : matrixFlow.propagatorKernelAt time field * matrixFlow.regulatedHessianKernelAt time field = 1 /\ matrixFlow.regulatedHessianKernelAt time field * matrixFlow.propagatorKernelAt time field = 1 := matrixFlow.exact_flow_uses_a_genuine_two_sided_inverse matrix_control_uses_the_public_inverse_substitution time field example {T : WetterichTypes} (context : SemanticContext T) (source : context.interpretAtom .inverseHessianIdentity) (time : Real) (configuration : T.FormulaConfiguration) : context.framework.exactFlowFormulaReadout.ConnectedInverseIdentityAt time configuration := context.inverse_source_atom_retains_actual_kernel_identities source time configuration example {T : WetterichTypes} (framework : Framework T) (flow : framework.toSharedExactFlowContext.EstablishedExactRegulatedScaleFlowAt framework.sharedExactFlowModel) : framework.ExactFlowEquationFormulaAt := framework.established_shared_flow_retains_the_selected_formula flow example : Fidelity.countermodelMatrix .connectedTraceWithoutInverseNotExactFlow := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .totalizedKernelInverseNotAdmissibleFlow := Fidelity.countermodel_matrix_is_realized _ end WetterichInverseReuse namespace BoseCurrentResponseReuse open MeasureTheory PhysicsLogic.Papers.FeynmanTwoFluid1954 open BoseCurrentResponseReadout abbrev Modes := Fin 3 × Bool /-- A fresh finite gas with doubled momentum magnitude and atom mass. -/ noncomputable def response : BoseCurrentResponseReadout Modes where inverseTemperatureValue := 1 heliumAtomMassValue := 2 momentumMeasure := Measure.count excitationEnergyValueAt := fun _ => Real.log 2 momentumComponentValueAt := fun axis mode => if mode.1 = axis then (if mode.2 then 2 else -2) else 0 theorem momentum_squared_is_four (mode : Modes) : response.momentumSquaredValueAt mode = 4 := by rcases mode with ⟨direction, sign⟩ fin_cases direction <;> cases sign <;> norm_num [momentumSquaredValueAt, response, Fin.sum_univ_three] <;> rfl theorem occupation_is_one (mode : Modes) : response.occupationValueAt mode = 1 := by dsimp [occupationValueAt, response] rw [one_mul, Real.exp_log (by norm_num)] norm_num theorem printed_weight_is_one (mode : Modes) : response.printedWeightValueAt mode = 1 := by simp [printedWeightValueAt, occupation_is_one] theorem actual_response_weight_is_two (mode : Modes) : response.responseWeightValueAt mode = 2 := by rw [response.response_weight_is_occupation_plus_occupation_squared, printed_weight_is_one, occupation_is_one] norm_num theorem actual_calculation_domain : response.CalculationDomainAt := by refine ⟨by norm_num [response], by norm_num [response], ?_, ?_, ?_, ?_⟩ · exact Filter.Eventually.of_forall fun _ => Real.log_pos (by norm_num) · exact Integrable.of_finite (μ := Measure.count) · exact Integrable.of_finite (μ := Measure.count) · intro axis driftAxis exact Integrable.of_finite (μ := Measure.count) theorem current_allows_differentiation_under_the_integral : response.DifferentiationUnderIntegralAt := by intro axis driftAxis have calculation := HasDerivAt.fun_sum (u := (Finset.univ : Finset Modes)) (fun mode _ => response.each_mode_has_the_actual_current_derivative axis driftAxis mode (by norm_num [response]) (Real.log_pos (by norm_num))) simpa only [response, integral_count] using calculation theorem counting_measure_has_mass_six : (Measure.count : Measure Modes).real Set.univ = 6 := by have calculation := integral_count (fun _ : Modes => (1 : Real)) norm_num [Modes] at calculation exact calculation theorem current_has_an_isotropic_response_tensor : response.IsotropicResponseIntegralAt := by intro axis driftAxis simp only [actual_response_weight_is_two, momentum_squared_is_four] fin_cases axis <;> fin_cases driftAxis <;> norm_num [response, integral_count, Fintype.sum_prod_type, Fin.sum_univ_three, counting_measure_has_mass_six] <;> rfl theorem occupied_momentum_moment_is_twenty_four : response.occupiedMomentumMomentValue = 24 := by unfold occupiedMomentumMomentValue simp only [momentum_squared_is_four, occupation_is_one, mul_one] norm_num [response, integral_count, counting_measure_has_mass_six] theorem inputs_realize_a_nonvacuous_bose_response : response.NonvacuousResponseInputsAt := by refine ⟨actual_calculation_domain, current_allows_differentiation_under_the_integral, current_has_an_isotropic_response_tensor, ?_⟩ rw [occupied_momentum_moment_is_twenty_four] norm_num theorem the_mass_and_momentum_normalization_is_retained : response.printedNormalDensityValue = 4 / (2 * Real.pi) ^ 3 /\ response.responseNormalDensityValue = 8 / (2 * Real.pi) ^ 3 := by dsimp [printedNormalDensityValue, responseNormalDensityValue] simp only [momentum_squared_is_four, printed_weight_is_one, actual_response_weight_is_two] norm_num [response, integral_count, counting_measure_has_mass_six] exact ⟨rfl, rfl⟩ theorem the_client_derives_actual_response_through_the_public_api : (∀ axis driftAxis, HasDerivAt (response.currentValueAt axis driftAxis) (if axis = driftAxis then response.responseNormalDensityValue else 0) 0) /\ (∀ axis, Not (HasDerivAt (response.currentValueAt axis axis) response.printedNormalDensityValue 0)) := ⟨response.integral_differentiation_and_isotropy_derive_the_normal_density_response actual_calculation_domain current_allows_differentiation_under_the_integral current_has_an_isotropic_response_tensor, response.the_printed_density_cannot_be_the_same_current_derivative inputs_realize_a_nonvacuous_bose_response⟩ example {T : FeynmanTypes} (framework : AtomicTwoFluidFramework T) (result : framework.EstablishedTwoFluidDescription) : framework.movingFluidFormulaReadout.PrintedNormalDensityResponseBoundaryAt := framework.the_two_fluid_result_retains_the_printed_density_response_boundary result example {T : FeynmanTypes} (context : SemanticContext T) (result : context.interpretAtom .establishedTwoFluidDescription) : context.framework.movingFluidFormulaReadout.PrintedNormalDensityResponseBoundaryAt := context.two_fluid_result_atom_retains_the_printed_density_response_boundary result example : Fidelity.countermodelMatrix .printedNormalDensityNotBoseCurrentResponse := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .boseCurrentResponseNotLiteralNormalDensity := Fidelity.countermodel_matrix_is_realized _ end BoseCurrentResponseReuse namespace LandauTransportReuse open PhysicsLogic.Papers.LandauSuperfluidity1941 open OriginalTwoMotionCounterflowReadout def independentTransport : OriginalTwoMotionCounterflowReadout where totalDensityValue := 7 normalDensityValue := 2 superfluidDensityValue := 5 temperatureValue := 3 specificEntropyValue := 2 normalVelocityValue := 5 superfluidVelocityValue := -2 wallAngularVelocityValue := 3 normalAngularVelocityValue := 3 superfluidAngularVelocityValue := 0 geometryMomentValue := 4 absoluteZeroTotalDensityValue := 8 absoluteZeroNormalDensityValue := 0 lambdaPointTotalDensityValue := 6 lambdaPointNormalDensityValue := 6 theorem the_client_has_a_physical_transport_domain : independentTransport.TransportDomainAt := by norm_num [TransportDomainAt, HasTwoMotionMassFractions, independentTransport] theorem the_client_has_separate_rotating_inputs : independentTransport.RotatingInputsAt := by norm_num [RotatingInputsAt, RotatingVesselSetupAt, HasTwoMotionMassFractions, TemperatureEndpointsAt, independentTransport] theorem the_client_currents_retain_the_temperature_factor : independentTransport.massCurrentValue = 0 /\ independentTransport.entropyCurrentValue = 70 /\ independentTransport.normalHeatCurrentValue = 210 := by norm_num [massCurrentValue, massCurrentValueAt, entropyCurrentValue, entropyCurrentValueAt, normalHeatCurrentValue, thermalCurrentValueAt, entropyDensityValue, independentTransport] theorem the_client_uses_the_actual_mass_response_derivatives (normalSpeed superSpeed : Real) : HasDerivAt (fun velocity => independentTransport.massCurrentValueAt velocity superSpeed) 2 normalSpeed /\ HasDerivAt (fun velocity => independentTransport.massCurrentValueAt normalSpeed velocity) 5 superSpeed := independentTransport.mass_response_coefficients_are_actual_derivatives normalSpeed superSpeed theorem the_client_derives_carried_inertia_from_angular_momentum : independentTransport.RetainsRotatingMassFractions := independentTransport.rotating_inputs_derive_the_retained_mass_fractions the_client_has_separate_rotating_inputs theorem the_client_has_the_correct_carried_and_full_inertias : independentTransport.carriedMomentOfInertiaValue = 8 /\ independentTransport.fullMomentOfInertiaValue = 28 := by rw [independentTransport.the_measured_carried_inertia_is_computed_from_angular_momentum the_client_has_separate_rotating_inputs.1] norm_num [fullMomentOfInertiaValue, independentTransport] theorem the_client_heat_relative_to_mass_flow_is_frame_invariant (frameSpeed : Real) : (independentTransport.boosted frameSpeed).heatCurrentRelativeToMassFlowValue = independentTransport.heatCurrentRelativeToMassFlowValue := independentTransport.heat_relative_to_mass_flow_is_galilean_invariant the_client_has_a_physical_transport_domain.1 frameSpeed theorem the_client_zero_mass_flow_gives_temperature_times_entropy_transport : independentTransport.heatCurrentRelativeToMassFlowValue = independentTransport.temperatureValue * independentTransport.entropyCurrentValue := independentTransport.zero_mass_flux_identifies_heat_with_temperature_times_entropy_current the_client_currents_retain_the_temperature_factor.1 theorem the_client_admits_heat_counterflow_in_any_small_speed_window (bound : Real) (positiveBound : 0 < bound) : (independentTransport.smallCounterflowAt bound).HasZeroMassCounterflowWithNonzeroHeat /\ |(independentTransport.smallCounterflowAt bound).normalVelocityValue| < bound /\ |(independentTransport.smallCounterflowAt bound).superfluidVelocityValue| < bound := independentTransport.small_counterflow_realizes_heat_transport_within_any_positive_speed_bound the_client_has_a_physical_transport_domain bound positiveBound theorem an_arithmetic_partition_can_fit_the_client_coefficients : independentTransport.ExactAtomPartitionAt 7 1 1 := by refine ⟨by norm_num, by norm_num, {0, 1}, ?_, ?_⟩ · norm_num [independentTransport] · norm_num [independentTransport, Finset.card_compl] example {T : LandauTypes} (framework : HeliumTwoMotionFramework T) (statistics : framework.EstablishedTwoMotionStatistics) (wallSpeed : Real) : HasDerivAt (framework.wallCarriedAngularMomentumValueAt framework.helium framework.lowTemperatureRegime) (framework.originalTwoMotionCounterflowReadout.geometryMomentValue * framework.densityValueAt framework.normalDensity framework.helium framework.lowTemperatureRegime) wallSpeed := framework.the_selected_vessel_angular_momentum_has_the_actual_normal_response statistics wallSpeed example {T : LandauTypes} (framework : HeliumTwoMotionFramework T) (hydrodynamics : framework.EstablishedHydrodynamics) (normalSpeed superSpeed : Real) : framework.linearizedEntropyCurrentValueAt framework.helium framework.hydrodynamicRegime normalSpeed superSpeed = (framework.totalDensityValueAt framework.helium framework.hydrodynamicRegime * framework.specificEntropyValueAt framework.helium framework.hydrodynamicRegime) * normalSpeed := framework.the_selected_linearized_entropy_current_uses_total_mass_and_specific_entropy hydrodynamics normalSpeed superSpeed example {T : LandauTypes} (context : SemanticContext T) (hydrodynamics : context.interpretAtom .establishedHydrodynamics) (bound : Real) (positiveBound : 0 < bound) : (context.framework.originalTwoMotionCounterflowReadout.smallCounterflowAt bound).HasZeroMassCounterflowWithNonzeroHeat /\ |(context.framework.originalTwoMotionCounterflowReadout.smallCounterflowAt bound).normalVelocityValue| < bound /\ |(context.framework.originalTwoMotionCounterflowReadout.smallCounterflowAt bound).superfluidVelocityValue| < bound := context.hydrodynamic_atom_supports_arbitrarily_small_constitutive_counterflow hydrodynamics bound positiveBound example : Fidelity.countermodelMatrix .integerResponseNotAbsenceOfEveryExactPartition := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .heatCurrentNotEntropyCurrent := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .rotatingResponseNotAxialCounterflow := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .wallFrameHeatlessNotEveryFrame := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .wallFrameHeatlessNotMassFrameHeatless := Fidelity.countermodel_matrix_is_realized _ end LandauTransportReuse namespace LandauSoundFamilyClient open PhysicsLogic.Papers.LandauSuperfluidity1941.SoundPropagation open Filter open scoped Topology /-- Independently constructed constitutive data. No phonon-response matching or physical helium equation of state is asserted for this diagnostic. -/ noncomputable def thermalFreeEnergy (temperature : Real) : Real := -temperature ^ 4 / 32 noncomputable def potential : ThermodynamicPotential := ThermodynamicPotential.separablePotential 9 thermalFreeEnergy theorem the_client_free_energy_has_its_actual_derivative (temperature : Real) : HasDerivAt thermalFreeEnergy (-temperature ^ 3 / 8) temperature := by convert (((hasDerivAt_id temperature).pow 4).neg.div_const 32) using 1 simp only [id_eq] ring theorem the_client_entropy_is_from_its_own_potential (density temperature : Real) : potential.entropy density temperature = temperature ^ 3 / 8 := by rw [potential, ThermodynamicPotential.the_separable_entropy_is_the_actual_thermal_derivative, (the_client_free_energy_has_its_actual_derivative temperature).deriv] ring theorem the_client_energy_is_from_its_own_potential (density temperature : Real) : potential.energy density temperature = 9 * Real.log density + 3 * temperature ^ 4 / 32 := by rw [ThermodynamicPotential.energy, the_client_entropy_is_from_its_own_potential] dsimp [potential, ThermodynamicPotential.separablePotential, thermalFreeEnergy] ring theorem the_client_heat_is_the_derivative_of_its_own_energy (density temperature : Real) : potential.heat density temperature = 3 * temperature ^ 3 / 8 := by have derivative := ((((hasDerivAt_id temperature).pow 4).const_mul 3).div_const 32).const_add (9 * Real.log density) have actual : HasDerivAt (potential.energy density) (3 * temperature ^ 3 / 8) temperature := by convert derivative.congr_of_eventuallyEq (Filter.Eventually.of_forall (the_client_energy_is_from_its_own_potential density)) using 1 simp only [id_eq] ring exact actual.deriv noncomputable def calculatedState (temperature : Real) : ThermodynamicState where normalDensity := temperature ^ 4 / 4 superfluidDensity := 28 - temperature ^ 4 / 4 temperature := temperature entropyPerMass := temperature ^ 3 / 8 heatPerMass := 3 * temperature ^ 3 / 8 pressureDensityDerivative := 9 pressureTemperatureDerivative := 0 entropyDensityDerivative := 0 noncomputable def family : AcousticFamily where stateAt := fun temperature => potential.state 28 (temperature ^ 4 / 4) temperature firstModeAt := fun temperature => LongitudinalMode.densityBranch (potential.state 28 (temperature ^ 4 / 4) temperature) secondModeAt := fun temperature => LongitudinalMode.entropyBranch (potential.state 28 (temperature ^ 4 / 4) temperature) theorem the_client_state_coefficients_are_actually_realized (temperature : Real) : potential.state 28 (temperature ^ 4 / 4) temperature = calculatedState temperature := by have coefficients := ThermodynamicPotential.a_separable_free_energy_derives_the_decoupled_acoustic_coefficients 9 thermalFreeEnergy 28 (temperature ^ 4 / 4) temperature (by norm_num) apply ThermodynamicState.ext · rfl · rfl · rfl · exact the_client_entropy_is_from_its_own_potential 28 temperature · exact the_client_heat_is_the_derivative_of_its_own_energy 28 temperature · exact coefficients.1 · exact coefficients.2.1 · exact coefficients.2.2 theorem the_client_has_actual_thermodynamic_response_regularity (temperature : Real) : potential.ResponseRegularityAt 28 temperature := by have freeEnergy := (the_client_free_energy_has_its_actual_derivative temperature).const_add (9 * Real.log 28) have entropy := (((hasDerivAt_id temperature).pow 3).div_const 8).congr_of_eventuallyEq (Filter.Eventually.of_forall (the_client_entropy_is_from_its_own_potential 28)) have derivatives := ThermodynamicPotential.a_separable_free_energy_has_actual_pressure_and_density_entropy_derivatives 9 thermalFreeEnergy 28 temperature (by norm_num) exact ⟨⟨freeEnergy.differentiableAt, entropy.differentiableAt⟩, derivatives.1.differentiableAt, derivatives.2.1.differentiableAt, derivatives.2.2.differentiableAt⟩ theorem the_client_has_actual_modes_at_positive_subcritical_temperatures (temperature : Real) (positiveTemperature : 0 < temperature) (normalBound : temperature ^ 4 < 112) : family.PhysicalModesAt temperature := by have domain : (calculatedState temperature).PhysicalDomainAt := by refine ⟨div_pos (pow_pos positiveTemperature 4) (by norm_num), ?_, positiveTemperature, div_pos (pow_pos positiveTemperature 3) (by norm_num), div_pos (mul_pos (by norm_num) (pow_pos positiveTemperature 3)) (by norm_num), by norm_num [calculatedState]⟩ dsimp [calculatedState] linarith simp only [family, AcousticFamily.PhysicalModesAt, the_client_state_coefficients_are_actually_realized] exact ⟨LongitudinalMode.decoupled_thermodynamics_constructs_a_nonzero_density_mode _ domain rfl rfl, LongitudinalMode.decoupled_thermodynamics_constructs_a_nonzero_entropy_mode _ domain rfl rfl, by norm_num [LongitudinalMode.densityBranch], by norm_num [LongitudinalMode.entropyBranch], rfl⟩ theorem the_client_has_two_nonvacuous_physical_modes : family.PhysicalModesAt 2 := the_client_has_actual_modes_at_positive_subcritical_temperatures 2 (by norm_num) (by norm_num) theorem the_client_has_actual_modes_throughout_a_low_temperature_neighborhood : ∀ᶠ temperature in nhdsWithin (0 : Real) (Set.Ioi 0), family.PhysicalModesAt temperature := by have temperatureLimit : Tendsto (fun temperature : Real => temperature) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0) := tendsto_id.mono_left inf_le_left filter_upwards [self_mem_nhdsWithin, (tendsto_order.mp temperatureLimit).2 2 (by norm_num)] with temperature positive belowTwo have temperaturePositive : 0 < temperature := positive have squareBelow : temperature ^ 2 < 4 := by nlinarith have fourthBelow : temperature ^ 4 < 112 := by nlinarith [mul_self_lt_mul_self (sq_nonneg temperature) squareBelow] exact the_client_has_actual_modes_at_positive_subcritical_temperatures temperature temperaturePositive fourthBelow theorem the_client_sound_speeds_are_three_and_two : (family.firstModeAt 2).speed = 3 /\ (family.secondModeAt 2).speed = 2 := by norm_num [family, the_client_state_coefficients_are_actually_realized, calculatedState, LongitudinalMode.densityBranch, LongitudinalMode.entropyBranch, ThermodynamicState.secondSoundEstimateSquared] all_goals rfl theorem the_library_realizes_both_client_waves_at_nonzero_wave_number : (family.firstModeAt 2).PropagatingWaveAt (family.stateAt 2) 2 /\ (family.secondModeAt 2).PropagatingWaveAt (family.stateAt 2) 2 := ⟨(LongitudinalMode.propagating_modes_are_realized_by_the_actual_wave_fields _ _ 2 (by norm_num)).mp the_client_has_two_nonvacuous_physical_modes.1, (LongitudinalMode.propagating_modes_are_realized_by_the_actual_wave_fields _ _ 2 (by norm_num)).mp the_client_has_two_nonvacuous_physical_modes.2.1⟩ theorem the_library_checks_both_client_dispersion_relations : (family.stateAt 2).characteristic ((family.firstModeAt 2).speed ^ 2) = 0 /\ (family.stateAt 2).characteristic ((family.secondModeAt 2).speed ^ 2) = 0 := ⟨LongitudinalMode.a_nonzero_propagating_mode_satisfies_the_coupled_sound_characteristic _ _ the_client_has_two_nonvacuous_physical_modes.1, LongitudinalMode.a_nonzero_propagating_mode_satisfies_the_coupled_sound_characteristic _ _ the_client_has_two_nonvacuous_physical_modes.2.1⟩ theorem the_library_exhibits_an_actual_nonconstant_client_field : ((family.secondModeAt 2).densityAmplitude * (family.secondModeAt 2).profile 2 0 0 ≠ (family.secondModeAt 2).densityAmplitude * (family.secondModeAt 2).profile 2 0 (Real.pi / 2)) \/ ((family.secondModeAt 2).temperatureAmplitude * (family.secondModeAt 2).profile 2 0 0 ≠ (family.secondModeAt 2).temperatureAmplitude * (family.secondModeAt 2).profile 2 0 (Real.pi / 2)) := LongitudinalMode.a_propagating_wave_has_an_actual_nonconstant_density_or_temperature_field _ _ 2 the_library_realizes_both_client_waves_at_nonzero_wave_number.2 theorem the_client_normal_response_changes_the_actual_sound_family (temperature : Real) (positiveTemperature : 0 < temperature) : (family.secondModeAt temperature).speed = Real.sqrt ((28 - temperature ^ 4 / 4) / 6) := by simp only [family, the_client_state_coefficients_are_actually_realized, calculatedState, LongitudinalMode.entropyBranch, ThermodynamicState.secondSoundEstimateSquared] congr 1 field_simp [ne_of_gt positiveTemperature] ring theorem the_client_sound_limit_retains_its_own_response_normalization : Tendsto (fun temperature => (family.secondModeAt temperature).speed) (nhdsWithin 0 (Set.Ioi 0)) (nhds (Real.sqrt (14 / 3 : Real))) := by have temperatureLimit : Tendsto (fun temperature : Real => temperature) (nhdsWithin 0 (Set.Ioi 0)) (nhds 0) := tendsto_id.mono_left inf_le_left have squares := (((tendsto_const_nhds (x := (28 : Real))).sub ((temperatureLimit.pow 4).div_const 4)).div_const 6) norm_num at squares have roots := Real.continuous_sqrt.continuousAt.tendsto.comp squares apply roots.congr' filter_upwards [self_mem_nhdsWithin] with temperature positive exact (the_client_normal_response_changes_the_actual_sound_family temperature positive).symm theorem the_client_normal_velocity_has_its_actual_amplitude : (family.firstModeAt 2).normalVelocityAmplitude = 3 / 28 := by norm_num [family, the_client_state_coefficients_are_actually_realized, calculatedState, LongitudinalMode.densityBranch, ThermodynamicState.totalDensity] theorem the_library_retains_viscosity_and_wave_number_in_the_actual_residual : (family.firstModeAt 2).viscousMomentumResidual (family.stateAt 2) 3 2 0 0 = 9 / 7 := by rw [LongitudinalMode.the_viscous_correction_is_from_the_actual_spatial_curvature, LongitudinalMode.the_actual_differential_residual_is_the_amplitude_balance, the_client_normal_velocity_has_its_actual_amplitude] norm_num [LongitudinalMode.profile] theorem the_library_rejects_a_positive_viscosity_solution_for_the_client_wave : (family.firstModeAt 2).LinearWaveEquationsAt (family.stateAt 2) 2 /\ ¬(∀ time position, (family.firstModeAt 2).viscousMomentumResidual (family.stateAt 2) 3 2 time position = 0) := by refine ⟨the_library_realizes_both_client_waves_at_nonzero_wave_number.1.2.2.2.2, ?_⟩ intro equation have selected := equation 0 0 rw [the_library_retains_viscosity_and_wave_number_in_the_actual_residual] at selected norm_num at selected theorem the_library_preserves_phase_speed_for_arbitrarily_small_client_fields (velocityBound : Real) (positiveVelocityBound : 0 < velocityBound) : ∃ small : LongitudinalMode, small.speed = 3 /\ small.PropagatingWaveAt (family.stateAt 2) 2 /\ ∀ time position, |small.densityAmplitude * small.profile 2 time position| < 1 / 10 /\ |small.temperatureAmplitude * small.profile 2 time position| < 1 / 20 /\ |small.normalVelocityAmplitude * small.profile 2 time position| < velocityBound /\ |small.superfluidVelocityAmplitude * small.profile 2 time position| < velocityBound := by obtain ⟨small, speed, wave, density, temperature, normal, superfluid, _firstCorrection, _secondCorrection⟩ := (family.firstModeAt 2).actual_propagating_waves_admit_arbitrarily_small_perturbations (family.stateAt 2) 2 the_client_has_two_nonvacuous_physical_modes.1 (by norm_num) (1 / 10) (1 / 20) velocityBound (by norm_num) (by norm_num) positiveVelocityBound refine ⟨small, speed.trans the_client_sound_speeds_are_three_and_two.1, wave, ?_⟩ intro time position exact ⟨lt_of_le_of_lt (small.the_actual_field_is_bounded_by_its_amplitude _ 2 time position) density, lt_of_le_of_lt (small.the_actual_field_is_bounded_by_its_amplitude _ 2 time position) temperature, lt_of_le_of_lt (small.the_actual_field_is_bounded_by_its_amplitude _ 2 time position) normal, lt_of_le_of_lt (small.the_actual_field_is_bounded_by_its_amplitude _ 2 time position) superfluid⟩ end LandauSoundFamilyClient section LandauActualSoundFidelityReuse open PhysicsLogic.Papers.LandauSuperfluidity1941 example : Fidelity.countermodelMatrix .constantWaveProfileNotPropagationBalance := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .onePotentialNotIdenticalHeldVariableConventions := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .coupledWavesNotDecoupledSoundEstimates := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .vanishingSuperfluidDensityNotZeroSoundWithoutRegularity := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .idealSoundWaveNotPositiveViscositySolution := Fidelity.countermodel_matrix_is_realized _ example {T : LandauTypes} (context : SemanticContext T) (holds : context.interpretAtom .establishedHydrodynamics) : context.framework.originalTwoSoundHydrodynamicReadout.ThermodynamicCoefficientsAt := context.hydrodynamic_atom_retains_actual_thermodynamic_derivatives holds example {T : LandauTypes} (context : SemanticContext T) (holds : context.interpretAtom .establishedHydrodynamics) : context.framework.originalTwoSoundHydrodynamicReadout.HasOriginalZeroTemperatureAndLambdaPointLimits := context.hydrodynamic_atom_retains_actual_one_sided_sound_limits holds example {T : LandauTypes} (context : SemanticContext T) (holds : context.interpretAtom .establishedHydrodynamics) (mode : SoundPropagation.LongitudinalMode) : mode.massCurrentAmplitude context.framework.originalTwoSoundHydrodynamicReadout.selectedState = context.framework.linearizedMassCurrentValueAt context.framework.helium context.framework.hydrodynamicRegime mode.normalVelocityAmplitude mode.superfluidVelocityAmplitude := context.hydrodynamic_atom_uses_the_same_actual_mass_current holds mode end LandauActualSoundFidelityReuse namespace LandauMicroscopicOperatorClient open PhysicsLogic.Papers.LandauSuperfluidity1941.MicroscopicMotion open FiniteDiagonalModel noncomputable section /-- A separately chosen three-state system tests the sector API. This is not a realization of the full liquid's distributional commutators. -/ def dynamics : SectorDynamics Unit Unit (State (Fin 3)) := diagonalDynamics ![5, 5, 11] ![0, 0, 2] def levels : SectorLevels dynamics where potentialState := ket 1 vortexState := ket 2 potentialEnergy := 5 vortexEnergy := 11 theorem the_client_potential_sector_has_its_actual_kernel (state : State (Fin 3)) : state ∈ dynamics.potentialSector ↔ state 2 = 0 := by change (∀ _ : Unit, diagonal (fun index => ((![0, 0, 2] : Fin 3 -> Real) index : Complex)) state = 0) ↔ state 2 = 0 constructor · intro zero have coordinate := congrArg (fun vector : State (Fin 3) => vector 2) (zero ()) change (2 : Complex) * state 2 = 0 at coordinate exact (mul_eq_zero.mp coordinate).resolve_left (by norm_num) · intro zero _probe apply PiLp.ext intro index fin_cases index <;> simp [zero] theorem the_client_has_distinct_normalized_potential_states : dynamics.NormalizedPotentialStateAt (ket 0) /\ dynamics.NormalizedPotentialStateAt (ket 1) /\ (ket 0 : State (Fin 3)) ≠ ket 1 := by refine ⟨⟨each_basis_ket_is_an_actual_normalized_state 0, ?_⟩, ⟨each_basis_ket_is_an_actual_normalized_state 1, ?_⟩, ?_⟩ · rw [the_client_potential_sector_has_its_actual_kernel] simp [ket, EuclideanSpace.single_apply] · rw [the_client_potential_sector_has_its_actual_kernel] simp [ket, EuclideanSpace.single_apply] · intro same have coordinate := congrArg (fun vector : State (Fin 3) => vector 0) same norm_num [ket, EuclideanSpace.single_apply] at coordinate theorem the_client_has_a_normalized_nonpotential_state : dynamics.VortexStateAt (ket 2) := by refine ⟨each_basis_ket_is_an_actual_normalized_state 2, ?_⟩ rw [the_client_potential_sector_has_its_actual_kernel] simp [ket] theorem the_client_pure_vortex_sector_has_its_actual_orthogonality (state : State (Fin 3)) : state ∈ dynamics.potentialSectorᗮ ↔ state 0 = 0 /\ state 1 = 0 := by rw [Submodule.mem_orthogonal] constructor · intro orthogonal exact ⟨by simpa [ket, EuclideanSpace.inner_single_left] using orthogonal (ket 0) the_client_has_distinct_normalized_potential_states.1.2, by simpa [ket, EuclideanSpace.inner_single_left] using orthogonal (ket 1) the_client_has_distinct_normalized_potential_states.2.1.2⟩ · rintro ⟨first, second⟩ vector potential have zero := (the_client_potential_sector_has_its_actual_kernel vector).mp potential simp [PiLp.inner_apply, Fin.sum_univ_succ, RCLike.inner_apply, first, second, zero] theorem every_normalized_client_potential_state_has_energy_five (state : State (Fin 3)) (potential : dynamics.NormalizedPotentialStateAt state) : dynamics.StationaryAt 5 state := by refine ⟨potential.1, ?_⟩ have zero := (the_client_potential_sector_has_its_actual_kernel state).mp potential.2 apply PiLp.ext intro index change ((![5, 5, 11] : Fin 3 -> Real) index : Complex) * state index = (5 : Complex) * state index fin_cases index <;> simp [zero] theorem every_normalized_client_pure_vortex_state_has_energy_eleven (state : State (Fin 3)) (normalized : ‖state‖ = 1) (vortex : state ∈ dynamics.potentialSectorᗮ) : dynamics.StationaryAt 11 state := by refine ⟨normalized, ?_⟩ obtain ⟨first, second⟩ := (the_client_pure_vortex_sector_has_its_actual_orthogonality state).mp vortex apply PiLp.ext intro index change ((![5, 5, 11] : Fin 3 -> Real) index : Complex) * state index = (11 : Complex) * state index fin_cases index <;> simp [first, second] theorem the_client_ground_states_are_actual_sector_energy_minima : levels.ActualSectorGroundStatesAt := by have potential : dynamics.NormalizedPotentialStateAt (ket 1) := the_client_has_distinct_normalized_potential_states.2.1 have pureVortex : ket 2 ∈ dynamics.potentialSectorᗮ := by rw [the_client_pure_vortex_sector_has_its_actual_orthogonality] simp [ket, EuclideanSpace.single_apply] have stationaryLevels : levels.ActualLowestStationaryLevelsAt := by refine ⟨potential, every_normalized_client_potential_state_has_energy_five _ potential, the_client_has_a_normalized_nonpotential_state, every_normalized_client_pure_vortex_state_has_energy_eleven _ (each_basis_ket_is_an_actual_normalized_state 2) pureVortex, ?_, ?_⟩ · intro energy state stationary zero have expected := dynamics.a_normalized_stationary_state_has_its_actual_energy_expectation 5 state (every_normalized_client_potential_state_has_energy_five state ⟨stationary.1, zero⟩) have actual := dynamics.a_normalized_stationary_state_has_its_actual_energy_expectation energy state stationary exact le_of_eq (expected.symm.trans actual) · intro energy state stationary nonzero have coordinate : state 2 ≠ 0 := by simpa only [the_client_potential_sector_has_its_actual_kernel] using nonzero exact le_of_eq (an_actual_nonzero_eigenstate_coordinate_fixes_the_energy ![5, 5, 11] ![0, 0, 2] energy state stationary 2 coordinate) refine ⟨stationaryLevels, pureVortex, ?_, ?_⟩ · intro state potential exact le_of_eq (dynamics.a_normalized_stationary_state_has_its_actual_energy_expectation 5 state (every_normalized_client_potential_state_has_energy_five state potential)).symm · intro state normalized vortex exact le_of_eq (dynamics.a_normalized_stationary_state_has_its_actual_energy_expectation 11 state (every_normalized_client_pure_vortex_state_has_energy_eleven state normalized vortex)).symm theorem the_client_has_a_symmetric_hamiltonian_and_actual_schrodinger_evolution : dynamics.hamiltonian.IsSymmetric /\ dynamics.TimeEvolutionApproximationAt /\ dynamics.EvolutionGroupAt /\ dynamics.NormPreservingEvolutionAt := ⟨the_actual_real_diagonal_hamiltonian_is_symmetric _, the_finite_schrodinger_series_converges_to_the_actual_phase_evolution _ _, the_actual_phase_evolution_has_the_group_law _, the_actual_phase_evolution_preserves_the_state_norm _⟩ theorem the_library_derives_conservation_for_the_client_operator_model : dynamics.PreservesPotentialMotionAt := dynamics.actual_hamiltonian_and_propagator_limits_derive_potential_motion_conservation (the_diagonal_models_have_actual_curl_commutator_closure _ _) (the_actual_finite_zero_vorticity_sector_is_closed _ _) (the_diagonal_model_uses_its_actual_hamiltonian _ _) the_client_has_a_symmetric_hamiltonian_and_actual_schrodinger_evolution.2.1 the_client_has_a_symmetric_hamiltonian_and_actual_schrodinger_evolution.2.2.2 theorem the_client_potential_state_remains_normalized_and_nonzero (time : Real) : dynamics.NormalizedPotentialStateAt (dynamics.evolution time levels.potentialState) /\ dynamics.evolution time levels.potentialState ≠ 0 := by have conserved := the_library_derives_conservation_for_the_client_operator_model time (ket 1) the_client_has_distinct_normalized_potential_states.2.1 exact ⟨conserved, dynamics.a_normalized_potential_state_is_a_nonzero_vector _ conserved⟩ theorem the_client_excludes_an_actual_potential_to_vortex_evolution : ¬dynamics.PotentialToVortexEvolutionExistsAt := dynamics.conservation_excludes_an_actual_potential_to_vortex_time_evolution the_library_derives_conservation_for_the_client_operator_model theorem the_client_gap_uses_the_actual_six_unit_energy_difference : levels.ActualSectorGroundStatesAt /\ levels.excitationGap = 6 /\ 0 < levels.excitationGap := by refine ⟨the_client_ground_states_are_actual_sector_energy_minima, ?_, ?_⟩ · change (11 : Real) - 5 = 6 norm_num · change (0 : Real) < 11 - 5 norm_num theorem the_client_pure_vortex_states_have_actual_curl_norm_two (state : State (Fin 3)) (normalized : ‖state‖ = 1) (vortex : state ∈ dynamics.potentialSectorᗮ) : ‖dynamics.vorticity () state‖ = 2 := by obtain ⟨first, second⟩ := (the_client_pure_vortex_sector_has_its_actual_orthogonality state).mp vortex have same : dynamics.vorticity () state = (2 : Complex) • state := by apply PiLp.ext intro index change ((![0, 0, 2] : Fin 3 -> Real) index : Complex) * state index = (2 : Complex) * state index fin_cases index <;> simp [first, second] rw [same, norm_smul, normalized] norm_num theorem the_client_separated_vorticity_domain_is_explicit : dynamics.SeparatedVorticityStatesAt .pureMotionSectors (fun _ => 1) := by refine ⟨2, by norm_num, ?_⟩ rintro state ⟨normalized, pure⟩ nonzero small have bound := small () (le_refl 1) rw [the_client_pure_vortex_states_have_actual_curl_norm_two state normalized (pure.resolve_left nonzero)] at bound exact (lt_irrefl 2) bound theorem the_library_retains_the_client_selected_vorticity_domain : ¬dynamics.ArbitrarilyWeakVortexStatesAt .pureMotionSectors (fun _ => 1) := dynamics.a_selected_vorticity_separation_excludes_arbitrarily_weak_states_in_that_scope .pureMotionSectors (fun _ => 1) the_client_separated_vorticity_domain_is_explicit theorem the_client_vortex_evolution_has_an_actual_nontrivial_phase : dynamics.evolution (Real.pi / 11) levels.vortexState = -levels.vortexState := by have exponent : -Complex.I * ((Real.pi / 11 : Real) : Complex) * (11 : Complex) = -((Real.pi : Complex) * Complex.I) := by push_cast ring have phase : Complex.exp (-Complex.I * ((Real.pi / 11 : Real) : Complex) * (11 : Complex)) = -1 := by rw [exponent, Complex.exp_neg, Complex.exp_pi_mul_I] norm_num simp only [Complex.ofReal_div, Complex.ofReal_ofNat, neg_mul] at phase apply PiLp.ext intro index fin_cases index <;> simp [dynamics, diagonalDynamics, phaseEvolution, levels, ket, EuclideanSpace.single_apply, phase] theorem the_client_propagator_is_not_an_assigned_identity : dynamics.evolution (Real.pi / 11) ≠ 1 := by intro identity have evolution := the_client_vortex_evolution_has_an_actual_nontrivial_phase rw [identity, Module.End.one_apply] at evolution have coordinate := congrArg (fun vector : State (Fin 3) => vector 2) evolution norm_num [levels, ket] at coordinate end end LandauMicroscopicOperatorClient section LandauMicroscopicSourceReuse open PhysicsLogic.Papers.LandauSuperfluidity1941 example : Fidelity.countermodelMatrix .conservedSeparatedMotionSectorsNotEnergyGap := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .operatorConservationNotPotentialStateExistence := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .hamiltonianCommutationNotZeroVorticity := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .zeroVorticityExpectationNotPotentialSector := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .stationaryVorticitySeparationNotAllSuperpositions := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .symmetrizedVelocityNotOneSidedQuotient := Fidelity.countermodel_matrix_is_realized _ example : Fidelity.countermodelMatrix .microscopicVelocityNotRatioOfMeans := Fidelity.countermodel_matrix_is_realized _ example {T : LandauTypes} (context : SemanticContext T) (holds : context.interpretAtom .establishedQuantumLiquidSpectrum) : context.framework.selectedMicroscopicLiquid.levels.ActualSectorGroundStatesAt := context.microscopic_atom_retains_actual_sector_energy_minima holds example {T : LandauTypes} (context : SemanticContext T) (holds : context.interpretAtom .establishedQuantumLiquidSpectrum) : ¬context.framework.selectedMicroscopicLiquid.dynamics.ArbitrarilyWeakVortexStatesAt context.framework.selectedMicroscopicLiquid.vorticityStateScope MicroscopicMotion.MicroscopicLiquidData.probeSize := context.microscopic_atom_retains_the_selected_vorticity_state_domain holds end LandauMicroscopicSourceReuse namespace RabiBeamReadoutClient open PhysicsLogic.Papers.Rabi1939 BeamRefocusing MeasureTheory noncomputable section def geometry : Geometry := ⟨1, 2, 1, 2, 2, 1⟩ def apparatus : Apparatus := ⟨geometry, 4, 5, 1 / 2⟩ def firstPacket : Packet := (2, 1 / 2, -1 / 2) def secondPacket : Packet := (4, 1 / 2, -1 / 2) def particle (speed : Real) : ParticleRay := ⟨2, speed, 1 / 2, -1 / 2⟩ def sourceMeasure : Measure Packet := Measure.dirac firstPacket + Measure.dirac firstPacket + Measure.dirac secondPacket def pulse (speed : Real) : ResonanceFormulaReadout where staticFieldMagnitude := 100 oscillatingFieldAmplitude := 2 angularMomentumMagnitude := 1 / 2 planckConstant := 1 magneticMoment := 1 / 2 signedMagneticMoment := 1 / 2 larmorFrequency := 100 driveFrequency := 100 interactionTime := 1 / speed transitionProbability := Real.sin (Real.pi / speed) ^ 2 theorem the_velocity_resolved_pulse_uses_the_original_selected_formulas (speed : Real) (positive : 0 < speed) : (pulse speed).EquationOneLarmorFrequencyAt ∧ (pulse speed).EquationTwoWeakOscillatingTransitionAt := by have domain : (pulse speed).PhysicalFieldFrequencyDomainAt := by norm_num [ResonanceFormulaReadout.PhysicalFieldFrequencyDomainAt, pulse, le_of_lt (one_div_pos.mpr positive)] exact le_of_lt positive have delta : (pulse speed).equationTwoDeltaValueAt = (1 : Real) / 100 := by norm_num [ResonanceFormulaReadout.equationTwoDeltaValueAt, pulse] have factor : (pulse speed).equationTwoDetuningFactorValueAt = ((1 : Real) / 100) ^ 2 := by unfold ResonanceFormulaReadout.equationTwoDetuningFactorValueAt rw [delta] norm_num [ResonanceFormulaReadout.equationTwoFrequencyRatioValueAt, pulse] constructor · refine ⟨domain, by norm_num [pulse], ?_⟩ change (100 : Real) = (1 / 2 * 100) / (1 / 2 * 1) norm_num · refine ⟨domain, ?_⟩ rw [delta, factor, Real.sqrt_sq (by norm_num)] change Real.sin (Real.pi / speed) ^ 2 = ((1 / 100 : Real) ^ 2 / (1 / 100 : Real) ^ 2) * Real.sin (Real.pi * (1 / speed) * 100 * (1 / 100 : Real)) ^ 2 have phase : Real.pi * (1 / speed) * 100 * (1 / 100 : Real) = Real.pi / speed := by ring rw [phase, div_self (by norm_num : ((1 / 100 : Real) ^ 2) ≠ 0), one_mul] theorem the_velocity_resolved_pulses_are_actual_normalized_coherent_transitions (speed : Real) (positive : 0 < speed) : PhysicsLogic.Framework.CoherentTwoState.NonvacuousCoherentTransitionAt (pulse speed).printedPulseMatrix PhysicsLogic.Framework.CoherentTwoState.lowerState ((pulse speed).printedPulseMatrix.mulVec PhysicsLogic.Framework.CoherentTwoState.lowerState) (pulse speed).transitionProbability := (pulse speed).equation_two_realizes_actual_coherent_pulse (the_velocity_resolved_pulse_uses_the_original_selected_formulas speed positive).2 theorem the_two_selected_velocities_have_different_actual_transition_probabilities : (pulse 2).transitionProbability = 1 ∧ (pulse 4).transitionProbability = 1 / 2 := by change Real.sin (Real.pi / 2) ^ 2 = 1 ∧ Real.sin (Real.pi / 4) ^ 2 = (1 : Real) / 2 norm_num [Real.sin_pi_div_four, div_pow, Real.sq_sqrt] theorem the_client_interaction_times_are_computed_from_one_unit_drive_length : (pulse 2).interactionTime = 1 / 2 ∧ (pulse 4).interactionTime = 1 / 4 := by norm_num [pulse] theorem the_client_gradients_match_its_distinct_geometric_factors : apparatus.PhysicalAt ∧ apparatus.MatchedGradientGeometryAt ∧ geometry.firstFactor = 10 ∧ geometry.secondFactor = 8 := by norm_num [apparatus, geometry, Apparatus.PhysicalAt, Apparatus.MatchedGradientGeometryAt, Geometry.PhysicalAt, Geometry.firstFactor, Geometry.secondFactor, Geometry.slitDistance, Geometry.postSlitDistance] change (4 : Real) * 10 = 40 ∧ (10 : Real) = 10 norm_num theorem the_library_computes_the_client_detector_position (speed : Real) (positive : 0 < speed) : (apparatus.atDetector (particle speed)).position = 10 / speed ^ 2 := by have formula := apparatus.matched_gradient_geometry_computes_the_projection_difference (particle speed) the_client_gradients_match_its_distinct_geometric_factors.1 ⟨by norm_num [particle], positive⟩ the_client_gradients_match_its_distinct_geometric_factors.2.1 rw [formula] change ((1 / 2 : Real) - (-1 / 2)) * (4 * geometry.firstFactor) / (2 * 2 * speed ^ 2) = 10 / speed ^ 2 rw [the_client_gradients_match_its_distinct_geometric_factors.2.2.1] field_simp [ne_of_gt positive] ring theorem both_client_transition_branches_are_resolved_by_the_actual_aperture : apparatus.detectorHalfWidth < |(apparatus.atDetector (particle 2)).position| ∧ apparatus.detectorHalfWidth < |(apparatus.atDetector (particle 4)).position| := by rw [the_library_computes_the_client_detector_position 2 (by norm_num), the_library_computes_the_client_detector_position 4 (by norm_num)] norm_num [apparatus] theorem the_client_unchanged_rays_are_transmitted_at_every_positive_speed (speed : Real) (positive : 0 < speed) : apparatus.acceptance (particle speed).unchanged = 1 := apparatus.matched_fields_transmit_each_unchanged_positive_speed_ray (particle speed) the_client_gradients_match_its_distinct_geometric_factors.1 ⟨by norm_num [particle], positive⟩ the_client_gradients_match_its_distinct_geometric_factors.2.1 theorem the_client_aperture_rejects_both_changed_rays : apparatus.acceptance (particle 2) = 0 ∧ apparatus.acceptance (particle 4) = 0 := ⟨apparatus.a_resolved_projection_change_misses_the_actual_detector (particle 2) both_client_transition_branches_are_resolved_by_the_actual_aperture.1, apparatus.a_resolved_projection_change_misses_the_actual_detector (particle 4) both_client_transition_branches_are_resolved_by_the_actual_aperture.2⟩ theorem each_client_integrand_is_integrable (f : Packet -> Real) : Integrable f sourceMeasure := ((integrable_dirac (by simp)).add_measure (integrable_dirac (by simp))).add_measure (integrable_dirac (by simp)) theorem the_client_source_integral_keeps_its_unequal_velocity_weights (f : Packet -> Real) : (∫ packet, f packet ∂sourceMeasure) = 2 * f firstPacket + f secondPacket := by rw [sourceMeasure, integral_add_measure ((integrable_dirac (by simp)).add_measure (integrable_dirac (by simp))) (integrable_dirac (by simp)), integral_add_measure (integrable_dirac (by simp)) (integrable_dirac (by simp))] simp only [integral_dirac] ring def beam : BeamData where apparatus := apparatus molecularMass := 2 sourceMeasure := sourceMeasure transmission := fun _ => 4 / 5 transitionProbability := fun packet => (pulse packet.1).transitionProbability detectorGain := 3 baselineBackground := 7 drivenBackground := 7 theorem the_client_beam_has_actual_positive_flux_and_bounded_probabilities : beam.PhysicalAt ∧ beam.IntegralDomainAt ∧ beam.ControlledDetectorAt := by refine ⟨⟨the_client_gradients_match_its_distinct_geometric_factors.1, by norm_num [beam], ?_, ?_, ?_⟩, ⟨each_client_integrand_is_integrable _, each_client_integrand_is_integrable _, each_client_integrand_is_integrable _⟩, ?_⟩ · dsimp [beam, sourceMeasure] infer_instance · change 0 < ∫ _packet : Packet, (1 : Real) ∂sourceMeasure rw [the_client_source_integral_keeps_its_unequal_velocity_weights] norm_num · simp [beam, sourceMeasure, ae_add_measure_iff, ae_dirac_eq, firstPacket, secondPacket, the_two_selected_velocities_have_different_actual_transition_probabilities.1, the_two_selected_velocities_have_different_actual_transition_probabilities.2] norm_num · norm_num [BeamData.ControlledDetectorAt, beam] theorem the_client_packet_detection_weights_come_from_its_pulses_and_aperture : beam.baselineDensity firstPacket = 4 / 5 ∧ beam.baselineDensity secondPacket = 4 / 5 ∧ beam.drivenDensity firstPacket = 0 ∧ beam.drivenDensity secondPacket = 2 / 5 ∧ beam.rejectedTransitionDensity firstPacket = 4 / 5 ∧ beam.rejectedTransitionDensity secondPacket = 2 / 5 := by change (4 / 5 : Real) * apparatus.acceptance (particle 2).unchanged = 4 / 5 ∧ (4 / 5 : Real) * apparatus.acceptance (particle 4).unchanged = 4 / 5 ∧ (4 / 5 : Real) * ((1 - (pulse 2).transitionProbability) * apparatus.acceptance (particle 2).unchanged + (pulse 2).transitionProbability * apparatus.acceptance (particle 2)) = 0 ∧ (4 / 5 : Real) * ((1 - (pulse 4).transitionProbability) * apparatus.acceptance (particle 4).unchanged + (pulse 4).transitionProbability * apparatus.acceptance (particle 4)) = 2 / 5 ∧ (4 / 5 : Real) * (pulse 2).transitionProbability * (1 - apparatus.acceptance (particle 2)) = 4 / 5 ∧ (4 / 5 : Real) * (pulse 4).transitionProbability * (1 - apparatus.acceptance (particle 4)) = 2 / 5 rw [the_client_unchanged_rays_are_transmitted_at_every_positive_speed 2 (by norm_num), the_client_unchanged_rays_are_transmitted_at_every_positive_speed 4 (by norm_num), the_client_aperture_rejects_both_changed_rays.1, the_client_aperture_rejects_both_changed_rays.2, the_two_selected_velocities_have_different_actual_transition_probabilities.1, the_two_selected_velocities_have_different_actual_transition_probabilities.2] norm_num theorem the_client_integrates_actual_counts_instead_of_assigning_a_signal_drop : beam.referenceIntensity = 3 ∧ beam.baselineIntensity = 12 / 5 ∧ beam.drivenIntensity = 2 / 5 ∧ (∫ packet, beam.rejectedTransitionDensity packet ∂beam.sourceMeasure) = 2 := by have weights := the_client_packet_detection_weights_come_from_its_pulses_and_aperture change (∫ _packet : Packet, (1 : Real) ∂sourceMeasure) = 3 ∧ (∫ packet, beam.baselineDensity packet ∂sourceMeasure) = 12 / 5 ∧ (∫ packet, beam.drivenDensity packet ∂sourceMeasure) = 2 / 5 ∧ (∫ packet, beam.rejectedTransitionDensity packet ∂sourceMeasure) = 2 simp only [the_client_source_integral_keeps_its_unequal_velocity_weights] rw [weights.1, weights.2.1, weights.2.2.1, weights.2.2.2.1, weights.2.2.2.2.1, weights.2.2.2.2.2] norm_num theorem the_client_has_a_populated_resolvable_transition : beam.ResponsivePopulationAt := by have member : firstPacket ∈ beam.responsivePackets := by refine ⟨by norm_num [beam], ?_, both_client_transition_branches_are_resolved_by_the_actual_aperture.1⟩ change 0 < (pulse 2).transitionProbability rw [the_two_selected_velocities_have_different_actual_transition_probabilities.1] norm_num change 0 < (Measure.dirac firstPacket + Measure.dirac firstPacket + Measure.dirac secondPacket) beam.responsivePackets simp only [Measure.add_apply] have first : (Measure.dirac firstPacket) beam.responsivePackets = 1 := by simp [member] rw [first] positivity theorem the_library_derives_the_client_signal_decrease_and_transition_inference : beam.drivenIntensity < beam.baselineIntensity ∧ beam.drivenSignal < beam.baselineSignal ∧ ¬(∀ᵐ packet ∂beam.sourceMeasure, beam.transitionProbability packet = 0) := by have outcome := beam.populated_resolved_transitions_lower_the_controlled_detector_signal the_client_beam_has_actual_positive_flux_and_bounded_probabilities.1 the_client_gradients_match_its_distinct_geometric_factors.2.1 the_client_beam_has_actual_positive_flux_and_bounded_probabilities.2.1 the_client_has_a_populated_resolvable_transition the_client_beam_has_actual_positive_flux_and_bounded_probabilities.2.2 exact ⟨outcome.1, outcome.2, beam.a_controlled_signal_drop_excludes_an_almost_everywhere_absent_transition the_client_beam_has_actual_positive_flux_and_bounded_probabilities.2.2 outcome.2⟩ theorem the_client_electronic_readout_uses_the_same_gain_and_background : beam.baselineSignal = 71 / 5 ∧ beam.drivenSignal = 41 / 5 := by unfold BeamData.baselineSignal BeamData.drivenSignal rw [the_client_integrates_actual_counts_instead_of_assigning_a_signal_drop.2.1, the_client_integrates_actual_counts_instead_of_assigning_a_signal_drop.2.2.1] norm_num [beam] theorem the_library_preserves_the_client_baseline_velocity_law_without_perfect_transmission : beam.BaselineVelocityLawPreservedAt ∧ beam.baselineIntensity < beam.referenceIntensity := by have uniform : beam.UniformBaselineTransmissionAt (4 / 5) := by norm_num [BeamData.UniformBaselineTransmissionAt, beam] refine ⟨beam.uniform_baseline_loss_preserves_the_actual_velocity_law the_client_beam_has_actual_positive_flux_and_bounded_probabilities.1 the_client_gradients_match_its_distinct_geometric_factors.2.1 (4 / 5) uniform, ?_⟩ rw [the_client_integrates_actual_counts_instead_of_assigning_a_signal_drop.1, the_client_integrates_actual_counts_instead_of_assigning_a_signal_drop.2.1] norm_num def observedReadout : PhysicsLogic.Papers.Rabi1939.BeamRefocusing.SignalObservation := ⟨14, 17 / 2, 1 / 4, 1 / 3⟩ theorem the_client_separates_mean_predictions_from_accurate_observed_readings : observedReadout.AccurateFor beam ∧ observedReadout.ResolvedMeanLossAt beam ∧ observedReadout.ResolvedObservedLossAt := by unfold BeamRefocusing.SignalObservation.AccurateFor BeamRefocusing.SignalObservation.ResolvedMeanLossAt BeamRefocusing.SignalObservation.ResolvedObservedLossAt rw [the_client_electronic_readout_uses_the_same_gain_and_background.1, the_client_electronic_readout_uses_the_same_gain_and_background.2] norm_num [observedReadout] change (7 / 12 : Real) < 6 norm_num theorem the_client_observed_readout_has_a_resolved_decrease_and_implies_a_transition : observedReadout.drivenReading < observedReadout.baselineReading ∧ ¬(∀ᵐ packet ∂beam.sourceMeasure, beam.transitionProbability packet = 0) := by have measured := the_client_separates_mean_predictions_from_accurate_observed_readings exact ⟨observedReadout.a_mean_loss_larger_than_the_error_budget_forces_observed_ordering beam measured.1 measured.2.1, observedReadout.a_controlled_resolved_observed_loss_excludes_absent_transitions beam measured.1 measured.2.2 the_client_beam_has_actual_positive_flux_and_bounded_probabilities.2.2⟩ end end RabiBeamReadoutClient namespace RabiMultipletClient open PhysicsLogic.Papers.Rabi1939 BeamRefocusing MagneticSublevels MeasureTheory open scoped BigOperators noncomputable section set_option backward.isDefEq.respectTransparency false /- Independent numerical beam, not a reconstruction of the historical data. The incoming state is an incoherent mixture, and the pulse is the source's weak-field rotating-component formula, not exact laboratory evolution. -/ def geometry : Geometry := ⟨1, 1, 1, 2, 1, 1⟩ def apparatus : Apparatus := ⟨geometry, 3, 4, 1 / 5⟩ def sourceMeasure : Measure Real := Measure.dirac 1 + Measure.dirac 5 + Measure.dirac 5 def pulse (speed : Real) : ResonanceFormulaReadout where staticFieldMagnitude := 100 oscillatingFieldAmplitude := 2 angularMomentumMagnitude := 3 / 2 planckConstant := 1 magneticMoment := 3 / 2 signedMagneticMoment := 3 / 2 larmorFrequency := 100 driveFrequency := 100 interactionTime := 5 / (6 * speed) transitionProbability := Real.sin (5 * Real.pi / (6 * speed)) ^ 2 def beam : MultipletBeam.Data 3 where apparatus := apparatus molecularMass := 2 signedMoment := 3 / 2 sourceMeasure := sourceMeasure initialPopulation := fun _ initial => if initial.val = 0 then 2 / 3 else if initial.val = 1 then 1 / 3 else 0 transmission := fun _ _ => 3 / 4 auxiliaryProbability := fun speed => (pulse speed).transitionProbability detectorGain := 2 baselineBackground := 5 drivenBackground := 5 theorem the_same_field_history_supplies_the_velocity_resolved_source_formula (speed : Real) (positive : 0 < speed) : (pulse speed).EquationOneLarmorFrequencyAt ∧ (pulse speed).EquationTwoWeakOscillatingTransitionAt := by have domain : (pulse speed).PhysicalFieldFrequencyDomainAt := by norm_num [ResonanceFormulaReadout.PhysicalFieldFrequencyDomainAt, pulse] positivity have delta : (pulse speed).equationTwoDeltaValueAt = (1 : Real) / 100 := by norm_num [ResonanceFormulaReadout.equationTwoDeltaValueAt, pulse] have factor : (pulse speed).equationTwoDetuningFactorValueAt = ((1 : Real) / 100) ^ 2 := by unfold ResonanceFormulaReadout.equationTwoDetuningFactorValueAt rw [delta] norm_num [ResonanceFormulaReadout.equationTwoFrequencyRatioValueAt, pulse] constructor · refine ⟨domain, by norm_num [pulse], ?_⟩ change (100 : Real) = (3 / 2 * 100) / (3 / 2 * 1) norm_num · refine ⟨domain, ?_⟩ rw [delta, factor, Real.sqrt_sq (by norm_num)] change Real.sin (5 * Real.pi / (6 * speed)) ^ 2 = ((1 / 100 : Real) ^ 2 / (1 / 100 : Real) ^ 2) * Real.sin (Real.pi * (5 / (6 * speed)) * 100 * (1 / 100 : Real)) ^ 2 have phase : Real.pi * (5 / (6 * speed)) * 100 * (1 / 100 : Real) = 5 * Real.pi / (6 * speed) := by ring rw [phase, div_self (by norm_num : ((1 / 100 : Real) ^ 2) ≠ 0), one_mul] theorem the_auxiliary_half_spin_matches_the_gyromagnetic_ratio_not_the_target_moment (speed : Real) (positive : 0 < speed) : (pulse speed).spinHalfReference.EquationOneLarmorFrequencyAt ∧ (pulse speed).spinHalfReference.magneticMoment = 1 / 2 ∧ (pulse speed).spinHalfReference.magneticMoment ≠ (pulse speed).magneticMoment ∧ (pulse speed).interactionTime * speed = 5 / 6 := by refine ⟨(pulse speed).the_spin_half_reference_has_the_same_larmor_frequency (the_same_field_history_supplies_the_velocity_resolved_source_formula speed positive).1, ?_⟩ norm_num [pulse, ResonanceFormulaReadout.spinHalfReference] field_simp [ne_of_gt positive] theorem the_two_selected_speeds_have_the_same_computed_auxiliary_probability : (pulse 1).transitionProbability = 1 / 4 ∧ (pulse 5).transitionProbability = 1 / 4 := by have slow : 5 * Real.pi / (6 * (1 : Real)) = Real.pi - Real.pi / 6 := by ring have fast : 5 * Real.pi / (6 * (5 : Real)) = Real.pi / 6 := by ring change Real.sin (5 * Real.pi / (6 * (1 : Real))) ^ 2 = 1 / 4 ∧ Real.sin (5 * Real.pi / (6 * (5 : Real))) ^ 2 = 1 / 4 rw [slow, fast, Real.sin_pi_sub, Real.sin_pi_div_six] norm_num theorem each_integrand_is_integrable (f : Real -> Real) : Integrable f sourceMeasure := ((integrable_dirac (by simp)).add_measure (integrable_dirac (by simp))).add_measure (integrable_dirac (by simp)) theorem the_source_integral_retains_the_unequal_speed_weights (f : Real -> Real) : (∫ speed, f speed ∂sourceMeasure) = f 1 + 2 * f 5 := by rw [sourceMeasure, integral_add_measure ((integrable_dirac (by simp)).add_measure (integrable_dirac (by simp))) (integrable_dirac (by simp)), integral_add_measure (integrable_dirac (by simp)) (integrable_dirac (by simp))] simp only [integral_dirac] ring theorem the_distinct_geometric_factors_match_the_actual_gradients : apparatus.PhysicalAt ∧ apparatus.MatchedGradientGeometryAt ∧ geometry.firstFactor = 4 ∧ geometry.secondFactor = 3 := by norm_num [apparatus, geometry, Apparatus.PhysicalAt, Apparatus.MatchedGradientGeometryAt, Geometry.PhysicalAt, Geometry.firstFactor, Geometry.secondFactor, Geometry.slitDistance, Geometry.postSlitDistance] theorem the_initial_mixture_and_pulse_are_normalized_at_every_positive_speed (speed : Real) (positive : 0 < speed) : beam.ChannelDomainAt speed := by have bounds := (pulse speed).equation_two_derives_admissible_transition_probability (the_same_field_history_supplies_the_velocity_resolved_source_formula speed positive).2 refine ⟨positive, ?_, ?_, ?_, bounds.2.1, bounds.2.2⟩ · intro initial dsimp [beam] split_ifs <;> norm_num · norm_num [beam, Sublevel, Fin.sum_univ_succ] · intro initial norm_num [beam] theorem the_beam_has_physical_normalized_and_integrable_data : beam.PhysicalAt ∧ beam.IntegralDomainAt ∧ beam.NormalizedTransitionsAt ∧ beam.ControlledDetectorAt := by have physical : beam.PhysicalAt := by refine ⟨the_distinct_geometric_factors_match_the_actual_gradients.1, by norm_num [beam], by norm_num, by norm_num [beam], ?_, ?_, ?_⟩ · dsimp [beam, sourceMeasure] infer_instance · change 0 < ∫ _speed : Real, (1 : Real) ∂sourceMeasure rw [the_source_integral_retains_the_unequal_speed_weights] norm_num · change ∀ᵐ speed ∂sourceMeasure, beam.ChannelDomainAt speed simp only [sourceMeasure, ae_add_measure_iff, ae_dirac_eq] exact ⟨⟨the_initial_mixture_and_pulse_are_normalized_at_every_positive_speed 1 (by norm_num), the_initial_mixture_and_pulse_are_normalized_at_every_positive_speed 5 (by norm_num)⟩, the_initial_mixture_and_pulse_are_normalized_at_every_positive_speed 5 (by norm_num)⟩ exact ⟨physical, ⟨each_integrand_is_integrable _, each_integrand_is_integrable _, each_integrand_is_integrable _⟩, beam.the_source_spins_have_actual_normalized_multiplet_transitions (by simp [SourceNuclearSpinAt]) physical, by norm_num [MultipletBeam.Data.ControlledDetectorAt, beam]⟩ theorem the_library_computes_the_sublevel_dependent_ray_position (speed : Real) (positive : 0 < speed) (initial final : Fin 4) : (apparatus.atDetector (beam.ray speed initial final)).position = 3 * ((final.val : Real) - initial.val) / speed ^ 2 := by rw [apparatus.matched_gradient_geometry_computes_the_projection_difference (beam.ray speed initial final) the_distinct_geometric_factors_match_the_actual_gradients.1 ⟨by norm_num [MultipletBeam.Data.ray, beam], positive⟩ the_distinct_geometric_factors_match_the_actual_gradients.2.1] norm_num [MultipletBeam.Data.ray, beam, magneticProjection, magneticQuantumNumber, apparatus, geometry, Geometry.firstFactor, Geometry.slitDistance, Geometry.postSlitDistance] ring theorem the_slow_aperture_accepts_only_the_unchanged_sublevel (initial final : Fin 4) : apparatus.acceptance (beam.ray 1 initial final) = if initial = final then 1 else 0 := by unfold Apparatus.acceptance Apparatus.AcceptedAt rw [the_library_computes_the_sublevel_dependent_ray_position 1 (by norm_num)] fin_cases initial <;> fin_cases final <;> norm_num [apparatus] theorem the_fast_aperture_accepts_exactly_the_nearest_sublevels (initial final : Fin 4) : apparatus.acceptance (beam.ray 5 initial final) = if |(final.val : Int) - initial.val| ≤ 1 then 1 else 0 := by unfold Apparatus.acceptance Apparatus.AcceptedAt rw [the_library_computes_the_sublevel_dependent_ray_position 5 (by norm_num)] fin_cases initial <;> fin_cases final <;> norm_num [apparatus] theorem the_actual_transition_matrix_uses_the_computed_auxiliary_probability (speed : Real) (selected : speed = 1 ∨ speed = 5) (initial final : Fin 4) : beam.transition speed initial final = spinThreeHalfEntry (3 / 4) (1 / 4) initial final := by have probability : beam.auxiliaryProbability speed = 1 / 4 := by rcases selected with rfl | rfl · exact the_two_selected_speeds_have_the_same_computed_auxiliary_probability.1 · exact the_two_selected_speeds_have_the_same_computed_auxiliary_probability.2 unfold MultipletBeam.Data.transition rw [probability, spin_three_half_probabilities_depend_on_the_auxiliary_probability (1 / 4) (by norm_num) (by norm_num)] norm_num theorem the_full_population_and_aperture_compute_both_speed_densities : beam.baselineDensity 1 = 3 / 4 ∧ beam.baselineDensity 5 = 3 / 4 ∧ beam.drivenDensity 1 = 57 / 256 ∧ beam.drivenDensity 5 = 163 / 256 := by have slow := the_actual_transition_matrix_uses_the_computed_auxiliary_probability 1 (Or.inl rfl) have fast := the_actual_transition_matrix_uses_the_computed_auxiliary_probability 5 (Or.inr rfl) unfold MultipletBeam.Data.baselineDensity MultipletBeam.Data.drivenDensity simp only [slow, fast, show beam.apparatus = apparatus from rfl, the_slow_aperture_accepts_only_the_unchanged_sublevel, the_fast_aperture_accepts_exactly_the_nearest_sublevels] norm_num [beam, Sublevel, spinThreeHalfEntry, Fin.sum_univ_succ] theorem the_integrated_counts_come_from_the_mixed_population : beam.referenceIntensity = 3 ∧ beam.baselineIntensity = 9 / 4 ∧ beam.drivenIntensity = 383 / 256 ∧ (∫ speed, beam.rejectedTransitionDensity speed ∂beam.sourceMeasure) = 193 / 256 := by have densities := the_full_population_and_aperture_compute_both_speed_densities have baseline : beam.baselineIntensity = 9 / 4 := by change (∫ speed, beam.baselineDensity speed ∂sourceMeasure) = 9 / 4 rw [the_source_integral_retains_the_unequal_speed_weights, densities.1, densities.2.1] norm_num have driven : beam.drivenIntensity = 383 / 256 := by change (∫ speed, beam.drivenDensity speed ∂sourceMeasure) = 383 / 256 rw [the_source_integral_retains_the_unequal_speed_weights, densities.2.2.1, densities.2.2.2] norm_num refine ⟨?_, baseline, driven, ?_⟩ · change (∫ _speed : Real, (1 : Real) ∂sourceMeasure) = 3 rw [the_source_integral_retains_the_unequal_speed_weights] norm_num · rw [← beam.the_actual_signal_loss_sums_over_all_initial_and_final_sublevels the_beam_has_physical_normalized_and_integrable_data.1 the_distinct_geometric_factors_match_the_actual_gradients.2.1 the_beam_has_physical_normalized_and_integrable_data.2.2.1 the_beam_has_physical_normalized_and_integrable_data.2.1, baseline, driven] norm_num theorem the_beam_has_a_populated_resolvable_off_diagonal_channel : beam.ResponsivePopulationAt := by have member : (1 : Real) ∈ beam.responsiveSpeeds := by refine ⟨0, 1, by norm_num [beam], by norm_num [beam], ?_, ?_⟩ · rw [the_actual_transition_matrix_uses_the_computed_auxiliary_probability 1 (Or.inl rfl)] norm_num [spinThreeHalfEntry] · change apparatus.detectorHalfWidth < |(apparatus.atDetector (beam.ray 1 0 1)).position| rw [the_library_computes_the_sublevel_dependent_ray_position 1 (by norm_num)] norm_num [apparatus] change 0 < sourceMeasure beam.responsiveSpeeds simp only [sourceMeasure, Measure.add_apply] have first : (Measure.dirac 1) beam.responsiveSpeeds = 1 := by simp [member] rw [first] positivity theorem the_library_derives_signal_loss_and_some_populated_reorientation : beam.drivenIntensity < beam.baselineIntensity ∧ beam.drivenSignal < beam.baselineSignal ∧ ¬(∀ᵐ speed ∂beam.sourceMeasure, beam.PopulatedReorientationAbsentAt speed) := by have data := the_beam_has_physical_normalized_and_integrable_data have loss := beam.a_populated_resolved_multiplet_lowers_the_controlled_signal data.1 the_distinct_geometric_factors_match_the_actual_gradients.2.1 data.2.2.1 data.2.1 the_beam_has_a_populated_resolvable_off_diagonal_channel data.2.2.2 exact ⟨loss.1, loss.2, beam.a_controlled_multiplet_signal_drop_requires_populated_reorientation data.1 the_distinct_geometric_factors_match_the_actual_gradients.2.1 data.2.2.1 data.2.1 data.2.2.2 loss.2⟩ theorem the_predicted_mean_signals_use_the_same_gain_and_background : beam.baselineSignal = 19 / 2 ∧ beam.drivenSignal = 1023 / 128 := by unfold MultipletBeam.Data.baselineSignal MultipletBeam.Data.drivenSignal rw [the_integrated_counts_come_from_the_mixed_population.2.1, the_integrated_counts_come_from_the_mixed_population.2.2.1] norm_num [beam] theorem the_library_preserves_the_baseline_velocity_law_of_the_mixed_population : beam.BaselineVelocityLawPreservedAt ∧ beam.baselineIntensity < beam.referenceIntensity := by have uniform : beam.UniformBaselineTransmissionAt (3 / 4) := by norm_num [MultipletBeam.Data.UniformBaselineTransmissionAt, beam] refine ⟨beam.an_incoherent_normalized_population_preserves_the_uniform_baseline_velocity_law the_beam_has_physical_normalized_and_integrable_data.1 the_distinct_geometric_factors_match_the_actual_gradients.2.1 (3 / 4) uniform, ?_⟩ rw [the_integrated_counts_come_from_the_mixed_population.1, the_integrated_counts_come_from_the_mixed_population.2.1] norm_num def observedReadout : SignalObservation := ⟨47 / 5, 81 / 10, 1 / 8, 1 / 8⟩ theorem finite_readings_have_explicit_errors_separate_from_the_multiplet_means : beam.ObservationAccurateAt observedReadout ∧ beam.ObservationResolvesMeanLossAt observedReadout ∧ observedReadout.ResolvedObservedLossAt := by unfold MultipletBeam.Data.ObservationAccurateAt MultipletBeam.Data.ObservationResolvesMeanLossAt SignalObservation.ResolvedObservedLossAt rw [the_predicted_mean_signals_use_the_same_gain_and_background.1, the_predicted_mean_signals_use_the_same_gain_and_background.2] norm_num [observedReadout] theorem the_library_derives_a_resolved_observed_multiplet_transition : observedReadout.drivenReading < observedReadout.baselineReading ∧ ¬(∀ᵐ speed ∂beam.sourceMeasure, beam.PopulatedReorientationAbsentAt speed) := by have data := the_beam_has_physical_normalized_and_integrable_data have readings := finite_readings_have_explicit_errors_separate_from_the_multiplet_means exact ⟨beam.a_resolved_multiplet_mean_loss_orders_accurate_readings observedReadout readings.1 readings.2.1, beam.a_controlled_resolved_multiplet_readout_requires_populated_reorientation observedReadout data.1 the_distinct_geometric_factors_match_the_actual_gradients.2.1 data.2.2.1 data.2.1 data.2.2.2 readings.1 readings.2.2⟩ #print axioms the_same_field_history_supplies_the_velocity_resolved_source_formula #print axioms the_beam_has_physical_normalized_and_integrable_data #print axioms the_library_computes_the_sublevel_dependent_ray_position #print axioms the_full_population_and_aperture_compute_both_speed_densities #print axioms the_integrated_counts_come_from_the_mixed_population #print axioms the_library_derives_signal_loss_and_some_populated_reorientation #print axioms the_library_preserves_the_baseline_velocity_law_of_the_mixed_population #print axioms the_library_derives_a_resolved_observed_multiplet_transition end end RabiMultipletClient namespace RabiMolecularClient open PhysicsLogic.Papers.Rabi1939 MolecularRegime MagneticSublevels noncomputable section set_option backward.isDefEq.respectTransparency false def spectatorProjection (spectator : Fin 3) : Real := 1 - (spectator.val : Real) def spectrum : EnergySpectrum 3 (Fin 3) where planckConstant := 3 signedGyromagneticRatio := 2 spectatorEnergy := fun field spectator => -7 * field * spectatorProjection spectator interactionEnergy := fun _ level spectator => -6 * magneticQuantumNumber 3 level * spectatorProjection spectator + magneticQuantumNumber 3 level ^ 2 def electronicProjector : Matrix (Fin 3) (Fin 3) Complex := !![1, 0, 0; 0, 1, 0; 0, 0, 0] def electronicMoment : Matrix (Fin 3) (Fin 3) Complex := !![0, 0, 2; 0, 0, 1; 2, 1, 0] def electronic : ElectronicSector := ⟨3, electronicProjector, electronicMoment⟩ def experiment : HighFieldCase 3 (Fin 3) where spectrum := spectrum populatedSpectator := 0 electronicSector := electronic field := 1000 frequencyBudget := 8 / 3 relativeTolerance := 1 / 700 validField := fun field => 0 < field ∧ field < 2000 theorem the_nine_interaction_gaps_are_computed_from_the_two_couplings (field : Real) (branch : Fin 3) (spectator : Fin 3) : spectrum.interactionGap field branch spectator = 6 * spectatorProjection spectator + 2 * (branch.val : Real) - 2 := by dsimp [EnergySpectrum.interactionGap, spectrum, magneticQuantumNumber] push_cast ring theorem the_internal_energy_calculation_bounds_every_adjacent_gap (field : Real) : spectrum.UniformInteractionGapBoundAt field (8 / 3) := by refine ⟨by norm_num, ?_⟩ intro branch spectator rw [the_nine_interaction_gaps_are_computed_from_the_two_couplings] fin_cases branch <;> fin_cases spectator <;> norm_num [spectrum, spectatorProjection] theorem the_independent_electronic_sector_has_zero_restricted_first_order_moment : electronic.FirstOrderNeutralAt := by change electronicProjector.IsHermitian ∧ electronicProjector * electronicProjector = electronicProjector ∧ electronicProjector ≠ 0 ∧ electronicMoment.IsHermitian ∧ electronicProjector * electronicMoment * electronicProjector = 0 refine ⟨?_, ?_, ?_, ?_, ?_⟩ · ext i j fin_cases i <;> fin_cases j <;> norm_num [electronicProjector, Matrix.conjTranspose] · ext i j fin_cases i <;> fin_cases j <;> norm_num [electronicProjector, Matrix.mul_apply, Fin.sum_univ_succ] · intro zero have entry := congrArg (fun matrix => matrix 0 0) zero norm_num [electronicProjector] at entry · ext i j fin_cases i <;> fin_cases j <;> norm_num [electronicMoment, Matrix.conjTranspose, map_ofNat] · ext i j fin_cases i <;> fin_cases j <;> norm_num [electronicProjector, electronicMoment, Matrix.mul_apply, Fin.sum_univ_succ] theorem the_electronic_sector_retains_a_nonzero_coupling_outside_the_selected_space : ¬electronicMoment * electronicProjector = 0 := by intro zero have entry := congrArg (fun matrix => matrix 2 0) zero norm_num [electronicProjector, electronicMoment, Matrix.mul_apply, Fin.sum_univ_succ, Matrix.cons_val_two, Matrix.vecHead, Matrix.vecTail] at entry theorem the_selected_field_is_in_the_finite_valid_physical_regime : spectrum.PhysicalAt experiment.field ∧ experiment.validField experiment.field ∧ ¬experiment.validField 2000 := by norm_num [EnergySpectrum.PhysicalAt, spectrum, experiment] theorem the_requested_precision_threshold_is_satisfied_without_exact_decoupling : experiment.SetupAt := by refine ⟨the_selected_field_is_in_the_finite_valid_physical_regime.1, the_selected_field_is_in_the_finite_valid_physical_regime.2.1, the_independent_electronic_sector_has_zero_restricted_first_order_moment, ?_, ?_⟩ · norm_num [experiment] · norm_num [experiment, spectrum] theorem the_library_establishes_the_independent_molecular_moment_precision : experiment.EstablishedAt := by apply experiment.electronic_suppression_and_controlled_gap_calculations_establish_moment_precision the_requested_precision_threshold_is_satisfied_without_exact_decoupling exact the_internal_energy_calculation_bounds_every_adjacent_gap 1000 theorem the_nuclear_moment_scale_comes_from_h_spin_and_gyromagnetic_ratio : spectrum.actualMomentMagnitude = 9 := by norm_num [EnergySpectrum.actualMomentMagnitude, spectrum] theorem the_library_retains_all_moment_bounds_with_the_finite_regime : experiment.validField experiment.field ∧ ∀ branch spectator, |spectrum.inferredMomentMagnitude 1000 branch spectator - 9| / 9 < 1 / 700 := by have established := the_library_establishes_the_independent_molecular_moment_precision refine ⟨established.1.2.1, ?_⟩ simpa only [HighFieldCase.RelativeMomentPrecisionAt, experiment, the_nuclear_moment_scale_comes_from_h_spin_and_gyromagnetic_ratio] using established.2.2 theorem the_library_retains_the_selected_spectator_without_erasing_other_configurations : experiment.validField experiment.field ∧ ∀ branch, |spectrum.inferredMomentMagnitude 1000 branch 0 - 9| / 9 < 1 / 700 := by simpa only [experiment, the_nuclear_moment_scale_comes_from_h_spin_and_gyromagnetic_ratio] using experiment.the_established_precision_retains_a_populated_spectator_and_valid_regime the_library_establishes_the_independent_molecular_moment_precision theorem the_largest_selected_internal_gap_has_a_nonzero_inferred_moment_error : spectrum.interactionGap 1000 2 0 = 8 ∧ spectrum.inferredMomentMagnitude 1000 2 0 = 2253 / 250 ∧ |spectrum.inferredMomentMagnitude 1000 2 0 - spectrum.actualMomentMagnitude| / spectrum.actualMomentMagnitude = 1 / 750 := by norm_num [EnergySpectrum.interactionGap, EnergySpectrum.inferredMomentMagnitude, EnergySpectrum.actualMomentMagnitude, EnergySpectrum.positiveFrequency, EnergySpectrum.gapFrequency, EnergySpectrum.energy, spectrum, spectatorProjection, magneticQuantumNumber] theorem established_molecular_precision_does_not_mean_the_free_spin_value_is_exact : experiment.EstablishedAt ∧ ¬spectrum.inferredMomentMagnitude 1000 2 0 = spectrum.actualMomentMagnitude := by refine ⟨the_library_establishes_the_independent_molecular_moment_precision, ?_⟩ rw [the_largest_selected_internal_gap_has_a_nonzero_inferred_moment_error.2.1, the_nuclear_moment_scale_comes_from_h_spin_and_gyromagnetic_ratio] norm_num #print axioms the_nine_interaction_gaps_are_computed_from_the_two_couplings #print axioms the_internal_energy_calculation_bounds_every_adjacent_gap #print axioms the_independent_electronic_sector_has_zero_restricted_first_order_moment #print axioms the_electronic_sector_retains_a_nonzero_coupling_outside_the_selected_space #print axioms the_selected_field_is_in_the_finite_valid_physical_regime #print axioms the_requested_precision_threshold_is_satisfied_without_exact_decoupling #print axioms the_library_establishes_the_independent_molecular_moment_precision #print axioms the_nuclear_moment_scale_comes_from_h_spin_and_gyromagnetic_ratio #print axioms the_library_retains_all_moment_bounds_with_the_finite_regime #print axioms the_library_retains_the_selected_spectator_without_erasing_other_configurations #print axioms the_largest_selected_internal_gap_has_a_nonzero_inferred_moment_error #print axioms established_molecular_precision_does_not_mean_the_free_spin_value_is_exact end end RabiMolecularClient namespace OnsagerDissipationClient open PhysicsLogic.Papers.Onsager1931.DissipationFunctional noncomputable section /-! Independent finite-component clients, not continuum crystal realizations. Their temperature, resistance, force, and selected current are constructed here; no production countermodel or variational conclusion is imported as an input. -/ def transport : Data (Fin 2 -> Real) := Data.ofResistance 5 !![3, 1; 1, 2] ![5, 0] def physical : Fin 2 -> Real := ![2, -1] theorem the_client_pairing_and_entropy_follow_its_coefficients (first second : Fin 2 -> Real) : transport.resistancePairing first second = (3 * first 0 * second 0 + first 0 * second 1 + first 1 * second 0 + 2 * first 1 * second 1) / 5 ∧ transport.dissipation first = (2 * first 0 ^ 2 + (first 0 + first 1) ^ 2 + first 1 ^ 2) / 10 ∧ transport.entropySupply first = first 0 := by have pairing (a b : Fin 2 -> Real) : transport.resistancePairing a b = (3 * a 0 * b 0 + a 0 * b 1 + a 1 * b 0 + 2 * a 1 * b 1) / 5 := by rw [transport, Data.the_finite_pairing_retains_the_resistance_and_temperature] simp [Fin.sum_univ_two] ring refine ⟨pairing first second, ?_, ?_⟩ · rw [Data.dissipation, pairing first first] ring · rw [transport, Data.the_finite_entropy_supply_is_the_actual_force_flux_pairing] simp [Fin.sum_univ_two] theorem the_client_has_actual_reciprocity_and_strict_dissipation : transport.ReciprocalAt ∧ transport.StrictDissipationAt := by constructor · intro first second rw [(the_client_pairing_and_entropy_follow_its_coefficients first second).1, (the_client_pairing_and_entropy_follow_its_coefficients second first).1] ring · intro flow nonzero rw [(the_client_pairing_and_entropy_follow_its_coefficients flow flow).2.1] have component : flow 0 ≠ 0 ∨ flow 1 ≠ 0 := by by_contra none push_neg at none apply nonzero funext axis fin_cases axis <;> simp [none.1, none.2] rcases component with first | second · nlinarith [sq_pos_of_ne_zero first, sq_nonneg (flow 1), sq_nonneg (flow 0 + flow 1)] · nlinarith [sq_pos_of_ne_zero second, sq_nonneg (flow 0), sq_nonneg (flow 0 + flow 1)] theorem the_selected_client_flow_satisfies_its_constitutive_equation : transport.ConstitutiveAt physical := by intro variation rw [(the_client_pairing_and_entropy_follow_its_coefficients variation variation).2.2, (the_client_pairing_and_entropy_follow_its_coefficients variation physical).1] simp [physical] ring theorem the_library_derives_the_client_nonvacuous_unique_maximum : transport.NonvacuousDissipationMaximumAt physical := by exact transport.an_active_constitutive_flow_has_a_nonvacuous_variational_maximum the_client_has_actual_reciprocity_and_strict_dissipation.1 the_client_has_actual_reciprocity_and_strict_dissipation.2 physical (by intro zero; have coordinate := congrFun zero 0; norm_num [physical] at coordinate) the_selected_client_flow_satisfies_its_constitutive_equation theorem the_client_physical_functionals_are_computed : transport.dissipation physical = 1 ∧ transport.entropySupply physical = 2 ∧ transport.functional physical = 1 := by have values := the_client_pairing_and_entropy_follow_its_coefficients physical physical rw [Data.functional, values.2.1, values.2.2] norm_num [physical] change ((8 + (1 : Real) ^ 2 + 1) / 10 = 1) ∧ (2 - (8 + (1 : Real) ^ 2 + 1) / 10 = 1) norm_num theorem the_library_computes_the_client_variation_for_every_trial (variation : Fin 2 -> Real) : transport.functional (physical + variation) = 1 - (2 * variation 0 ^ 2 + (variation 0 + variation 1) ^ 2 + variation 1 ^ 2) / 10 := by rw [transport.reciprocity_and_the_constitutive_equation_give_the_exact_variation_gap the_client_has_actual_reciprocity_and_strict_dissipation.1 physical the_selected_client_flow_satisfies_its_constitutive_equation, the_client_physical_functionals_are_computed.2.2, (the_client_pairing_and_entropy_follow_its_coefficients variation variation).2.1] theorem the_library_excludes_the_client_on_shell_substitution_for_all_trials : transport.entropySupply physical = 2 * transport.dissipation physical ∧ transport.entropySupply ((2 : Real) • physical) ≠ 2 * transport.dissipation ((2 : Real) • physical) := transport.the_physical_entropy_identity_is_not_an_identity_in_varied_flow physical the_selected_client_flow_satisfies_its_constitutive_equation the_library_derives_the_client_nonvacuous_unique_maximum.2.1 theorem the_client_extremum_is_not_an_unconstrained_entropy_maximum : transport.MaximumAt physical ∧ transport.entropySupply physical < transport.entropySupply ((2 : Real) • physical) := by refine ⟨the_library_derives_the_client_nonvacuous_unique_maximum.2.2.2.1, ?_⟩ rw [map_smul, the_client_physical_functionals_are_computed.2.1] norm_num def nonsymmetric : Data (Fin 2 -> Real) := Data.ofResistance 3 !![2, 1; -1, 2] ![2, -1] def nonsymmetricPhysical : Fin 2 -> Real := ![1, 0] theorem the_second_client_pairing_exposes_its_antisymmetric_part (first second : Fin 2 -> Real) : nonsymmetric.resistancePairing first second = (2 * first 0 * second 0 + first 0 * second 1 - first 1 * second 0 + 2 * first 1 * second 1) / 3 ∧ nonsymmetric.dissipation first = (first 0 ^ 2 + first 1 ^ 2) / 3 ∧ nonsymmetric.entropySupply first = (2 * first 0 - first 1) / 3 := by have pairing (a b : Fin 2 -> Real) : nonsymmetric.resistancePairing a b = (2 * a 0 * b 0 + a 0 * b 1 - a 1 * b 0 + 2 * a 1 * b 1) / 3 := by rw [nonsymmetric, Data.the_finite_pairing_retains_the_resistance_and_temperature] simp [Fin.sum_univ_two] ring refine ⟨pairing first second, ?_, ?_⟩ · rw [Data.dissipation, pairing first first] ring · rw [nonsymmetric, Data.the_finite_entropy_supply_is_the_actual_force_flux_pairing] simp [Fin.sum_univ_two] ring theorem the_second_client_is_strictly_dissipative_and_constitutive : nonsymmetric.StrictDissipationAt ∧ nonsymmetric.ConstitutiveAt nonsymmetricPhysical := by constructor · intro flow nonzero rw [(the_second_client_pairing_exposes_its_antisymmetric_part flow flow).2.1] have component : flow 0 ≠ 0 ∨ flow 1 ≠ 0 := by by_contra none push_neg at none apply nonzero funext axis fin_cases axis <;> simp [none.1, none.2] rcases component with first | second · nlinarith [sq_pos_of_ne_zero first, sq_nonneg (flow 1)] · nlinarith [sq_pos_of_ne_zero second, sq_nonneg (flow 0)] · intro variation rw [(the_second_client_pairing_exposes_its_antisymmetric_part variation variation).2.2, (the_second_client_pairing_exposes_its_antisymmetric_part variation nonsymmetricPhysical).1] simp [nonsymmetricPhysical] theorem the_second_client_is_not_reciprocal : ¬nonsymmetric.ReciprocalAt := by intro reciprocal have equality := reciprocal (![1, 0] : Fin 2 -> Real) ![0, 1] rw [(the_second_client_pairing_exposes_its_antisymmetric_part ![1, 0] ![0, 1]).1, (the_second_client_pairing_exposes_its_antisymmetric_part ![0, 1] ![1, 0]).1] at equality norm_num at equality theorem the_second_client_constitutive_flow_does_not_maximize_the_actual_functional : nonsymmetric.StrictDissipationAt ∧ nonsymmetric.ConstitutiveAt nonsymmetricPhysical ∧ ¬nonsymmetric.MaximumAt nonsymmetricPhysical := by refine ⟨the_second_client_is_strictly_dissipative_and_constitutive.1, the_second_client_is_strictly_dissipative_and_constitutive.2, ?_⟩ intro maximum have comparison := maximum (![1, -(1 / 2)] : Fin 2 -> Real) have dissipation (flow : Fin 2 -> Real) := (the_second_client_pairing_exposes_its_antisymmetric_part flow flow).2.1 have entropy (flow : Fin 2 -> Real) := (the_second_client_pairing_exposes_its_antisymmetric_part flow flow).2.2 simp only [Data.functional, dissipation, entropy] at comparison norm_num [nonsymmetricPhysical] at comparison end end OnsagerDissipationClient namespace AndersonSpectralReturnClient /-! Independently constructed three-mode and isolated-level spectral data. No production countermodel witness is imported into these calculations. The source motivation is Anderson1958 Section V, pp. 1504-1505; these are finite spectral controls, not random-lattice localization proofs. -/ open PhysicsLogic.Papers.AndersonLocalization1958 open Filter Set open scoped BigOperators Topology noncomputable section def spectrum : CoherentReturn.FiniteSpectralReadout (Fin 3) where energy := ![-2, 0, 3] initialOverlap := ![(Real.sqrt (1 / 2) : Real), (1 : Complex) / 2, Complex.I / 2] theorem the_independent_three_mode_weights_are_computed (mode : Fin 3) : spectrum.weight mode = ![1 / 2, 1 / 4, 1 / 4] mode := by fin_cases mode <;> dsimp [CoherentReturn.FiniteSpectralReadout.weight, spectrum] <;> norm_num [Complex.normSq_apply] theorem the_independent_initial_state_is_normalized : spectrum.NormalizedAt := by change (∑ mode, spectrum.weight mode) = 1 rw [Fin.sum_univ_three] have zero : spectrum.weight 0 = 1 / 2 := the_independent_three_mode_weights_are_computed 0 have one : spectrum.weight 1 = 1 / 4 := the_independent_three_mode_weights_are_computed 1 have two : spectrum.weight 2 = 1 / 4 := the_independent_three_mode_weights_are_computed 2 rw [zero, one, two] norm_num theorem the_library_preserves_the_actual_evolved_probability (time : Real) : (∑ mode, Complex.normSq (spectrum.evolved time mode)) = 1 := spectrum.a_normalized_initial_state_stays_normalized the_independent_initial_state_is_normalized time theorem the_library_recovers_unit_initial_return : spectrum.returnAmplitude 0 = 1 := spectrum.a_normalized_state_has_unit_initial_return the_independent_initial_state_is_normalized theorem the_library_identifies_the_actual_return_overlap (time : Real) : spectrum.returnAmplitude time = ∑ mode, star (spectrum.initialOverlap mode) * spectrum.evolved time mode := spectrum.return_is_the_actual_initial_evolved_overlap time theorem the_independent_pole_weights_follow_the_three_distinct_energies : spectrum.poleWeight (-2) = 1 / 2 ∧ spectrum.poleWeight 0 = 1 / 4 ∧ spectrum.poleWeight 3 = 1 / 4 ∧ spectrum.poleWeight 5 = 0 := by simp only [CoherentReturn.FiniteSpectralReadout.poleWeight, the_independent_three_mode_weights_are_computed, Fin.sum_univ_succ] dsimp [spectrum] norm_num theorem the_library_extracts_the_selected_quarter_weight_from_the_actual_green_function : Tendsto (fun sigma : Real => Complex.I * (sigma : Complex) * spectrum.diagonalGreen 0 sigma) (nhdsWithin 0 (Ioi 0)) (nhds ((1 / 4 : Real) : Complex)) := by have extraction := spectrum.the_green_function_extracts_the_grouped_energy_weight 0 rw [the_independent_pole_weights_follow_the_three_distinct_energies.2.1] at extraction exact extraction theorem the_library_finds_no_pole_at_the_absent_energy : Tendsto (fun sigma : Real => Complex.I * (sigma : Complex) * spectrum.diagonalGreen 5 sigma) (nhdsWithin 0 (Ioi 0)) (nhds 0) := by have extraction := spectrum.the_green_function_extracts_the_grouped_energy_weight 5 rw [the_independent_pole_weights_follow_the_three_distinct_energies.2.2.2] at extraction exact extraction def degenerateSpectrum : CoherentReturn.FiniteSpectralReadout (Fin 3) := { spectrum with energy := ![0, 0, 3] } theorem degeneracy_changes_the_grouped_pole_without_changing_any_mode_weight : (∀ mode, degenerateSpectrum.weight mode = spectrum.weight mode) ∧ degenerateSpectrum.poleWeight 0 = 3 / 4 ∧ spectrum.poleWeight 0 = 1 / 4 := by refine ⟨fun _ => rfl, ?_, the_independent_pole_weights_follow_the_three_distinct_energies.2.1⟩ simp only [CoherentReturn.FiniteSpectralReadout.poleWeight, show ∀ mode, degenerateSpectrum.weight mode = spectrum.weight mode from fun _ => rfl, the_independent_three_mode_weights_are_computed, Fin.sum_univ_succ] dsimp [degenerateSpectrum] norm_num def isolatedSpectrum : CoherentReturn.FiniteSpectralReadout Unit where energy := fun _ => 7 initialOverlap := fun _ => 1 def isolatedPole : DiagonalPoleReadout where energy := 7 bareSiteEnergy := 7 diagonalGreenAt := isolatedSpectrum.diagonalGreen 7 selfEnergyAt := fun _ => 0 normalizedLocatorAt := fun _ => 0 locatorLimit := 0 poleWeight := 1 theorem the_independent_single_mode_resolvent_is_actually_realized : isolatedPole.DiagonalResolventAt ∧ isolatedPole.PoleExtractionAt := by constructor · intro sigma _positive dsimp [isolatedPole] simp [CoherentReturn.FiniteSpectralReadout.diagonalGreen, CoherentReturn.FiniteSpectralReadout.weight, isolatedSpectrum] · have extraction := isolatedSpectrum.the_green_function_extracts_the_grouped_energy_weight 7 simpa [CoherentReturn.FiniteSpectralReadout.poleWeight, CoherentReturn.FiniteSpectralReadout.weight, isolatedSpectrum] using extraction theorem the_independent_selected_pole_has_zero_normalized_locator : isolatedPole.NormalizedImaginaryLocatorAt ∧ isolatedPole.FiniteLocatorLimitAt := by refine ⟨?_, by norm_num [isolatedPole], tendsto_const_nhds⟩ intro sigma _positive simp [DiagonalPoleReadout.imaginarySelfEnergyAt, isolatedPole] theorem the_independent_data_realize_all_source_calculation_inputs : isolatedPole.SourceCalculationAt := by have resolvent := the_independent_single_mode_resolvent_is_actually_realized have locator := the_independent_selected_pole_has_zero_normalized_locator refine ⟨by norm_num [DiagonalPoleReadout.PoleEquationAt, isolatedPole], resolvent.1, resolvent.2, ?_, locator.1, locator.2⟩ have ratio := Complex.continuous_ofReal.continuousAt.tendsto.comp (isolatedPole.the_imaginary_pole_ratio_has_the_computed_limit locator.1 locator.2) have difference := resolvent.2.sub ratio have zero : (isolatedPole.poleWeight : Complex) - ((1 / (1 + isolatedPole.locatorLimit) : Real) : Complex) = 0 := by dsimp [isolatedPole] have scalar : (1 : Real) / (1 + 0) = 1 := by norm_num rw [scalar] exact sub_self _ rw [zero] at difference exact difference theorem the_library_computes_the_positive_independent_pole_weight : isolatedPole.poleWeight = 1 / (1 + isolatedPole.locatorLimit) ∧ 0 < isolatedPole.poleWeight ∧ isolatedPole.poleWeight ≤ 1 := isolatedPole.a_finite_nonnegative_locator_gives_a_positive_bounded_pole_weight the_independent_data_realize_all_source_calculation_inputs def volumes : SpectralPoleVolumeFamily where poleAt := fun _ => isolatedPole uniformLocatorBound := 0 limitingPoleWeight := 1 theorem the_independent_volume_family_retains_the_actual_finite_calculations : volumes.FinitePoleCalculationsAt := fun _ => the_independent_data_realize_all_source_calculation_inputs theorem the_independent_volume_family_has_a_uniform_locator_bound_and_a_limit : volumes.InfiniteVolumeInputsAt := ⟨le_refl _, Filter.Eventually.of_forall fun _ => le_refl _, tendsto_const_nhds⟩ theorem the_library_carries_the_computed_weight_through_the_controlled_volume_limit : 0 < 1 / (1 + volumes.uniformLocatorBound) ∧ 1 / (1 + volumes.uniformLocatorBound) ≤ volumes.limitingPoleWeight ∧ volumes.limitingPoleWeight ≤ 1 := volumes.uniform_locator_control_preserves_a_positive_infinite_volume_weight (volumes.finite_source_calculations_determine_the_finite_pole_weights the_independent_volume_family_retains_the_actual_finite_calculations) the_independent_volume_family_has_a_uniform_locator_bound_and_a_limit end end AndersonSpectralReturnClient /-! Independent complex-hopping realization of Anderson's p. 1493 dynamics API. -/ namespace AndersonSiteEvolutionClient open PhysicsLogic.Papers.AndersonLocalization1958 open SiteEvolutionReadout Set open scoped BigOperators noncomputable section def model : SiteEvolutionReadout (Fin 3) where initialSite := 0 energyFrequency := ![0, 0, 7] hopping := fun site other => if site = 0 ∧ other = 1 then 3 * Complex.I else if site = 1 ∧ other = 0 then -3 * Complex.I else 0 amplitude := fun site time => if site = 0 then (Real.cos (3 * time) : Complex) else if site = 1 then -(Real.sin (3 * time) : Complex) else 0 theorem the_independent_complex_hopping_is_hermitian : model.HermitianHoppingAt := by intro site other _ fin_cases site <;> fin_cases other <;> norm_num [model] theorem the_independent_hopping_has_conjugate_not_equal_reverse_entries : model.hopping 0 1 = 3 * Complex.I ∧ model.hopping 1 0 = -3 * Complex.I ∧ model.hopping 0 1 ≠ model.hopping 1 0 := by refine ⟨rfl, rfl, ?_⟩ intro equality have impossible := congrArg Complex.im equality norm_num [model] at impossible theorem the_independent_initial_amplitudes_are_the_actual_single_site_state : model.InitialSingleSiteAt := by refine ⟨?_, ?_⟩ · norm_num [model] · intro site notInitial change site ≠ 0 at notInitial fin_cases site <;> simp [model] at * theorem the_independent_sine_cosine_amplitudes_solve_the_actual_source_equation : model.EquationOneOnFutureAt := by intro site time _ refine ⟨summable_of_ne_finset_zero (s := Finset.univ) (by simp), ?_⟩ have linear := (hasDerivAt_id time).const_mul (3 : Real) fin_cases site · convert (linear.cos.ofReal_comp).hasDerivWithinAt (s := Ici 0) using 1 simp [hamiltonianAction, hoppingAction, hoppingTerm, model, tsum_fintype, Fin.sum_univ_three] ring_nf simp · convert (linear.sin.ofReal_comp.neg).hasDerivWithinAt (s := Ici 0) using 1 simp [hamiltonianAction, hoppingAction, hoppingTerm, model, tsum_fintype, Fin.sum_univ_three] ring_nf simp · simpa [hamiltonianAction, hoppingAction, hoppingTerm, model, tsum_fintype, Fin.sum_univ_three] using (hasDerivAt_const time (0 : Complex)).hasDerivWithinAt (s := Ici 0) theorem the_library_derives_unit_probability_for_the_independent_solution (time : Real) (future : 0 ≤ time) : model.totalProbability time = 1 := model.finite_closed_single_site_evolution_stays_normalized the_independent_sine_cosine_amplitudes_solve_the_actual_source_equation the_independent_complex_hopping_is_hermitian the_independent_initial_amplitudes_are_the_actual_single_site_state time future theorem the_library_derives_the_independent_local_probability_flow (site : Fin 3) (time : Real) (future : 0 ≤ time) : HasDerivWithinAt (model.probability site) (model.incomingCurrent site time) (Ici 0) time := model.the_probability_derivative_is_the_actual_incoming_current the_independent_sine_cosine_amplitudes_solve_the_actual_source_equation site time future theorem the_independent_bond_currents_have_the_computed_signs (time : Real) : model.bondCurrent 0 1 time = -6 * Real.cos (3 * time) * Real.sin (3 * time) ∧ model.bondCurrent 1 0 time = 6 * Real.cos (3 * time) * Real.sin (3 * time) := by have reverse := model.hermitian_hopping_gives_opposite_bond_currents the_independent_complex_hopping_is_hermitian 1 0 time have forward : model.bondCurrent 0 1 time = -6 * Real.cos (3 * time) * Real.sin (3 * time) := by dsimp [bondCurrent, model] simp only [show (1 : Fin 3) ≠ 0 from by decide, if_false] ring exact ⟨forward, by rw [reverse, forward]; ring⟩ theorem the_library_cancels_the_independent_total_incoming_current (time : Real) : (∑ site : Fin 3, model.incomingCurrent site time) = 0 := model.finite_hermitian_bond_currents_cancel the_independent_complex_hopping_is_hermitian time end #print axioms PhysicsLogic.IndependentReuse.AndersonSiteEvolutionClient.the_independent_bond_currents_have_the_computed_signs #print axioms PhysicsLogic.IndependentReuse.AndersonSiteEvolutionClient.the_independent_complex_hopping_is_hermitian #print axioms PhysicsLogic.IndependentReuse.AndersonSiteEvolutionClient.the_independent_hopping_has_conjugate_not_equal_reverse_entries #print axioms PhysicsLogic.IndependentReuse.AndersonSiteEvolutionClient.the_independent_initial_amplitudes_are_the_actual_single_site_state #print axioms PhysicsLogic.IndependentReuse.AndersonSiteEvolutionClient.the_independent_sine_cosine_amplitudes_solve_the_actual_source_equation #print axioms PhysicsLogic.IndependentReuse.AndersonSiteEvolutionClient.the_library_cancels_the_independent_total_incoming_current #print axioms PhysicsLogic.IndependentReuse.AndersonSiteEvolutionClient.the_library_derives_the_independent_local_probability_flow #print axioms PhysicsLogic.IndependentReuse.AndersonSiteEvolutionClient.the_library_derives_unit_probability_for_the_independent_solution end AndersonSiteEvolutionClient namespace LandauOperationalClient open MeasureTheory Set Filter open PhysicsLogic.Papers.LandauDamping1946.OperationalIntegrals noncomputable section /-- An independent damped Fourier signal and its measured future history. These test transform semantics, not a self-consistent kinetic/Poisson plasma. -/ def signal (time : Real) : Complex := (3 + 2 * Complex.I) * Complex.exp (-(2 + 4 * Complex.I) * time) def recordedSignal (time : Real) : Complex := if time < 0 then 7 else signal time def transform (parameter : Complex) : Complex := (3 + 2 * Complex.I) / (parameter + 2 + 4 * Complex.I) theorem the_recorded_future_has_a_nonzero_initial_datum : recordedSignal 0 = 3 + 2 * Complex.I ∧ recordedSignal 0 ≠ 0 := by constructor · simp [recordedSignal, signal] · intro zero have realPart := congrArg Complex.re zero norm_num [recordedSignal, signal] at realPart theorem the_recorded_future_has_the_same_actual_laplace_value (parameter : Complex) : laplaceValue recordedSignal parameter = laplaceValue signal parameter := actual_laplace_values_depend_only_on_the_future _ _ (fun time future => by simp [recordedSignal, not_lt.mpr future]) parameter theorem the_independent_integrand_has_the_correct_shift (parameter : Complex) (time : Real) : laplaceIntegrand signal parameter time = (3 + 2 * Complex.I) * laplaceIntegrand (ballisticPhase 4) (parameter + 2) time := by dsimp [signal, laplaceIntegrand, ballisticPhase] rw [mul_assoc, ← Complex.exp_add, ← Complex.exp_add] congr 2 ring theorem the_independent_damped_transform_exists_on_its_larger_domain (parameter : Complex) (domain : -2 < parameter.re) : LaplaceExistsAt recordedSignal parameter ∧ laplaceValue recordedSignal parameter = transform parameter := by have shifted : 0 < (parameter + 2).re := by simp only [Complex.add_re, Complex.re_ofNat] linarith have integrable := constant_amplitudes_preserve_laplace_integrability (ballisticPhase 4) (parameter + 2) (3 + 2 * Complex.I) (ballistic_laplace_exists_on_the_right 4 (parameter + 2) shifted) have equality : laplaceIntegrand signal parameter = laplaceIntegrand (fun time => (3 + 2 * Complex.I) * ballisticPhase 4 time) (parameter + 2) := by funext time rw [the_independent_integrand_has_the_correct_shift] dsimp [laplaceIntegrand] ring have signalIntegrable : LaplaceExistsAt signal parameter := by unfold LaplaceExistsAt rw [equality] exact integrable have value : laplaceValue signal parameter = transform parameter := by change (∫ time in Ici (0 : Real), laplaceIntegrand signal parameter time) = _ rw [equality] change laplaceValue (fun time => (3 + 2 * Complex.I) * ballisticPhase 4 time) (parameter + 2) = _ rw [actual_laplace_values_respect_constant_amplitudes, ballistic_laplace_is_the_actual_resolvent 4 (parameter + 2) shifted] dsimp [transform] ring refine ⟨?_, (the_recorded_future_has_the_same_actual_laplace_value parameter).trans value⟩ exact (laplace_integrability_depends_only_on_the_future _ _ (fun time future => by simp [recordedSignal, not_lt.mpr future]) parameter).mpr signalIntegrable theorem this_damped_signal_still_has_an_actual_transform_at_minus_one : LaplaceExistsAt recordedSignal (-1) ∧ laplaceValue recordedSignal (-1) = (3 + 2 * Complex.I) / (1 + 4 * Complex.I) := by have result := the_independent_damped_transform_exists_on_its_larger_domain (-1) (by norm_num) refine ⟨result.1, ?_⟩ rw [result.2] dsimp [transform] congr 1 ring def clockSignal (reading : Real) : Complex := recordedSignal (2 * reading) theorem calibrating_the_clock_preserves_the_actual_time_measure (parameter : Complex) : laplaceValue (fun time => clockSignal (time / 2)) parameter = laplaceValue recordedSignal parameter := by apply congrArg (fun function => laplaceValue function parameter) funext time dsimp [clockSignal] congr 1 ring /-- A rationalized expression with a removable exceptional point away from the chosen inverse contour; no global equality is assumed. -/ def rationalizedTransform (parameter : Complex) : Complex := (3 + 2 * Complex.I) * (parameter + 2 - 4 * Complex.I) / ((parameter + 2) ^ 2 + 16) theorem both_resolvent_factors_are_nonzero_on_the_actual_inverse_contour (coordinate : Real) : (1 + Complex.I * coordinate + 2 + 4 * Complex.I) ≠ 0 ∧ (1 + Complex.I * coordinate + 2 - 4 * Complex.I) ≠ 0 := by constructor <;> intro zero all_goals have realPart := congrArg Complex.re zero norm_num at realPart theorem rationalization_preserves_the_actual_contour_transform (coordinate : Real) : LaplaceExistsAt recordedSignal (1 + Complex.I * coordinate) ∧ laplaceValue recordedSignal (1 + Complex.I * coordinate) = rationalizedTransform (1 + Complex.I * coordinate) := by have value := the_independent_damped_transform_exists_on_its_larger_domain (1 + Complex.I * coordinate) (by norm_num) refine ⟨value.1, value.2.trans ?_⟩ have factors := both_resolvent_factors_are_nonzero_on_the_actual_inverse_contour coordinate have denominator : (1 + Complex.I * coordinate + 2) ^ 2 + 16 = (1 + Complex.I * coordinate + 2 + 4 * Complex.I) * (1 + Complex.I * coordinate + 2 - 4 * Complex.I) := by calc _ = (1 + Complex.I * coordinate + 2) ^ 2 - 16 * Complex.I ^ 2 := by rw [Complex.I_sq]; ring _ = _ := by ring unfold transform rationalizedTransform rw [denominator] exact (mul_div_mul_right (3 + 2 * Complex.I) (1 + Complex.I * coordinate + 2 + 4 * Complex.I) factors.2).symm theorem the_contour_representations_need_not_agree_at_the_removed_point : rationalizedTransform (-2 + 4 * Complex.I) = 0 ∧ transform (-2 + 4 * Complex.I) ≠ 0 := by constructor · norm_num [rationalizedTransform] · apply div_ne_zero · intro zero have realPart := congrArg Complex.re zero norm_num at realPart · intro zero have imaginaryPart := congrArg Complex.im zero norm_num at imaginaryPart theorem common_contour_data_make_the_independent_positive_time_reconstruction_unique (firstHistory secondHistory : Real -> Complex) (firstInverse : PositiveTimeInversionAt firstHistory transform 1) (secondInverse : PositiveTimeInversionAt secondHistory rationalizedTransform 1) (time : Real) (future : 0 < time) : firstHistory time = secondHistory time := by apply a_common_inverse_contour_determines_the_positive_time_value transform rationalizedTransform 1 time (firstHistory time) (secondHistory time) · intro coordinate have original := the_independent_damped_transform_exists_on_its_larger_domain (1 + Complex.I * coordinate) (by norm_num) exact original.2.symm.trans (rationalization_preserves_the_actual_contour_transform coordinate).2 · exact (firstInverse.2 time future).2 · exact (secondInverse.2 time future).2 end #print axioms PhysicsLogic.IndependentReuse.LandauOperationalClient.the_recorded_future_has_a_nonzero_initial_datum #print axioms PhysicsLogic.IndependentReuse.LandauOperationalClient.the_recorded_future_has_the_same_actual_laplace_value #print axioms PhysicsLogic.IndependentReuse.LandauOperationalClient.the_independent_integrand_has_the_correct_shift #print axioms PhysicsLogic.IndependentReuse.LandauOperationalClient.the_independent_damped_transform_exists_on_its_larger_domain #print axioms PhysicsLogic.IndependentReuse.LandauOperationalClient.this_damped_signal_still_has_an_actual_transform_at_minus_one #print axioms PhysicsLogic.IndependentReuse.LandauOperationalClient.calibrating_the_clock_preserves_the_actual_time_measure #print axioms PhysicsLogic.IndependentReuse.LandauOperationalClient.both_resolvent_factors_are_nonzero_on_the_actual_inverse_contour #print axioms PhysicsLogic.IndependentReuse.LandauOperationalClient.rationalization_preserves_the_actual_contour_transform #print axioms PhysicsLogic.IndependentReuse.LandauOperationalClient.the_contour_representations_need_not_agree_at_the_removed_point #print axioms PhysicsLogic.IndependentReuse.LandauOperationalClient.common_contour_data_make_the_independent_positive_time_reconstruction_unique end LandauOperationalClient /-! Independent Bethe error-control client. Its real phase and elastic unitary scattering amplitude are constructed here; no production control witness is used and no fixed-potential inverse-scattering realization is asserted. -/ namespace BetheRadialClient open PhysicsLogic.Papers.Bethe1949.RadialComparison open Approximation Observation PhysicsLogic.Paradigm.ScaleAndRegime noncomputable section /-- Independent elastic phase data, not an asserted fixed-potential solution. -/ def physicalPhase (momentum : Real) : Real := -3 + 2 * momentum ^ 2 + 2 * momentum ^ 4 def runningRange (momentum : Real) : Real := 4 + 4 * momentum ^ 2 def reducedPhase (momentum : Real) : Real := twoParameterPhase 3 4 (momentum ^ 2) def elasticS (momentum : Real) : Complex := ((physicalPhase momentum : Complex) + Complex.I * momentum) / ((physicalPhase momentum : Complex) - Complex.I * momentum) theorem the_independent_range_and_phase_obey_the_exact_comparison (momentum : Real) : ExactComparisonAt (physicalPhase momentum) 3 (runningRange momentum) (momentum ^ 2) := by dsimp [ExactComparisonAt, physicalPhase, runningRange] ring theorem the_independent_range_has_a_uniform_quadratic_replacement_error : UniformQuadraticRangeControlOn runningRange 4 1 2 := by refine ⟨by norm_num, by norm_num, ?_⟩ intro momentum _positive _small refine ⟨by positivity, ?_⟩ have residual : runningRange momentum / 2 - 4 / 2 = 2 * momentum ^ 2 := by dsimp [runningRange] ring rw [residual, abs_of_nonneg (by positivity : 0 <= 2 * momentum ^ 2)] theorem the_library_derives_the_uniform_quartic_phase_control : UniformQuarticPhaseErrorAt physicalPhase 3 4 := uniform_quadratic_range_control_derives_quartic_accuracy _ _ _ _ 1 2 (fun momentum _ _ => the_independent_range_and_phase_obey_the_exact_comparison momentum) the_independent_range_has_a_uniform_quadratic_replacement_error theorem the_independent_elastic_amplitude_is_actually_unitary (momentum : Real) (positive : 0 < momentum) : Complex.normSq (elasticS momentum) = 1 := by have denominator : Complex.normSq ((physicalPhase momentum : Complex) - Complex.I * momentum) = physicalPhase momentum ^ 2 + momentum ^ 2 := by simp [Complex.normSq_apply] ring have numerator : Complex.normSq ((physicalPhase momentum : Complex) + Complex.I * momentum) = physicalPhase momentum ^ 2 + momentum ^ 2 := by simp [Complex.normSq_apply] ring dsimp [elasticS] rw [Complex.normSq_div, numerator, denominator] exact div_self (ne_of_gt (by nlinarith [sq_nonneg (physicalPhase momentum)])) theorem the_elastic_scattering_amplitude_retains_the_phase_function (momentum : Real) (positive : 0 < momentum) : ((physicalPhase momentum : Complex) - Complex.I * momentum) * elasticS momentum = (physicalPhase momentum : Complex) + Complex.I * momentum := by have nonzero : (physicalPhase momentum : Complex) - Complex.I * momentum ≠ 0 := by intro zero have imaginary := congrArg Complex.im zero simp at imaginary linarith dsimp [elasticS] field_simp theorem the_library_keeps_the_selected_nonzero_phase_error : PhaseApproximationAt (physicalPhase (1 / 2)) 3 4 ((1 / 2) ^ 2) (1 / 2) ∧ physicalPhase (1 / 2) - reducedPhase (1 / 2) = 1 / 8 := by refine ⟨controlled_range_replacement_gives_a_phase_error_bound _ _ _ _ _ _ (the_independent_range_and_phase_obey_the_exact_comparison _) ?_, ?_⟩ · have replacement := the_independent_range_has_a_uniform_quadratic_replacement_error.2.2 (1 / 2) (by norm_num) (by norm_num) norm_num at replacement exact replacement · norm_num [physicalPhase, reducedPhase, twoParameterPhase] theorem the_library_control_has_an_actual_matched_phase_observation : phaseContext.ControlledApproximation (reducedPhase (1 / 2)) (physicalPhase (1 / 2)) (1 / 4) (fun _ => True) (1 / 8) := by have controlled := the_phase_error_is_a_controlled_observable_approximation (physicalPhase (1 / 2)) 3 4 ((1 / 2) ^ 2) (1 / 2) (by norm_num) the_library_keeps_the_selected_nonzero_phase_error.1 norm_num [reducedPhase] at controlled ⊢ exact controlled theorem an_independent_readout_error_composes_with_the_physics_truncation : phaseContext.ControlledApproximation (reducedPhase (1 / 2) + 1 / 16) (physicalPhase (1 / 2)) (1 / 4) (fun _ => True) (3 / 16) := by have readout : phaseContext.ControlledApproximation (reducedPhase (1 / 2) + 1 / 16) (reducedPhase (1 / 2)) (1 / 4) (fun _ => True) (1 / 16) := by refine ⟨by norm_num [phaseContext], by norm_num [phaseContext], ⟨(), reducedPhase (1 / 2), trivial, by norm_num [phaseContext], rfl⟩, ?_⟩ intro _observable _selected _physical value predicted change value = reducedPhase (1 / 2) at predicted subst value refine ⟨reducedPhase (1 / 2) + 1 / 16, rfl, ?_⟩ change |reducedPhase (1 / 2) + 1 / 16 - reducedPhase (1 / 2)| <= 1 / 16 norm_num [reducedPhase, twoParameterPhase] exact controlled_approximations_compose_with_an_explicit_accuracy_rule phaseContext _ _ _ _ _ (1 / 16) (1 / 8) (3 / 16) readout the_library_control_has_an_actual_matched_phase_observation (fun first middle last firstBound secondBound => by change |first - last| <= 3 / 16 change |first - middle| <= 1 / 16 at firstBound change |middle - last| <= 1 / 8 at secondBound linarith [abs_sub_le first middle last]) theorem elastic_unitarity_does_not_make_the_two_parameter_truncation_exact : Complex.normSq (elasticS (1 / 2)) = 1 ∧ UniformQuarticPhaseErrorAt physicalPhase 3 4 ∧ physicalPhase (1 / 2) ≠ reducedPhase (1 / 2) := by refine ⟨the_independent_elastic_amplitude_is_actually_unitary _ (by norm_num), the_library_derives_the_uniform_quartic_phase_control, ?_⟩ have residual := the_library_keeps_the_selected_nonzero_phase_error.2 intro equality rw [equality, sub_self] at residual norm_num at residual end #print axioms PhysicsLogic.IndependentReuse.BetheRadialClient.the_independent_range_and_phase_obey_the_exact_comparison #print axioms PhysicsLogic.IndependentReuse.BetheRadialClient.the_independent_range_has_a_uniform_quadratic_replacement_error #print axioms PhysicsLogic.IndependentReuse.BetheRadialClient.the_library_derives_the_uniform_quartic_phase_control #print axioms PhysicsLogic.IndependentReuse.BetheRadialClient.the_independent_elastic_amplitude_is_actually_unitary #print axioms PhysicsLogic.IndependentReuse.BetheRadialClient.the_elastic_scattering_amplitude_retains_the_phase_function #print axioms PhysicsLogic.IndependentReuse.BetheRadialClient.the_library_keeps_the_selected_nonzero_phase_error #print axioms PhysicsLogic.IndependentReuse.BetheRadialClient.the_library_control_has_an_actual_matched_phase_observation #print axioms PhysicsLogic.IndependentReuse.BetheRadialClient.an_independent_readout_error_composes_with_the_physics_truncation #print axioms PhysicsLogic.IndependentReuse.BetheRadialClient.elastic_unitarity_does_not_make_the_two_parameter_truncation_exact end BetheRadialClient namespace BCSThermalClient set_option backward.isDefEq.respectTransparency false open PhysicsLogic.Papers.BardeenCooperSchrieffer1957.Thermal open Filter Set open scoped Topology noncomputable section /-! Independently specified near-critical trial data exercise the thermal limit and variational interfaces. No production control is imported or used. This is not claimed to solve the microscopic BCS gap integral or to realize the source's full free-energy functional. -/ def data : Data where boltzmannConstant := 3 densityOfStates := 2 attractiveCoupling := 1 phononCutoff := 12 criticalTemperature := 2 temperatureCeiling := 3 magneticField := 0 gap := fun temperature => 15 * Real.sqrt (1 - temperature / 2) freeEnergy := fun temperature gap => -200 * temperature ^ 2 + Real.tanh 1 / 225 * (gap ^ 2 - 225 * (1 - temperature / 2)) ^ 2 criticalCoefficient := 5 / 2 criticalCoefficientError := 1 criticalSlopeCoefficient := 25 / 4 criticalSlopeCoefficientError := 4 theorem the_independent_parameters_define_a_positive_temperature_window : data.PhysicalDomainAt ∧ data.criticalTemperature < data.temperatureCeiling := by norm_num [Data.PhysicalDomainAt, data] theorem the_independent_gap_has_its_actual_leading_coefficient : data.HasSquareRootLeadingProfile := by refine ⟨by norm_num [data], ?_⟩ apply tendsto_const_nhds.congr' filter_upwards [show ∀ᶠ value in data.criticalApproach, value < 2 from self_mem_nhdsWithin] with value below have positive : 0 < 1 - value / 2 := by linarith have nonzero := (Real.sqrt_pos.mpr positive).ne' dsimp [data, Data.criticalScale] rw [show (3 : Real) * 2 = 6 by norm_num, mul_div_mul_right _ _ nonzero] norm_num theorem the_library_derives_closure_of_the_independent_gap : Tendsto data.gap data.criticalApproach (𝓝 0) := data.a_square_root_leading_profile_derives_actual_gap_closure (by norm_num [data]) (by norm_num [data]) the_independent_gap_has_its_actual_leading_coefficient theorem the_independent_printed_coefficient_has_a_nonzero_error_budget : data.PrintedGapCoefficientControlledAt ∧ data.PrintedSlopeCoefficientControlledAt := by norm_num [Data.PrintedGapCoefficientControlledAt, Data.PrintedSlopeCoefficientControlledAt, data] theorem the_library_keeps_the_eventual_temperature_regime_and_gap_error (remainder : Real) (positive : 0 < remainder) : ∀ᶠ temperature in data.criticalApproach, 0 < temperature ∧ temperature < data.criticalTemperature ∧ |data.gap temperature - data.printedGapPrediction temperature| <= (data.criticalCoefficientError + remainder) * data.criticalScale temperature := data.the_printed_leading_law_has_an_explicit_eventual_error_budget (by norm_num [data]) (by norm_num [data]) the_independent_gap_has_its_actual_leading_coefficient the_independent_printed_coefficient_has_a_nonzero_error_budget.1 remainder positive theorem the_independent_gap_is_not_the_exact_printed_prediction : data.gap (3 / 2) = 15 / 2 ∧ data.gap (3 / 2) - data.printedGapPrediction (3 / 2) = -21 / 10 := by norm_num [data, Data.printedGapPrediction, Data.criticalScale, Real.sqrt_div] theorem the_inverse_temperature_curve_has_its_actual_squared_gap_derivative : HasDerivWithinAt (fun inverse => data.gapAtInverseTemperature inverse ^ 2) 1350 (Ici data.criticalInverseTemperature) data.criticalInverseTemperature := by have smooth : HasDerivAt (fun inverse : Real => 225 * (1 - (1 / inverse) / 6)) 1350 (1 / 6) := by have inverse : HasDerivAt (fun value : Real => value⁻¹) (-36) (1 / 6) := by convert hasDerivAt_inv (x := (1 / 6 : Real)) (by norm_num) using 1 norm_num have result := ((inverse.div_const 6).const_sub 1).const_mul 225 norm_num [one_div] at result ⊢ exact result have represented : ∀ inverse ∈ Ici (1 / 6 : Real), data.gapAtInverseTemperature inverse ^ 2 = 225 * (1 - (1 / inverse) / 6) := by intro inverse above change (1 / 6 : Real) <= inverse at above have positive : 0 < inverse := lt_of_lt_of_le (by norm_num) above have ratio : (1 / inverse) / 6 <= (1 : Real) := by rw [div_le_iff₀ (by norm_num : (0 : Real) < 6), div_le_iff₀ positive] linarith have square := Real.sq_sqrt (sub_nonneg.mpr ratio) have temperature : (1 / (3 * inverse)) / 2 = (1 / inverse) / 6 := by field_simp; ring dsimp [Data.gapAtInverseTemperature, data] rw [temperature, mul_pow, square] norm_num convert smooth.hasDerivWithinAt.congr represented (represented (1 / 6) (le_refl (1 / 6 : Real))) using 1 <;> norm_num [Data.criticalInverseTemperature, Data.inverseTemperature, data] theorem the_independent_curve_realizes_the_slope_input : data.CriticalGapSlopeAt := by refine ⟨by norm_num [data], ?_⟩ convert the_inverse_temperature_curve_has_its_actual_squared_gap_derivative using 1 norm_num [Data.criticalInverseTemperature, Data.inverseTemperature, data] theorem the_library_preserves_the_cutoff_correction_in_the_heat_readout : data.leadingCriticalHeatJump = 225 ∧ data.criticalHeatJump = 225 * Real.tanh 1 ∧ data.criticalHeatJump < data.leadingCriticalHeatJump := by have slope := (data.the_actual_squared_gap_slope_gives_a_positive_heat_jump the_independent_parameters_define_a_positive_temperature_window.1 the_independent_curve_realizes_the_slope_input).1 have leading : data.leadingCriticalHeatJump = 225 := by rw [Data.leadingCriticalHeatJump, slope] norm_num [data, Data.criticalInverseTemperature, Data.inverseTemperature] refine ⟨leading, ?_, (data.retaining_a_finite_shell_does_not_give_the_exact_infinite_limit_heat_jump the_independent_parameters_define_a_positive_temperature_window.1 the_independent_curve_realizes_the_slope_input).2.1⟩ rw [Data.criticalHeatJump, leading] norm_num [Data.criticalThermalShellFactor, Data.criticalInverseTemperature, Data.inverseTemperature, data] theorem the_selected_gap_minimizes_the_independent_trial_family : data.FreeEnergyMinimumAt (3 / 2) (15 / 2) := by have shell := data.the_finite_thermal_shell_factor_is_positive_and_strictly_below_one the_independent_parameters_define_a_positive_temperature_window.1 have positive : 0 < Real.tanh 1 := by norm_num [Data.criticalThermalShellFactor, Data.criticalInverseTemperature, Data.inverseTemperature, data] at shell exact shell.1 refine ⟨by norm_num, ?_⟩ intro gap _physical have bound := mul_nonneg (div_nonneg positive.le (by norm_num : (0 : Real) <= 225)) (sq_nonneg (gap ^ 2 - 225 * (1 - (3 / 2 : Real) / 2))) dsimp [data] norm_num at bound ⊢ nlinarith abbrev State := Real × Real def admissible (state : State) : Prop := 0 <= state.1 ∧ 0 <= state.2 def freeEnergy (state : State) : Real := data.freeEnergy (3 / 2) state.1 + state.2 def prepared : State := (15 / 2, 0) theorem every_admissible_state_has_an_actual_lower_bound_in_the_trial_family : ∀ state, admissible state -> ∃ gap, 0 <= gap ∧ data.freeEnergy (3 / 2) gap <= freeEnergy state := by intro state allowed exact ⟨state.1, allowed.1, le_add_of_nonneg_right allowed.2⟩ theorem the_library_transfers_the_minimum_to_all_admissible_states : admissible prepared ∧ ∀ state, admissible state -> freeEnergy prepared <= freeEnergy state := by exact data.a_gap_family_minimum_bounds_every_admissible_variational_state (3 / 2) (15 / 2) freeEnergy admissible prepared (by norm_num [admissible, prepared]) (by simp [freeEnergy, prepared]) every_admissible_state_has_an_actual_lower_bound_in_the_trial_family the_selected_gap_minimizes_the_independent_trial_family theorem the_admissibility_condition_cannot_be_dropped_from_the_minimum : (∀ state, admissible state -> freeEnergy prepared <= freeEnergy state) ∧ ¬(∀ state, freeEnergy prepared <= freeEnergy state) := by refine ⟨the_library_transfers_the_minimum_to_all_admissible_states.2, ?_⟩ intro unrestricted have lower := unrestricted (15 / 2, -1) simp only [freeEnergy, prepared, add_zero] at lower linarith theorem the_library_recovers_actual_normal_occupations_at_the_closed_gap (energy : Real) : data.electronOccupation data.criticalTemperature energy (data.gap data.criticalTemperature) = data.fermiOccupation data.criticalTemperature energy := by have closed : data.gap data.criticalTemperature = 0 := by norm_num [data] rw [closed] exact data.a_zero_gap_recovers_the_actual_normal_fermi_occupation _ energy end #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_independent_parameters_define_a_positive_temperature_window #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_independent_gap_has_its_actual_leading_coefficient #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_library_derives_closure_of_the_independent_gap #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_independent_printed_coefficient_has_a_nonzero_error_budget #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_library_keeps_the_eventual_temperature_regime_and_gap_error #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_independent_gap_is_not_the_exact_printed_prediction #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_inverse_temperature_curve_has_its_actual_squared_gap_derivative #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_independent_curve_realizes_the_slope_input #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_library_preserves_the_cutoff_correction_in_the_heat_readout #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_selected_gap_minimizes_the_independent_trial_family #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.every_admissible_state_has_an_actual_lower_bound_in_the_trial_family #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_library_transfers_the_minimum_to_all_admissible_states #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_admissibility_condition_cannot_be_dropped_from_the_minimum #print axioms PhysicsLogic.IndependentReuse.BCSThermalClient.the_library_recovers_actual_normal_occupations_at_the_closed_gap end BCSThermalClient namespace FPUIntegrationClockClient open PhysicsLogic.Papers.FermiPastaUlam1955.IntegrationClock noncomputable def step : Real := Real.sqrt 2 / 4 theorem the_independent_step_realizes_the_squared_caption : FigureOneStepAt step := by have square := Real.sq_sqrt (show (0 : Real) ≤ 2 by norm_num) constructor · dsimp [step] positivity · dsimp [step] nlinarith theorem the_library_converts_cycles_using_the_actual_step : elapsedTime step 4033 - elapsedTime step 4032 = Real.sqrt 2 / 4 := by exact cycle_time_increases_by_the_actual_step step 4032 theorem the_library_rejects_using_the_caption_square_as_elapsed_time : elapsedTime step 4032 ≠ 504 := by have reject := a_positive_cycle_count_does_not_turn_the_step_square_into_elapsed_time step the_independent_step_realizes_the_squared_caption 4032 (by norm_num) norm_num at reject ⊢ exact reject theorem the_library_separates_distinct_executed_cycle_counts : elapsedTime step 4032 < elapsedTime step 12000 := by exact a_positive_step_makes_the_computational_clock_strictly_increasing step the_independent_step_realizes_the_squared_caption.1 (by norm_num) #print axioms the_independent_step_realizes_the_squared_caption #print axioms the_library_converts_cycles_using_the_actual_step #print axioms the_library_rejects_using_the_caption_square_as_elapsed_time #print axioms the_library_separates_distinct_executed_cycle_counts end FPUIntegrationClockClient namespace PenroseAreaGeometryClient noncomputable section open PhysicsLogic.Papers.Penrose1965 def reference : SchwarzschildReference.Data := ⟨5, 7⟩ def nearby : SchwarzschildReference.Data := ⟨19 / 4, 7⟩ theorem independent_metric_readouts_realize_a_strict_relative_area_budget : SchwarzschildReference.ControlledAreaDepartureAt reference nearby.toSurfaceGeometry := by refine ⟨1 / 4, by norm_num, by norm_num, ?_⟩ intro point family rw [SchwarzschildReference.the_reference_expansion_is_the_same_computed_area_derivative _ (by norm_num [nearby])] cases family <;> norm_num [SchwarzschildReference.Data.referenceExpansion, SchwarzschildReference.Data.radialRate, SchwarzschildReference.schwarzschildFactor, reference, nearby] theorem actual_reference_and_control_derive_trapping_for_the_other_metric : nearby.toSurfaceGeometry.SourceGeometryAt := SchwarzschildReference.controlled_area_departure_preserves_both_future_null_families reference (by norm_num [SchwarzschildReference.Data.PhysicalAt, reference]) (by norm_num [SchwarzschildReference.Data.InsideReferenceRadiusAt, reference]) nearby.toSurfaceGeometry (SchwarzschildReference.the_reference_readout_has_regular_geometry_at_positive_radius _ (by norm_num [nearby])) independent_metric_readouts_realize_a_strict_relative_area_budget theorem the_independent_initial_convergence_is_a_computed_quantity (point : SurfaceGeometry.RoundPoint) : nearby.toSurfaceGeometry.convergenceAt point false = 5 / 196 := by rw [SurfaceGeometry.NullSurfaceReadout.convergenceAt, SchwarzschildReference.the_reference_expansion_is_the_same_computed_area_derivative _ (by norm_num [nearby])] norm_num [SchwarzschildReference.Data.referenceExpansion, SchwarzschildReference.Data.radialRate, SchwarzschildReference.schwarzschildFactor, nearby] theorem both_independent_normal_families_have_a_computed_positive_margin (point : SurfaceGeometry.RoundPoint) (family : Bool) : (5 / 196 : Real) <= nearby.toSurfaceGeometry.convergenceAt point family := by cases family · rw [the_independent_initial_convergence_is_a_computed_quantity] · rw [SurfaceGeometry.NullSurfaceReadout.convergenceAt, SchwarzschildReference.the_reference_expansion_is_the_same_computed_area_derivative _ (by norm_num [nearby])] norm_num [SchwarzschildReference.Data.referenceExpansion, SchwarzschildReference.Data.radialRate, nearby] theorem the_independent_convergence_is_continuous_across_the_surface : nearby.toSurfaceGeometry.ContinuousInitialConvergenceAt := by intro family change Continuous (fun point : SurfaceGeometry.RoundPoint => nearby.toSurfaceGeometry.convergenceAt point family) have computed : (fun point : SurfaceGeometry.RoundPoint => nearby.toSurfaceGeometry.convergenceAt point family) = fun _ => -nearby.referenceExpansion family / 2 := by funext point rw [SurfaceGeometry.NullSurfaceReadout.convergenceAt, SchwarzschildReference.the_reference_expansion_is_the_same_computed_area_derivative _ (by norm_num [nearby])] rw [computed] exact continuous_const theorem compactness_and_the_independent_area_geometry_supply_a_uniform_margin : nearby.toSurfaceGeometry.UniformInitialConvergenceMarginAt := nearby.toSurfaceGeometry.compactness_and_continuous_trapping_supply_a_uniform_positive_margin actual_reference_and_control_derive_trapping_for_the_other_metric the_independent_convergence_is_continuous_across_the_surface /-- These are zeros of the represented local normal-area curves, not a claim that the reference has been extended to a complete ambient spacetime. -/ theorem every_independent_normal_area_has_a_zero_with_one_common_bound : (0 : Real) < 196 / 5 ∧ ∀ (point : SurfaceGeometry.RoundPoint) (family : Bool), ∃ parameter : Real, parameter ∈ Set.Ioc 0 (196 / 5) ∧ nearby.toSurfaceGeometry.areaAt point family parameter = 0 := by refine ⟨by norm_num, ?_⟩ intro point family change ∃ parameter : Real, parameter ∈ Set.Ioc 0 (196 / 5) ∧ (SurfaceGeometry.roundReadout nearby.radius (fun _ family => nearby.radialRate family)).areaAt point family parameter = 0 rw [SurfaceGeometry.the_round_area_profile_is_a_pullback_along_its_actual_normal_curve] cases family · refine ⟨196 / 5, by norm_num [Set.mem_Ioc], ?_⟩ norm_num [nearby, SchwarzschildReference.Data.radialRate, SchwarzschildReference.schwarzschildFactor] · refine ⟨7, by norm_num [Set.mem_Ioc], ?_⟩ norm_num [nearby, SchwarzschildReference.Data.radialRate] theorem the_paper_profile_must_match_its_actual_initial_geometry {T : PenroseTypes} (framework : CollapseFramework T) (geometry : framework.SelectedFutureTrappedSurfaceGeometryAt) (evolution : framework.PrintedOpticalScalarEvolutionAt framework.nullCrossSectionalArea framework.orthogonalFutureNullCongruence framework.futureDirection framework.initialAffineParameter framework.selectedSpacetime framework.selectedRegime) (identification : framework.InitialAreaReadoutMatchesSelectedProfileAt) : framework.convergenceValueAt framework.orthogonalFutureNullCongruence framework.futureDirection (framework.affineParameterValue framework.initialAffineParameter) = framework.trappedSurfaceNullGeometryReadout.convergenceAt framework.selectedSurfacePoint framework.selectedNullFamily := framework.the_initial_optical_convergence_is_computed_from_the_selected_surface geometry evolution identification #print axioms independent_metric_readouts_realize_a_strict_relative_area_budget #print axioms actual_reference_and_control_derive_trapping_for_the_other_metric #print axioms the_independent_initial_convergence_is_a_computed_quantity #print axioms both_independent_normal_families_have_a_computed_positive_margin #print axioms the_independent_convergence_is_continuous_across_the_surface #print axioms compactness_and_the_independent_area_geometry_supply_a_uniform_margin #print axioms every_independent_normal_area_has_a_zero_with_one_common_bound #print axioms the_paper_profile_must_match_its_actual_initial_geometry end end PenroseAreaGeometryClient namespace BlasiusSimilarityClient open PhysicsLogic.Papers.Blasius1908 theorem selected_physical_fields_derive_the_profile_problem {T : Blasius1908Types} (framework : BlasiusFramework T) (inputs : framework.SelectedFlatPlateSimilarityInputs) : framework.PrintedFlatPlateBoundaryValueProblemFormulaAt ∧ framework.ActualFlatPlateFieldRealizationAt := ⟨framework.selected_similarity_inputs_derive_the_printed_boundary_value_problem inputs, framework.selected_similarity_inputs_derive_actual_field_realization inputs⟩ theorem actual_profile_kinematics_do_not_supply_the_missing_momentum_equation : (FlatPlateProfileControls.RetainedSimilarityFieldControl (2653 / 2000) ∧ FlatPlateProfileControls.reportedBracket (2653 / 2000)) ∧ ¬(FlatPlateProfileControls.profileGeometry (2653 / 2000)).InteriorMomentumEquation := FlatPlateProfileControls.a_reported_bracket_and_actual_similarity_fields_still_do_not_solve_momentum theorem local_law_covariance_does_not_assert_selected_flow_invariance : (SimilaritySelectionControl.ScalingClosedLocalWallFamilyAt ∧ SimilaritySelectionControl.LocalWallFlowAt 1 ∧ (SimilaritySelectionControl.shear 1).tangential 0 1 1 = 1 ∧ (SimilaritySelectionControl.shear 2).tangential 0 1 1 = 2) ∧ ¬SimilaritySelectionControl.SelectedFlowSelfSimilarAt 1 := SimilaritySelectionControl.local_equation_covariance_does_not_select_a_self_similar_flow theorem the_nonunit_trial_preserves_boundaries_but_fails_actual_dynamics : ((FlatPlateProfileControls.nonunitTrial.tangential 1 0 = 0 ∧ FlatPlateProfileControls.nonunitTrial.normal 1 0 = 0) ∧ Filter.Tendsto (FlatPlateProfileControls.nonunitTrial.tangential 1) Filter.atTop (nhds 6)) ∧ FlatPlateProfileControls.nonunitTrial.fields.continuityResidual 0 1 (4 / 3) = 0 ∧ FlatPlateProfileControls.nonunitTrial.fields.retainedTangentialResidual 0 1 (4 / 3) ≠ 0 := by obtain ⟨_regular, continuity, residual⟩ := FlatPlateProfileControls.the_nonunit_trial_has_a_strictly_nonzero_actual_momentum_residual exact ⟨FlatPlateProfileControls.the_nonunit_trial_matches_the_actual_wall_and_outer_stream 1 (by norm_num), continuity, ne_of_lt residual⟩ #print axioms selected_physical_fields_derive_the_profile_problem #print axioms actual_profile_kinematics_do_not_supply_the_missing_momentum_equation #print axioms local_law_covariance_does_not_assert_selected_flow_invariance #print axioms the_nonunit_trial_preserves_boundaries_but_fails_actual_dynamics end BlasiusSimilarityClient namespace RabiForceClient open MeasureTheory PhysicsLogic.Papers.Rabi1939 open MagneticSublevels MolecularRegime MolecularForce BeamRefocusing open PhysicsLogic.Framework open scoped BigOperators noncomputable section set_option backward.isDefEq.respectTransparency false /-- Fresh masses, fields, geometry, spin, spectator mixture, slit weights, and readout calibration. No production countermodel witness is used. -/ def geometry : Geometry := ⟨2, 1, 1, 1, 1, 2⟩ def apparatus : Apparatus := ⟨geometry, 2, 2, 1 / 2⟩ def spectatorMoment (spectator : Fin 3) : Real := 2 * magneticQuantumNumber 2 spectator def totalMoment (level : Sublevel 3) (spectator : Fin 3) : Real := magneticQuantumNumber 3 level + spectatorMoment spectator def spectrum : EnergySpectrum 3 (Fin 3) where planckConstant := 2 signedGyromagneticRatio := 1 / 2 spectatorEnergy := fun field spectator => -spectatorMoment spectator * field interactionEnergy := fun _ _ _ => 0 def reference : MultipletBeam.Data 3 where apparatus := apparatus molecularMass := 2 signedMoment := 3 / 2 sourceMeasure := Measure.dirac 2 initialPopulation := fun _ initial => if initial = 0 then 1 else 0 transmission := fun _ _ => 2 / 3 auxiliaryProbability := fun _ => 1 / 4 detectorGain := 6 baselineBackground := 2 drivenBackground := 2 def spectatorWeight : Fin 3 -> Real := ![1 / 2, 1 / 3, 1 / 6] def slitTransmission : Fin 3 -> Real := ![1, 1 / 2, 0] theorem the_client_spectator_distribution_and_transmission_are_physical : (∀ spectator, 0 < spectatorWeight spectator) ∧ (∑ spectator, spectatorWeight spectator) = 1 ∧ (∀ spectator, 0 ≤ slitTransmission spectator ∧ slitTransmission spectator ≤ 1) := by refine ⟨?_, by norm_num [spectatorWeight, Fin.sum_univ_succ], ?_⟩ · intro spectator fin_cases spectator <;> norm_num [spectatorWeight] · intro spectator fin_cases spectator <;> norm_num [slitTransmission] def spectatorDistribution : PMF (Fin 3) := PMF.ofFintype (fun spectator => ENNReal.ofReal (spectatorWeight spectator)) (by rw [← ENNReal.ofReal_sum_of_nonneg (fun spectator _ => (the_client_spectator_distribution_and_transmission_are_physical.1 spectator).le), the_client_spectator_distribution_and_transmission_are_physical.2.1, ENNReal.ofReal_one]) def nuclearWeight (initial final : Sublevel 3) : Real := spinThreeHalfEntry (3 / 4) (1 / 4) initial final theorem the_client_nuclear_transition_weights_are_normalized (initial : Sublevel 3) : (∀ final, 0 ≤ nuclearWeight initial final) ∧ (∑ final, nuclearWeight initial final) = 1 := by constructor · intro final fin_cases initial <;> fin_cases final <;> norm_num [nuclearWeight, spinThreeHalfEntry] · fin_cases initial <;> norm_num [nuclearWeight, spinThreeHalfEntry, Fin.sum_univ_succ] def nuclearDistribution (initial : Sublevel 3) : PMF (Sublevel 3) := PMF.ofFintype (fun final => ENNReal.ofReal (nuclearWeight initial final)) (by rw [← ENNReal.ofReal_sum_of_nonneg (fun final _ => (the_client_nuclear_transition_weights_are_normalized initial).1 final), (the_client_nuclear_transition_weights_are_normalized initial).2, ENNReal.ofReal_one]) def pulse : CoherentTwoState.Operator := CoherentTwoState.pulseMatrix 1 0 (Real.pi / 6) theorem the_client_auxiliary_probability_is_an_actual_coherent_transition : CoherentTwoState.NonvacuousCoherentTransitionAt pulse CoherentTwoState.lowerState (pulse.mulVec CoherentTwoState.lowerState) (1 / 4) := by refine ⟨CoherentTwoState.pulse_is_unitary 1 0 _ (by norm_num), CoherentTwoState.lower_state_is_normalized, rfl, ?_⟩ rw [pulse, CoherentTwoState.single_pulse_transition_probability, Real.sin_pi_div_six] norm_num theorem the_client_pulse_has_an_actual_effective_schrodinger_generator : CoherentTwoState.SolvesSchrodingerAt (CoherentTwoState.rotatingGenerator 1 0 (Real.pi / 3)) (fun time => (CoherentTwoState.pulseMatrix 1 0 ((Real.pi / 3) * time / 2)).mulVec CoherentTwoState.lowerState) := CoherentTwoState.pulse_solves_schrodinger 1 0 (Real.pi / 3) (by norm_num) _ theorem the_client_magnetic_kernel_has_the_calculated_nuclear_probabilities (speed : Real) (initial final : Sublevel 3) : reference.transition speed initial final = nuclearWeight initial final := by change fromAuxiliaryProbability 3 (1 / 4) initial final = _ rw [spin_three_half_probabilities_depend_on_the_auxiliary_probability (1 / 4) (by norm_num) (by norm_num)] norm_num [nuclearWeight] def molecular : MolecularForce.Data reference spectrum where firstField := 5000 secondField := 7000 firstSpectatorSlope := fun spectator => -spectatorMoment spectator secondSpectatorSlope := fun spectator => -spectatorMoment spectator firstInteractionSlope := fun _ _ => 0 secondInteractionSlope := fun _ _ => 0 firstForce := fun level spectator => 2 * totalMoment level spectator secondForce := fun level spectator => -2 * totalMoment level spectator spectatorPopulation := fun _ _ => spectatorDistribution transmission := fun _ _ => slitTransmission jointTransition := fun _ initial spectator => (nuclearDistribution initial).map (fun final => (final, spectator)) theorem the_client_reference_geometry_and_beam_are_physical : reference.PhysicalAt ∧ reference.IntegralDomainAt ∧ reference.apparatus.MatchedGradientGeometryAt ∧ reference.ControlledDetectorAt := by have physicalApparatus : apparatus.PhysicalAt := by norm_num [Apparatus.PhysicalAt, Geometry.PhysicalAt, apparatus, geometry] refine ⟨?_, ⟨integrable_dirac (by simp), integrable_dirac (by simp), integrable_dirac (by simp)⟩, ?_, by norm_num [MultipletBeam.Data.ControlledDetectorAt, reference]⟩ · refine ⟨physicalApparatus, by norm_num [reference], by norm_num, by norm_num [reference], ?_, ?_, ?_⟩ · dsimp [reference] infer_instance · simp [MultipletBeam.Data.referenceIntensity, reference] · simp only [reference, ae_dirac_eq] norm_num [MultipletBeam.Data.ChannelDomainAt, reference] intro initial split_ifs <;> norm_num · norm_num [Apparatus.MatchedGradientGeometryAt, reference, apparatus, geometry, Geometry.firstFactor, Geometry.secondFactor, Geometry.slitDistance, Geometry.postSlitDistance] theorem the_client_total_projection_is_the_actual_molecular_energy_slope (field : Real) (level : Sublevel 3) (spectator : Fin 3) : EnergySlopeAt spectrum field level spectator (totalMoment level spectator) := by have calculated := differentiating_the_actual_spectrum_retains_both_corrections spectrum field level spectator (-spectatorMoment spectator) 0 (by simpa [spectrum] using (hasDerivAt_id field).const_mul (-spectatorMoment spectator)) (hasDerivAt_const field 0) convert calculated using 1 dsimp [spectrum, totalMoment] ring theorem the_client_spectral_and_moment_calculations_supply_the_force_controls : molecular.ForceCalculationAt := by refine ⟨by norm_num [MolecularForce.Data.NuclearMomentCalibrationAt, reference, spectrum], ?_, fun _ => rfl⟩ refine ⟨?_, ?_, fun _ _ => hasDerivAt_const _ 0, fun _ _ => hasDerivAt_const _ 0⟩ · intro spectator simpa [molecular, spectrum] using (hasDerivAt_id (5000 : Real)).const_mul (-spectatorMoment spectator) · intro spectator simpa [molecular, spectrum] using (hasDerivAt_id (7000 : Real)).const_mul (-spectatorMoment spectator) theorem the_client_forces_satisfy_the_adiabatic_law_in_both_magnets : molecular.SecularForceSetupAt := by refine ⟨by norm_num [EnergySpectrum.PhysicalAt, molecular, spectrum], by norm_num [EnergySpectrum.PhysicalAt, molecular, spectrum], ⟨?_, ?_⟩, fun _ _ => ⟨rfl, rfl⟩⟩ · intro level spectator have slope := the_client_total_projection_is_the_actual_molecular_energy_slope (molecular.firstFieldProfile 0) level spectator have field : HasDerivAt molecular.firstFieldProfile 2 0 := by change HasDerivAt (fun coordinate : Real => 5000 + 2 * coordinate) 2 0 simpa only [mul_one, id_eq] using ((hasDerivAt_id (0 : Real)).const_mul 2).const_add (5000 : Real) unfold AdiabaticForceAt convert slope.comp 0 field using 1 dsimp [molecular] ring · intro level spectator have slope := the_client_total_projection_is_the_actual_molecular_energy_slope (molecular.secondFieldProfile 0) level spectator have field : HasDerivAt molecular.secondFieldProfile (-2) 0 := by change HasDerivAt (fun coordinate : Real => 7000 - 2 * coordinate) (-2) 0 simpa only [mul_one, id_eq, Pi.sub_apply, zero_sub] using (hasDerivAt_const (0 : Real) (7000 : Real)).sub ((hasDerivAt_id 0).const_mul 2) unfold AdiabaticForceAt convert slope.comp 0 field using 1 dsimp [molecular] ring theorem the_source_preparation_is_spectator_dependent_and_has_computed_mean_efficiency : molecular.transmission 2 0 0 = 1 ∧ molecular.transmission 2 0 1 = 1 / 2 ∧ molecular.transmission 2 0 2 = 0 ∧ average (molecular.spectatorPopulation 2 0) (molecular.transmission 2 0) = 2 / 3 := by refine ⟨rfl, rfl, rfl, ?_⟩ rw [average_finite] norm_num [molecular, spectatorDistribution, PMF.ofFintype_apply, spectatorWeight, slitTransmission, Fin.sum_univ_succ] theorem the_joint_kernel_preserves_spectator_projection_and_has_the_source_nuclear_marginal : molecular.JointReadoutCalculationAt := by apply ae_of_all intro speed refine ⟨fun _ spectator => the_client_spectator_distribution_and_transmission_are_physical.2.2 spectator, ?_, ?_, ?_⟩ · intro initial exact the_source_preparation_is_spectator_dependent_and_has_computed_mean_efficiency.2.2.2 · intro initial spectator _populated final change (((nuclearDistribution initial).map (fun level => (level, spectator))).map Prod.fst final).toReal = _ rw [PMF.map_comp] change ((nuclearDistribution initial).map id final).toReal = _ rw [PMF.map_id, the_client_magnetic_kernel_has_the_calculated_nuclear_probabilities] exact ENNReal.toReal_ofReal ((the_client_nuclear_transition_weights_are_normalized initial).1 final) · intro initial spectator _populated outcome supported obtain ⟨final, _weight, equal⟩ := (PMF.mem_support_map_iff _ _ _).1 supported rw [← equal] rfl theorem the_actual_aperture_resolves_the_client_nuclear_flip (initial final : Sublevel 3) : reference.apparatus.acceptance (reference.ray 2 initial final) = if initial = final then 1 else 0 := by unfold Apparatus.acceptance Apparatus.AcceptedAt rw [reference.apparatus.matched_gradient_geometry_computes_the_projection_difference _ the_client_reference_geometry_and_beam_are_physical.1.1 ⟨by norm_num [MultipletBeam.Data.ray, reference], by norm_num [MultipletBeam.Data.ray]⟩ the_client_reference_geometry_and_beam_are_physical.2.2.1] fin_cases initial <;> fin_cases final <;> norm_num [MultipletBeam.Data.ray, reference, magneticProjection, magneticQuantumNumber, apparatus, geometry, Geometry.firstFactor, Geometry.slitDistance, Geometry.postSlitDistance] theorem the_client_nuclear_reference_fluxes_and_signals_are_computed : reference.baselineIntensity = 2 / 3 ∧ reference.drivenIntensity = 9 / 32 ∧ reference.baselineSignal = 6 ∧ reference.drivenSignal = 59 / 16 := by have baselineDensity : reference.baselineDensity 2 = 2 / 3 := by unfold MultipletBeam.Data.baselineDensity simp only [the_actual_aperture_resolves_the_client_nuclear_flip, ite_true, mul_one] norm_num [reference] have drivenDensity : reference.drivenDensity 2 = 9 / 32 := by unfold MultipletBeam.Data.drivenDensity simp only [the_actual_aperture_resolves_the_client_nuclear_flip, the_client_magnetic_kernel_has_the_calculated_nuclear_probabilities] norm_num [reference, nuclearWeight, spinThreeHalfEntry, Fin.sum_univ_succ] have baseline : reference.baselineIntensity = 2 / 3 := by change (∫ speed, reference.baselineDensity speed ∂Measure.dirac 2) = _ simpa only [integral_dirac] using baselineDensity have driven : reference.drivenIntensity = 9 / 32 := by change (∫ speed, reference.drivenDensity speed ∂Measure.dirac 2) = _ simpa only [integral_dirac] using drivenDensity refine ⟨baseline, driven, ?_, ?_⟩ · rw [MultipletBeam.Data.baselineSignal, baseline] norm_num [reference] · rw [MultipletBeam.Data.drivenSignal, driven] norm_num [reference] theorem the_library_derives_the_actual_joint_molecular_signals : molecular.baselineSignal = 6 ∧ molecular.drivenSignal = 59 / 16 := by have signals := molecular.the_spectator_resolved_beam_has_the_derived_nuclear_only_mean_signals the_client_reference_geometry_and_beam_are_physical.1 the_client_reference_geometry_and_beam_are_physical.2.2.1 the_client_forces_satisfy_the_adiabatic_law_in_both_magnets the_client_spectral_and_moment_calculations_supply_the_force_controls the_joint_kernel_preserves_spectator_projection_and_has_the_source_nuclear_marginal exact ⟨signals.1.trans the_client_nuclear_reference_fluxes_and_signals_are_computed.2.2.1, signals.2.trans the_client_nuclear_reference_fluxes_and_signals_are_computed.2.2.2⟩ def observation : SignalObservation where baselineReading := 6 drivenReading := 59 / 16 baselineError := 1 / 10 drivenError := 1 / 10 theorem the_client_readings_are_accurate_and_resolve_the_molecular_mean_loss : molecular.ObservationAccurateAt observation ∧ observation.ResolvedObservedLossAt := by norm_num [MolecularForce.Data.ObservationAccurateAt, observation, the_library_derives_the_actual_joint_molecular_signals.1, the_library_derives_the_actual_joint_molecular_signals.2, SignalObservation.ResolvedObservedLossAt] theorem the_library_infers_nuclear_reorientation_from_the_controlled_joint_readout : ¬(∀ᵐ speed ∂reference.sourceMeasure, reference.PopulatedReorientationAbsentAt speed) := by exact molecular.a_resolved_observed_molecular_loss_requires_populated_nuclear_reorientation the_client_reference_geometry_and_beam_are_physical.1 the_client_reference_geometry_and_beam_are_physical.2.2.1 the_client_forces_satisfy_the_adiabatic_law_in_both_magnets the_client_spectral_and_moment_calculations_supply_the_force_controls the_joint_kernel_preserves_spectator_projection_and_has_the_source_nuclear_marginal (reference.the_source_spins_have_actual_normalized_multiplet_transitions (by simp [SourceNuclearSpinAt]) the_client_reference_geometry_and_beam_are_physical.1) the_client_reference_geometry_and_beam_are_physical.2.1 the_client_reference_geometry_and_beam_are_physical.2.2.2 observation the_client_readings_are_accurate_and_resolve_the_molecular_mean_loss.1 the_client_readings_are_accurate_and_resolve_the_molecular_mean_loss.2 #print axioms the_actual_aperture_resolves_the_client_nuclear_flip #print axioms the_client_auxiliary_probability_is_an_actual_coherent_transition #print axioms the_client_forces_satisfy_the_adiabatic_law_in_both_magnets #print axioms the_client_magnetic_kernel_has_the_calculated_nuclear_probabilities #print axioms the_client_nuclear_reference_fluxes_and_signals_are_computed #print axioms the_client_nuclear_transition_weights_are_normalized #print axioms the_client_pulse_has_an_actual_effective_schrodinger_generator #print axioms the_client_readings_are_accurate_and_resolve_the_molecular_mean_loss #print axioms the_client_reference_geometry_and_beam_are_physical #print axioms the_client_spectator_distribution_and_transmission_are_physical #print axioms the_client_spectral_and_moment_calculations_supply_the_force_controls #print axioms the_client_total_projection_is_the_actual_molecular_energy_slope #print axioms the_joint_kernel_preserves_spectator_projection_and_has_the_source_nuclear_marginal #print axioms the_library_derives_the_actual_joint_molecular_signals #print axioms the_library_infers_nuclear_reorientation_from_the_controlled_joint_readout #print axioms the_source_preparation_is_spectator_dependent_and_has_computed_mean_efficiency end end RabiForceClient end PhysicsLogic.IndependentReuse /-! A fresh cubic-chain phase record tests the paper-local energy API. No production countermodel, historical run, or exact trajectory is reused. -/ namespace PhysicsLogic.IndependentReuse.FPUEnergyClient open PhysicsLogic.Papers.FermiPastaUlam1955 open SineCoordinateConvention HamiltonianEnergy Finset noncomputable section def interaction : Interaction where kind := .cubic quadraticCoefficient := 0 cubicCoefficient := 1 / 8 threshold := 0 slopeIncrement := 0 def phase (scale : Real) : FixedEndPhase where span := 4 displacement := fun site => if site = 1 then scale else if site = 3 then -scale else 0 velocity := fun site => if site = 1 then scale else if site = 2 then 2 * scale else if site = 3 then scale else 0 theorem the_client_grid_and_energy_components_are_computed (scale : Real) : (phase scale).FixedEndpointsAt ∧ (phase scale).linearHamiltonian = 5 * scale ^ 2 ∧ interaction.nonlinearEnergy (phase scale) = scale ^ 4 / 8 ∧ interaction.hamiltonian (phase scale) = 5 * scale ^ 2 + scale ^ 4 / 8 := by refine ⟨by simp [phase, FixedEndPhase.FixedEndpointsAt], ?_, ?_, ?_⟩ all_goals simp [phase, interaction, FixedEndPhase.linearHamiltonian, FixedEndPhase.particleKineticEnergy, FixedEndPhase.bondPotentialEnergy, Interaction.nonlinearEnergy, Interaction.nonlinearPotential, Interaction.hamiltonian, Interaction.potential, Interaction.bondStretch, sum_range_succ] ring theorem the_client_potential_has_the_same_cubic_force (stretch : Real) : HasDerivAt interaction.potential (stretch + stretch ^ 3 / 8) stretch := by convert interaction.the_three_bond_potentials_differentiate_to_the_same_bond_forces stretch using 1 simp [Interaction.force, interaction] ring def recordedPhase (cycle : Nat) : FixedEndPhase := phase (if cycle = 0 then 1 else -1) def record : EnergyCheckData where interaction := interaction endpointSpan := 4 displacement := fun cycle => (recordedPhase cycle).displacement velocity := fun cycle => (recordedPhase cycle).velocity samples := {1} reportedEnergy := fun cycle => if cycle = 0 then 83 / 16 else 81 / 16 evaluationError := fun _ => 1 / 16 driftTolerance := 1 / 40 theorem the_client_record_really_uses_the_selected_phase (cycle : Nat) : record.phase cycle = recordedPhase cycle := rfl theorem the_client_record_has_computed_linear_nonlinear_and_total_energy (cycle : Nat) : (record.phase cycle).linearHamiltonian = 5 ∧ interaction.nonlinearEnergy (record.phase cycle) = 1 / 8 ∧ record.actualEnergy cycle = 41 / 8 := by have energy := the_client_grid_and_energy_components_are_computed (if cycle = 0 then 1 else -1) change (phase (if cycle = 0 then 1 else -1)).linearHamiltonian = 5 ∧ interaction.nonlinearEnergy (phase (if cycle = 0 then 1 else -1)) = 1 / 8 ∧ interaction.hamiltonian (phase (if cycle = 0 then 1 else -1)) = 41 / 8 rw [energy.2.1, energy.2.2.1, energy.2.2.2] split_ifs <;> norm_num theorem the_client_energy_domain_is_inhabited : record.CoordinateDomainAt := by refine ⟨by simp [record, Interaction.AdmissibleAt, interaction], by decide, by decide, by simp [record], ?_⟩ intro cycle selected exact (the_client_grid_and_energy_components_are_computed _).1 theorem the_client_controls_the_two_energy_components : record.ComponentsAccurateAt (fun cycle => record.reportedEnergy cycle - 1 / 8) (fun _ => 1 / 8) (fun _ => 1 / 16) (fun _ => 0) := by intro cycle selected have energy := the_client_record_has_computed_linear_nonlinear_and_total_energy cycle change record.reportedEnergy cycle = (record.reportedEnergy cycle - 1 / 8) + 1 / 8 ∧ 0 ≤ (1 / 16 : Real) ∧ 0 ≤ (0 : Real) ∧ |(record.reportedEnergy cycle - 1 / 8) - (record.phase cycle).linearHamiltonian| ≤ 1 / 16 ∧ |(1 / 8 : Real) - interaction.nonlinearEnergy (record.phase cycle)| ≤ 0 ∧ (1 / 16 : Real) = 1 / 16 + 0 rw [energy.1, energy.2.1] by_cases initial : cycle = 0 <;> simp [record, initial] <;> norm_num theorem the_library_derives_the_client_total_energy_evaluation_bound : record.EvaluationAccurateAt := record.controlling_both_computed_energy_components_controls_the_total_readout _ _ _ _ the_client_controls_the_two_energy_components theorem the_client_reported_energy_check_is_computed : record.ReportedConstancyAt := by norm_num [EnergyCheckData.ReportedConstancyAt, record] theorem the_library_bounds_the_clients_actual_hamiltonian_drift : record.ActualDriftBoundAt := record.a_reported_check_and_evaluation_control_bound_the_actual_hamiltonian_drift the_client_reported_energy_check_is_computed the_library_derives_the_client_total_energy_evaluation_bound theorem the_client_drift_budget_is_not_just_the_reported_budget : record.driftTolerance * record.reportedEnergy 0 = 83 / 640 ∧ record.driftTolerance * record.reportedEnergy 0 + record.evaluationError 1 + record.evaluationError 0 = 163 / 640 ∧ record.driftTolerance * record.reportedEnergy 0 < record.driftTolerance * record.reportedEnergy 0 + record.evaluationError 1 + record.evaluationError 0 := by norm_num [record] theorem the_library_derives_a_resolved_physical_reference_energy : 0 < record.actualEnergy 0 := record.a_resolved_positive_reference_report_gives_positive_physical_energy the_library_derives_the_client_total_energy_evaluation_bound (by norm_num [record]) theorem the_library_derives_the_client_five_percent_fractional_bound : 0 < record.actualEnergy 0 ∧ ∀ cycle ∈ record.samples, |record.actualEnergy cycle - record.actualEnergy 0| / record.actualEnergy 0 ≤ 1 / 20 := by apply record.the_actual_fractional_drift_needs_the_reference_energy_and_both_evaluation_errors the_client_reported_energy_check_is_computed the_library_derives_the_client_total_energy_evaluation_bound (by norm_num [record]) (1 / 20) · intro cycle selected norm_num [record] · norm_num theorem the_client_has_apparent_energy_drift_without_physical_energy_drift : record.actualEnergy 0 = record.actualEnergy 1 ∧ record.reportedEnergy 0 ≠ record.reportedEnergy 1 := by rw [(the_client_record_has_computed_linear_nonlinear_and_total_energy 0).2.2, (the_client_record_has_computed_linear_nonlinear_and_total_energy 1).2.2] norm_num [record] theorem the_library_retains_the_client_nonlinear_fraction_bound : |interaction.nonlinearEnergy (record.phase 0)| ≤ (1 / 40) * |record.reportedEnergy 0| := by simpa only [add_zero] using record.a_reported_nonlinear_fraction_retains_its_component_evaluation_error _ _ _ _ the_client_controls_the_two_energy_components 0 (by simp) (1 / 40) (by norm_num [record]) def mismatchedForce : Interaction := { interaction with cubicCoefficient := 1 / 4 } theorem changing_only_the_force_coefficient_breaks_the_potential_identification : ¬HasDerivAt interaction.potential (mismatchedForce.force 1) 1 := by intro wrong have different := wrong.unique (the_client_potential_has_the_same_cubic_force 1) norm_num [mismatchedForce, interaction, Interaction.force] at different end end PhysicsLogic.IndependentReuse.FPUEnergyClient -- Independent diagnostic proof-axiom inventory. #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_client_grid_and_energy_components_are_computed #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_client_potential_has_the_same_cubic_force #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_client_record_really_uses_the_selected_phase #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_client_record_has_computed_linear_nonlinear_and_total_energy #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_client_energy_domain_is_inhabited #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_client_controls_the_two_energy_components #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_library_derives_the_client_total_energy_evaluation_bound #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_client_reported_energy_check_is_computed #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_library_bounds_the_clients_actual_hamiltonian_drift #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_client_drift_budget_is_not_just_the_reported_budget #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_library_derives_a_resolved_physical_reference_energy #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_library_derives_the_client_five_percent_fractional_bound #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_client_has_apparent_energy_drift_without_physical_energy_drift #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.the_library_retains_the_client_nonlinear_fraction_bound #print axioms PhysicsLogic.IndependentReuse.FPUEnergyClient.changing_only_the_force_coefficient_breaks_the_potential_identification namespace PhysicsLogic.IndependentReuse.CosmologyPosteriorRouteClient open PhysicsLogic.Framework.StandardCandlePosterior theorem the_riess_main_result_keeps_both_actual_method_probabilities {T : PhysicsLogic.Papers.RiessEtAl1998.RiessTypes} (framework : PhysicsLogic.Papers.RiessEtAl1998.SupernovaAccelerationFramework T) (result : framework.EstablishedCosmologicalInference) : framework.PosteriorModelSetup ∧ NonvacuousAccelerationPosteriorAt (framework.cosmologicalPosterior .mlcs) (995 / 1000) ∧ (999 / 1000 : ENNReal) < framework.cosmologicalPosterior .template accelerating := by have posterior := framework.established_inference_retains_actual_acceleration_posterior_evidence result exact ⟨framework.established_inference_retains_the_method_likelihood_representation result, posterior.1, posterior.2.2.1⟩ theorem the_perlmutter_main_result_keeps_two_distinct_region_inputs {T : PhysicsLogic.Papers.PerlmutterEtAl1999.PerlmutterTypes} (framework : PhysicsLogic.Papers.PerlmutterEtAl1999.SupernovaCosmologyFramework T) (result : framework.EstablishedPrimaryCosmologicalFit) : framework.PrimaryPosteriorModelSetup ∧ NonvacuousAccelerationPosteriorAt framework.primaryCosmologicalPosterior framework.accelerationConfidenceThreshold ∧ (998 / 1000 : ENNReal) ≤ framework.primaryCosmologicalPosterior positiveLambda := by have posterior := framework.established_fit_retains_actual_marginal_posterior_evidence result exact ⟨framework.established_fit_retains_the_joint_density_and_parameter_domain result, posterior.1, posterior.2.1⟩ theorem the_riess_posterior_cannot_have_more_acceleration_than_positive_lambda_mass {T : PhysicsLogic.Papers.RiessEtAl1998.RiessTypes} (framework : PhysicsLogic.Papers.RiessEtAl1998.SupernovaAccelerationFramework T) (result : framework.EstablishedCosmologicalInference) : framework.cosmologicalPosterior .mlcs accelerating ≤ framework.cosmologicalPosterior .mlcs positiveLambda := the_same_physical_posterior_gives_no_more_acceleration_mass_than_positive_lambda_mass _ (framework.established_inference_retains_actual_acceleration_posterior_evidence result).1.1.2 theorem the_perlmutter_declared_acceleration_threshold_is_a_genuine_probability_bound {T : PhysicsLogic.Papers.PerlmutterEtAl1999.PerlmutterTypes} (framework : PhysicsLogic.Papers.PerlmutterEtAl1999.SupernovaCosmologyFramework T) (result : framework.EstablishedPrimaryCosmologicalFit) : (1 / 2 : ENNReal) < framework.primaryCosmologicalPosterior accelerating ∧ framework.accelerationConfidenceThreshold ≤ 1 := actual_acceleration_evidence_requires_a_probability_above_one_half _ _ (framework.established_fit_retains_actual_marginal_posterior_evidence result).1 end PhysicsLogic.IndependentReuse.CosmologyPosteriorRouteClient #print axioms PhysicsLogic.IndependentReuse.CosmologyPosteriorRouteClient.the_riess_main_result_keeps_both_actual_method_probabilities #print axioms PhysicsLogic.IndependentReuse.CosmologyPosteriorRouteClient.the_perlmutter_main_result_keeps_two_distinct_region_inputs #print axioms PhysicsLogic.IndependentReuse.CosmologyPosteriorRouteClient.the_riess_posterior_cannot_have_more_acceleration_than_positive_lambda_mass #print axioms PhysicsLogic.IndependentReuse.CosmologyPosteriorRouteClient.the_perlmutter_declared_acceleration_threshold_is_a_genuine_probability_bound /-! An independently constructed dimensional binding model and the original paper's shared source handoff. These are not historical parameter fits. -/ namespace PhysicsLogic.IndependentReuse.BlattBindingIterationClient open PhysicsLogic.Papers.BlattJackson1949.BindingIteration open PhysicsLogic.Framework.NeutralBoundState noncomputable section def data : DimensionalData where inverseLength := 2 effectiveRange := 1 / 10 bindingMomentum := 9 / 4 shapeRemainder := -1 / 320 coefficientBound := 1 def phase (momentum : Complex) : Complex := -2 + momentum ^ 2 / 20 + 4 / 32805 * momentum ^ 4 def boundMomentum : Complex := -Complex.I * (9 / 4) theorem the_clients_continued_phase_has_a_genuine_bound_zero : phase boundMomentum = -9 / 4 ∧ phase boundMomentum - Complex.I * boundMomentum ≠ 0 ∧ (phase boundMomentum + Complex.I * boundMomentum) / (phase boundMomentum - Complex.I * boundMomentum) = 0 := by norm_num [phase, boundMomentum, pow_succ, Complex.ext_iff, Complex.mul_re, Complex.mul_im, Complex.div_re, Complex.div_im, Complex.normSq_apply] theorem the_clients_binding_residual_is_computed_from_the_actual_momentum : effectiveRangeResidual data.inverseLength data.effectiveRange data.bindingMomentum = -1 / 320 := by norm_num [data, effectiveRangeResidual] theorem the_client_meets_the_raw_iteration_inputs : data.Inputs := by norm_num [DimensionalData.Inputs, DimensionalData.smallRange, data, effectiveRangeResidual] theorem the_clients_two_corrections_and_error_are_computed : data.iteratedMomentum = 56 / 25 ∧ data.iterationError = 1 / 100 := by norm_num [data, DimensionalData.iteratedMomentum, DimensionalData.iterationError] theorem the_library_derives_the_clients_controlled_binding_formula : data.bindingMomentum = data.inverseLength + data.effectiveRange / 2 * data.inverseLength ^ 2 + data.effectiveRange ^ 2 / 2 * data.inverseLength ^ 3 + data.iterationError ∧ |data.iterationError| ≤ (5 + 48 * data.coefficientBound) * data.effectiveRange ^ 3 * data.inverseLength ^ 4 := data.the_iteration_formula_retains_its_computed_remainder the_client_meets_the_raw_iteration_inputs theorem the_library_derives_the_nonzero_inverse_range_error : extractedRange data.inverseLength data.bindingMomentum - data.effectiveRange = -1 / 810 := by rw [range_extraction_error_is_scaled_binding_residual _ _ _ (by norm_num [data]), the_clients_binding_residual_is_computed_from_the_actual_momentum] norm_num [data] def largeRoot : DimensionalData where inverseLength := 5 / 18 effectiveRange := 1 bindingMomentum := 5 / 3 shapeRemainder := 0 coefficientBound := 0 theorem a_small_range_parameter_and_exact_quadratic_root_do_not_select_the_required_branch : 0 < largeRoot.bindingMomentum ∧ largeRoot.smallRange ≤ 1 / 2 ∧ effectiveRangeResidual largeRoot.inverseLength largeRoot.effectiveRange largeRoot.bindingMomentum = 0 ∧ ¬largeRoot.Inputs := by norm_num [largeRoot, DimensionalData.smallRange, DimensionalData.Inputs, effectiveRangeResidual] theorem a_source_handoff_retains_the_computed_iteration_and_inverse_errors {T : PhysicsLogic.Papers.BlattJackson1949.BlattJacksonTypes} (framework : PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework T) (reductionSource : framework.BlattJacksonEffectiveRangeDerivationObtains .stationaryFunctionalAndZeroEnergyTrial) (extractionSource : framework.BlattJacksonBoundStateExtractionDerivationObtains .scatteringMatrixContinuationAndEmpiricalExtraction) : |framework.equationFiveFiveRemainderValueAt framework.neutronProtonTriplet framework.deuteronBoundEnergy| ≤ (5 + 48 * framework.boundStateShapeCoefficientBound framework.neutronProtonTriplet) * framework.effectiveRangeValue framework.neutronProtonTriplet ^ 3 * framework.reciprocalScatteringLength framework.neutronProtonTriplet ^ 4 ∧ |extractedRange (framework.reciprocalScatteringLength framework.neutronProtonTriplet) (framework.bindingMomentumValue framework.deuteronBoundEnergy) - framework.effectiveRangeValue framework.neutronProtonTriplet| ≤ 2 * framework.boundStateShapeCoefficientBound framework.neutronProtonTriplet * framework.effectiveRangeValue framework.neutronProtonTriplet ^ 3 * framework.bindingMomentumValue framework.deuteronBoundEnergy ^ 2 := by have composition := framework.blatt_jackson_stationary_functional_pole_and_triplet_evidence_realize_source_sensitive_bound_state_extraction reductionSource extractionSource have joinedSource := framework.toEffectiveRangeBoundStateExtractionContext.source_sensitive_bound_state_extraction_exposes_handoff_derivation composition rcases joinedSource with ⟨_reduction, _extraction, _law, boundRelation, _printed, _triplet, _extracted, _qualified⟩ exact ⟨framework.bound_state_route_retains_a_quantitative_iteration_error boundRelation, framework.bound_state_route_quantifies_the_inverse_range_extraction_error boundRelation⟩ end end PhysicsLogic.IndependentReuse.BlattBindingIterationClient #print axioms PhysicsLogic.IndependentReuse.BlattBindingIterationClient.the_library_derives_the_clients_controlled_binding_formula #print axioms PhysicsLogic.IndependentReuse.BlattBindingIterationClient.the_library_derives_the_nonzero_inverse_range_error #print axioms PhysicsLogic.IndependentReuse.BlattBindingIterationClient.a_source_handoff_retains_the_computed_iteration_and_inverse_errors /-! Independent bounded field response at temperature three. The client constructs its own function and derivative, supplies both corrected bounds, and uses paper-local public lemmas without production diagnostic witnesses. It is not a microscopic Heisenberg realization or a new paper benchmark. -/ set_option autoImplicit false namespace PhysicsLogic.IndependentReuse.MerminSmallFieldResponseClient open Filter Topology PhysicsLogic.Papers.MerminWagner1966.FieldResponse noncomputable section def response (field : Real) : Real := field / (3 * (1 + |field|)) def susceptibility (field : Real) : Real := 1 / (3 * (1 + field) ^ 2) theorem the_clients_response_is_bounded_by_the_field (field : Real) : |response field| ≤ |field| / 3 := by have positive : 0 < 3 * (1 + |field|) := by positivity rw [response, abs_div, abs_of_pos positive] apply (div_le_iff₀ positive).mpr nlinarith [sq_nonneg |field|] theorem the_clients_positive_field_magnetization_is_not_zero (field : Real) (positive : 0 < field) : 0 < response field := by exact div_pos positive (by positivity) theorem the_clients_response_is_globally_bounded (field : Real) : |response field| < 1 / 3 := by have positive : 0 < 3 * (1 + |field|) := by positivity rw [response, abs_div, abs_of_pos positive] apply (div_lt_iff₀ positive).mpr nlinarith [abs_nonneg field] theorem the_client_satisfies_the_one_dimensional_bound_at_temperature_three : OneDimensionalBound response 3 := by refine ⟨1, 1, by norm_num, by norm_num, ?_⟩ intro field positive small have square : |field| ^ 2 ≤ 1 := by simpa using pow_le_pow_left₀ positive.le small.le 2 have cube : |field| ^ 3 ≤ |field| := by nlinarith [mul_le_mul_of_nonneg_left square positive.le] have powered := pow_le_pow_left₀ (abs_nonneg (response field)) (the_clients_response_is_bounded_by_the_field field) 3 rw [div_pow] at powered norm_num at powered ⊢ nlinarith theorem the_client_satisfies_the_two_dimensional_bound_at_temperature_three : TwoDimensionalBound response 3 := by refine ⟨1, 1, by norm_num, by norm_num, ?_⟩ intro field positive small have rootBound : Real.sqrt |field| ≤ 1 := by simpa using Real.sqrt_le_sqrt small.le have fieldBelowRoot : |field| ≤ Real.sqrt |field| := by have square := Real.sq_sqrt (abs_nonneg field) nlinarith [mul_nonneg (Real.sqrt_nonneg |field|) (sub_nonneg.mpr rootBound)] have temperatureRoot : Real.sqrt (3 : Real) ≤ 2 := by nlinarith [Real.sqrt_nonneg (3 : Real), Real.sq_sqrt (by norm_num : (0 : Real) ≤ 3)] have responseBound : |response field| ≤ Real.sqrt |field| / 3 := (the_clients_response_is_bounded_by_the_field field).trans (by linarith) calc |response field| * Real.sqrt (3 : Real) * Real.sqrt (abs (Real.log (abs field))) ≤ (Real.sqrt |field| / 3) * 2 * Real.sqrt (abs (Real.log (abs field))) := mul_le_mul_of_nonneg_right (mul_le_mul responseBound temperatureRoot (Real.sqrt_nonneg _) (by positivity)) (Real.sqrt_nonneg _) _ = (2 / 3) * (Real.sqrt |field| * Real.sqrt (abs (Real.log (abs field)))) := by ring _ ≤ (2 / 3) * 1 := mul_le_mul_of_nonneg_left (the_sqrt_log_weight_is_bounded_after_the_field_factor field positive small) (by norm_num) _ < 1 := by norm_num theorem the_library_derives_the_clients_limit_from_the_one_dimensional_bound : Tendsto response positiveFieldApproach (nhds 0) := the_one_dimensional_bound_forces_zero_magnetization response 3 (by norm_num) the_client_satisfies_the_one_dimensional_bound_at_temperature_three theorem the_library_derives_the_clients_limit_from_the_two_dimensional_bound : Tendsto response positiveFieldApproach (nhds 0) := the_two_dimensional_bound_forces_zero_magnetization response 3 (by norm_num) the_client_satisfies_the_two_dimensional_bound_at_temperature_three theorem the_clients_positive_field_derivative_is_computed (field : Real) (positive : 0 < field) : HasDerivAt response (susceptibility field) field := by have denominator : (1 : Real) + field ≠ 0 := by linarith have quotient := (hasDerivAt_id field).div ((hasDerivAt_id field).const_add 1) denominator have base : HasDerivAt (fun x : Real => x / (3 * (1 + x))) (susceptibility field) field := by convert quotient.div_const 3 using 1 · funext x dsimp rw [div_div] congr 1 ring · unfold susceptibility dsimp field_simp [denominator] ring apply base.congr_of_eventuallyEq filter_upwards [eventually_gt_nhds positive] with x xpos simp [response, abs_of_pos xpos] theorem the_clients_actual_susceptibility_has_finite_limit : Tendsto susceptibility positiveFieldApproach (nhds (1 / 3)) := by have base : ContinuousAt susceptibility 0 := by unfold susceptibility exact continuousAt_const.div (continuousAt_const.mul ((continuousAt_const.add continuousAt_id).pow 2)) (by norm_num) have value : susceptibility 0 = 1 / 3 := by norm_num [susceptibility] rw [← value] exact base.tendsto.mono_left nhdsWithin_le_nhds theorem the_client_has_finite_susceptibility_at_zero_field : FiniteSusceptibilityAt response := by refine ⟨susceptibility, 1 / 3, ?_, the_clients_actual_susceptibility_has_finite_limit⟩ filter_upwards [positive_fields_are_eventually_below_any_fixed_radius 1 (by norm_num)] with field selected exact the_clients_positive_field_derivative_is_computed field selected.1 theorem the_library_excludes_a_divergent_susceptibility_for_this_client : ¬DivergentSusceptibilityAt response := by intro divergent exact divergent_susceptibility_excludes_a_finite_susceptibility_limit response divergent the_client_has_finite_susceptibility_at_zero_field end end PhysicsLogic.IndependentReuse.MerminSmallFieldResponseClient #print axioms PhysicsLogic.IndependentReuse.MerminSmallFieldResponseClient.the_library_derives_the_clients_limit_from_the_one_dimensional_bound #print axioms PhysicsLogic.IndependentReuse.MerminSmallFieldResponseClient.the_library_derives_the_clients_limit_from_the_two_dimensional_bound #print axioms PhysicsLogic.IndependentReuse.MerminSmallFieldResponseClient.the_library_excludes_a_divergent_susceptibility_for_this_client open Lean Elab Command run_cmd do let environment ← getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.MerminSmallFieldResponseClient let mut traversal : Lean.CollectAxioms.State := {} let mut declarations := 0 let mut theorems := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue declarations := declarations + 1 if let .thmInfo _ := info then theorems := theorems + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theorems ≥ 11 do throwError "Missing independent response theorems" for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational response-client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Response client depends on a production diagnostic {dependency}" let uses : Array (Name × Name) := #[ (``PhysicsLogic.IndependentReuse.MerminSmallFieldResponseClient.the_client_satisfies_the_two_dimensional_bound_at_temperature_three, ``PhysicsLogic.Papers.MerminWagner1966.FieldResponse.the_sqrt_log_weight_is_bounded_after_the_field_factor), (``PhysicsLogic.IndependentReuse.MerminSmallFieldResponseClient.the_library_derives_the_clients_limit_from_the_one_dimensional_bound, ``PhysicsLogic.Papers.MerminWagner1966.FieldResponse.the_one_dimensional_bound_forces_zero_magnetization), (``PhysicsLogic.IndependentReuse.MerminSmallFieldResponseClient.the_library_derives_the_clients_limit_from_the_two_dimensional_bound, ``PhysicsLogic.Papers.MerminWagner1966.FieldResponse.the_two_dimensional_bound_forces_zero_magnetization), (``PhysicsLogic.IndependentReuse.MerminSmallFieldResponseClient.the_library_excludes_a_divergent_susceptibility_for_this_client, ``PhysicsLogic.Papers.MerminWagner1966.FieldResponse.divergent_susceptibility_excludes_a_finite_susceptibility_limit) ] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Mermin-Wagner response client audit passed: {declarations} declarations, {theorems} elaborated theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, no production diagnostic dependencies." namespace PhysicsLogic.IndependentReuse.ChoptuikThresholdClient open Filter Topology PhysicsLogic.Papers.Choptuik1993 noncomputable section def curve : ThresholdMassCurve where massAtGap := fun gap => 3 * |gap| ^ (2 / 5 : Real) * (1 + gap / (2 + |gap|)) familyCoefficient := 3 exponent := 2 / 5 theorem the_clients_correction_is_positive_and_bounded (gap : Real) (positive : 0 < gap) : 0 < gap / (2 + |gap|) /\ gap / (2 + |gap|) < 1 := by have denominator : 0 < 2 + |gap| := by positivity refine ⟨div_pos positive denominator, (div_lt_one denominator).mpr ?_⟩ rw [abs_of_pos positive] linarith theorem the_clients_supercritical_masses_are_positive (gap : Real) (positive : 0 < gap) : 0 < curve.massAtGap gap := by have correction := the_clients_correction_is_positive_and_bounded gap positive have power := Real.rpow_pos_of_pos (abs_pos.mpr positive.ne') (2 / 5 : Real) change 0 < 3 * |gap| ^ (2 / 5 : Real) * (1 + gap / (2 + |gap|)) positivity theorem the_clients_relative_residual_has_one_common_bound (gap : Real) (positive : 0 < gap) : |curve.massAtGap gap - curve.fittedMass gap| ≤ |curve.fittedMass gap| := by have correction := the_clients_correction_is_positive_and_bounded gap positive have scale : 0 ≤ 3 * |gap| ^ (2 / 5 : Real) := by positivity calc |curve.massAtGap gap - curve.fittedMass gap| = |(3 * |gap| ^ (2 / 5 : Real)) * (gap / (2 + |gap|))| := by congr 1 dsimp [curve, ThresholdMassCurve.fittedMass, PhysicsLogic.Framework.CriticalThresholdMassCurve.fittedMass] ring _ = (3 * |gap| ^ (2 / 5 : Real)) * (gap / (2 + |gap|)) := abs_of_nonneg (mul_nonneg scale correction.1.le) _ ≤ (3 * |gap| ^ (2 / 5 : Real)) * 1 := mul_le_mul_of_nonneg_left correction.2.le scale _ = |curve.fittedMass gap| := by change (3 * |gap| ^ (2 / 5 : Real)) * 1 = |3 * |gap| ^ (2 / 5 : Real)| rw [abs_of_nonneg scale] ring theorem the_client_supplies_continuum_control_without_an_exact_power_law : curve.ControlledRelativeFit := by refine ⟨2, 1, by norm_num, by norm_num, ?_⟩ intro gap positive _small simpa only [one_mul] using the_clients_relative_residual_has_one_common_bound gap positive theorem the_library_derives_the_clients_actual_mass_limit : curve.VanishesAtThreshold := curve.controlled_relative_fit_forces_zero_threshold_mass (by norm_num [curve]) the_client_supplies_continuum_control_without_an_exact_power_law theorem the_library_excludes_a_positive_mass_floor_for_the_client : ¬curve.HasPositiveMassFloor := curve.vanishing_mass_excludes_a_positive_mass_floor the_library_derives_the_clients_actual_mass_limit theorem the_library_and_positive_masses_give_arbitrarily_small_positive_masses : curve.ArbitrarilySmallPositiveMass := by intro epsilon positive obtain ⟨gap, positiveGap, _smallGap, smallMass⟩ := curve.vanishing_mass_supplies_small_positive_gaps the_library_derives_the_clients_actual_mass_limit 1 epsilon (by norm_num) positive have positiveMass := the_clients_supercritical_masses_are_positive gap positiveGap exact ⟨gap, positiveGap, positiveMass, by simpa [abs_of_pos positiveMass] using smallMass⟩ abbrev types : ChoptuikTypes where Family := Unit Parameter := Real Evolution := Real Mass := Real Exponent := Real Regime := Unit NumericalRun := Unit Profile := Unit Scale := Real Theory := Unit World := Unit def readout : MassScalingFormulaReadout types where family := () criticalParameter := 7 supercriticalParameter := 8 blackHoleMass := 4 exponent := 2 / 5 regime := () criticalParameterValue := 7 supercriticalParameterValue := 8 blackHoleMassValue := 4 printedHorizonMassValue := 16 apparentHorizonRadiusValue := 8 familyCoefficientValue := 3 exponentValue := 2 / 5 massFitToleranceValue := 1 theorem the_clients_readout_has_a_physical_horizon_and_positive_fit_domain : readout.FormulaDomainAt /\ readout.ApparentHorizonThresholdAt := by norm_num [readout, MassScalingFormulaReadout.FormulaDomainAt, MassScalingFormulaReadout.ApparentHorizonThresholdAt] theorem the_selected_mass_comes_from_the_same_nonexact_curve : curve.massAtGap (readout.supercriticalParameterValue - readout.criticalParameterValue) = readout.blackHoleMassValue /\ curve.familyCoefficient = readout.familyCoefficientValue /\ curve.exponent = readout.exponentValue := by have gap : readout.supercriticalParameterValue - readout.criticalParameterValue = 1 := by change (8 : Real) - 7 = 1 norm_num rw [gap] refine ⟨?_, rfl, rfl⟩ change (3 : Real) * |(1 : Real)| ^ (2 / 5 : Real) * (1 + 1 / (2 + |(1 : Real)|)) = 4 rw [abs_of_pos (by norm_num : (0 : Real) < 1), Real.one_rpow] norm_num theorem the_library_normalizes_the_clients_horizon_mass : readout.blackHoleMassValue = readout.apparentHorizonRadiusValue / 2 := MassScalingFormulaReadout.apparent_horizon_threshold_gives_half_radius readout the_clients_readout_has_a_physical_horizon_and_positive_fit_domain.2 theorem the_client_has_an_actual_nonzero_fit_residual : readout.EquationTwelvePowerLawFitAt /\ |curve.massAtGap 1 - curve.fittedMass 1| = 1 := by constructor · change |(4 : Real) - 3 * Real.rpow |(8 : Real) - 7| (2 / 5)| ≤ 1 have gap : |(8 : Real) - 7| = 1 := by norm_num have onePower : Real.rpow (1 : Real) (2 / 5 : Real) = 1 := Real.one_rpow _ rw [gap, onePower] rw [abs_of_nonneg (by norm_num : (0 : Real) ≤ 4 - 3 * 1)] norm_num · change |(3 : Real) * |(1 : Real)| ^ (2 / 5 : Real) * (1 + 1 / (2 + |(1 : Real)|)) - 3 * |(1 : Real)| ^ (2 / 5 : Real)| = 1 rw [abs_of_pos (by norm_num : (0 : Real) < 1), Real.one_rpow] rw [abs_of_nonneg (by norm_num : (0 : Real) ≤ 3 * 1 * (1 + 1 / (2 + 1)) - 3 * 1)] norm_num end end PhysicsLogic.IndependentReuse.ChoptuikThresholdClient #print axioms PhysicsLogic.IndependentReuse.ChoptuikThresholdClient.the_library_derives_the_clients_actual_mass_limit #print axioms PhysicsLogic.IndependentReuse.ChoptuikThresholdClient.the_library_and_positive_masses_give_arbitrarily_small_positive_masses #print axioms PhysicsLogic.IndependentReuse.ChoptuikThresholdClient.the_library_normalizes_the_clients_horizon_mass open Lean Elab Command run_cmd do let environment ← getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.ChoptuikThresholdClient let mut traversal : Lean.CollectAxioms.State := {} let mut declarations := 0 let mut theorems := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue declarations := declarations + 1 if let .thmInfo _ := info then theorems := theorems + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theorems ≥ 11 do throwError "Missing independent threshold-client theorems" for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational threshold-client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Threshold client depends on a production diagnostic {dependency}" let uses : Array (Name × Name) := #[ (``PhysicsLogic.IndependentReuse.ChoptuikThresholdClient.the_library_derives_the_clients_actual_mass_limit, ``PhysicsLogic.Papers.Choptuik1993.ThresholdMassCurve.controlled_relative_fit_forces_zero_threshold_mass), (``PhysicsLogic.IndependentReuse.ChoptuikThresholdClient.the_library_excludes_a_positive_mass_floor_for_the_client, ``PhysicsLogic.Papers.Choptuik1993.ThresholdMassCurve.vanishing_mass_excludes_a_positive_mass_floor), (``PhysicsLogic.IndependentReuse.ChoptuikThresholdClient.the_library_and_positive_masses_give_arbitrarily_small_positive_masses, ``PhysicsLogic.Papers.Choptuik1993.ThresholdMassCurve.vanishing_mass_supplies_small_positive_gaps), (``PhysicsLogic.IndependentReuse.ChoptuikThresholdClient.the_library_normalizes_the_clients_horizon_mass, ``PhysicsLogic.Papers.Choptuik1993.MassScalingFormulaReadout.apparent_horizon_threshold_gives_half_radius) ] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Choptuik threshold client audit passed: {declarations} declarations, {theorems} elaborated theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, no production diagnostic dependencies." section set_option maxRecDepth 4096 set_option maxHeartbeats 1600000 namespace PhysicsLogic.IndependentReuse.EvansCurvatureClient noncomputable section open PhysicsLogic.Framework.FiniteMetricGeometry open PhysicsLogic.Papers.EvansColeman1994 open Filter Topology -- The pointwise jet of -4 dt^2 + (1+t^2)(dx^2+dy^2+dz^2) at t=0. def curvedJet : MetricTwoJet Real where metric := fun a b => if a = b then (if a = 0 then -4 else 1) else 0 inverseMetric := fun a b => if a = b then (if a = 0 then -(1 / 4) else 1) else 0 firstDerivative := fun _ _ _ => 0 secondDerivative := fun d e a b => if d = 0 ∧ e = 0 ∧ a = b ∧ a ≠ 0 then 2 else 0 theorem curved_jet_is_coherent : curvedJet.Coherent := by refine ⟨?_, ?_, ?_, ?_, ?_, ?_⟩ · intro a b fin_cases a <;> fin_cases b <;> norm_num [curvedJet, Matrix.diagonal] · intro a b fin_cases a <;> fin_cases b <;> norm_num [curvedJet, Matrix.diagonal] · intro a b fin_cases a <;> fin_cases b <;> norm_num only [coordinateSum, curvedJet, kronecker, Fin.ext_iff] <;> norm_num <;> rfl · intros rfl · intro d e a b simp only [curvedJet] split_ifs <;> simp_all · intro d e a b simp only [curvedJet] split_ifs <;> simp_all theorem actual_ricci_components : ∀ a b, curvedJet.modernRicci a b = (if a = b then (if a = 0 then -3 else 1/4) else 0) := by intro a b fin_cases a <;> fin_cases b <;> norm_num only [MetricTwoJet.modernRicci, MetricTwoJet.riemann, MetricTwoJet.christoffel, MetricTwoJet.christoffelDerivative, MetricTwoJet.inverseDerivative, coordinateSum, curvedJet, Fin.ext_iff] <;> norm_num [Fin.ext_iff] <;> ring_nf <;> change (0 : Real) * (-1 / 2) = 0 all_goals norm_num theorem inverse_metric_contraction_is_not_a_euclidean_sum : curvedJet.ricciSquare = 3/4 ∧ coordinateSum (fun a => coordinateSum (fun b => curvedJet.modernRicci a b ^ 2)) = 147/16 := by simp only [MetricTwoJet.ricciSquare, actual_ricci_components] norm_num only [coordinateSum, curvedJet, Fin.ext_iff] norm_num [Fin.ext_iff] ring_nf change (3 / 4 : Real) + 0 * (63 / 4) = 3 / 4 norm_num theorem independent_profile_has_zero_actual_curvature (t r : Real) (ht : t < 0) (hr : 0 < r) : Curvature.ricciSquare (Regularity.flatKinematicProfile (fun x => 7 * x ^ 2)) t r = 0 := Curvature.flat_profile_has_zero_geometric_curvature _ t r ht hr theorem the_main_result_preserves_metric_binding {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (curvature : context.SelectedCurvatureCalculationAt) (calibration : context.SelectedIndexCalibrationAt) (lapse : context.SelectedLapseCalculationAt) : Curvature.InteriorGeometryRegular context.selectedSimilarityProfile ∧ Curvature.ReadoutMatches context.selectedSingularityReadout.limits context.selectedSimilarityProfile context.selectedSingularityReadout.centralClock := (context.calculated_curvature_and_lapse_asymptotics_establish_the_selected_limits curvature calibration lapse).2.2.2.2.2.2.2.2 theorem independent_central_limit_consumer (readout : Singularity.Readout) (profile : Similarity.Profile) (trace : SingularityFields.CentralTrace) (clock : SingularityFields.CentralClock) (traceMatches : trace.MatchesProfile profile) (proper : clock.IsProperFor trace) (densityMatches : SingularityFields.DensityMatches readout trace clock) (curvatureMatches : Curvature.ReadoutMatches readout profile clock) (identity : Curvature.RadiationIdentity profile) : readout.RadiationCurvatureIdentity := Curvature.geometric_radiation_identity_passes_to_the_bound_central_readout readout profile trace clock traceMatches proper densityMatches curvatureMatches identity end end PhysicsLogic.IndependentReuse.EvansCurvatureClient end run_cmd do let environment <- Lean.getEnv let declarations : Array Lean.Name := #[ ``PhysicsLogic.IndependentReuse.EvansCurvatureClient.curved_jet_is_coherent, ``PhysicsLogic.IndependentReuse.EvansCurvatureClient.actual_ricci_components, ``PhysicsLogic.IndependentReuse.EvansCurvatureClient.inverse_metric_contraction_is_not_a_euclidean_sum, ``PhysicsLogic.IndependentReuse.EvansCurvatureClient.independent_profile_has_zero_actual_curvature, ``PhysicsLogic.IndependentReuse.EvansCurvatureClient.the_main_result_preserves_metric_binding, ``PhysicsLogic.IndependentReuse.EvansCurvatureClient.independent_central_limit_consumer] for declaration in declarations do let (_, dependencies) := ((Lean.CollectAxioms.collect declaration).run environment).run {} for axiomName in dependencies.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational curvature-client axiom {axiomName}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.EvansCurvatureClient.independent_profile_has_zero_actual_curvature, ``PhysicsLogic.Papers.EvansColeman1994.Curvature.flat_profile_has_zero_geometric_curvature), (``PhysicsLogic.IndependentReuse.EvansCurvatureClient.the_main_result_preserves_metric_binding, ``PhysicsLogic.Papers.EvansColeman1994.RadiationFluidCollapseContext.calculated_curvature_and_lapse_asymptotics_establish_the_selected_limits), (``PhysicsLogic.IndependentReuse.EvansCurvatureClient.independent_central_limit_consumer, ``PhysicsLogic.Papers.EvansColeman1994.Curvature.geometric_radiation_identity_passes_to_the_bound_central_readout)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Evans-Coleman curvature client audit passed: {declarations.size} checked theorems, {uses.size} public theorem uses, real metric jet and non-Euclidean contraction, foundational axioms only." section set_option autoImplicit false namespace PhysicsLogic.IndependentReuse.EvansPhotonClient noncomputable section open Filter Topology open PhysicsLogic.Papers.EvansColeman1994 open PhotonPropagation def spacetime : PolarSpacetime where lapse := fun _ => 3 radialMetric := fun _ => 2 def profile : Similarity.Profile where index := 1 scale := 1 radialMetric := fun _ => 2 radialVelocity := fun _ => 0 scaledDensity := fun x => 4 * x ^ 2 lapseProfile := fun x => -3 / x def photons (radius : Real) : RadialPhoton where position := fun parameter => ((2 / 3) * (2 - radius) * parameter, radius + (2 - radius) * parameter) coordinateEnergy := fun _ => 6 * (2 - radius) theorem the_continued_metric_matches_the_independent_profile : spacetime.ExtendsProfile profile := by intro time radius negativeTime positiveRadius constructor · dsimp [spacetime, profile, Similarity.Profile.lapse, Similarity.Profile.coordinate] rw [Real.rpow_one] field_simp [negativeTime.ne, positiveRadius.ne'] · rfl theorem independent_photon_propagation (radius : Real) (positive : 0 < radius) (below : radius < 2) : (photons radius).PropagatesIn spacetime := by constructor · intro parameter inSegment refine ⟨?_, ?_, ?_, ?_, ?_⟩ · dsimp [photons] nlinarith [mul_nonneg (show 0 ≤ 2 - radius by linarith) inSegment.1] · dsimp [photons] exact mul_pos (by norm_num) (sub_pos.mpr below) · exact ⟨by norm_num [spacetime], by norm_num [spacetime], contDiffAt_const, contDiffAt_const⟩ · convert (hasDerivAt_id parameter).const_mul ((2 / 3 : Real) * (2 - radius)) using 1 norm_num [photons, RadialPhoton.tangent, RadialPhoton.endpointAt, spacetime, Singularity.RadialPhotonEndpoint.outwardPhoton] ring · convert ((hasDerivAt_id parameter).const_mul (2 - radius)).const_add radius using 1 norm_num [photons, RadialPhoton.tangent, RadialPhoton.endpointAt, spacetime, Singularity.RadialPhotonEndpoint.outwardPhoton] · intro parameter inSegment have temporal : (photons radius).temporalForce spacetime parameter = 0 := by simp [RadialPhoton.temporalForce, timeDerivative, spacetime, PolarSpacetime.timeMetric, PolarSpacetime.radialMetricSquared] have radial : (photons radius).radialForce spacetime parameter = 0 := by simp [RadialPhoton.radialForce, radialDerivative, spacetime, PolarSpacetime.timeMetric, PolarSpacetime.radialMetricSquared] constructor · rw [temporal] exact hasDerivAt_const parameter (-(6 * (2 - radius))) · have momentum : (photons radius).radialMomentum spacetime = fun _ => 4 * (2 - radius) := by funext value norm_num [RadialPhoton.radialMomentum, PolarSpacetime.radialMetricSquared, photons, spacetime, RadialPhoton.tangent, RadialPhoton.endpointAt, Singularity.RadialPhotonEndpoint.outwardPhoton] rw [radial, momentum] exact hasDerivAt_const parameter (4 * (2 - radius)) theorem the_independent_family_reaches_radius_two : FamilyPropagates spacetime photons 2 := by refine ⟨by norm_num, ?_⟩ have small : ∀ᶠ radius : Real in Singularity.positiveApproach, radius < 2 := (eventually_lt_nhds (by norm_num : (0 : Real) < 2)).filter_mono nhdsWithin_le_nhds filter_upwards [self_mem_nhdsWithin, small] with radius positive below exact ⟨below, independent_photon_propagation radius positive below, by simp [photons], by simp [photons]⟩ def limits : Singularity.Readout where centralDensity := fun _ => 0 ricciSquare := fun _ => 0 boundaryLapse := fun _ => 3 similarityIndex := 1 theorem actual_profile_boundary_lapse : SingularityFields.BoundaryLapseMatches limits profile := by intro radius positiveRadius apply tendsto_const_nhds.congr' filter_upwards [self_mem_nhdsWithin] with time positiveTime exact (the_continued_metric_matches_the_independent_profile (-time) radius (neg_neg_of_pos positiveTime) positiveRadius).1 theorem the_library_derives_physical_endpoints : (endpoints spacetime photons).PhysicalEndpoints := propagated_endpoints_are_physical spacetime photons 2 the_independent_family_reaches_radius_two theorem the_library_derives_actual_emission_matching : (endpoints spacetime photons).EmissionMatches limits := propagated_emission_uses_the_selected_boundary_lapse spacetime photons 2 profile limits the_independent_family_reaches_radius_two the_continued_metric_matches_the_independent_profile actual_profile_boundary_lapse theorem the_library_places_reception_in_the_continued_region : ∀ᶠ radius in Singularity.positiveApproach, 0 < ((photons radius).position 1).1 := reception_from_the_similarity_boundary_requires_future_geometry spacetime photons 2 the_independent_family_reaches_radius_two theorem the_library_conserves_the_clients_coordinate_energy (radius : Real) (positive : 0 < radius) (below : radius < 2) : (photons radius).coordinateEnergy 1 = (photons radius).coordinateEnergy 0 := by apply (photons radius).stationary_geometry_conserves_the_actual_coordinate_energy spacetime (independent_photon_propagation radius positive below) intro parameter inSegment constructor <;> simp [timeDerivative, PolarSpacetime.timeMetric, PolarSpacetime.radialMetricSquared, spacetime] theorem the_receiving_clock_is_not_normalized_to_one : (endpoints spacetime photons).TransferControlled (1 / 3) := by refine ⟨by norm_num, ?_⟩ apply tendsto_const_nhds.congr' have physical := the_library_derives_physical_endpoints filter_upwards [physical] with radius admissible have nonzero : 6 * (2 - radius) ≠ 0 := admissible.1.2.2.ne' simp [Singularity.ObserverFrequencyFamily.transferFactor, endpoints, RadialPhoton.endpointAt, spacetime, photons, nonzero] theorem the_main_redshift_result_retains_actual_propagation {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (calibration : context.SelectedIndexCalibrationAt) (lapse : context.SelectedLapseCalculationAt) (transport : context.SelectedObserverTransportAt) : context.SelectedPhotonPropagationAt := (context.lapse_and_observer_transport_establish_infinite_redshift calibration lapse transport).2.2 /-- A time-dependent lapse in flat spacetime coordinates; proper time is t+t^2/2 on the regular patch t>-1. Coordinate energy is not observer frequency. -/ def changingClock : PolarSpacetime where lapse := fun point => point.1 + 1 radialMetric := fun _ => 1 def changingPhoton : RadialPhoton where position := fun parameter => (Real.sqrt (1 + 2 * parameter) - 1, 1 + parameter) coordinateEnergy := fun parameter => Real.sqrt (1 + 2 * parameter) theorem changing_clock_tangent (parameter : Real) (positive : 0 < 1 + 2 * parameter) : changingPhoton.tangent changingClock parameter = (1 / Real.sqrt (1 + 2 * parameter), 1) := by have nonzero := (Real.sqrt_pos.2 positive).ne' ext <;> simp [RadialPhoton.tangent, RadialPhoton.endpointAt, Singularity.RadialPhotonEndpoint.outwardPhoton, changingPhoton, changingClock, nonzero] all_goals field_simp [nonzero] theorem changing_clock_forces (parameter : Real) (positive : 0 < 1 + 2 * parameter) : changingPhoton.temporalForce changingClock parameter = -1 / Real.sqrt (1 + 2 * parameter) /\ changingPhoton.radialForce changingClock parameter = 0 := by have temporal : timeDerivative changingClock.timeMetric (changingPhoton.position parameter) = -2 * Real.sqrt (1 + 2 * parameter) := by change deriv (fun time : Real => -(time + 1) ^ 2) (Real.sqrt (1 + 2 * parameter) - 1) = _ have derivative := (((hasDerivAt_id (Real.sqrt (1 + 2 * parameter) - 1)).add_const 1).pow 2).neg simpa only [id_eq, Nat.cast_ofNat, Nat.reduceSub, pow_one, mul_one, sub_add_cancel, neg_mul] using derivative.deriv have tangent := changing_clock_tangent parameter positive constructor · dsimp only [RadialPhoton.temporalForce] rw [temporal, tangent] simp only [timeDerivative, PolarSpacetime.radialMetricSquared, changingClock, one_pow, deriv_const, zero_mul, add_zero] field_simp [(Real.sqrt_pos.2 positive).ne'] · simp [RadialPhoton.radialForce, radialDerivative, PolarSpacetime.timeMetric, PolarSpacetime.radialMetricSquared, changingClock] theorem a_changing_coordinate_energy_can_satisfy_actual_photon_transport : changingPhoton.PropagatesIn changingClock := by have rootDerivative (parameter : Real) (nonnegative : 0 ≤ parameter) : HasDerivAt (fun value : Real => Real.sqrt (1 + 2 * value)) (1 / Real.sqrt (1 + 2 * parameter)) parameter := by have positive : 0 < 1 + 2 * parameter := by linarith have derivative := (((hasDerivAt_id parameter).const_mul 2).const_add 1).sqrt positive.ne' simp only [id_eq] at derivative convert derivative using 1 field_simp [(Real.sqrt_pos.2 positive).ne'] constructor · intro parameter inSegment have positive : 0 < 1 + 2 * parameter := by linarith [inSegment.1] have rootPositive := Real.sqrt_pos.2 positive refine ⟨by dsimp [changingPhoton]; linarith [inSegment.1], rootPositive, ?_, ?_, ?_⟩ · refine ⟨by simpa [changingClock, changingPhoton] using rootPositive, by norm_num [changingClock], ?_, contDiffAt_const⟩ unfold changingClock fun_prop · rw [changing_clock_tangent parameter positive] exact (rootDerivative parameter inSegment.1).sub_const 1 · rw [changing_clock_tangent parameter positive] exact (hasDerivAt_id parameter).const_add 1 · intro parameter inSegment have positive : 0 < 1 + 2 * parameter := by linarith [inSegment.1] have forces := changing_clock_forces parameter positive constructor · rw [forces.1] change HasDerivAt (fun value : Real => -Real.sqrt (1 + 2 * value)) (-1 / Real.sqrt (1 + 2 * parameter)) parameter simpa only [neg_div] using (rootDerivative parameter inSegment.1).neg · have nearby : ∀ᶠ value : Real in nhds parameter, 0 < 1 + 2 * value := by have continuous : Continuous (fun value : Real => 1 + 2 * value) := by fun_prop exact continuous.continuousAt.eventually (eventually_gt_nhds positive) have momentum : changingPhoton.radialMomentum changingClock =ᶠ[nhds parameter] (fun _ => 1) := by filter_upwards [nearby] with value positiveValue simp [RadialPhoton.radialMomentum, PolarSpacetime.radialMetricSquared, changingPhoton, changingClock, RadialPhoton.tangent, RadialPhoton.endpointAt, Singularity.RadialPhotonEndpoint.outwardPhoton, (Real.sqrt_pos.2 positiveValue).ne'] rw [forces.2] exact (hasDerivAt_const parameter (1 : Real)).congr_of_eventuallyEq momentum theorem coordinate_energy_changes_without_a_measured_frequency_shift : changingPhoton.coordinateEnergy 0 < changingPhoton.coordinateEnergy 1 /\ (changingPhoton.endpointAt changingClock 0).observedFrequency = (changingPhoton.endpointAt changingClock 1).observedFrequency := by constructor · dsimp [changingPhoton] norm_num nlinarith [Real.sq_sqrt (by norm_num : (0 : Real) ≤ 3), Real.sqrt_nonneg (3 : Real)] · have frequency (parameter : Real) (positive : 0 < 1 + 2 * parameter) : (changingPhoton.endpointAt changingClock parameter).observedFrequency = 1 := by have nonzero := (Real.sqrt_pos.2 positive).ne' rw [Singularity.RadialPhotonEndpoint.measured_frequency_is_energy_over_lapse _ (by simpa [RadialPhoton.endpointAt, changingClock, changingPhoton] using nonzero)] simp [RadialPhoton.endpointAt, changingClock, changingPhoton, nonzero] rw [frequency 0 (by norm_num), frequency 1 (by norm_num)] theorem the_library_does_not_impose_stationarity : changingPhoton.PropagatesIn changingClock /\ ¬ changingPhoton.StationaryAlong changingClock := by refine ⟨a_changing_coordinate_energy_can_satisfy_actual_photon_transport, ?_⟩ intro stationary have conserved := changingPhoton.stationary_geometry_conserves_the_actual_coordinate_energy changingClock a_changing_coordinate_energy_can_satisfy_actual_photon_transport stationary exact (ne_of_lt coordinate_energy_changes_without_a_measured_frequency_shift.1) conserved.symm end end PhysicsLogic.IndependentReuse.EvansPhotonClient end run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.EvansPhotonClient let mut traversal : Lean.CollectAxioms.State := {} let mut declarations := 0 let mut theorems := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue declarations := declarations + 1 if let .thmInfo _ := info then theorems := theorems + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theorems >= 15 do throwError "Missing independent photon-client theorems" for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational photon-client axiom {axiomName}" let diagnostics : Array Lean.Name := #[ ``PhysicsLogic.Papers.EvansColeman1994.PhotonPropagation.flatSpacetime, ``PhysicsLogic.Papers.EvansColeman1994.PhotonPropagation.flatPhoton, ``PhysicsLogic.Papers.EvansColeman1994.PhotonPropagation.nonaffinePhoton, ``PhysicsLogic.Papers.EvansColeman1994.Similarity.Profile.homotheticNonSolution] for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if diagnostics.any (fun diagnostic => diagnostic.isPrefixOf dependency) || (parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls"))) then throwError "Photon client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.EvansPhotonClient.the_library_derives_physical_endpoints, ``PhysicsLogic.Papers.EvansColeman1994.PhotonPropagation.propagated_endpoints_are_physical), (``PhysicsLogic.IndependentReuse.EvansPhotonClient.the_library_derives_actual_emission_matching, ``PhysicsLogic.Papers.EvansColeman1994.PhotonPropagation.propagated_emission_uses_the_selected_boundary_lapse), (``PhysicsLogic.IndependentReuse.EvansPhotonClient.the_library_places_reception_in_the_continued_region, ``PhysicsLogic.Papers.EvansColeman1994.PhotonPropagation.reception_from_the_similarity_boundary_requires_future_geometry), (``PhysicsLogic.IndependentReuse.EvansPhotonClient.the_library_conserves_the_clients_coordinate_energy, ``PhysicsLogic.Papers.EvansColeman1994.PhotonPropagation.RadialPhoton.stationary_geometry_conserves_the_actual_coordinate_energy), (``PhysicsLogic.IndependentReuse.EvansPhotonClient.the_main_redshift_result_retains_actual_propagation, ``PhysicsLogic.Papers.EvansColeman1994.RadiationFluidCollapseContext.lapse_and_observer_transport_establish_infinite_redshift), (``PhysicsLogic.IndependentReuse.EvansPhotonClient.coordinate_energy_changes_without_a_measured_frequency_shift, ``PhysicsLogic.Papers.EvansColeman1994.Singularity.RadialPhotonEndpoint.measured_frequency_is_energy_over_lapse), (``PhysicsLogic.IndependentReuse.EvansPhotonClient.the_library_does_not_impose_stationarity, ``PhysicsLogic.Papers.EvansColeman1994.PhotonPropagation.RadialPhoton.stationary_geometry_conserves_the_actual_coordinate_energy)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Evans-Coleman photon client audit passed: {declarations} declarations, {theorems} elaborated theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, nonunit receiving clock and nonstationary propagation, no production diagnostic dependencies." section set_option autoImplicit false namespace PhysicsLogic.IndependentReuse.EvansAcousticClient noncomputable section open PhysicsLogic.Papers.EvansColeman1994 open AcousticCharacteristics def movingRadiation : FluidState where lapse := 4 radialMetric := 3 density := 6 velocity := 3 / 4 theorem the_independent_moving_fluid_is_physical : movingRadiation.Admissible := by norm_num [FluidState.Admissible, movingRadiation] theorem the_actual_time_flux_jacobian_is_computed : movingRadiation.timeJacobian = !![(15 / 4 : Real), 72 / 5; 45 / 4, 612 / 5] := by rw [FluidState.time_jacobian_is_the_derivative_of_the_printed_fluxes] norm_num [movingRadiation, FluidState.enthalpy, FluidState.pressure, lorentz] -- Normalize residual real-number instances produced by the pinned norm_num. change (3 : Real) * (6 + 2) * (3 / 4) / (5 / 4) = 72 / 5 /\ (12 : Real) * (5 / 4) * (3 / 4) = 45 / 4 /\ (9 : Real) * (6 + 2) * (17 / 10) = 612 / 5 norm_num theorem the_actual_radial_flux_jacobian_is_computed : movingRadiation.radialJacobian = !![(3 : Real), 32; 13, 144] := by rw [FluidState.radial_jacobian_is_the_derivative_of_the_printed_fluxes] norm_num [movingRadiation, FluidState.enthalpy, FluidState.pressure, lorentz] change ((3 : Real) = 3 /\ (4 : Real) * (6 + 2) = 32) /\ (13 : Real) = 13 /\ (24 : Real) * (6 + 2) * (3 / 4) = 144 norm_num theorem the_actual_characteristic_polynomial_is_computed (speed : Real) : (movingRadiation.principalSymbol speed).det = 297 * speed ^ 2 - 360 * speed + 16 := by rw [FluidState.principalSymbol, the_actual_time_flux_jacobian_is_computed, the_actual_radial_flux_jacobian_is_computed, Matrix.det_fin_two] dsimp ring theorem both_computed_branches_are_characteristic : movingRadiation.CharacteristicAt movingRadiation.ingoingSpeed /\ movingRadiation.CharacteristicAt movingRadiation.outgoingSpeed := by constructor · exact (movingRadiation.a_positive_radiation_fluid_has_exactly_the_two_acoustic_characteristics the_independent_moving_fluid_is_physical _).mpr (Or.inl rfl) · exact (movingRadiation.a_positive_radiation_fluid_has_exactly_the_two_acoustic_characteristics the_independent_moving_fluid_is_physical _).mpr (Or.inr rfl) theorem both_branches_have_nonzero_fluid_amplitudes : movingRadiation.HasNonzeroPrincipalAmplitudeAt movingRadiation.ingoingSpeed /\ movingRadiation.HasNonzeroPrincipalAmplitudeAt movingRadiation.outgoingSpeed := ⟨(movingRadiation.characteristic_iff_a_nonzero_fluid_principal_amplitude _).mp both_computed_branches_are_characteristic.1, (movingRadiation.characteristic_iff_a_nonzero_fluid_principal_amplitude _).mp both_computed_branches_are_characteristic.2⟩ theorem an_ingoing_fluid_frame_mode_can_move_outward_in_coordinates : 0 < movingRadiation.ingoingSpeed := by unfold FluidState.ingoingSpeed apply div_pos _ (mul_pos the_independent_moving_fluid_is_physical.2.1 movingRadiation.acoustic_denominators_are_positive.1) have soundSquared : Similarity.Profile.radiationSoundSpeed ^ 2 = (1 / 3 : Real) := Real.sq_sqrt (by norm_num) have soundPositive := Similarity.Profile.the_radiation_sound_speed_is_positive_and_subluminal.1 norm_num [movingRadiation, lorentz] nlinarith [soundSquared] theorem both_modes_move_outward_but_remain_inside_the_metric_light_cone : 0 < movingRadiation.ingoingSpeed /\ movingRadiation.ingoingSpeed < movingRadiation.outgoingSpeed /\ movingRadiation.outgoingSpeed < movingRadiation.lapse / movingRadiation.radialMetric := by have causal := movingRadiation.acoustic_characteristics_are_distinct_and_inside_the_metric_light_cone the_independent_moving_fluid_is_physical exact ⟨an_ingoing_fluid_frame_mode_can_move_outward_in_coordinates, causal.2⟩ theorem the_main_sonic_result_retains_a_nonzero_fluid_amplitude {T : EvansColemanTypes} (context : RadiationFluidCollapseContext T) (setup : context.SelfSimilarConstructionSetup) (calculation : context.SelfSimilarConstructionCalculation) (time radius : Real) (point : Similarity.Profile.CollapsePoint time radius) (selected : context.selectedSimilarityProfile.coordinate time radius = 1) : (profileState context.selectedSimilarityProfile time radius).HasNonzeroPrincipalAmplitudeAt (context.selectedSimilarityProfile.index * radius / time) := ((context.the_constructed_sonic_surface_is_a_fluid_characteristic setup calculation).2 time radius point selected).2.2.1 end end PhysicsLogic.IndependentReuse.EvansAcousticClient end run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.EvansAcousticClient let mut traversal : Lean.CollectAxioms.State := {} let mut declarations := 0 let mut theorems := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue declarations := declarations + 1 if let .thmInfo _ := info then theorems := theorems + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theorems >= 9 do throwError "Missing independent acoustic-client theorems" for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational acoustic-client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Acoustic client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.EvansAcousticClient.the_actual_time_flux_jacobian_is_computed, ``PhysicsLogic.Papers.EvansColeman1994.AcousticCharacteristics.FluidState.time_jacobian_is_the_derivative_of_the_printed_fluxes), (``PhysicsLogic.IndependentReuse.EvansAcousticClient.the_actual_radial_flux_jacobian_is_computed, ``PhysicsLogic.Papers.EvansColeman1994.AcousticCharacteristics.FluidState.radial_jacobian_is_the_derivative_of_the_printed_fluxes), (``PhysicsLogic.IndependentReuse.EvansAcousticClient.both_computed_branches_are_characteristic, ``PhysicsLogic.Papers.EvansColeman1994.AcousticCharacteristics.FluidState.a_positive_radiation_fluid_has_exactly_the_two_acoustic_characteristics), (``PhysicsLogic.IndependentReuse.EvansAcousticClient.both_branches_have_nonzero_fluid_amplitudes, ``PhysicsLogic.Papers.EvansColeman1994.AcousticCharacteristics.FluidState.characteristic_iff_a_nonzero_fluid_principal_amplitude), (``PhysicsLogic.IndependentReuse.EvansAcousticClient.an_ingoing_fluid_frame_mode_can_move_outward_in_coordinates, ``PhysicsLogic.Papers.EvansColeman1994.AcousticCharacteristics.FluidState.acoustic_denominators_are_positive), (``PhysicsLogic.IndependentReuse.EvansAcousticClient.both_modes_move_outward_but_remain_inside_the_metric_light_cone, ``PhysicsLogic.Papers.EvansColeman1994.AcousticCharacteristics.FluidState.acoustic_characteristics_are_distinct_and_inside_the_metric_light_cone), (``PhysicsLogic.IndependentReuse.EvansAcousticClient.the_main_sonic_result_retains_a_nonzero_fluid_amplitude, ``PhysicsLogic.Papers.EvansColeman1994.RadiationFluidCollapseContext.the_constructed_sonic_surface_is_a_fluid_characteristic)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Evans-Coleman acoustic client audit passed: {declarations} declarations, {theorems} elaborated theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, moving-fluid flux matrices and nonzero causal amplitudes, no production diagnostic dependencies." set_option autoImplicit false namespace PhysicsLogic.IndependentReuse.BerryTwoLevelClient noncomputable section open PhysicsLogic.Papers.Berry1984.TwoLevelGeometry def offAxisPoint : Parameters := ⟨3, 4, 12⟩ def offAxisEigenvector : PhysicsLogic.Papers.Berry1984.TwoLevelGeometry.State := ![25, 3 + 4 * Complex.I] theorem the_independent_point_has_radius_thirteen : radius offAxisPoint = 13 := by norm_num [radius, offAxisPoint] theorem the_independent_point_is_nondegenerate : Regular offAxisPoint := by rw [Regular, the_independent_point_has_radius_thirteen] norm_num theorem the_independent_hamiltonian_is_hermitian : (hamiltonian offAxisPoint).IsHermitian := hamiltonian_is_hermitian offAxisPoint theorem the_client_vector_has_nonzero_squared_norm : overlap offAxisEigenvector offAxisEigenvector = 650 := by apply Complex.ext <;> norm_num [overlap, offAxisEigenvector, Complex.mul_re, Complex.mul_im] theorem the_actual_matrix_selects_the_client_vector : (hamiltonian offAxisPoint).mulVec offAxisEigenvector = (13 / 2 : Complex) • offAxisEigenvector := by ext i fin_cases i <;> apply Complex.ext <;> simp [hamiltonian, offAxisPoint, offAxisEigenvector, Matrix.mulVec, dotProduct, Fin.sum_univ_two, Complex.mul_re, Complex.mul_im] <;> norm_num theorem both_computed_projectors_are_nonzero : upperProjector offAxisPoint ≠ 0 /\ lowerProjector offAxisPoint ≠ 0 := ⟨upper_projector_nonzero _, lower_projector_nonzero _⟩ theorem the_client_projectors_select_distinct_eigenvalues : hamiltonian offAxisPoint * upperProjector offAxisPoint = (13 / 2 : Complex) • upperProjector offAxisPoint /\ hamiltonian offAxisPoint * lowerProjector offAxisPoint = (-13 / 2 : Complex) • lowerProjector offAxisPoint := by constructor · simpa [the_independent_point_has_radius_thirteen] using upper_projector_selects_eigenvalue offAxisPoint the_independent_point_is_nondegenerate · simpa [the_independent_point_has_radius_thirteen] using lower_projector_selects_eigenvalue offAxisPoint the_independent_point_is_nondegenerate theorem the_actual_y_derivative_retains_the_imaginary_pauli_entries : hamiltonianDerivative offAxisPoint 1 = !![0, -Complex.I / 2; Complex.I / 2, 0] := by rw [hamiltonian_derivative_is_half_pauli] rfl theorem the_off_axis_curvature_has_the_computed_radius_cubed_denominator : spectralCurvature offAxisPoint = ![3 / 4394, 4 / 4394, 12 / 4394] := by ext axis rw [the_actual_spectral_curvature_is_the_half_monopole offAxisPoint the_independent_point_is_nondegenerate axis] rw [the_independent_point_has_radius_thirteen] fin_cases axis <;> norm_num [coordinate, offAxisPoint] theorem the_lower_branch_reverses_the_actual_spectral_numerator : lowerSpectralNumerator offAxisPoint 0 1 = -spectralNumerator offAxisPoint 0 1 := lower_spectral_numerator_is_opposite offAxisPoint 0 1 theorem the_independent_point_has_actual_spectral_geometry : SpectralGeometry offAxisPoint := regular_parameters_realize_actual_spectral_geometry offAxisPoint the_independent_point_is_nondegenerate def operatorReadout : PhysicsLogic.Papers.Berry1984.TwoLevelDegeneracyFormulaReadout where xValue := 3 yValue := 4 zValue := 12 radiusValue := 13 hamiltonianEntryZeroZeroValue := 6 hamiltonianEntryZeroOneValue := (3 - Complex.I * 4) / 2 hamiltonianEntryOneZeroValue := (3 + Complex.I * 4) / 2 hamiltonianEntryOneOneValue := -6 upperEnergyValue := 13 / 2 lowerEnergyValue := -(13 / 2) upperCurvatureVectorCoefficientValue := upperCurvatureAtGap offAxisPoint 13 0 / 3 lowerCurvatureVectorCoefficientValue := lowerCurvatureAtGap offAxisPoint 13 0 / 3 solidAngleValue := 0 upperPhaseAngleValue := 0 lowerPhaseAngleValue := 0 encirclingRealPhaseFactorValue := -1 nonencirclingRealPhaseFactorValue := 1 theorem the_operator_readout_has_the_canonical_hamiltonian : operatorReadout.CanonicalHamiltonianAt := by refine ⟨the_independent_point_is_nondegenerate, the_independent_point_has_radius_thirteen.symm, ?_⟩ simp [PhysicsLogic.Papers.Berry1984.TwoLevelDegeneracyFormulaReadout.EquationTwelveAt, operatorReadout] norm_num theorem the_operator_readout_has_the_actual_branch_eigenvalue_equations : operatorReadout.EigenvalueReadoutAt := by rw [PhysicsLogic.Papers.Berry1984.TwoLevelDegeneracyFormulaReadout.EigenvalueReadoutAt, PhysicsLogic.Papers.Berry1984.TwoLevelDegeneracyFormulaReadout.canonical_entries_identify_the_actual_hamiltonian _ the_operator_readout_has_the_canonical_hamiltonian.2.2] simpa [operatorReadout, PhysicsLogic.Papers.Berry1984.TwoLevelDegeneracyFormulaReadout.parameters, offAxisPoint, neg_div] using the_client_projectors_select_distinct_eigenvalues theorem the_operator_readout_matches_the_actual_spectral_traces : operatorReadout.SpectralCurvatureReadoutAt := by have upper : upperCurvatureAtGap offAxisPoint 13 = ![3 / 4394, 4 / 4394, 12 / 4394] := by rw [← the_independent_point_has_radius_thirteen, upper_curvature_at_radius_is_spectral_curvature, the_off_axis_curvature_has_the_computed_radius_cubed_denominator] have lower := lower_curvature_at_gap_is_opposite offAxisPoint 13 rw [upper] at lower change (∀ axis, (upperCurvatureAtGap offAxisPoint 13 0 / 3) * coordinate offAxisPoint axis = upperCurvatureAtGap offAxisPoint ((13 / 2) - (-(13 / 2))) axis) /\ (∀ axis, (lowerCurvatureAtGap offAxisPoint 13 0 / 3) * coordinate offAxisPoint axis = lowerCurvatureAtGap offAxisPoint ((13 / 2) - (-(13 / 2))) axis) have gap : (13 / 2 : Real) - (-(13 / 2)) = 13 := by norm_num rw [gap, upper, lower] constructor <;> intro axis <;> fin_cases axis <;> norm_num [coordinate, offAxisPoint] theorem the_independent_readout_realizes_the_operator_inputs : operatorReadout.OperatorCalculationAt := by refine ⟨the_operator_readout_has_the_canonical_hamiltonian, the_operator_readout_has_the_actual_branch_eigenvalue_equations, the_operator_readout_matches_the_actual_spectral_traces, ?_, ?_⟩ · simp [PhysicsLogic.Papers.Berry1984.TwoLevelDegeneracyFormulaReadout.EquationEighteenAt, operatorReadout] · norm_num [PhysicsLogic.Papers.Berry1984.TwoLevelDegeneracyFormulaReadout.EquationNineteenAt, operatorReadout] theorem the_operator_inputs_derive_the_printed_formulas : operatorReadout.SourceFormulaAt := operatorReadout.operator_calculation_derives_the_printed_formula_package the_independent_readout_realizes_the_operator_inputs theorem the_recovered_branch_energy_gap_is_thirteen : operatorReadout.upperEnergyValue - operatorReadout.lowerEnergyValue = 13 := operatorReadout.equation_thirteen_energy_gap_equals_radius the_operator_inputs_derive_the_printed_formulas.2.2.1 theorem the_spectral_trace_readouts_recover_both_monopole_coefficients : operatorReadout.upperCurvatureVectorCoefficientValue = 1 / 4394 /\ operatorReadout.lowerCurvatureVectorCoefficientValue = -(1 / 4394) := by have coefficients := operatorReadout.actual_spectral_curvature_readouts_derive_equation_seventeen the_operator_readout_has_the_canonical_hamiltonian the_operator_readout_has_the_actual_branch_eigenvalue_equations the_operator_readout_matches_the_actual_spectral_traces change operatorReadout.upperCurvatureVectorCoefficientValue = 1 / (2 * (13 : Real) ^ 3) /\ operatorReadout.lowerCurvatureVectorCoefficientValue = -(1 / (2 * (13 : Real) ^ 3)) at coefficients norm_num at coefficients exact coefficients end end PhysicsLogic.IndependentReuse.BerryTwoLevelClient run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.BerryTwoLevelClient let mut traversal : Lean.CollectAxioms.State := {} let mut declarations := 0 let mut theorems := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue declarations := declarations + 1 if let .thmInfo _ := info then theorems := theorems + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theorems >= 18 do throwError "Missing independent Berry two-level client theorems" for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational Berry two-level client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Berry two-level client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.BerryTwoLevelClient.the_independent_hamiltonian_is_hermitian, ``PhysicsLogic.Papers.Berry1984.TwoLevelGeometry.hamiltonian_is_hermitian), (``PhysicsLogic.IndependentReuse.BerryTwoLevelClient.both_computed_projectors_are_nonzero, ``PhysicsLogic.Papers.Berry1984.TwoLevelGeometry.upper_projector_nonzero), (``PhysicsLogic.IndependentReuse.BerryTwoLevelClient.the_client_projectors_select_distinct_eigenvalues, ``PhysicsLogic.Papers.Berry1984.TwoLevelGeometry.upper_projector_selects_eigenvalue), (``PhysicsLogic.IndependentReuse.BerryTwoLevelClient.the_client_projectors_select_distinct_eigenvalues, ``PhysicsLogic.Papers.Berry1984.TwoLevelGeometry.lower_projector_selects_eigenvalue), (``PhysicsLogic.IndependentReuse.BerryTwoLevelClient.the_actual_y_derivative_retains_the_imaginary_pauli_entries, ``PhysicsLogic.Papers.Berry1984.TwoLevelGeometry.hamiltonian_derivative_is_half_pauli), (``PhysicsLogic.IndependentReuse.BerryTwoLevelClient.the_off_axis_curvature_has_the_computed_radius_cubed_denominator, ``PhysicsLogic.Papers.Berry1984.TwoLevelGeometry.the_actual_spectral_curvature_is_the_half_monopole), (``PhysicsLogic.IndependentReuse.BerryTwoLevelClient.the_lower_branch_reverses_the_actual_spectral_numerator, ``PhysicsLogic.Papers.Berry1984.TwoLevelGeometry.lower_spectral_numerator_is_opposite), (``PhysicsLogic.IndependentReuse.BerryTwoLevelClient.the_independent_point_has_actual_spectral_geometry, ``PhysicsLogic.Papers.Berry1984.TwoLevelGeometry.regular_parameters_realize_actual_spectral_geometry), (``PhysicsLogic.IndependentReuse.BerryTwoLevelClient.the_operator_inputs_derive_the_printed_formulas, ``PhysicsLogic.Papers.Berry1984.TwoLevelDegeneracyFormulaReadout.operator_calculation_derives_the_printed_formula_package), (``PhysicsLogic.IndependentReuse.BerryTwoLevelClient.the_recovered_branch_energy_gap_is_thirteen, ``PhysicsLogic.Papers.Berry1984.TwoLevelDegeneracyFormulaReadout.equation_thirteen_energy_gap_equals_radius), (``PhysicsLogic.IndependentReuse.BerryTwoLevelClient.the_spectral_trace_readouts_recover_both_monopole_coefficients, ``PhysicsLogic.Papers.Berry1984.TwoLevelDegeneracyFormulaReadout.actual_spectral_curvature_readouts_derive_equation_seventeen)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Berry two-level client audit passed: {declarations} declarations, {theorems} elaborated theorems, {uses.size} public API uses, {traversal.axioms.size} foundational axioms, off-axis Hamiltonian and eigenvector with computed spectral curvature, no production diagnostic dependencies." namespace PhysicsLogic.IndependentReuse.LeeYangFiniteChainClient open PhysicsLogic.Papers.LeeYang1952.OneDimensionalRoots set_option backward.isDefEq.respectTransparency false noncomputable def root : ℂ := ⟨1 / 3, 2 * Real.sqrt 2 / 3⟩ noncomputable def partition (z : ℂ) : ℂ := z ^ 3 + z ^ 2 / 3 + z / 3 + 1 theorem the_three_spin_root_obeys_equation_forty_three : FiniteRootFormula (1 / 3) 3 root := by refine ⟨by norm_num, 1, by norm_num, by norm_num, ?_⟩ norm_num [root, Real.cos_pi_div_three] theorem the_independent_root_has_unit_modulus : OnUnitCircle root := by have square := Real.sq_sqrt (show (0 : ℝ) ≤ 2 by norm_num) dsimp [OnUnitCircle, root] nlinarith theorem the_actual_three_spin_partition_vanishes : partition root = 0 := by have square := Real.sq_sqrt (show (0 : ℝ) ≤ 2 by norm_num) have quadratic : root ^ 2 - (2 / 3 : ℂ) * root + 1 = 0 := by apply Complex.ext <;> simp [root, pow_two, Complex.mul_re, Complex.mul_im] <;> nlinarith calc partition root = (root + 1) * (root ^ 2 - (2 / 3 : ℂ) * root + 1) := by dsimp [partition] ring _ = 0 := by rw [quadratic, mul_zero] theorem the_library_bounds_the_roots_real_part : root.re ≤ 7 / 9 := by have bound := finite_root_formula_bounds_real_part (1 / 3) 3 root (by norm_num) (by norm_num) the_three_spin_root_obeys_equation_forty_three norm_num at bound ⊢ exact bound theorem the_library_gives_a_volume_uniform_gap_at_one : 4 / 9 ≤ Complex.normSq (root - 1) := by have bound := root_gap_at_positive_real_activity (1 / 3) 1 3 root (by norm_num) (by norm_num) (by norm_num) the_three_spin_root_obeys_equation_forty_three the_independent_root_has_unit_modulus norm_num at bound ⊢ exact bound theorem the_library_also_excludes_the_positive_activity_two : 8 / 9 ≤ Complex.normSq (root - 2) := by have bound := root_gap_at_positive_real_activity (1 / 3) 2 3 root (by norm_num) (by norm_num) (by norm_num) the_three_spin_root_obeys_equation_forty_three the_independent_root_has_unit_modulus norm_num at bound ⊢ exact bound theorem the_arc_edge_gap_is_not_the_selected_finite_root_gap : Complex.normSq (upperEndpoint (1 / 3) - 1) = 4 / 9 ∧ Complex.normSq (root - 1) = 4 / 3 := by constructor · have exactGap := (endpoints_have_unit_modulus_and_exact_gap (1 / 3) (by norm_num) (by norm_num)).2.2.1 norm_num at exactGap ⊢ exact exactGap · have square := Real.sq_sqrt (show (0 : ℝ) ≤ 2 by norm_num) dsimp [Complex.normSq, root] nlinarith theorem the_client_factor_has_a_fixed_interaction_thermal_realization : ∃ parameters : ThermalParameters, parameters.FiniteTemperatureFerromagnet ∧ parameters.interactionEnergy = 1 ∧ parameters.boltzmannFactor = 1 / 3 := every_positive_factor_below_one_has_thermal_realization (1 / 3) (by norm_num) (by norm_num) end PhysicsLogic.IndependentReuse.LeeYangFiniteChainClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.LeeYangFiniteChainClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theoremCount >= 8 do throwError "Missing independent Lee-Yang finite-chain theorems" for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational Lee-Yang finite-chain client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Lee-Yang finite-chain client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.LeeYangFiniteChainClient.the_library_bounds_the_roots_real_part, ``PhysicsLogic.Papers.LeeYang1952.OneDimensionalRoots.finite_root_formula_bounds_real_part), (``PhysicsLogic.IndependentReuse.LeeYangFiniteChainClient.the_library_gives_a_volume_uniform_gap_at_one, ``PhysicsLogic.Papers.LeeYang1952.OneDimensionalRoots.root_gap_at_positive_real_activity), (``PhysicsLogic.IndependentReuse.LeeYangFiniteChainClient.the_library_also_excludes_the_positive_activity_two, ``PhysicsLogic.Papers.LeeYang1952.OneDimensionalRoots.root_gap_at_positive_real_activity), (``PhysicsLogic.IndependentReuse.LeeYangFiniteChainClient.the_arc_edge_gap_is_not_the_selected_finite_root_gap, ``PhysicsLogic.Papers.LeeYang1952.OneDimensionalRoots.endpoints_have_unit_modulus_and_exact_gap), (``PhysicsLogic.IndependentReuse.LeeYangFiniteChainClient.the_client_factor_has_a_fixed_interaction_thermal_realization, ``PhysicsLogic.Papers.LeeYang1952.OneDimensionalRoots.every_positive_factor_below_one_has_thermal_realization)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Lee-Yang finite-chain client audit passed: {theoremCount} theorems, {uses.size} public API uses, actual three-spin polynomial root and fixed-interaction thermal realization, no production diagnostics." namespace PhysicsLogic.IndependentReuse.LeeYangFugacityClient open PhysicsLogic.Papers.LeeYang1952.ActivityCoordinates set_option backward.isDefEq.respectTransparency false noncomputable def parameters : Parameters := ⟨2, Real.log 2, 1⟩ noncomputable def normalizedRoot : ℂ := ⟨1 / 8, 3 * Real.sqrt 7 / 8⟩ noncomputable def normalizedPolynomial (z : ℂ) : ℂ := 1 + 3 * z / 4 + 3 * z ^ 2 / 4 + z ^ 3 noncomputable def gasPolynomial (y : ℂ) : ℂ := 1 + 3 * y + 12 * y ^ 2 + 64 * y ^ 3 theorem the_chain_has_physical_thermal_parameters : parameters.PhysicalDomain := by constructor <;> norm_num [parameters] theorem the_scale_is_one_quarter : parameters.fugacityScale = 1 / 4 := by have scale := chain_scale_is_the_square_of_the_root_parameter (Real.log 2) 1 norm_num [Real.exp_neg, Real.exp_log] at scale exact scale theorem the_zero_field_fugacity_is_one_quarter : parameters.gasFugacity 0 = 1 / 4 := by rw [zero_field_gas_fugacity_is_the_scale, the_scale_is_one_quarter] theorem the_normalized_polynomial_is_the_rescaled_gas_polynomial (z : ℂ) : gasPolynomial (parameters.toFugacity z) = normalizedPolynomial z := by simp only [Parameters.toFugacity, the_scale_is_one_quarter] push_cast dsimp [gasPolynomial, normalizedPolynomial] ring theorem the_selected_point_is_on_the_normalized_circle : Complex.normSq normalizedRoot = 1 := by have square := Real.sq_sqrt (show (0 : ℝ) ≤ 7 by norm_num) dsimp [normalizedRoot, Complex.normSq] nlinarith theorem the_selected_point_is_an_actual_normalized_partition_zero : normalizedPolynomial normalizedRoot = 0 := by have square := Real.sq_sqrt (show (0 : ℝ) ≤ 7 by norm_num) have quadratic : normalizedRoot ^ 2 - normalizedRoot / 4 + 1 = 0 := by apply Complex.ext <;> simp [normalizedRoot, pow_two, Complex.mul_re, Complex.mul_im] <;> nlinarith calc normalizedPolynomial normalizedRoot = (normalizedRoot + 1) * (normalizedRoot ^ 2 - normalizedRoot / 4 + 1) := by dsimp [normalizedPolynomial] ring _ = 0 := by rw [quadratic, mul_zero] theorem the_actual_gas_partition_zero_has_squared_modulus_one_sixteenth : gasPolynomial (parameters.toFugacity normalizedRoot) = 0 ∧ Complex.normSq (parameters.toFugacity normalizedRoot) = 1 / 16 := by constructor · rw [the_normalized_polynomial_is_the_rescaled_gas_polynomial] exact the_selected_point_is_an_actual_normalized_partition_zero · rw [unit_circle_becomes_the_fugacity_circle parameters normalizedRoot the_selected_point_is_on_the_normalized_circle, the_scale_is_one_quarter] norm_num theorem changing_coordinate_does_not_merge_distinct_states : parameters.toFugacity normalizedRoot ≠ parameters.toFugacity 1 := by intro equal have collapsed := complex_rescaling_is_injective parameters equal have realParts := congrArg Complex.re collapsed norm_num [normalizedRoot] at realParts theorem the_same_bond_weight_on_a_cubic_lattice_has_a_different_scale : (Parameters.mk 6 (Real.log 2) 1).fugacityScale = 1 / 64 := by have scale := cubic_scale_is_the_sixth_power_of_the_root_parameter (Real.log 2) 1 norm_num [Real.exp_neg, Real.exp_log] at scale exact scale end PhysicsLogic.IndependentReuse.LeeYangFugacityClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.LeeYangFugacityClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theoremCount >= 9 do throwError "Missing independent Lee-Yang fugacity theorems" for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational Lee-Yang fugacity client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Lee-Yang fugacity client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.LeeYangFugacityClient.the_scale_is_one_quarter, ``PhysicsLogic.Papers.LeeYang1952.ActivityCoordinates.chain_scale_is_the_square_of_the_root_parameter), (``PhysicsLogic.IndependentReuse.LeeYangFugacityClient.the_zero_field_fugacity_is_one_quarter, ``PhysicsLogic.Papers.LeeYang1952.ActivityCoordinates.zero_field_gas_fugacity_is_the_scale), (``PhysicsLogic.IndependentReuse.LeeYangFugacityClient.the_actual_gas_partition_zero_has_squared_modulus_one_sixteenth, ``PhysicsLogic.Papers.LeeYang1952.ActivityCoordinates.unit_circle_becomes_the_fugacity_circle), (``PhysicsLogic.IndependentReuse.LeeYangFugacityClient.changing_coordinate_does_not_merge_distinct_states, ``PhysicsLogic.Papers.LeeYang1952.ActivityCoordinates.complex_rescaling_is_injective), (``PhysicsLogic.IndependentReuse.LeeYangFugacityClient.the_same_bond_weight_on_a_cubic_lattice_has_a_different_scale, ``PhysicsLogic.Papers.LeeYang1952.ActivityCoordinates.cubic_scale_is_the_sixth_power_of_the_root_parameter)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Lee-Yang fugacity client audit passed: {theoremCount} theorems, {uses.size} public API uses, actual raw and normalized partition polynomials, no production diagnostics." namespace PhysicsLogic.IndependentReuse.LeeYangSpinGasClient open PhysicsLogic.Papers.LeeYang1952.SpinGasDictionary set_option backward.isDefEq.respectTransparency false noncomputable def ringModel : FiniteModel (Fin 4) where bonds := {(0, 1), (1, 2), (2, 3), (3, 0)} bondEnergy := fun _ => Real.log 2 field := Real.log 2 thermalEnergy := 1 def alternating : Configuration (Fin 4) := fun site => decide (site.val % 2 = 0) theorem the_four_site_ring_is_physical : ringModel.PhysicalDomain := by norm_num [FiniteModel.PhysicalDomain, ringModel] decide +kernel theorem the_alternating_state_has_two_gas_particles : FiniteModel.particleNumber (spinToGas alternating) = 2 := by change (∑ site : Fin 4, occupation (spinToGas alternating) site) = 2 rw [Fin.sum_univ_succ, Fin.sum_univ_succ, Fin.sum_univ_succ, Fin.sum_univ_succ] norm_num [spinToGas, occupation, alternating] theorem the_alternating_spin_energy_is_four_log_two : ringModel.spinEnergy alternating = 4 * Real.log 2 := by have sum : FiniteModel.spinSum alternating = 0 := by change (∑ site : Fin 4, spinValue alternating site) = 0 rw [Fin.sum_univ_succ, Fin.sum_univ_succ, Fin.sum_univ_succ, Fin.sum_univ_succ] norm_num [spinValue, alternating] simp only [FiniteModel.spinEnergy, sum, mul_zero, zero_add, ringModel] simp (disch := decide) only [Finset.sum_insert, Finset.sum_singleton] norm_num [alternating] ring theorem the_library_gives_the_grand_energy_shift : ringModel.grandEnergy (spinToGas alternating) = 8 * Real.log 2 := by rw [ringModel.grand_energy_is_shifted_spin_energy, the_alternating_spin_energy_is_four_log_two] norm_num [ringModel] ring theorem the_configuration_weight_ratio_is_one_sixteenth : ringModel.gasWeight (spinToGas alternating) = (1 / 16) * ringModel.spinWeight alternating := by rw [ringModel.configuration_weights_correspond] have factor : ringModel.weightFactor = 1 / 16 := by norm_num only [FiniteModel.weightFactor, ringModel, Fintype.card_fin, div_one] rw [neg_mul, Real.exp_neg] rw [show (4 : ℝ) * Real.log 2 = (4 : ℕ) * Real.log 2 by norm_num, Real.exp_nat_mul] norm_num [Real.exp_log] rw [factor] theorem the_total_partition_ratio_has_the_same_normalization : ringModel.gasPartition = ringModel.weightFactor * ringModel.spinPartition := ringModel.partition_sums_correspond theorem the_pressure_contains_the_field_offset : ringModel.pressure = -ringModel.freeEnergyPerSite - Real.log 2 := ringModel.pressure_is_minus_free_energy_minus_field the_four_site_ring_is_physical theorem the_average_density_has_the_magnetization_offset : ringModel.density = (1 - ringModel.magnetization) / 2 := ringModel.density_is_one_minus_magnetization_over_two the_four_site_ring_is_physical theorem a_two_point_occupancy_observable_transports : ringModel.gasExpectation (fun gas => occupation gas 0 * occupation gas 2) = ringModel.spinExpectation (fun spin => occupation (spinToGas spin) 0 * occupation (spinToGas spin) 2) := ringModel.every_observable_expectation_transports _ theorem the_regular_ring_recovers_a_single_fugacity (site : Fin 4) : ringModel.localFugacity site = (PhysicsLogic.Papers.LeeYang1952.ActivityCoordinates.Parameters.mk 2 (Real.log 2) 1).gasFugacity (Real.log 2) := by apply ringModel.regular_bond_dictionary_recovers_activity_coordinates _ rfl intro candidate simp only [FiniteModel.incidentEnergy, ringModel] simp (disch := decide) only [Finset.sum_insert, Finset.sum_singleton] fin_cases candidate <;> simp only [Fin.ext_iff] <;> norm_num <;> ring end PhysicsLogic.IndependentReuse.LeeYangSpinGasClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.LeeYangSpinGasClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theoremCount >= 10 do throwError "Missing independent Lee-Yang spin/gas theorems" for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational Lee-Yang spin/gas client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Lee-Yang spin/gas client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.LeeYangSpinGasClient.the_library_gives_the_grand_energy_shift, ``PhysicsLogic.Papers.LeeYang1952.SpinGasDictionary.FiniteModel.grand_energy_is_shifted_spin_energy), (``PhysicsLogic.IndependentReuse.LeeYangSpinGasClient.the_configuration_weight_ratio_is_one_sixteenth, ``PhysicsLogic.Papers.LeeYang1952.SpinGasDictionary.FiniteModel.configuration_weights_correspond), (``PhysicsLogic.IndependentReuse.LeeYangSpinGasClient.the_total_partition_ratio_has_the_same_normalization, ``PhysicsLogic.Papers.LeeYang1952.SpinGasDictionary.FiniteModel.partition_sums_correspond), (``PhysicsLogic.IndependentReuse.LeeYangSpinGasClient.the_pressure_contains_the_field_offset, ``PhysicsLogic.Papers.LeeYang1952.SpinGasDictionary.FiniteModel.pressure_is_minus_free_energy_minus_field), (``PhysicsLogic.IndependentReuse.LeeYangSpinGasClient.the_average_density_has_the_magnetization_offset, ``PhysicsLogic.Papers.LeeYang1952.SpinGasDictionary.FiniteModel.density_is_one_minus_magnetization_over_two), (``PhysicsLogic.IndependentReuse.LeeYangSpinGasClient.a_two_point_occupancy_observable_transports, ``PhysicsLogic.Papers.LeeYang1952.SpinGasDictionary.FiniteModel.every_observable_expectation_transports), (``PhysicsLogic.IndependentReuse.LeeYangSpinGasClient.the_regular_ring_recovers_a_single_fugacity, ``PhysicsLogic.Papers.LeeYang1952.SpinGasDictionary.FiniteModel.regular_bond_dictionary_recovers_activity_coordinates)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Lee-Yang microscopic dictionary client audit passed: {theoremCount} theorems, {uses.size} public API uses, actual four-site spin states, energies, correlations and normalization, no production diagnostics." namespace PhysicsLogic.IndependentReuse.LeeYangMicroscopicPolynomialClient open PhysicsLogic.Papers.LeeYang1952.SpinGasDictionary open PhysicsLogic.Papers.LeeYang1952.SpinGasPartition set_option backward.isDefEq.respectTransparency false noncomputable def repulsiveGas : FiniteModel (Fin 2) where bonds := {(0, 1)} bondEnergy := fun _ => -Real.log 2 field := 0 thermalEnergy := 1 private theorem configuration_universe : (Finset.univ : Finset (Configuration (Fin 2))) = {![false, false], ![false, true], ![true, false], ![true, true]} := by decide +kernel private theorem configuration_counts : particleCount (![false, false] : Configuration (Fin 2)) = 0 ∧ particleCount (![false, true] : Configuration (Fin 2)) = 1 ∧ particleCount (![true, false] : Configuration (Fin 2)) = 1 ∧ particleCount (![true, true] : Configuration (Fin 2)) = 2 := by decide +kernel theorem the_repulsive_gas_has_a_physical_finite_dictionary : repulsiveGas.PhysicalDomain := by norm_num [FiniteModel.PhysicalDomain, repulsiveGas] theorem its_spin_coupling_is_strictly_antiferromagnetic : repulsiveGas.bondEnergy (0, 1) < 0 := by exact neg_neg_of_pos (Real.log_pos (by norm_num)) theorem actual_zero_field_sectors_give_the_normalized_polynomial (z : ℂ) : normalizedPolynomial repulsiveGas z = 1 + 4 * z + z ^ 2 := by have cancel : Real.log 2 + Real.log 2 - 2 * Real.log 2 = 0 := by ring simp only [normalizedPolynomial, configuration_universe] simp (disch := decide) only [Finset.sum_insert, Finset.sum_singleton] simp only [configuration_counts.1, configuration_counts.2.1, configuration_counts.2.2.1, configuration_counts.2.2.2] norm_num [zeroFieldWeight, zeroField, repulsiveGas, FiniteModel.gasWeight, FiniteModel.grandEnergy, FiniteModel.gasInteractionEnergy, FiniteModel.occupiedBondEnergy, FiniteModel.chemicalEnergy, FiniteModel.incidentEnergy, occupation, Fin.sum_univ_two, cancel, Real.exp_log] ring theorem actual_pair_energies_give_the_raw_fugacity_polynomial (y : ℂ) : rawFugacityPolynomial repulsiveGas y = 1 + 2 * y + y ^ 2 / 4 := by have weight : Real.exp (-(2 * Real.log 2)) = 1 / 4 := by rw [Real.exp_neg, show (2 : ℝ) * Real.log 2 = Real.log 2 + Real.log 2 by ring, Real.exp_add] norm_num [Real.exp_log] simp only [rawFugacityPolynomial, configuration_universe] simp (disch := decide) only [Finset.sum_insert, Finset.sum_singleton] simp only [configuration_counts.1, configuration_counts.2.1, configuration_counts.2.2.1, configuration_counts.2.2.2] norm_num [pairWeight, repulsiveGas, FiniteModel.gasInteractionEnergy, FiniteModel.occupiedBondEnergy, occupation, weight] ring theorem the_library_rescales_every_complex_activity (z : ℂ) : normalizedPolynomial repulsiveGas z = rawFugacityPolynomial repulsiveGas (2 * z) := by have incident : repulsiveGas.ConstantIncidentEnergy (-Real.log 2) := by intro site fin_cases site <;> norm_num [FiniteModel.incidentEnergy, repulsiveGas] have identity := homogeneous_dictionary_rescales_the_full_complex_polynomial repulsiveGas (-Real.log 2) incident z simpa [repulsiveGas, Real.exp_log] using identity theorem every_allowed_equation_four_coefficient_is_positive (number : ℕ) (allowed : number ≤ 2) : 0 < labelledConfigurationalIntegral repulsiveGas number / (number.factorial : ℝ) := microscopic_equation_four_coefficients_are_strictly_positive repulsiveGas number allowed noncomputable def offCircleRoot : ℂ := ((-2 + Real.sqrt 3 : ℝ) : ℂ) theorem positive_coefficients_do_not_put_an_antiferromagnetic_root_on_the_circle : normalizedPolynomial repulsiveGas offCircleRoot = 0 ∧ Complex.normSq offCircleRoot < 1 := by have square := Real.sq_sqrt (show (0 : ℝ) ≤ 3 by norm_num) have nonnegative := Real.sqrt_nonneg (3 : ℝ) have lower : (1 : ℝ) < Real.sqrt 3 := by nlinarith have upper : Real.sqrt 3 < (2 : ℝ) := by nlinarith constructor · rw [actual_zero_field_sectors_give_the_normalized_polynomial] apply Complex.ext <;> simp [offCircleRoot, Complex.mul_re, Complex.mul_im, pow_two] <;> nlinarith · simp only [offCircleRoot, Complex.normSq_ofReal] nlinarith theorem the_same_root_is_a_zero_of_the_correctly_rescaled_gas_polynomial : rawFugacityPolynomial repulsiveGas (2 * offCircleRoot) = 0 := by rw [← the_library_rescales_every_complex_activity] exact positive_coefficients_do_not_put_an_antiferromagnetic_root_on_the_circle.1 theorem the_physical_nonzero_field_evaluates_the_same_microscopic_ensemble : ({repulsiveGas with field := Real.log 2} : FiniteModel (Fin 2)).gasPartition = 33 / 16 := by have activity : Real.exp (-2 * Real.log 2) = 1 / 4 := by rw [neg_mul, Real.exp_neg, show (2 : ℝ) * Real.log 2 = Real.log 2 + Real.log 2 by ring, Real.exp_add] norm_num [Real.exp_log] have value := normalized_polynomial_evaluates_the_actual_gas_partition ({repulsiveGas with field := Real.log 2} : FiniteModel (Fin 2)) rw [normalized_polynomial_is_independent_of_field, actual_zero_field_sectors_give_the_normalized_polynomial] at value simp only [repulsiveGas, div_one] at value rw [activity] at value norm_num at value apply Complex.ofReal_injective push_cast exact value.symm theorem the_labelled_integrals_reconstruct_the_raw_polynomial (y : ℂ) : rawFugacityPolynomial repulsiveGas y = ∑ number ∈ Finset.range 3, ((labelledConfigurationalIntegral repulsiveGas number : ℂ) / (number.factorial : ℂ)) * y ^ number := microscopic_raw_polynomial_has_equation_four_coefficients repulsiveGas y end PhysicsLogic.IndependentReuse.LeeYangMicroscopicPolynomialClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.LeeYangMicroscopicPolynomialClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theoremCount >= 10 do throwError "Missing independent microscopic-polynomial theorems" for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational microscopic-polynomial client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Microscopic-polynomial client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.LeeYangMicroscopicPolynomialClient.the_library_rescales_every_complex_activity, ``PhysicsLogic.Papers.LeeYang1952.SpinGasPartition.homogeneous_dictionary_rescales_the_full_complex_polynomial), (``PhysicsLogic.IndependentReuse.LeeYangMicroscopicPolynomialClient.every_allowed_equation_four_coefficient_is_positive, ``PhysicsLogic.Papers.LeeYang1952.SpinGasPartition.microscopic_equation_four_coefficients_are_strictly_positive), (``PhysicsLogic.IndependentReuse.LeeYangMicroscopicPolynomialClient.the_same_root_is_a_zero_of_the_correctly_rescaled_gas_polynomial, ``PhysicsLogic.Papers.LeeYang1952.SpinGasPartition.homogeneous_dictionary_rescales_the_full_complex_polynomial), (``PhysicsLogic.IndependentReuse.LeeYangMicroscopicPolynomialClient.the_physical_nonzero_field_evaluates_the_same_microscopic_ensemble, ``PhysicsLogic.Papers.LeeYang1952.SpinGasPartition.normalized_polynomial_evaluates_the_actual_gas_partition), (``PhysicsLogic.IndependentReuse.LeeYangMicroscopicPolynomialClient.the_physical_nonzero_field_evaluates_the_same_microscopic_ensemble, ``PhysicsLogic.Papers.LeeYang1952.SpinGasPartition.normalized_polynomial_is_independent_of_field), (``PhysicsLogic.IndependentReuse.LeeYangMicroscopicPolynomialClient.the_labelled_integrals_reconstruct_the_raw_polynomial, ``PhysicsLogic.Papers.LeeYang1952.SpinGasPartition.microscopic_raw_polynomial_has_equation_four_coefficients)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Lee-Yang microscopic polynomial client audit passed: {theoremCount} theorems, {uses.size} public API uses, independently reconstructed repulsive gas, actual off-circle root and finite-field partition, no production diagnostics." namespace PhysicsLogic.IndependentReuse.LeeYangCircleDomainClient open PhysicsLogic.Papers.LeeYang1952.SpinGasDictionary open PhysicsLogic.Papers.LeeYang1952.SpinGasPartition open PhysicsLogic.Papers.LeeYang1952.FerromagneticCircle set_option backward.isDefEq.respectTransparency false noncomputable def magnet : FiniteModel (Fin 2) where bonds := {(0, 1)} bondEnergy := fun _ => Real.log 3 field := 0 thermalEnergy := 1 theorem the_interacting_two_site_model_is_physical : magnet.PhysicalDomain := by norm_num [FiniteModel.PhysicalDomain, magnet] theorem its_actual_bond_is_ferromagnetic : FerromagneticCouplings magnet := by intro bond member exact le_of_lt (Real.log_pos (by norm_num)) theorem its_gas_pair_energy_is_attractive : ∀ bond ∈ magnet.bonds, gasPairEnergy magnet bond ≤ 0 := (ferromagnetic_couplings_are_exactly_nonpositive_gas_pair_energies magnet).mp its_actual_bond_is_ferromagnetic theorem the_actual_mixed_spin_factor_is_one_third : mixedSpinFactor magnet (0, 1) = 1 / 3 := by norm_num [mixedSpinFactor, magnet, Real.exp_neg, Real.exp_log] theorem positive_temperature_and_bond_signs_supply_the_appendix_domain : ContractiveBondFactors magnet := (ferromagnetism_is_equivalent_to_the_physical_appendix_weight_bound magnet the_interacting_two_site_model_is_physical).mp its_actual_bond_is_ferromagnetic private theorem configuration_universe : (Finset.univ : Finset (Configuration (Fin 2))) = {![false, false], ![false, true], ![true, false], ![true, true]} := by decide +kernel private theorem configuration_counts : particleCount (![false, false] : Configuration (Fin 2)) = 0 ∧ particleCount (![false, true] : Configuration (Fin 2)) = 1 ∧ particleCount (![true, false] : Configuration (Fin 2)) = 1 ∧ particleCount (![true, true] : Configuration (Fin 2)) = 2 := by decide +kernel theorem the_cut_polynomial_is_the_source_two_spin_base_case (factor : Fin 2 × Fin 2 → ℝ) (z : ℂ) : cutPolynomial magnet.bonds factor z = 1 + 2 * (factor (0, 1) : ℂ) * z + z ^ 2 := by simp only [cutPolynomial, configuration_universe] simp (disch := decide) only [Finset.sum_insert, Finset.sum_singleton] simp only [configuration_counts.1, configuration_counts.2.1, configuration_counts.2.2.1, configuration_counts.2.2.2] norm_num [cutWeight, magnet] ring theorem the_appendix_calculation_is_inhabited_by_an_actual_two_spin_proof : AppendixTwoCircleCalculation magnet := by intro physical factor bounded have bound := bounded (0, 1) (by simp [magnet]) obtain ⟨lower, upper⟩ := abs_le.mp bound have product : 0 ≤ (1 - factor (0, 1)) * (1 + factor (0, 1)) := mul_nonneg (by linarith) (by linarith) have squareBound : (factor (0, 1)) ^ 2 ≤ 1 := by nlinarith have square := Real.sq_sqrt (show 0 ≤ 1 - (factor (0, 1)) ^ 2 by linarith) simp only [pow_two] at square constructor · refine ⟨⟨-factor (0, 1), Real.sqrt (1 - (factor (0, 1)) ^ 2)⟩, ?_⟩ rw [the_cut_polynomial_is_the_source_two_spin_base_case] apply Complex.ext <;> simp [Complex.mul_re, Complex.mul_im, pow_two] <;> nlinarith · intro z zero rw [the_cut_polynomial_is_the_source_two_spin_base_case] at zero have realZero := congrArg Complex.re zero have imagZero := congrArg Complex.im zero simp [Complex.mul_re, Complex.mul_im, pow_two] at realZero imagZero by_cases imaginary : z.im = 0 · have sumSquares : (z.re + factor (0, 1)) ^ 2 + (1 - (factor (0, 1)) ^ 2) = 0 := by rw [imaginary] at realZero nlinarith have squareZero : (z.re + factor (0, 1)) ^ 2 = 0 := by nlinarith [sq_nonneg (z.re + factor (0, 1))] have linear := eq_zero_of_pow_eq_zero squareZero simp only [Complex.normSq_apply, imaginary] nlinarith · have productZero : z.im * (2 * factor (0, 1) + 2 * z.re) = 0 := by nlinarith have linear := (mul_eq_zero.mp productZero).resolve_left imaginary have realPart : z.re = -factor (0, 1) := by linarith simp only [Complex.normSq_apply] nlinarith [realZero, realPart] theorem the_library_derives_actual_microscopic_roots_on_the_circle : (∃ z : ℂ, normalizedPolynomial magnet z = 0) ∧ ∀ z : ℂ, normalizedPolynomial magnet z = 0 → Complex.normSq z = 1 := physical_ferromagnet_realizes_the_microscopic_circle_theorem magnet the_interacting_two_site_model_is_physical its_actual_bond_is_ferromagnetic the_appendix_calculation_is_inhabited_by_an_actual_two_spin_proof theorem the_actual_zero_field_weights_are_the_cut_weights (gas : Configuration (Fin 2)) : zeroFieldWeight magnet gas = cutWeight magnet.bonds (mixedSpinFactor magnet) gas := zero_field_weights_are_products_over_mixed_bonds magnet gas end PhysicsLogic.IndependentReuse.LeeYangCircleDomainClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.LeeYangCircleDomainClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theoremCount >= 9 do throwError "Missing independent ferromagnetic-domain theorems" for axiomName in traversal.axioms do unless axiomName == ``propext || axiomName == ``Classical.choice || axiomName == ``Quot.sound do throwError "Nonfoundational ferromagnetic-domain client axiom {axiomName}" for dependency in traversal.visited.toList do let parts := dependency.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Ferromagnetic-domain client depends on a production diagnostic {dependency}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.LeeYangCircleDomainClient.its_gas_pair_energy_is_attractive, ``PhysicsLogic.Papers.LeeYang1952.FerromagneticCircle.ferromagnetic_couplings_are_exactly_nonpositive_gas_pair_energies), (``PhysicsLogic.IndependentReuse.LeeYangCircleDomainClient.positive_temperature_and_bond_signs_supply_the_appendix_domain, ``PhysicsLogic.Papers.LeeYang1952.FerromagneticCircle.ferromagnetism_is_equivalent_to_the_physical_appendix_weight_bound), (``PhysicsLogic.IndependentReuse.LeeYangCircleDomainClient.the_library_derives_actual_microscopic_roots_on_the_circle, ``PhysicsLogic.Papers.LeeYang1952.FerromagneticCircle.physical_ferromagnet_realizes_the_microscopic_circle_theorem), (``PhysicsLogic.IndependentReuse.LeeYangCircleDomainClient.the_actual_zero_field_weights_are_the_cut_weights, ``PhysicsLogic.Papers.LeeYang1952.FerromagneticCircle.zero_field_weights_are_products_over_mixed_bonds)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Lee-Yang ferromagnetic-domain client audit passed: {theoremCount} theorems, {uses.size} public API uses, actual interacting model and kernel-checked two-spin Appendix II calculation, no production diagnostics." namespace PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient open PhysicsLogic.Papers.LeeYang1952.Coexistence Filter Topology set_option backward.isDefEq.respectTransparency false noncomputable def amplitude : ℝ := Real.sqrt (Real.sqrt (Real.sqrt (175 / 256))) noncomputable def state : MagneticData where magnetization := fun field => if 0 < field then amplitude else if field < 0 then -amplitude else 0 spontaneous := amplitude angularRootDensity := fun _ => amplitude / (2 * Real.pi) theorem the_independent_amplitude_satisfies_the_printed_exact_equation : Equation47Calculation (1 / 3) amplitude := by have outer : amplitude ^ 2 = Real.sqrt (Real.sqrt (175 / 256)) := Real.sq_sqrt (by positivity) have middle : (Real.sqrt (Real.sqrt (175 / 256))) ^ 2 = Real.sqrt (175 / 256) := Real.sq_sqrt (by positivity) have inner : (Real.sqrt (175 / 256)) ^ 2 = (175 : ℝ) / 256 := Real.sq_sqrt (by norm_num) have eighth : amplitude ^ 8 = (175 : ℝ) / 256 := by calc amplitude ^ 8 = ((amplitude ^ 2) ^ 2) ^ 2 := by ring _ = (175 : ℝ) / 256 := by rw [outer, middle, inner] refine ⟨by dsimp [amplitude]; positivity, ?_⟩ norm_num [eighth] theorem the_selected_boltzmann_factor_is_strictly_subcritical : BelowCritical (1 / 3) := by have square := Real.sq_sqrt (show (0 : ℝ) ≤ 2 by norm_num) have positive := Real.sqrt_nonneg (2 : ℝ) constructor · norm_num · nlinarith theorem the_library_derives_a_positive_unsaturated_exact_amplitude : 0 < amplitude ∧ amplitude < 1 := the_exact_square_lattice_amplitude_is_positive_and_unsaturated (1 / 3) amplitude the_selected_boltzmann_factor_is_strictly_subcritical the_independent_amplitude_satisfies_the_printed_exact_equation theorem the_selected_profile_has_the_positive_field_limit : PositiveFieldMagnetizationCalculation state := by apply tendsto_const_nhds.congr' apply eventually_nhdsWithin_of_forall intro field positive simp [state, show 0 < field from positive] theorem the_selected_profile_has_actual_field_reversal : OddMagnetizationAwayFromZero state := by intro field nonzero rcases lt_or_gt_of_ne nonzero with negative | positive · simp [state, negative, not_lt.mpr negative.le, neg_pos.mpr negative] · simp [state, positive, not_lt.mpr (le_of_lt (neg_neg_of_pos positive))] theorem the_selected_profile_has_opposite_one_sided_limits : OneSidedMagnetizationCalculation state := spin_reversal_and_the_positive_limit_determine_both_one_sided_limits state the_selected_profile_has_actual_field_reversal the_selected_profile_has_the_positive_field_limit open PhysicsLogic.Papers.LeeYang1952.SpinGasDictionary open PhysicsLogic.Papers.LeeYang1952.SpinReversal noncomputable def mixedBondModel : FiniteModel (Fin 3) where bonds := {(0, 1), (1, 2)} bondEnergy := fun bond => if bond = (0, 1) then 2 else -1 field := 3 thermalEnergy := 4 theorem the_independent_mixed_bond_model_is_physical : mixedBondModel.PhysicalDomain := by norm_num [mixedBondModel, FiniteModel.PhysicalDomain] decide theorem the_independent_interactions_include_both_signs : mixedBondModel.bondEnergy (0, 1) = 2 ∧ mixedBondModel.bondEnergy (1, 2) = -1 := by norm_num [mixedBondModel] theorem the_library_matches_actual_reversed_energies (spin : Configuration (Fin 3)) : (reversedField mixedBondModel).spinEnergy (configurationEquivalence spin) = mixedBondModel.spinEnergy spin := simultaneous_field_and_spin_reversal_preserves_energy mixedBondModel spin theorem the_library_matches_the_independent_finite_partition : (reversedField mixedBondModel).spinPartition = mixedBondModel.spinPartition := the_finite_partition_function_is_even_in_field mixedBondModel theorem the_library_reverses_the_actual_finite_magnetization : (reversedField mixedBondModel).magnetization = -mixedBondModel.magnetization := finite_magnetization_is_odd_in_field mixedBondModel theorem the_library_keeps_the_gas_density_and_pressure_transformations : (reversedField mixedBondModel).density + mixedBondModel.density = 1 ∧ (reversedField mixedBondModel).pressure = mixedBondModel.pressure + 6 := by refine ⟨reversed_field_densities_are_complementary mixedBondModel the_independent_mixed_bond_model_is_physical, ?_⟩ simpa only [show mixedBondModel.field = 3 from rfl, show (2 : ℝ) * 3 = 6 by norm_num] using reversed_field_pressure_retains_the_linear_offset mixedBondModel the_independent_mixed_bond_model_is_physical theorem the_library_keeps_the_nonunit_reciprocal_fugacity_scale : let parameters := PhysicsLogic.Papers.LeeYang1952.ActivityCoordinates.Parameters.mk 4 2 3 parameters.gasFugacity 1 * parameters.gasFugacity (-1) = parameters.fugacityScale ^ 2 ∧ parameters.fugacityScale < 1 := by dsimp only constructor · exact opposite_fields_have_scaled_reciprocal_gas_fugacities _ _ · exact PhysicsLogic.Papers.LeeYang1952.ActivityCoordinates.ferromagnetic_scale_is_below_one _ (by norm_num [PhysicsLogic.Papers.LeeYang1952.ActivityCoordinates.Parameters.PhysicalDomain]) (by norm_num) theorem the_library_detects_discontinuity_despite_zero_field_value_zero : state.magnetization 0 = 0 ∧ ¬ContinuousAt state.magnetization 0 := ⟨by simp [state], nonzero_spontaneous_magnetization_excludes_continuity state the_library_derives_a_positive_unsaturated_exact_amplitude.1 the_selected_profile_has_opposite_one_sided_limits⟩ theorem the_library_transports_the_actual_gas_and_liquid_limits : DensityCoexistenceLimits state := one_sided_magnetization_limits_determine_gas_and_liquid_densities state the_selected_profile_has_opposite_one_sided_limits theorem the_library_gives_distinct_physical_densities : 0 < gasDensity state ∧ gasDensity state < liquidDensity state ∧ liquidDensity state < 1 := physical_spontaneous_amplitude_gives_distinct_positive_phase_densities state the_library_derives_a_positive_unsaturated_exact_amplitude.1 the_library_derives_a_positive_unsaturated_exact_amplitude.2 theorem the_library_transports_both_finite_specific_volumes : Tendsto (specificVolume state) (𝓝[>] 0) (𝓝 (1 / gasDensity state)) ∧ Tendsto (specificVolume state) (𝓝[<] 0) (𝓝 (1 / liquidDensity state)) := positive_phase_densities_transport_the_specific_volume_limits state the_library_transports_the_actual_gas_and_liquid_limits the_library_gives_distinct_physical_densities.1 (lt_trans the_library_gives_distinct_physical_densities.1 the_library_gives_distinct_physical_densities.2.1) theorem the_root_density_readout_uses_the_equation_thirty_eight_normalization : state.spontaneous = 2 * Real.pi * state.angularRootDensity 0 := by dsimp [state] field_simp theorem the_library_derives_the_density_jump_without_doubling_it : liquidDensity state - gasDensity state = 2 * Real.pi * state.angularRootDensity 0 ∧ 1 / (1 / liquidDensity state) - 1 / (1 / gasDensity state) = 2 * Real.pi * state.angularRootDensity 0 := root_boundary_and_spin_gas_dictionary_give_the_density_jump state the_root_density_readout_uses_the_equation_thirty_eight_normalization theorem the_library_preserves_the_field_activity_orientation : Tendsto (normalizedActivity 3) (𝓝[>] 0) (𝓝[<] 1) ∧ Tendsto (normalizedActivity 3) (𝓝[<] 0) (𝓝[>] 1) := positive_temperature_reverses_the_two_sides_of_zero_field 3 (by norm_num) theorem the_library_requires_actual_unit_activity_to_select_zero_field : normalizedActivity 3 0 = 1 ∧ normalizedActivity 3 1 ≠ 1 := by constructor · exact (unit_normalized_activity_is_exactly_zero_field 3 0 (by norm_num)).mpr rfl · intro unit have zero := (unit_normalized_activity_is_exactly_zero_field 3 1 (by norm_num)).mp unit norm_num at zero end PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 21 do throwError "Missing independent coexistence and field-reversal theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent coexistence client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent coexistence client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_derives_a_positive_unsaturated_exact_amplitude, ``PhysicsLogic.Papers.LeeYang1952.Coexistence.the_exact_square_lattice_amplitude_is_positive_and_unsaturated), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_detects_discontinuity_despite_zero_field_value_zero, ``PhysicsLogic.Papers.LeeYang1952.Coexistence.nonzero_spontaneous_magnetization_excludes_continuity), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_transports_the_actual_gas_and_liquid_limits, ``PhysicsLogic.Papers.LeeYang1952.Coexistence.one_sided_magnetization_limits_determine_gas_and_liquid_densities), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_gives_distinct_physical_densities, ``PhysicsLogic.Papers.LeeYang1952.Coexistence.physical_spontaneous_amplitude_gives_distinct_positive_phase_densities), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_transports_both_finite_specific_volumes, ``PhysicsLogic.Papers.LeeYang1952.Coexistence.positive_phase_densities_transport_the_specific_volume_limits), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_derives_the_density_jump_without_doubling_it, ``PhysicsLogic.Papers.LeeYang1952.Coexistence.root_boundary_and_spin_gas_dictionary_give_the_density_jump), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_preserves_the_field_activity_orientation, ``PhysicsLogic.Papers.LeeYang1952.Coexistence.positive_temperature_reverses_the_two_sides_of_zero_field), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_requires_actual_unit_activity_to_select_zero_field, ``PhysicsLogic.Papers.LeeYang1952.Coexistence.unit_normalized_activity_is_exactly_zero_field), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_selected_profile_has_opposite_one_sided_limits, ``PhysicsLogic.Papers.LeeYang1952.Coexistence.spin_reversal_and_the_positive_limit_determine_both_one_sided_limits), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_matches_actual_reversed_energies, ``PhysicsLogic.Papers.LeeYang1952.SpinReversal.simultaneous_field_and_spin_reversal_preserves_energy), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_matches_the_independent_finite_partition, ``PhysicsLogic.Papers.LeeYang1952.SpinReversal.the_finite_partition_function_is_even_in_field), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_reverses_the_actual_finite_magnetization, ``PhysicsLogic.Papers.LeeYang1952.SpinReversal.finite_magnetization_is_odd_in_field), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_keeps_the_gas_density_and_pressure_transformations, ``PhysicsLogic.Papers.LeeYang1952.SpinReversal.reversed_field_densities_are_complementary), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_keeps_the_gas_density_and_pressure_transformations, ``PhysicsLogic.Papers.LeeYang1952.SpinReversal.reversed_field_pressure_retains_the_linear_offset), (``PhysicsLogic.IndependentReuse.LeeYangCoexistenceClient.the_library_keeps_the_nonunit_reciprocal_fugacity_scale, ``PhysicsLogic.Papers.LeeYang1952.SpinReversal.opposite_fields_have_scaled_reciprocal_gas_fugacities)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Lee-Yang coexistence client audit passed: {theoremCount} theorems, {uses.size} public API uses, independently reconstructed exact amplitude, one-sided profile and mixed-coupling finite ensemble, no production diagnostics." namespace PhysicsLogic.IndependentReuse.LeeYangSquareLatticeClient open PhysicsLogic.Papers.LeeYang1952 open SpinGasDictionary SquareLattice Filter Topology set_option backward.isDefEq.respectTransparency false def cylinder : Geometry (Fin 20) := canonicalGeometry 4 5 false true def model : FiniteModel (Fin 20) := cylinder.buildModel 2 1 3 theorem the_selected_cylinder_has_valid_lengths : cylinder.LengthsValid := by simp [Geometry.LengthsValid, cylinder, canonicalGeometry] theorem the_library_checks_the_selected_physical_model : model.PhysicalDomain := cylinder.the_constructed_model_is_physical the_selected_cylinder_has_valid_lengths 2 1 3 (by norm_num) theorem the_library_identifies_exactly_the_cylinder_bonds : cylinder.MatchesBonds model := cylinder.the_constructed_model_has_exactly_the_selected_square_bonds the_selected_cylinder_has_valid_lengths 2 1 3 theorem the_coordinate_tests_distinguish_wrap_bonds_and_diagonals : cylinder.Adjacent 0 4 ∧ cylinder.Adjacent 0 5 ∧ ¬cylinder.Adjacent 0 15 ∧ ¬cylinder.Adjacent 0 6 := by decide theorem the_library_exposes_a_real_wrap_bond_but_no_diagonal : ((0, 4) ∈ model.bonds ∨ (4, 0) ∈ model.bonds) ∧ ¬((0, 6) ∈ model.bonds ∨ (6, 0) ∈ model.bonds) := by refine ⟨(the_library_identifies_exactly_the_cylinder_bonds.2 0 4).mpr the_coordinate_tests_distinguish_wrap_bonds_and_diagonals.1, ?_⟩ intro diagonal exact the_coordinate_tests_distinguish_wrap_bonds_and_diagonals.2.2.2 ((the_library_identifies_exactly_the_cylinder_bonds.2 0 6).mp diagonal) def relabeledCylinder : Geometry (Fin 20) := { cylinder with coordinate := (Equiv.swap 0 19).trans cylinder.coordinate } theorem changing_site_labels_preserves_the_coordinate_meaning : relabeledCylinder.Adjacent 19 4 ∧ ¬relabeledCylinder.Adjacent 0 4 := by decide def family : ThermodynamicFamily ℕ where siteCount := fun size => (size + 4) * (size + 5) baseModel := fun size => (canonicalGeometry (size + 4) (size + 5) false true).buildModel 2 0 3 sizeFilter := atTop thermalEnergy := 3 freeEnergyLimit := fun _ => 0 def geometryAt (size : ℕ) : Geometry (Fin (family.siteCount size)) := canonicalGeometry (size + 4) (size + 5) false true theorem the_independent_family_grows_in_both_directions : TwoDimensionalExhaustion family geometryAt := by refine ⟨Eventually.of_forall (fun size => (geometryAt size).the_constructed_model_has_exactly_the_selected_square_bonds ?_ 2 0 3), ?_, ?_⟩ · simp [Geometry.LengthsValid, geometryAt, canonicalGeometry] · exact tendsto_add_atTop_nat 4 · exact tendsto_add_atTop_nat 5 theorem the_library_derives_unbounded_physical_volume : Tendsto family.siteCount family.sizeFilter atTop := growth_in_both_coordinate_directions_gives_unbounded_volume family geometryAt the_independent_family_grows_in_both_directions theorem the_independent_growing_family_is_physical : family.PhysicalDomain := by refine ⟨?_, the_library_derives_unbounded_physical_volume, ?_⟩ · change (atTop : Filter ℕ).NeBot infer_instance · intro size refine ⟨(geometryAt size).the_constructed_model_is_physical ?_ 2 0 3 (by norm_num), rfl⟩ simp [Geometry.LengthsValid, geometryAt, canonicalGeometry] noncomputable def inconsistentReadout : Coexistence.MagneticData where magnetization := fun _ => 1 / 2 spontaneous := 1 / 2 angularRootDensity := fun _ => 0 theorem the_library_rejects_a_bounded_readout_not_matched_to_the_ensemble : ¬Coexistence.ThermodynamicMagnetizationCalculation family inconsistentReadout := by intro convergence have odd := Coexistence.microscopic_spin_reversal_transports_to_the_nonzero_field_limit family inconsistentReadout the_independent_growing_family_is_physical convergence have contradiction := odd 2 (by norm_num) norm_num [inconsistentReadout] at contradiction end PhysicsLogic.IndependentReuse.LeeYangSquareLatticeClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.LeeYangSquareLatticeClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 10 do throwError "Missing independent square-lattice applicability theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent square-lattice client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent square-lattice client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.LeeYangSquareLatticeClient.the_library_checks_the_selected_physical_model, ``PhysicsLogic.Papers.LeeYang1952.SquareLattice.Geometry.the_constructed_model_is_physical), (``PhysicsLogic.IndependentReuse.LeeYangSquareLatticeClient.the_library_identifies_exactly_the_cylinder_bonds, ``PhysicsLogic.Papers.LeeYang1952.SquareLattice.Geometry.the_constructed_model_has_exactly_the_selected_square_bonds), (``PhysicsLogic.IndependentReuse.LeeYangSquareLatticeClient.the_library_exposes_a_real_wrap_bond_but_no_diagonal, ``PhysicsLogic.Papers.LeeYang1952.SquareLattice.Geometry.the_constructed_model_has_exactly_the_selected_square_bonds), (``PhysicsLogic.IndependentReuse.LeeYangSquareLatticeClient.the_library_derives_unbounded_physical_volume, ``PhysicsLogic.Papers.LeeYang1952.SquareLattice.growth_in_both_coordinate_directions_gives_unbounded_volume), (``PhysicsLogic.IndependentReuse.LeeYangSquareLatticeClient.the_library_rejects_a_bounded_readout_not_matched_to_the_ensemble, ``PhysicsLogic.Papers.LeeYang1952.Coexistence.microscopic_spin_reversal_transports_to_the_nonzero_field_limit)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Lee-Yang square-lattice client audit passed: {theoremCount} theorems, {uses.size} public API uses, independent cylindrical geometry, relabeling, two-axis exhaustion and rejected unmatched readout, no production diagnostics." namespace PhysicsLogic.IndependentReuse.CoordinateBalanceClient open MeasureTheory PhysicsLogic.Papers.Einstein1916.CoordinateBalance def stationary : Current := fun _ => ![0, 0, 0, 7] def source : DifferentialSource where coupling := 5 matter := stationary gravity := fun _ _ => 0 potential := fun _ _ _ => 0 domain := Set.univ theorem current_is_the_selected_total : source.total = stationary := by funext point axis simp [DifferentialSource.total, source] theorem differential_inputs_are_realized : source.DifferentialInput := by refine ⟨Set.univ_nonempty, by norm_num [source], ?_, ?_, ?_, ?_⟩ · intro point _ axis rw [current_is_the_selected_total] exact differentiableAt_const _ · intro point _ constructor · intro axis component exact differentiableAt_const _ · intro component simp [DifferentialSource.potentialDivergence, coordinateDerivative, source] · intro point _ rw [current_is_the_selected_total] simp [DifferentialSource.doubleDivergence, DifferentialSource.potentialDivergence, divergence, coordinateDerivative, stationary, source] · intro point _ simp [DifferentialSource.doubleDivergence, DifferentialSource.potentialDivergence, coordinateDerivative, source] theorem library_derives_local_balance : NonvacuousLocalBalance stationary Set.univ := by have balanced := source.field_equation_and_metric_identity_give_local_balance differential_inputs_are_realized simpa only [current_is_the_selected_total] using balanced /-- Boundary geometry can be chosen independently: this stationary current has zero spatial transport for any boundary measure and normal. -/ noncomputable def cube (surface : Measure Space) (normal : Space → Fin 3 → ℝ) : Region where interior := Set.Icc (0 : Space) 1 surfaceMeasure := surface inwardNormal := normal theorem charge_is_nonzero (surface : Measure Space) (normal : Space → Fin 3 → ℝ) (time : ℝ) : (cube surface normal).charge stationary time = 7 := by simp [Region.charge, cube, stationary, Measure.real, Real.volume_Icc_pi] theorem spatial_divergence_is_zero (time : ℝ) (position : Space) : spatialDivergence stationary time position = 0 := by simp [spatialDivergence, coordinateDerivative, stationary] theorem boundary_flux_is_zero (surface : Measure Space) (normal : Space → Fin 3 → ℝ) (time : ℝ) : (cube surface normal).inwardFlux stationary time = 0 := by unfold Region.inwardFlux have integrand : (fun position => ∑ axis : Fin 3, stationary (atTime time position) axis.castSucc * (cube surface normal).inwardNormal position axis) = (fun _ : Space => (0 : ℝ)) := by funext position simp only [Fin.sum_univ_three] change 0 * normal position 0 + 0 * normal position 1 + 0 * normal position 2 = 0 ring rw [integrand] simp theorem all_integral_inputs_are_realized (surface : Measure Space) (normal : Space → Fin 3 → ℝ) (time : ℝ) : (cube surface normal).IntegralInput stationary Set.univ time := by refine ⟨?_, fun _ _ => Set.mem_univ _, ?_, ?_, ?_, ?_, ?_, ?_⟩ · refine ⟨measurableSet_Icc, ?_, ?_⟩ <;> simp [cube, Real.volume_Icc_pi] · exact integrableOn_const (by simp [cube, Real.volume_Icc_pi]) · simp [coordinateDerivative, stationary] · rw [show spatialDivergence stationary time = (fun _ => 0) from funext (spatial_divergence_is_zero time)] exact integrableOn_zero · simp only [Fin.sum_univ_three] change Integrable (fun position => 0 * normal position 0 + 0 * normal position 1 + 0 * normal position 2) surface simp · unfold Region.VolumeDifferentiation have charge : (cube surface normal).charge stationary = fun _ => 7 := funext (charge_is_nonzero surface normal) rw [charge] simpa [coordinateDerivative, stationary] using (hasDerivAt_const time (7 : ℝ)) · unfold Region.GaussCalculation simp_rw [spatial_divergence_is_zero] rw [boundary_flux_is_zero] simp theorem library_derives_the_integral_balance (surface : Measure Space) (normal : Space → Fin 3 → ℝ) (time : ℝ) : (cube surface normal).IntegralBalance stationary time := (cube surface normal).local_balance_and_integral_inputs_give_equation_49a stationary Set.univ time library_derives_local_balance.2 (all_integral_inputs_are_realized surface normal time) theorem library_derives_constant_nonzero_charge (surface : Measure Space) (normal : Space → Fin 3 → ℝ) : (cube surface normal).charge stationary 0 = 7 ∧ ∀ time ∈ Set.Icc (0 : ℝ) 3, (cube surface normal).charge stationary time = (cube surface normal).charge stationary 0 := by refine ⟨charge_is_nonzero surface normal 0, ?_⟩ exact (cube surface normal).zero_flux_throughout_interval_makes_charge_constant stationary 0 3 (fun time _ => library_derives_the_integral_balance surface normal time) (fun time _ => boundary_flux_is_zero surface normal time) theorem certified_paper_conclusion_supplies_the_actual_boundary_balance {T : PhysicsLogic.Papers.Einstein1916.RelativityTypes} (context : PhysicsLogic.Papers.Einstein1916.SemanticContext T) (certificate : PhysicsLogic.Papers.Einstein1916.routedArgument.RouteSemanticallyCertified context.interpretAtom .conservationAndMatter) (component : Fin 4) (region : Region) (time : ℝ) (input : region.IntegralInput (context.framework.selectedTotalEnergyCurrent context.metric context.energy component) {point | context.framework.coordinatePointAdmissibleIn point context.fieldCoordinates} time) : region.IntegralBalance (context.framework.selectedTotalEnergyCurrent context.metric context.energy component) time := (context.certified_total_conservation_retains_volume_balance certificate).2 component region time input end PhysicsLogic.IndependentReuse.CoordinateBalanceClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.CoordinateBalanceClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 10 do throwError "Missing independent coordinate-balance theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent coordinate-balance client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent coordinate-balance client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.CoordinateBalanceClient.library_derives_local_balance, ``PhysicsLogic.Papers.Einstein1916.CoordinateBalance.DifferentialSource.field_equation_and_metric_identity_give_local_balance), (``PhysicsLogic.IndependentReuse.CoordinateBalanceClient.library_derives_the_integral_balance, ``PhysicsLogic.Papers.Einstein1916.CoordinateBalance.Region.local_balance_and_integral_inputs_give_equation_49a), (``PhysicsLogic.IndependentReuse.CoordinateBalanceClient.library_derives_constant_nonzero_charge, ``PhysicsLogic.Papers.Einstein1916.CoordinateBalance.Region.zero_flux_throughout_interval_makes_charge_constant), (``PhysicsLogic.IndependentReuse.CoordinateBalanceClient.certified_paper_conclusion_supplies_the_actual_boundary_balance, ``PhysicsLogic.Papers.Einstein1916.SemanticContext.certified_total_conservation_retains_volume_balance)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Coordinate-balance client audit passed: {theoremCount} theorems, {uses.size} public API uses, positive-volume nonzero charge and source-route integral consequence, no production diagnostics." namespace PhysicsLogic.IndependentReuse.NoetherCurrentClient open PhysicsLogic.Papers.Noether1918.FiniteCurrents open scoped BigOperators /-- A two-dimensional free scalar profile u(x,y)=x^2-y^2. -/ noncomputable def scalar : System 2 1 where domain := Set.univ configuration := fun _ point => (point 0) ^ 2 - (point 1) ^ 2 density := fun point => 2 * ((point 0) ^ 2 + (point 1) ^ 2) euler := fun _ _ => 0 noncomputable def fieldShift : Generator 2 1 where deltaX := fun _ _ => 0 deltaU := fun _ _ => 1 boundaryTerm := fun point => ![-2 * point 0, 2 * point 1] densityVariation := fun _ => 0 theorem scalar_domain_is_regular : scalar.RegularDomain := by refine ⟨by decide, by decide, Set.univ_nonempty, ?_⟩ intro point _ field axis fin_cases field fin_cases axis <;> simp [scalar, Function.update] theorem scalar_is_on_shell : scalar.OnShell := ⟨scalar_domain_is_regular, fun _ _ _ => rfl⟩ theorem shift_is_regular : fieldShift.RegularOn scalar := by intro point _ constructor · intro axis fin_cases axis <;> simp [fieldShift, Function.update] <;> fun_prop · intro axis simp [Generator.transport, fieldShift] theorem actual_gradient (point : Point 2) : coordinateDerivative (scalar.configuration 0) point 0 = 2 * point 0 ∧ coordinateDerivative (scalar.configuration 0) point 1 = -2 * point 1 := by constructor · change deriv (fun value : ℝ => value ^ 2 - (point 1) ^ 2) (point 0) = 2 * point 0 simp · change deriv (fun value : ℝ => (point 0) ^ 2 - value ^ 2) (point 1) = -2 * point 1 simp theorem density_is_the_actual_gradient_energy (point : Point 2) : scalar.density point = ((coordinateDerivative (scalar.configuration 0) point 0) ^ 2 + (coordinateDerivative (scalar.configuration 0) point 1) ^ 2) / 2 := by rw [(actual_gradient point).1, (actual_gradient point).2] dsimp [scalar] ring theorem boundary_is_the_negative_gradient (point : Point 2) (axis : Fin 2) : fieldShift.boundaryTerm point axis = -coordinateDerivative (scalar.configuration 0) point axis := by fin_cases axis · change -2 * point 0 = -coordinateDerivative (scalar.configuration 0) point 0 rw [(actual_gradient point).1] ring · change 2 * point 1 = -coordinateDerivative (scalar.configuration 0) point 1 rw [(actual_gradient point).2] ring theorem euler_is_the_actual_negative_laplacian (point : Point 2) : scalar.euler 0 point = -(coordinateDerivative (fun location => coordinateDerivative (scalar.configuration 0) location 0) point 0 + coordinateDerivative (fun location => coordinateDerivative (scalar.configuration 0) location 1) point 1) := by have firstGradient : (fun location => coordinateDerivative (scalar.configuration 0) location 0) = (fun location => 2 * location 0) := funext (fun location => (actual_gradient location).1) have secondGradient : (fun location => coordinateDerivative (scalar.configuration 0) location 1) = (fun location => -2 * location 1) := funext (fun location => (actual_gradient location).2) rw [firstGradient, secondGradient] change 0 = -(deriv (fun value : ℝ => 2 * value) (point 0) + deriv (fun value : ℝ => -2 * value) (point 1)) have first : deriv (fun value : ℝ => 2 * value) (point 0) = 2 := by simpa using ((hasDerivAt_id (point 0)).const_mul 2).deriv have second : deriv (fun value : ℝ => -2 * value) (point 1) = -2 := by simpa using ((hasDerivAt_id (point 1)).const_mul (-2)).deriv rw [first, second] norm_num theorem boundary_has_zero_divergence (point : Point 2) : divergence fieldShift.boundaryTerm point = 0 := by simp only [divergence, Fin.sum_univ_two] change deriv (fun value : ℝ => -2 * value) (point 0) + deriv (fun value : ℝ => 2 * value) (point 1) = 0 have first : deriv (fun value : ℝ => -2 * value) (point 0) = -2 := by simpa using ((hasDerivAt_id (point 0)).const_mul (-2)).deriv have second : deriv (fun value : ℝ => 2 * value) (point 1) = 2 := by simpa using ((hasDerivAt_id (point 1)).const_mul 2).deriv rw [first, second] norm_num theorem variation_and_invariance_hold : fieldShift.EquationThree scalar ∧ fieldShift.InfinitesimalInvariance scalar := by constructor · intro point _ rw [boundary_has_zero_divergence] simp [Generator.pairing, scalar, fieldShift] · intro point _ simp [Generator.transport, divergence, coordinateDerivative, fieldShift] theorem library_derives_the_current_identity : fieldShift.EquationThirteen scalar := fieldShift.variation_and_invariance_give_equation_thirteen scalar shift_is_regular variation_and_invariance_hold.1 variation_and_invariance_hold.2 theorem library_derives_local_conservation : scalar.LocalConservation (fieldShift.current scalar) := fieldShift.on_shell_equation_thirteen_gives_local_conservation scalar library_derives_the_current_identity shift_is_regular scalar_is_on_shell theorem current_is_nonzero_and_not_pointwise_constant : scalar.LocalConservation (fieldShift.current scalar) ∧ (fieldShift.current scalar) ![1, 1] 0 = -2 ∧ (fieldShift.current scalar) ![2, 1] 0 ≠ (fieldShift.current scalar) ![1, 1] 0 := by refine ⟨library_derives_local_conservation, ?_, ?_⟩ <;> norm_num [Generator.current, Generator.transport, fieldShift] theorem certified_paper_conservation_has_actual_coordinate_divergence {Problem Group History : Type} (context : PhysicsLogic.Papers.Noether1918.SemanticContext Problem Group History) (certificate : PhysicsLogic.Papers.Noether1918.routedArgument.RouteSemanticallyCertified context.interpretAtom .multidimensionalConservation) (relation : ℕ) (selected : relation < context.parameterCount) : (context.framework.selectedVariationalSystem context.problem context.selectedHistory).LocalConservation ((context.framework.finiteGenerator context.problem context.finiteGroup context.selectedHistory relation).current (context.framework.selectedVariationalSystem context.problem context.selectedHistory)) := (context.certified_conservation_retains_the_same_history_current certificate).2.2 relation selected end PhysicsLogic.IndependentReuse.NoetherCurrentClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.NoetherCurrentClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 12 do throwError "Missing independent Noether-current theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent Noether-current client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent Noether-current client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.NoetherCurrentClient.library_derives_the_current_identity, ``PhysicsLogic.Papers.Noether1918.FiniteCurrents.Generator.variation_and_invariance_give_equation_thirteen), (``PhysicsLogic.IndependentReuse.NoetherCurrentClient.library_derives_local_conservation, ``PhysicsLogic.Papers.Noether1918.FiniteCurrents.Generator.on_shell_equation_thirteen_gives_local_conservation), (``PhysicsLogic.IndependentReuse.NoetherCurrentClient.certified_paper_conservation_has_actual_coordinate_divergence, ``PhysicsLogic.Papers.Noether1918.SemanticContext.certified_conservation_retains_the_same_history_current)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" logInfo m!"Noether-current client audit passed: {theoremCount} theorems, {uses.size} public API uses, actual free-scalar gradient, nonzero nonconstant conserved current and certified paper projection, no production diagnostics." namespace PhysicsLogic.IndependentReuse.NoetherGaugeClient open PhysicsLogic.Papers.Noether1918.FiniteCurrents open PhysicsLogic.Papers.Noether1918.ArbitraryFunctions open MeasureTheory open scoped BigOperators ContDiff /-- Independent history u=t, v=0 for L=(u'-v)^2/2. -/ noncomputable def system : System 1 2 where domain := Set.univ configuration := ![fun point => point 0, fun _ => 0] density := fun _ => 1/2 euler := ![fun _ => 0, fun _ => -1] def variation : Operator 1 2 1 where termCount := 2 word := ![[], [0]] coefficient := fun field _ term _ => if field = term then 1 else 0 def boundary : Operator 1 1 1 where termCount := 1 word := fun _ => [] coefficient := fun _ _ _ _ => -1 noncomputable def generator (test : Fin 1 → Point 1 → ℝ) : Generator 1 2 where deltaX := fun _ _ => 0 deltaU := ![test 0, fun point => coordinateDerivative (test 0) point 0] boundaryTerm := fun point _ => -test 0 point densityVariation := fun _ => 0 noncomputable def family : Family 1 2 1 where variation := variation current := boundary gamma := boundary generator := generator surfaceMeasure := 0 normal := fun _ _ => 1 theorem regular_domain : system.RegularDomain := by refine ⟨by decide, by decide, Set.univ_nonempty, ?_⟩ intro point _ field axis fin_cases field <;> fin_cases axis <;> simp [system] theorem kinetic_density_and_euler (time : ℝ) : system.density ![time] = (deriv (fun t : ℝ => t) time)^2 / 2 ∧ system.euler 0 ![time] = -deriv (fun _ : ℝ => 1) time ∧ system.euler 1 ![time] = -(deriv (fun t : ℝ => t) time) := by simp [system] theorem density_is_unchanged_by_arbitrary_gauge_shift (test : ℝ → ℝ) (differentiable : Differentiable ℝ test) (epsilon time : ℝ) : (deriv (fun t => t + epsilon * test t) time - epsilon * deriv test time)^2 / 2 = 1/2 := by have derivative : deriv (fun t : ℝ => t + epsilon * test t) time = 1 + epsilon * deriv test time := by simpa using ((hasDerivAt_id time).add ((differentiable time).hasDerivAt.const_mul epsilon)).deriv rw [derivative] ring theorem variation_apply (test : Fin 1 → Point 1 → ℝ) (point : Point 1) : variation.apply test 0 point = test 0 point ∧ variation.apply test 1 point = coordinateDerivative (test 0) point 0 := by unfold Operator.apply change (∑ f : Fin 1, ∑ t : Fin 2, _) = _ ∧ (∑ f : Fin 1, ∑ t : Fin 2, _) = _ simp [variation, jet] theorem boundary_apply (test : Fin 1 → Point 1 → ℝ) (point : Point 1) (axis : Fin 1) : boundary.apply test axis point = -test 0 point := by unfold Operator.apply change (∑ f : Fin 1, ∑ t : Fin 1, _) = _ simp [boundary, jet] theorem actual_adjoint (point : Point 1) (function : Fin 1) : variation.adjoint system.euler function point = 0 := by unfold Operator.adjoint change (∑ field : Fin 2, ∑ term : Fin 2, _) = _ simp [variation, system, Fin.sum_univ_two, jet, coordinateDerivative] theorem operator_independent : variation.Independent Set.univ := by intro weights annihilates function have unitValue : variation.adjoint ![fun _ => 1, fun _ => 0] 0 ![0] = 1 := by unfold Operator.adjoint change (∑ field : Fin 2, ∑ term : Fin 2, _) = _ simp [variation, Fin.sum_univ_two, jet, coordinateDerivative] have smooth : ∀ field : Fin 2, ContDiff ℝ ∞ ((![fun _ : Point 1 => (1 : ℝ), fun _ => 0]) field) := by intro field fin_cases field <;> exact contDiff_const have evaluated := annihilates _ smooth ![0] (Set.mem_univ _) simp only [Fin.sum_univ_one, unitValue, mul_one] at evaluated fin_cases function exact evaluated theorem negative_test_divergence (test : Fin 1 → Point 1 → ℝ) (point : Point 1) : divergence (fun location (_ : Fin 1) => -test 0 location) point = -coordinateDerivative (test 0) point 0 := by simp [divergence, coordinateDerivative] theorem coordinate_path_regular (point : Point 1) (axis : Fin 1) : DifferentiableAt ℝ (fun value : ℝ => Function.update point axis value) (point axis) := by apply differentiableAt_pi.mpr intro index fin_cases axis fin_cases index simp theorem inputs_hold : family.DerivationInputs system 1 := by refine ⟨isOpen_univ, ?_, ?_, by simp [family], ?_, ?_, ?_, ?_, ?_⟩ · intro term fin_cases term <;> simp [family, variation] · constructor · intro field function term fin_cases field <;> fin_cases term <;> simp [family, variation, system] <;> exact contDiffOn_const · intro function have zero : family.variation.adjoint system.euler function = fun _ => 0 := funext fun point => actual_adjoint point function rw [zero] exact continuousOn_const · intro test _ point _ field fin_cases field <;> simp [family, generator, Generator.verticalVariation, variation_apply] · intro test _ funext point axis simp [family, generator, Generator.current, Generator.transport, boundary_apply] · intro test admissible have smooth := (admissible 0).1 have differentiable : Differentiable ℝ (test 0) := smooth.differentiable (by simp) refine ⟨?_, ?_, ?_, ?_⟩ · intro point _ constructor · intro axis change DifferentiableAt ℝ (fun value => -test 0 (Function.update point axis value)) (point axis) exact (differentiable.differentiableAt.comp _ (coordinate_path_regular point axis)).neg · intro axis simp [family, generator, Generator.transport] · intro point _ simp [family, generator, Generator.pairing, Generator.verticalVariation, system, Fin.sum_univ_two, negative_test_divergence] · intro point _ simp [family, generator, Generator.transport, divergence, coordinateDerivative] · intro point _ axis simp only [family, boundary_apply] exact (differentiable.differentiableAt.comp _ (coordinate_path_regular point axis)).neg · intro test _ point _ change (∑ field, system.euler field point * variation.apply test field point) = (∑ function, variation.adjoint system.euler function point * test function point) + divergence (fun location axis => boundary.apply test axis location) point simp only [actual_adjoint, zero_mul, Finset.sum_const_zero, zero_add] simp [variation_apply, boundary_apply, system, Fin.sum_univ_two, negative_test_divergence] · intro test _ have zero : family.residual test = fun _ _ => 0 := by funext point axis simp [Family.residual, family] have divzero : divergence (family.residual test) = fun _ => 0 := by rw [zero] funext point simp [divergence, coordinateDerivative] rw [divzero] exact ⟨integrableOn_zero, by simp [Family.surfaceFlux, family]⟩ theorem library_derives_equation_fifteen : family.EquationFifteen system := family.equation_fifteen_from_variation system 1 inputs_hold theorem library_derives_off_shell_identity : family.EquationSixteen system ∧ ¬system.OnShell := by refine ⟨family.equation_sixteen_from_arbitrary_tests system 1 inputs_hold, ?_⟩ intro onShell have impossible := onShell.2 ![0] (Set.mem_univ _) 1 norm_num [system] at impossible theorem library_derives_residual_balance (test : Fin 1 → Point 1 → ℝ) (admissible : Admissible system.domain test) : PhysicsLogic.Framework.CoordinateCurrent.NonvacuousLocalBalance (family.residual test) system.domain := ⟨regular_domain.1, regular_domain.2.2.1, family.residual_current_is_conserved system 1 inputs_hold test admissible⟩ theorem certificate_preserves_the_selected_operator_and_history {Problem Group History : Type} (context : PhysicsLogic.Papers.Noether1918.SemanticContext Problem Group History) (certificate : PhysicsLogic.Papers.Noether1918.routedArgument.RouteSemanticallyCertified context.interpretAtom .infiniteForward) (history : History) (regular : (context.framework.selectedVariationalSystem context.problem history).RegularDomain) (function : Fin context.functionCount) (point : Point (context.framework.independentVariableCount context.problem)) (inside : point ∈ (context.framework.selectedVariationalSystem context.problem history).domain) : (context.framework.arbitraryFunctionFamily context.problem context.infiniteGroup history context.functionCount).variation.adjoint (context.framework.selectedVariationalSystem context.problem history).euler function point = 0 := context.certified_arbitrary_function_route_has_pointwise_euler_identity certificate history regular function point inside open PhysicsLogic.Papers.Noether1918.EulerDependence noncomputable def equationRecovery : Recovery 1 2 1 where select := fun _ => 0 fromIdentities := id fromRemaining := fun residual _ point => coordinateDerivative (residual 1) point 0 initialCount := 0 initialWord := Fin.elim0 initialPoint := Fin.elim0 initialCoefficient := fun _ _ datum => Fin.elim0 datum theorem trial_adjoint (trial : Fields 1 2) (point : Point 1) : variation.adjoint trial 0 point = trial 0 point - coordinateDerivative (trial 1) point 0 := by unfold Operator.adjoint change (∑ field : Fin 2, ∑ term : Fin 2, _) = _ simp [variation, Fin.sum_univ_two, jet, coordinateDerivative, sub_eq_add_neg] theorem reconstruction_holds : equationRecovery.ReconstructionCalculation variation Set.univ := by intro trial _ selected point _ fin_cases selected have retained : equationRecovery.remainingField trial 1 = trial 1 := by funext location simp [Recovery.remainingField, equationRecovery] have initial : equationRecovery.initialContribution trial 0 point = 0 := by simp [Recovery.initialContribution, equationRecovery] change trial 0 point = variation.adjoint trial 0 point + coordinateDerivative (equationRecovery.remainingField trial 1) point 0 + equationRecovery.initialContribution trial 0 point rw [retained, initial, trial_adjoint] ring theorem recovery_inputs (selected : System 1 2) (domain : selected.domain = Set.univ) (smooth : SmoothOn selected.euler selected.domain) : equationRecovery.Inputs selected variation := by refine ⟨by rw [domain]; exact isOpen_univ, fun _ _ _ => Subsingleton.elim _ _, smooth, ?_, ?_, ?_, ?_, ?_⟩ · intro datum exact Fin.elim0 datum · intro residual zero exact zero · intro residual zero component point inside exact coordinate_derivative_zero_on_open (residual 1) selected.domain (by rw [domain]; exact isOpen_univ) (zero 1) point inside 0 · rw [domain] exact reconstruction_holds · intro component datum exact Fin.elim0 datum theorem remaining_equations (selected : System 1 2) (regular : selected.RegularDomain) (remaining : ∀ point ∈ selected.domain, selected.euler 1 point = 0) : equationRecovery.RemainingEquations selected := by refine ⟨regular, ?_⟩ intro field notSelected point inside fin_cases field · simp [equationRecovery] at notSelected · exact remaining point inside theorem library_eliminates_first_equation (selected : System 1 2) (domain : selected.domain = Set.univ) (regular : selected.RegularDomain) (smooth : SmoothOn selected.euler selected.domain) (identity : family.EquationSixteen selected) (remaining : ∀ point ∈ selected.domain, selected.euler 1 point = 0) : equationRecovery.SelectedEquations selected := equationRecovery.identities_and_recovery_give_selected_equations selected family (by decide) identity (remaining_equations selected regular remaining) (recovery_inputs selected domain smooth) theorem library_completes_all_equations (selected : System 1 2) (domain : selected.domain = Set.univ) (regular : selected.RegularDomain) (smooth : SmoothOn selected.euler selected.domain) (identity : family.EquationSixteen selected) (remaining : ∀ point ∈ selected.domain, selected.euler 1 point = 0) : selected.OnShell := equationRecovery.remaining_and_selected_equations_put_history_on_shell selected (remaining_equations selected regular remaining) (library_eliminates_first_equation selected domain regular smooth identity remaining) /-- Nonconstant history with u'=v; both Euler expressions really vanish. -/ noncomputable def solution : System 1 2 where domain := Set.univ configuration := ![fun point => point 0, fun _ => 1] density := fun _ => 0 euler := fun _ _ => 0 theorem solution_is_from_kinetic_density (time : ℝ) : solution.density ![time] = (deriv (fun t : ℝ => t) time - 1)^2 / 2 ∧ solution.euler 0 ![time] = -deriv (fun _ : ℝ => 0) time ∧ solution.euler 1 ![time] = -(deriv (fun t : ℝ => t) time - 1) := by simp [solution] theorem concrete_equation_recovery_is_nonvacuous : solution.OnShell := by apply library_completes_all_equations solution rfl · refine ⟨by decide, by decide, Set.univ_nonempty, ?_⟩ intro point _ field axis fin_cases field <;> fin_cases axis <;> simp [solution] · exact fun _ => contDiffOn_const · intro function point _ fin_cases function change variation.adjoint solution.euler 0 point = 0 rw [trial_adjoint] simp [solution, coordinateDerivative] · exact fun _ _ => rfl theorem library_propagates_reversed_equation_with_initial_value (euler : Fields 1 2) (smooth : SmoothOn euler Set.univ) (identity : ∀ point, variation.adjoint euler 0 point = 0) (remaining : ∀ point, euler 0 point = 0) (initialTime : ℝ) (initial : euler 1 ![initialTime] = 0) : ZeroOn euler Set.univ := derivative_identity_and_initial_value_put_both_equations_on_shell euler smooth (fun point => by simpa [trial_adjoint] using identity point) remaining initialTime initial theorem identity_and_first_equation_do_not_fix_initial_value : family.EquationSixteen system ∧ (∀ point, system.euler 0 point = 0) ∧ system.euler 1 ![0] ≠ 0 := ⟨library_derives_off_shell_identity.1, fun _ => rfl, by norm_num [system]⟩ noncomputable def changingHistory : System 1 2 where domain := Set.univ configuration := ![fun point => (point 0)^2, fun _ => 0] density := fun point => 2 * (point 0)^2 euler := ![fun _ => -2, fun point => -2 * point 0] theorem changing_history_is_from_kinetic_density (time : ℝ) : changingHistory.density ![time] = (deriv (fun t : ℝ => t^2) time)^2 / 2 ∧ changingHistory.euler 0 ![time] = -deriv (fun t : ℝ => 2*t) time ∧ changingHistory.euler 1 ![time] = -(deriv (fun t : ℝ => t^2) time) := by have linear : deriv (fun t : ℝ => 2*t) time = 2 := by simpa using ((hasDerivAt_id time).const_mul 2).deriv simp [changingHistory, linear] ring theorem pointwise_equation_does_not_supply_regional_equation : changingHistory.RegularDomain ∧ (∀ point, variation.adjoint changingHistory.euler 0 point = 0) ∧ changingHistory.euler 1 ![0] = 0 ∧ changingHistory.euler 0 ![0] ≠ 0 := by refine ⟨?_, ?_, by norm_num [changingHistory], by norm_num [changingHistory]⟩ · refine ⟨by decide, by decide, Set.univ_nonempty, ?_⟩ intro point _ field axis fin_cases field <;> fin_cases axis <;> simp [changingHistory] · intro point have linear : deriv (HMul.hMul (2 : ℝ)) (point 0) = 2 := by simpa using ((hasDerivAt_id (point 0)).const_mul 2).deriv rw [trial_adjoint] simp [changingHistory, coordinateDerivative, linear] theorem certificate_preserves_the_selected_euler_equation {Problem Group History : Type} (context : PhysicsLogic.Papers.Noether1918.SemanticContext Problem Group History) (certificate : PhysicsLogic.Papers.Noether1918.routedArgument.RouteSemanticallyCertified context.interpretAtom .infiniteEquationDependence) (selected : Fin context.functionCount) (point : Point (context.framework.independentVariableCount context.problem)) (inside : point ∈ (context.framework.selectedVariationalSystem context.problem context.selectedHistory).domain) : (context.framework.selectedVariationalSystem context.problem context.selectedHistory).euler ((context.framework.dependentEquationRecovery context.problem context.infiniteGroup context.selectedHistory context.functionCount).select selected) point = 0 := context.certified_equation_dependence_has_actual_selected_euler_zero certificate selected point inside end PhysicsLogic.IndependentReuse.NoetherGaugeClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.NoetherGaugeClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 10 do throwError "Missing independent arbitrary-function theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent gauge client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent gauge client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.NoetherGaugeClient.library_derives_equation_fifteen, ``PhysicsLogic.Papers.Noether1918.ArbitraryFunctions.Family.equation_fifteen_from_variation), (``PhysicsLogic.IndependentReuse.NoetherGaugeClient.library_derives_off_shell_identity, ``PhysicsLogic.Papers.Noether1918.ArbitraryFunctions.Family.equation_sixteen_from_arbitrary_tests), (``PhysicsLogic.IndependentReuse.NoetherGaugeClient.library_derives_residual_balance, ``PhysicsLogic.Papers.Noether1918.ArbitraryFunctions.Family.residual_current_is_conserved), (``PhysicsLogic.IndependentReuse.NoetherGaugeClient.certificate_preserves_the_selected_operator_and_history, ``PhysicsLogic.Papers.Noether1918.SemanticContext.certified_arbitrary_function_route_has_pointwise_euler_identity), (``PhysicsLogic.IndependentReuse.NoetherGaugeClient.library_eliminates_first_equation, ``PhysicsLogic.Papers.Noether1918.EulerDependence.Recovery.identities_and_recovery_give_selected_equations), (``PhysicsLogic.IndependentReuse.NoetherGaugeClient.library_completes_all_equations, ``PhysicsLogic.Papers.Noether1918.EulerDependence.Recovery.remaining_and_selected_equations_put_history_on_shell), (``PhysicsLogic.IndependentReuse.NoetherGaugeClient.recovery_inputs, ``PhysicsLogic.Papers.Noether1918.EulerDependence.coordinate_derivative_zero_on_open), (``PhysicsLogic.IndependentReuse.NoetherGaugeClient.library_propagates_reversed_equation_with_initial_value, ``PhysicsLogic.Papers.Noether1918.EulerDependence.derivative_identity_and_initial_value_put_both_equations_on_shell), (``PhysicsLogic.IndependentReuse.NoetherGaugeClient.certificate_preserves_the_selected_euler_equation, ``PhysicsLogic.Papers.Noether1918.SemanticContext.certified_equation_dependence_has_actual_selected_euler_zero)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Noether gauge client audit passed: {theoremCount} theorems, {uses.size} public API uses, arbitrary-function symmetry, off-shell identity, equation recovery and explicit initial constraint, no production diagnostics." namespace PhysicsLogic.IndependentReuse.NoetherFirstIntegralClient open PhysicsLogic.Papers.Noether1918.FiniteCurrents open PhysicsLogic.Papers.Noether1918.FirstIntegrals open scoped BigOperators /-- A different physical model from the production controls: q=cos(t) for the unit harmonic oscillator. Its density and boundary term vary. -/ noncomputable def oscillator : System 1 1 where domain := Set.univ configuration := fun _ point => Real.cos (point 0) density := fun point => ((Real.sin (point 0))^2 - (Real.cos (point 0))^2) / 2 euler := fun _ _ => 0 noncomputable def timeTranslation : Generator 1 1 where deltaX := fun _ _ => 1 deltaU := fun _ _ => 0 boundaryTerm := fun point _ => (Real.sin (point 0))^2 densityVariation := fun point => -2 * Real.sin (point 0) * Real.cos (point 0) noncomputable def energyCurrents : Fin 1 → Current 1 := fun _ => timeTranslation.current oscillator theorem kinetic_minus_potential_density_and_actual_euler (time : ℝ) : oscillator.density (line time) = ((deriv Real.cos time)^2 - (Real.cos time)^2) / 2 ∧ oscillator.euler 0 (line time) = -Real.cos time - deriv (fun t => -Real.sin t) time := by simp [oscillator, line] theorem regular_on_shell : oscillator.OnShell := by refine ⟨⟨by decide, by decide, Set.univ_nonempty, ?_⟩, fun _ _ _ => rfl⟩ intro point _ field axis fin_cases field fin_cases axis simp [oscillator] theorem generator_regular : timeTranslation.RegularOn oscillator := by intro point _ constructor · intro axis fin_cases axis simp [timeTranslation] · intro axis fin_cases axis simp [timeTranslation, Generator.transport, oscillator] fun_prop theorem boundary_divergence (point : Point 1) : divergence timeTranslation.boundaryTerm point = 2 * Real.sin (point 0) * Real.cos (point 0) := by simp [divergence, coordinateDerivative, timeTranslation] theorem transport_divergence (point : Point 1) : divergence (timeTranslation.transport oscillator) point = 2 * Real.sin (point 0) * Real.cos (point 0) := by have derivative : deriv (fun t : ℝ => ((Real.sin t)^2 - (Real.cos t)^2) / 2) (point 0) = 2 * Real.sin (point 0) * Real.cos (point 0) := by have calculated := (((Real.hasDerivAt_sin (point 0)).pow 2).sub ((Real.hasDerivAt_cos (point 0)).pow 2)).div_const 2 convert calculated.deriv using 1 simp ring simpa [divergence, coordinateDerivative, timeTranslation, Generator.transport, oscillator] using derivative theorem variation_identity : timeTranslation.EquationThree oscillator := by intro point _ rw [boundary_divergence] simp [Generator.pairing, oscillator, timeTranslation] theorem infinitesimal_invariance : timeTranslation.InfinitesimalInvariance oscillator := by intro point _ rw [transport_divergence] simp [timeTranslation] theorem library_derives_energy_conservation : oscillator.LocalConservation (timeTranslation.current oscillator) := timeTranslation.on_shell_equation_thirteen_gives_local_conservation oscillator (timeTranslation.variation_and_invariance_give_equation_thirteen oscillator generator_regular variation_identity infinitesimal_invariance) generator_regular regular_on_shell theorem library_derives_first_integral : FamilyResult oscillator energyCurrents := local_conservation_gives_first_integrals oscillator energyCurrents rfl (by decide) ⟨isOpen_univ, isPreconnected_univ⟩ (fun _ => library_derives_energy_conservation) theorem library_preserves_energy_at_arbitrary_times (first second : ℝ) : energyCurrents 0 (line first) 0 = energyCurrents 0 (line second) 0 := first_integrals_retain_values_at_two_times oscillator energyCurrents library_derives_first_integral 0 (line first) (line second) (Set.mem_univ _) (Set.mem_univ _) 0 theorem actual_energy_is_nonzero (time : ℝ) : energyCurrents 0 (line time) 0 = 1/2 := by have identity := Real.sin_sq_add_cos_sq time dsimp [energyCurrents, Generator.current, Generator.transport, timeTranslation, oscillator, line] nlinarith theorem boundary_term_alone_is_not_the_first_integral : timeTranslation.boundaryTerm (line 0) 0 ≠ timeTranslation.boundaryTerm (line (Real.pi/2)) 0 ∧ energyCurrents 0 (line 0) 0 = energyCurrents 0 (line (Real.pi/2)) 0 := by refine ⟨?_, library_preserves_energy_at_arbitrary_times 0 (Real.pi/2)⟩ norm_num [timeTranslation, line] theorem certified_paper_first_integral_preserves_the_actual_selected_current {Problem Group History : Type} (context : PhysicsLogic.Papers.Noether1918.SemanticContext Problem Group History) (certificate : PhysicsLogic.Papers.Noether1918.routedArgument.RouteSemanticallyCertified context.interpretAtom .oneDimensionalFirstIntegrals) (relation : Fin context.parameterCount) (first second : Point (context.framework.independentVariableCount context.problem)) (firstInside : first ∈ (context.framework.selectedVariationalSystem context.problem context.selectedHistory).domain) (secondInside : second ∈ (context.framework.selectedVariationalSystem context.problem context.selectedHistory).domain) (axis : Fin (context.framework.independentVariableCount context.problem)) : context.framework.selectedFirstIntegralCurrents context.problem context.finiteGroup context.selectedHistory context.parameterCount relation first axis = context.framework.selectedFirstIntegralCurrents context.problem context.finiteGroup context.selectedHistory context.parameterCount relation second axis := context.certified_first_integrals_retain_same_history_values certificate relation first second firstInside secondInside axis end PhysicsLogic.IndependentReuse.NoetherFirstIntegralClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.NoetherFirstIntegralClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 13 do throwError "Missing independent first-integral theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent first-integral client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent first-integral client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.NoetherFirstIntegralClient.library_derives_energy_conservation, ``PhysicsLogic.Papers.Noether1918.FiniteCurrents.Generator.variation_and_invariance_give_equation_thirteen), (``PhysicsLogic.IndependentReuse.NoetherFirstIntegralClient.library_derives_energy_conservation, ``PhysicsLogic.Papers.Noether1918.FiniteCurrents.Generator.on_shell_equation_thirteen_gives_local_conservation), (``PhysicsLogic.IndependentReuse.NoetherFirstIntegralClient.library_derives_first_integral, ``PhysicsLogic.Papers.Noether1918.FirstIntegrals.local_conservation_gives_first_integrals), (``PhysicsLogic.IndependentReuse.NoetherFirstIntegralClient.library_preserves_energy_at_arbitrary_times, ``PhysicsLogic.Papers.Noether1918.FirstIntegrals.first_integrals_retain_values_at_two_times), (``PhysicsLogic.IndependentReuse.NoetherFirstIntegralClient.certified_paper_first_integral_preserves_the_actual_selected_current, ``PhysicsLogic.Papers.Noether1918.SemanticContext.certified_first_integrals_retain_same_history_values)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Noether first-integral client audit passed: {theoremCount} theorems, {uses.size} public API uses, harmonic-oscillator variation, nonconstant boundary and density, conserved nonzero energy, no production diagnostics." namespace PhysicsLogic.IndependentReuse.BoundaryFluxClient open MeasureTheory PhysicsLogic.Papers.Einstein1916.CoordinateBalance open scoped BigOperators abbrev Plane := Fin 2 → ℝ abbrev Face := Fin 3 × Fin 2 def square : Set Plane := Set.pi Set.univ (fun _ => Set.Ioo 0 1) noncomputable def facePoint (face : Face) (point : Plane) : Space := if face.1 = 0 then ![(face.2.val : ℝ), point 0, point 1] else if face.1 = 1 then ![point 0, (face.2.val : ℝ), point 1] else ![point 0, point 1, (face.2.val : ℝ)] noncomputable def faceMeasure (face : Face) : Measure Space := Measure.map (facePoint face) (volume.restrict square) /-- Six ordinary area measures; omitted edges have two-dimensional measure zero. -/ noncomputable def cube : Region where interior := Set.Icc (0 : Space) 1 surfaceMeasure := ∑ face : Face, faceMeasure face inwardNormal := fun point axis => if point axis = 0 then 1 else if point axis = 1 then -1 else 0 theorem face_map_is_measurable (face : Face) : Measurable (facePoint face) := by apply measurable_pi_lambda intro axis fin_cases axis <;> unfold facePoint <;> split_ifs <;> simp <;> fun_prop theorem square_has_unit_area : volume square = 1 := by simp [square, Real.volume_pi_Ioo] theorem each_face_has_unit_area (face : Face) : faceMeasure face Set.univ = 1 := by simp [faceMeasure, Measure.map_apply (face_map_is_measurable face) MeasurableSet.univ, square_has_unit_area] theorem faces_lie_on_cube_boundary (face : Face) (point : Plane) (inside : point ∈ square) : facePoint face point ∈ cube.interior ∧ (facePoint face point face.1 = 0 ∨ facePoint face point face.1 = 1) := by have h0 : 0 < point 0 ∧ point 0 < 1 := inside 0 (Set.mem_univ _) have h1 : 0 < point 1 ∧ point 1 < 1 := inside 1 (Set.mem_univ _) rcases face with ⟨axis, side⟩ fin_cases axis <;> fin_cases side <;> simp [cube, facePoint, Set.mem_Icc, Pi.le_def, Fin.forall_fin_succ, h0.1.le, h0.2.le, h1.1.le, h1.2.le] theorem inward_normal_is_unit_on_faces (face : Face) (point : Plane) (inside : point ∈ square) : (∑ axis : Fin 3, (cube.inwardNormal (facePoint face point) axis) ^ 2) = 1 ∧ cube.inwardNormal (facePoint face point) face.1 = 1 - 2 * (face.2.val : ℝ) := by have h0 : 0 < point 0 ∧ point 0 < 1 := inside 0 (Set.mem_univ _) have h1 : 0 < point 1 ∧ point 1 < 1 := inside 1 (Set.mem_univ _) rcases face with ⟨axis, side⟩ fin_cases axis <;> fin_cases side <;> norm_num [cube, facePoint, Fin.sum_univ_three, ne_of_gt h0.1, ne_of_lt h0.2, ne_of_gt h1.1, ne_of_lt h1.2, Matrix.cons_val_two] <;> rfl /-- A test of the continuity equation, not a claim to solve Einstein's metric equations. -/ noncomputable def current : Current := fun point => ![-2 * (point 3 - 1) * point 0, 0, 0, 2 + (point 3 - 1) ^ 2] noncomputable def surfaceDensity (time : ℝ) (point : Space) : ℝ := -2 * (time - 1) * point 0 * (if point 0 = 0 then 1 else if point 0 = 1 then -1 else 0) noncomputable def faceFlux (time : ℝ) (face : Face) : ℝ := if face.1 = 0 ∧ face.2 = 1 then 2 * (time - 1) else 0 theorem surface_density_is_measurable (time : ℝ) : Measurable (surfaceDensity time) := by unfold surfaceDensity apply Measurable.mul (by fun_prop) exact Measurable.ite (measurableSet_eq_fun (measurable_pi_apply 0) measurable_const) measurable_const (Measurable.ite (measurableSet_eq_fun (measurable_pi_apply 0) measurable_const) measurable_const measurable_const) theorem surface_density_on_each_face (time : ℝ) (face : Face) (point : Plane) (inside : point ∈ square) : surfaceDensity time (facePoint face point) = faceFlux time face := by have h0 : 0 < point 0 ∧ point 0 < 1 := inside 0 (Set.mem_univ _) rcases face with ⟨axis, side⟩ fin_cases axis <;> fin_cases side <;> simp [surfaceDensity, facePoint, faceFlux, ne_of_gt h0.1, ne_of_lt h0.2] theorem surface_density_is_ae_constant_on_each_face (time : ℝ) (face : Face) : surfaceDensity time =ᵐ[faceMeasure face] (fun _ => faceFlux time face) := by apply (ae_map_iff (face_map_is_measurable face).aemeasurable (measurableSet_eq_fun (surface_density_is_measurable time) measurable_const)).2 filter_upwards [ae_restrict_mem (by exact MeasurableSet.univ_pi (fun _ => measurableSet_Ioo) : MeasurableSet square)] with point inside exact surface_density_on_each_face time face point inside theorem face_flux_is_integrable (time : ℝ) (face : Face) : Integrable (surfaceDensity time) (faceMeasure face) := by have finite : IsFiniteMeasure (faceMeasure face) := ⟨by rw [each_face_has_unit_area]; norm_num⟩ exact (integrable_const (faceFlux time face)).congr (surface_density_is_ae_constant_on_each_face time face).symm theorem actual_face_integral (time : ℝ) (face : Face) : (∫ point, surfaceDensity time point ∂faceMeasure face) = faceFlux time face := by rw [integral_congr_ae (surface_density_is_ae_constant_on_each_face time face)] simp [integral_const, Measure.real, each_face_has_unit_area] theorem current_surface_density (time : ℝ) (point : Space) : (∑ axis : Fin 3, current (atTime time point) axis.castSucc * cube.inwardNormal point axis) = surfaceDensity time point := by simp [Fin.sum_univ_three, current, atTime, cube, surfaceDensity] theorem actual_boundary_flux (time : ℝ) : cube.inwardFlux current time = 2 * (time - 1) := by unfold Region.inwardFlux simp_rw [current_surface_density] rw [show cube.surfaceMeasure = ∑ face : Face, faceMeasure face from rfl, integral_finset_sum_measure (fun face _ => face_flux_is_integrable time face)] simp_rw [actual_face_integral] norm_num [faceFlux, Fintype.sum_prod_type, Fin.sum_univ_three, Fin.sum_univ_two] theorem actual_charge (time : ℝ) : cube.charge current time = 2 + (time - 1) ^ 2 := by simp [Region.charge, cube, current, atTime, Measure.real, Real.volume_Icc_pi] theorem charge_stays_strictly_positive (time : ℝ) : 0 < cube.charge current time := by rw [actual_charge] positivity theorem current_is_regular (point : Point) : RegularAt current point := by intro axis fin_cases axis <;> simp [current, Function.update] all_goals fun_prop theorem actual_time_derivative (time : ℝ) (point : Space) : coordinateDerivative (fun location => current location 3) (atTime time point) 3 = 2 * (time - 1) := by change deriv (fun value : ℝ => 2 + (value - 1) ^ 2) time = _ convert ((((hasDerivAt_id time).sub_const 1).pow 2).const_add 2).deriv using 1 simp theorem actual_spatial_divergence (time : ℝ) (point : Space) : spatialDivergence current time point = -2 * (time - 1) := by simp only [spatialDivergence, Fin.sum_univ_three] change deriv (fun value : ℝ => -2 * (time - 1) * value) (point 0) + deriv (fun _ : ℝ => (0 : ℝ)) (point 1) + deriv (fun _ : ℝ => (0 : ℝ)) (point 2) = _ have calculated : deriv (fun value : ℝ => -2 * (time - 1) * value) (point 0) = -2 * (time - 1) := by simpa using ((hasDerivAt_id (point 0)).const_mul (-2 * (time - 1))).deriv rw [calculated] simp theorem library_current_has_local_balance : NonvacuousLocalBalance current Set.univ := by refine ⟨Set.univ_nonempty, fun point _ => ⟨current_is_regular point, ?_⟩⟩ have pointSplit : point = atTime (point 3) ![point 0, point 1, point 2] := by ext axis fin_cases axis <;> rfl rw [pointSplit, divergence_splits_into_space_and_time, actual_spatial_divergence, actual_time_derivative] ring theorem actual_integral_inputs (time : ℝ) : cube.IntegralInput current Set.univ time := by refine ⟨?_, fun _ _ => Set.mem_univ _, ?_, ?_, ?_, ?_, ?_, ?_⟩ · refine ⟨measurableSet_Icc, ?_, ?_⟩ <;> simp [cube, Real.volume_Icc_pi] · change IntegrableOn (fun _ : Space => 2 + (time - 1) ^ 2) cube.interior exact integrableOn_const (by simp [cube, Real.volume_Icc_pi]) · simp_rw [actual_time_derivative] exact integrableOn_const (by simp [cube, Real.volume_Icc_pi]) · rw [show spatialDivergence current time = (fun _ => -2 * (time - 1)) from funext (actual_spatial_divergence time)] exact integrableOn_const (by simp [cube, Real.volume_Icc_pi]) · simp_rw [current_surface_density] exact integrable_finset_sum_measure.mpr (fun face _ => face_flux_is_integrable time face) · unfold Region.VolumeDifferentiation simp_rw [actual_time_derivative] rw [show cube.charge current = (fun time => 2 + (time - 1) ^ 2) from funext actual_charge] simp only [cube, Measure.real, Real.volume_Icc_pi, integral_const, Measure.restrict_apply_univ] convert ((((hasDerivAt_id time).sub_const 1).pow 2).const_add 2) using 1 norm_num · unfold Region.GaussCalculation simp_rw [actual_spatial_divergence] rw [actual_boundary_flux] simp [cube, Measure.real, Real.volume_Icc_pi] theorem library_derives_boundary_balance (time : ℝ) : cube.IntegralBalance current time := cube.local_balance_and_integral_inputs_give_equation_49a current Set.univ time library_current_has_local_balance.2 (actual_integral_inputs time) theorem flux_is_time_integrable (start finish : ℝ) : IntervalIntegrable (cube.inwardFlux current) volume start finish := by simp_rw [show cube.inwardFlux current = (fun time => 2 * (time - 1)) from funext actual_boundary_flux] exact (by fun_prop : Continuous (fun time : ℝ => 2 * (time - 1))).intervalIntegrable _ _ theorem library_computes_finite_time_transport (start finish : ℝ) : cube.netInwardTransport current start finish = (finish - 1) ^ 2 - (start - 1) ^ 2 := by rw [← cube.integral_balance_gives_finite_time_budget current start finish (fun time _ => library_derives_boundary_balance time) (flux_is_time_integrable start finish), actual_charge, actual_charge] ring theorem instantaneous_zero_flux_does_not_give_interval_conservation : (∀ time ∈ Set.Icc (0 : ℝ) 2, cube.IntegralBalance current time) ∧ cube.inwardFlux current 1 = 0 ∧ ¬∀ time ∈ Set.Icc (0 : ℝ) 2, cube.charge current time = cube.charge current 0 := by refine ⟨fun time _ => library_derives_boundary_balance time, ?_, ?_⟩ · rw [actual_boundary_flux] norm_num · intro constant have impossible := constant 1 (by norm_num) rw [actual_charge, actual_charge] at impossible norm_num at impossible exact (zero_ne_one : (0 : ℝ) ≠ 1) impossible theorem zero_accumulated_flux_does_not_give_interval_conservation : cube.netInwardTransport current 0 2 = 0 ∧ cube.charge current 2 = cube.charge current 0 ∧ cube.charge current 1 ≠ cube.charge current 0 := by have zero : cube.netInwardTransport current 0 2 = 0 := by rw [library_computes_finite_time_transport] norm_num refine ⟨zero, ?_, ?_⟩ · exact (cube.zero_net_transport_iff_equal_endpoint_charges current 0 2 (fun time _ => library_derives_boundary_balance time) (flux_is_time_integrable 0 2)).mp zero · rw [actual_charge, actual_charge] norm_num exact (zero_ne_one : (0 : ℝ) ≠ 1) theorem outward_positive_transport_depletes_charge : cube.outwardFlux current 0 = 2 ∧ cube.netInwardTransport current 0 1 = -1 ∧ cube.charge current 1 - cube.charge current 0 = -1 := by rw [cube.reversing_normals_reverses_flux] simp only [actual_boundary_flux, library_computes_finite_time_transport, actual_charge] norm_num exact zero_sub (1 : ℝ) theorem reversed_time_budget_has_the_opposite_sign : cube.netInwardTransport current 1 0 = 1 := by rw [cube.reversing_time_reverses_net_transport, library_computes_finite_time_transport] norm_num theorem certified_paper_balance_controls_the_same_finite_time_charge {T : PhysicsLogic.Papers.Einstein1916.RelativityTypes} (context : PhysicsLogic.Papers.Einstein1916.SemanticContext T) (certificate : PhysicsLogic.Papers.Einstein1916.routedArgument.RouteSemanticallyCertified context.interpretAtom .conservationAndMatter) (component : Fin 4) (region : Region) (start finish : ℝ) (inputs : ∀ time ∈ Set.uIcc start finish, region.IntegralInput (context.framework.selectedTotalEnergyCurrent context.metric context.energy component) {point | context.framework.coordinatePointAdmissibleIn point context.fieldCoordinates} time) (integrableFlux : IntervalIntegrable (region.inwardFlux (context.framework.selectedTotalEnergyCurrent context.metric context.energy component)) volume start finish) : region.charge (context.framework.selectedTotalEnergyCurrent context.metric context.energy component) finish - region.charge (context.framework.selectedTotalEnergyCurrent context.metric context.energy component) start = region.netInwardTransport (context.framework.selectedTotalEnergyCurrent context.metric context.energy component) start finish := context.certified_total_conservation_retains_finite_time_budget certificate component region start finish inputs integrableFlux end PhysicsLogic.IndependentReuse.BoundaryFluxClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.BoundaryFluxClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 26 do throwError "Missing independent boundary-flux theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent boundary-flux client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent boundary-flux client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.BoundaryFluxClient.library_derives_boundary_balance, ``PhysicsLogic.Papers.Einstein1916.CoordinateBalance.Region.local_balance_and_integral_inputs_give_equation_49a), (``PhysicsLogic.IndependentReuse.BoundaryFluxClient.library_computes_finite_time_transport, ``PhysicsLogic.Papers.Einstein1916.CoordinateBalance.Region.integral_balance_gives_finite_time_budget), (``PhysicsLogic.IndependentReuse.BoundaryFluxClient.zero_accumulated_flux_does_not_give_interval_conservation, ``PhysicsLogic.Papers.Einstein1916.CoordinateBalance.Region.zero_net_transport_iff_equal_endpoint_charges), (``PhysicsLogic.IndependentReuse.BoundaryFluxClient.outward_positive_transport_depletes_charge, ``PhysicsLogic.Papers.Einstein1916.CoordinateBalance.Region.reversing_normals_reverses_flux), (``PhysicsLogic.IndependentReuse.BoundaryFluxClient.reversed_time_budget_has_the_opposite_sign, ``PhysicsLogic.Papers.Einstein1916.CoordinateBalance.Region.reversing_time_reverses_net_transport), (``PhysicsLogic.IndependentReuse.BoundaryFluxClient.certified_paper_balance_controls_the_same_finite_time_charge, ``PhysicsLogic.Papers.Einstein1916.SemanticContext.certified_total_conservation_retains_finite_time_budget)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Boundary-flux client audit passed: {theoremCount} theorems, {uses.size} public API uses, six unit-area cube faces, inward unit normals, nonzero transport, positive charge, endpoint-versus-interval counterexamples, no production diagnostics." namespace PhysicsLogic.IndependentReuse.BohmPolarCurrentClient open PhysicsLogic.Framework PhysicsLogic.Papers.Bohm1952.PolarCurrent /-- A local travelling polar profile. Global probability normalization is separate. -/ noncomputable def travelling : Model where mass := 2 hbar := 1 density := fun point => (1 + (point 0 - 3 * point 3) ^ 2) ^ 2 amplitude := fun point => 1 + (point 0 - 3 * point 3) ^ 2 phase := fun point => 6 * point 0 - 9 * point 3 wave := fun point => ((1 + (point 0 - 3 * point 3) ^ 2 : ℝ) : ℂ) * Complex.exp (((6 * point 0 - 9 * point 3 : ℝ) : ℂ) * Complex.I) domain := Set.univ theorem actual_phase_gradient (point : Point) (axis : Fin 3) : phaseGradient travelling.phase point axis = if axis = 0 then 6 else 0 := by fin_cases axis · change deriv (fun value : ℝ => 6 * value - 9 * point 3) (point 0) = 6 simpa using (((hasDerivAt_id (point 0)).const_mul 6).sub_const (9 * point 3)).deriv · change deriv (fun _ : ℝ => 6 * point 0 - 9 * point 3) (point 1) = 0 simp · change deriv (fun _ : ℝ => 6 * point 0 - 9 * point 3) (point 2) = 0 simp theorem actual_amplitude_derivative (point : Point) (axis : Fin 3) : CoordinateCurrent.coordinateDerivative travelling.amplitude point axis.castSucc = if axis = 0 then 2 * (point 0 - 3 * point 3) else 0 := by fin_cases axis <;> simp [travelling, CoordinateCurrent.coordinateDerivative, Function.update] theorem actual_probability_current (point : Point) : travelling.probabilityCurrent point = ![3 * travelling.density point, 0, 0, travelling.density point] := by unfold Model.probabilityCurrent current change ![travelling.density point * phaseGradient travelling.phase point 0 / 2, travelling.density point * phaseGradient travelling.phase point 1 / 2, travelling.density point * phaseGradient travelling.phase point 2 / 2, travelling.density point] = _ simp_rw [actual_phase_gradient] ext axis fin_cases axis <;> simp [Matrix.cons_val_two, Matrix.cons_val_three, show (1 : Fin 3) ≠ 0 by decide, show (2 : Fin 3) ≠ 0 by decide] all_goals ring theorem current_is_regular (point : Point) : CoordinateCurrent.RegularAt travelling.probabilityCurrent point := by intro axis simp_rw [actual_probability_current] fin_cases axis <;> simp [travelling, Function.update] all_goals fun_prop theorem all_field_derivatives_are_regular : travelling.RegularDomain := by refine ⟨isOpen_univ, Set.univ_nonempty, fun point _ => ⟨?_, ?_, current_is_regular point⟩⟩ · intro axis refine ⟨?_, ?_, ?_⟩ <;> fin_cases axis <;> dsimp [travelling, Function.update] all_goals fun_prop · intro axis simp_rw [actual_amplitude_derivative] fin_cases axis <;> simp [Function.update] theorem polar_representation_is_realized : travelling.PolarRepresentation := by intro point _ refine ⟨by dsimp [travelling]; positivity, ?_⟩ simp [travelling] theorem density_is_the_squared_amplitude : travelling.DensityAmplitudeIdentity := by intro point _ rfl theorem amplitude_has_no_nodes : travelling.NodeFree := by intro point _ have positive : 0 < travelling.amplitude point := by dsimp [travelling]; positivity exact positive.ne' theorem actual_density_time_derivative (point : Point) : CoordinateCurrent.coordinateDerivative travelling.density point 3 = -12 * (point 0 - 3 * point 3) * (1 + (point 0 - 3 * point 3) ^ 2) := by change deriv (fun value : ℝ => (1 + (point 0 - 3 * value) ^ 2) ^ 2) (point 3) = _ convert ((((((hasDerivAt_id (point 3)).const_mul 3).const_sub (point 0)).pow 2).const_add 1).pow 2).deriv using 1 simp ring theorem actual_spatial_current_divergence (point : Point) : spatialDivergence travelling.mass travelling.density travelling.phase point = 12 * (point 0 - 3 * point 3) * (1 + (point 0 - 3 * point 3) ^ 2) := by unfold spatialDivergence change (∑ axis : Fin 3, CoordinateCurrent.coordinateDerivative (fun location => travelling.probabilityCurrent location axis.castSucc) point axis.castSucc) = _ simp_rw [actual_probability_current] simp only [Fin.sum_univ_three] change deriv (fun value : ℝ => 3 * (1 + (value - 3 * point 3) ^ 2) ^ 2) (point 0) + deriv (fun _ : ℝ => (0 : ℝ)) (point 1) + deriv (fun _ : ℝ => (0 : ℝ)) (point 2) = _ simp only [deriv_const, add_zero] convert ((((((hasDerivAt_id (point 0)).sub_const (3 * point 3)).pow 2).const_add 1).pow 2).const_mul 3).deriv using 1 simp ring theorem actual_equation_five : travelling.EquationFive := by intro point _ rw [actual_density_time_derivative, actual_spatial_current_divergence] ring theorem library_derives_actual_born_current : travelling.BornCurrentResult := travelling.polar_equations_give_born_current (by norm_num [travelling]) (by norm_num [travelling]) all_field_derivatives_are_regular amplitude_has_no_nodes polar_representation_is_realized density_is_the_squared_amplitude actual_equation_five theorem library_identifies_actual_born_density (point : Point) : travelling.density point = Complex.normSq (travelling.wave point) := travelling.polar_representation_identifies_born_density polar_representation_is_realized density_is_the_squared_amplitude point (Set.mem_univ _) theorem library_retains_positive_density (point : Point) : 0 < travelling.density point := travelling.born_current_has_positive_density library_derives_actual_born_current point (Set.mem_univ _) theorem library_current_uses_mass_normalized_velocity (point : Point) : travelling.velocity point 0 = 3 ∧ travelling.probabilityCurrent point 0 = travelling.density point * travelling.velocity point 0 := by refine ⟨?_, travelling.current_is_density_times_guidance_velocity point 0⟩ simp only [Model.velocity, actual_phase_gradient] norm_num [travelling] rfl theorem density_evolves_despite_local_conservation : CoordinateCurrent.NonvacuousLocalBalance travelling.probabilityCurrent travelling.domain ∧ travelling.density (atTime 0 0) ≠ travelling.density (atTime 1 0) := by refine ⟨travelling.equation_five_gives_actual_local_balance all_field_derivatives_are_regular actual_equation_five, ?_⟩ norm_num [travelling, atTime, Matrix.cons_val_three] theorem amplitude_laplacian_is_computed (point : Point) : spatialLaplacian travelling.amplitude point = 2 := by unfold spatialLaplacian simp_rw [actual_amplitude_derivative] simp [Fin.sum_univ_three, CoordinateCurrent.coordinateDerivative, Function.update] theorem actual_hamilton_jacobi_balance (point : Point) : CoordinateCurrent.coordinateDerivative travelling.phase point 3 + (∑ axis : Fin 3, phaseGradient travelling.phase point axis ^ 2) / (2 * travelling.mass) + 1 / (2 * travelling.amplitude point) - travelling.hbar ^ 2 / (2 * travelling.mass) * (spatialLaplacian travelling.amplitude point / travelling.amplitude point) = 0 := by have timeDerivative : CoordinateCurrent.coordinateDerivative travelling.phase point 3 = -9 := by change deriv (fun value : ℝ => 6 * point 0 - 9 * value) (point 3) = -9 simpa using (((hasDerivAt_id (point 3)).const_mul 9).const_sub (6 * point 0)).deriv rw [amplitude_laplacian_is_computed, timeDerivative] simp_rw [actual_phase_gradient] simp [travelling] ring noncomputable def wrongMass : Model := { travelling with mass := 1 } theorem wrong_mass_changes_actual_continuity : ¬wrongMass.EquationFive := by intro equation have wrong := equation (atTime 0 ![1, 0, 0]) (Set.mem_univ _) have currentFormula : ∀ point, wrongMass.probabilityCurrent point = ![6 * travelling.density point, 0, 0, travelling.density point] := by intro point unfold Model.probabilityCurrent current change ![travelling.density point * phaseGradient travelling.phase point 0 / 1, travelling.density point * phaseGradient travelling.phase point 1 / 1, travelling.density point * phaseGradient travelling.phase point 2 / 1, travelling.density point] = _ simp_rw [actual_phase_gradient] ext axis fin_cases axis <;> simp [Matrix.cons_val_two, Matrix.cons_val_three, show (1 : Fin 3) ≠ 0 by decide, show (2 : Fin 3) ≠ 0 by decide] all_goals ring change CoordinateCurrent.coordinateDerivative travelling.density (atTime 0 ![1, 0, 0]) 3 + (∑ axis : Fin 3, CoordinateCurrent.coordinateDerivative (fun point => wrongMass.probabilityCurrent point axis.castSucc) (atTime 0 ![1, 0, 0]) axis.castSucc) = 0 at wrong rw [actual_density_time_derivative] at wrong simp_rw [currentFormula] at wrong simp only [Fin.sum_univ_three] at wrong change -12 * ((1 : ℝ) - 3 * 0) * (1 + ((1 : ℝ) - 3 * 0) ^ 2) + (deriv (fun value : ℝ => 6 * (1 + (value - 3 * 0) ^ 2) ^ 2) 1 + deriv (fun _ : ℝ => (0 : ℝ)) 0 + deriv (fun _ : ℝ => (0 : ℝ)) 0) = 0 at wrong simp only [mul_zero, sub_zero] at wrong have calculated : deriv (fun value : ℝ => 6 * (1 + value ^ 2) ^ 2) 1 = 48 := by convert (((((hasDerivAt_id (1 : ℝ)).pow 2).const_add 1).pow 2).const_mul 6).deriv using 1 norm_num rw [calculated] at wrong norm_num at wrong exact (by norm_num : (24 : ℝ) ≠ 0) wrong theorem certified_source_supplies_the_same_actual_current {T : PhysicsLogic.Papers.Bohm1952.BohmTypes} (context : PhysicsLogic.Papers.Bohm1952.SemanticContext T) (certificate : PhysicsLogic.Papers.Bohm1952.routedArgument.RouteSemanticallyCertified context.interpretAtom .schrodingerDecomposition) : context.framework.coordinatePolarModel.BornCurrentResult := context.certified_polar_route_retains_actual_born_current certificate end PhysicsLogic.IndependentReuse.BohmPolarCurrentClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.BohmPolarCurrentClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 19 do throwError "Missing independent Bohm polar-current theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent Bohm polar-current client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent Bohm polar-current client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.BohmPolarCurrentClient.library_derives_actual_born_current, ``PhysicsLogic.Papers.Bohm1952.PolarCurrent.Model.polar_equations_give_born_current), (``PhysicsLogic.IndependentReuse.BohmPolarCurrentClient.library_identifies_actual_born_density, ``PhysicsLogic.Papers.Bohm1952.PolarCurrent.Model.polar_representation_identifies_born_density), (``PhysicsLogic.IndependentReuse.BohmPolarCurrentClient.library_retains_positive_density, ``PhysicsLogic.Papers.Bohm1952.PolarCurrent.Model.born_current_has_positive_density), (``PhysicsLogic.IndependentReuse.BohmPolarCurrentClient.library_current_uses_mass_normalized_velocity, ``PhysicsLogic.Papers.Bohm1952.PolarCurrent.Model.current_is_density_times_guidance_velocity), (``PhysicsLogic.IndependentReuse.BohmPolarCurrentClient.density_evolves_despite_local_conservation, ``PhysicsLogic.Papers.Bohm1952.PolarCurrent.Model.equation_five_gives_actual_local_balance), (``PhysicsLogic.IndependentReuse.BohmPolarCurrentClient.certified_source_supplies_the_same_actual_current, ``PhysicsLogic.Papers.Bohm1952.SemanticContext.certified_polar_route_retains_actual_born_current), (``PhysicsLogic.IndependentReuse.BohmPolarCurrentClient.certified_source_supplies_the_same_actual_current, ``PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework.printed_polar_equations_give_actual_born_current)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Bohm polar-current client audit passed: {theoremCount} theorems, {uses.size} public API uses, actual time-dependent density, phase gradient, nonzero current, amplitude Laplacian, Hamilton-Jacobi balance and wrong-mass rejection, no production diagnostics." namespace PhysicsLogic.IndependentReuse.BohmGuidanceFlowClient open MeasureTheory ProbabilityTheory PhysicsLogic.Framework open PhysicsLogic.Papers.Bohm1952.GuidanceFlow /-- A one-dimensional normalized kinematic test, not an asserted free Schrodinger solution. -/ def phase (speed time position : ℝ) : ℝ := 2 * speed * position - speed ^ 2 * time noncomputable def wave (speed time position : ℝ) : ℂ := (Real.sqrt (gaussianPDFReal (speed * time) 1 position) : ℂ) * Complex.exp ((phase speed time position : ℂ) * Complex.I) theorem actual_born_density (speed time position : ℝ) : Complex.normSq (wave speed time position) = gaussianPDFReal (speed * time) 1 position := by rw [wave, Complex.normSq_mul, Complex.normSq_eq_norm_sq, Complex.norm_real, Complex.normSq_eq_norm_sq, Complex.norm_exp_ofReal_mul_I] simp only [Real.norm_eq_abs, sq_abs, one_pow, mul_one] exact Real.sq_sqrt (gaussianPDFReal_nonneg _ _ _) theorem wave_measurable (speed time : ℝ) : AEStronglyMeasurable (wave speed time) volume := by unfold wave phase fun_prop theorem actual_born_measure (speed time : ℝ) : GuidedDensityTransport.bornMeasure volume (wave speed time) = gaussianReal (speed * time) 1 := by rw [gaussianReal_of_var_ne_zero _ (by norm_num)] unfold GuidedDensityTransport.bornMeasure congr 1 funext position rw [actual_born_density] rfl noncomputable def model (speed : ℝ) : GuidedDensityTransport.Model ℝ ℝ where coordinateMeasure := volume wave := wave speed ensemble := fun time => gaussianReal (speed * time) 1 evolve := fun initial later position => position + speed * (later - initial) theorem born_law (speed time : ℝ) : (model speed).bornLaw time = gaussianReal (speed * time) 1 := actual_born_measure speed time theorem normalized (speed time : ℝ) : (model speed).NormalizedBornStateAt time := by have probability : IsProbabilityMeasure (GuidedDensityTransport.bornMeasure volume (wave speed time)) := by rw [actual_born_measure] infer_instance exact ⟨GuidedDensityTransport.measurable_wave_with_normalized_born_measure_is_square_integrable volume (wave speed time) (wave_measurable speed time) probability, probability⟩ theorem phase_determines_velocity (speed time position : ℝ) : deriv (phase speed time) position / 2 = speed := by have derivative := (((hasDerivAt_id position).const_mul (2 * speed)).sub_const (speed ^ 2 * time)).deriv simp only [id_eq, mul_one] at derivative change deriv (fun x => 2 * speed * x - speed ^ 2 * time) position / 2 = speed rw [derivative] ring noncomputable def history (speed initial : ℝ) : History 1 ℝ where position := fun time => (model speed).evolve 0 time initial coordinates := fun position _ => position fromCoordinates := fun coordinates => coordinates 0 velocity := fun time position _ => deriv (phase speed time) position / 2 admissible := fun time position => 0 < gaussianPDFReal (speed * time) 1 position theorem actual_history_solves_guidance (speed initial : ℝ) : (history speed initial).SolvesOn initial 0 1 := by refine ⟨by norm_num, by norm_num, by simp [history, model], ?_⟩ intro time _inside refine ⟨gaussianPDFReal_pos _ _ _ (by norm_num), rfl, ?_⟩ intro axis change HasDerivAt (fun value => initial + speed * (value - 0)) (deriv (phase speed time) _ / 2) time rw [phase_determines_velocity] simpa using (((hasDerivAt_id time).const_mul speed).const_add initial) theorem library_recovers_actual_motion (initial time : ℝ) (inside : time ∈ Set.uIcc (0 : ℝ) 1) : deriv (fun value => (model 1).evolve 0 value initial) time = 1 := by have result := (history 1 initial).guidance_is_the_actual_trajectory_derivative (actual_history_solves_guidance 1 initial) time inside 0 simpa only [history, phase_determines_velocity] using result theorem common_transport (speed initial later : ℝ) : (model speed).CommonTransportAt initial later := by have transported : (gaussianReal (speed * initial) 1).map ((model speed).evolve initial later) = gaussianReal (speed * later) 1 := by change (gaussianReal (speed * initial) 1).map (fun position => position + speed * (later - initial)) = _ rw [gaussianReal_map_add_const] congr 1 ring refine ⟨by unfold model; fun_prop, by unfold model; fun_prop, ?_, ?_⟩ · rw [born_law, born_law] exact transported.symm · exact transported.symm theorem library_transports_the_actual_born_law (speed initial later : ℝ) : (model speed).QuantumEquilibriumAt later := by apply (model speed).common_configuration_transport_preserves_quantum_equilibrium initial later (common_transport speed initial later) exact (born_law speed initial).symm theorem moving_law_is_not_stationary : (model 1).bornLaw 0 ≠ (model 1).bornLaw 1 := by intro same rw [born_law, born_law] at same have means := congrArg (fun law : Measure ℝ => ∫ position, position ∂law) same simp only [integral_id_gaussianReal, mul_zero, mul_one] at means exact (zero_ne_one : (0 : ℝ) ≠ 1) means noncomputable def excursionModel : GuidedDensityTransport.Model ℝ ℝ := { model 0 with evolve := fun initial later position => position + (later - initial) * (1 - (later - initial)) } noncomputable def excursionHistory (initial : ℝ) : History 1 ℝ := { history 0 initial with position := fun time => excursionModel.evolve 0 time initial } theorem correct_endpoint_pushforwards : excursionModel.CommonTransportAt 0 1 := by have same : excursionModel.evolve 0 1 = (model 0).evolve 0 1 := by funext position simp [excursionModel, model] unfold GuidedDensityTransport.Model.CommonTransportAt rw [same] exact common_transport 0 0 1 theorem library_rejects_the_wrong_intermediate_motion (initial : ℝ) : (excursionHistory initial).position 0 = initial ∧ (excursionHistory initial).position 1 = initial ∧ ¬(excursionHistory initial).SolvesOn initial 0 1 := by refine ⟨by simp [excursionHistory, excursionModel], by simp [excursionHistory, excursionModel], ?_⟩ intro solution have constant := (excursionHistory initial).zero_guidance_keeps_the_actual_configuration_fixed solution (by intros; exact phase_determines_velocity 0 _ _) (1 / 2) (by norm_num [Set.uIcc]) change initial + (1 / 2 - 0 : ℝ) * (1 - (1 / 2 - 0)) = initial at constant linarith theorem normalized_endpoint_agreement_is_not_a_guided_flow : excursionModel.NormalizedBornStateAt 0 ∧ excursionModel.NormalizedBornStateAt 1 ∧ excursionModel.CommonTransportAt 0 1 ∧ excursionModel.QuantumEquilibriumAt 1 ∧ ∀ initial, ¬(excursionHistory initial).SolvesOn initial 0 1 := by refine ⟨normalized 0 0, normalized 0 1, correct_endpoint_pushforwards, ?_, ?_⟩ · exact library_transports_the_actual_born_law 0 0 1 · exact fun initial => (library_rejects_the_wrong_intermediate_motion initial).2.2 theorem source_certificate_requires_actual_guidance {T : PhysicsLogic.Papers.Bohm1952.BohmTypes} (context : PhysicsLogic.Papers.Bohm1952.SemanticContext T) (certificate : PhysicsLogic.Papers.Bohm1952.routedArgument.RouteSemanticallyCertified context.interpretAtom .equilibriumStatistics) : context.framework.ActualGuidedEnsemble := context.certified_equilibrium_route_retains_actual_guidance certificate theorem source_calculation_links_the_actual_flow_and_current {T : PhysicsLogic.Papers.Bohm1952.BohmTypes} (framework : PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework T) (setup : framework.EquivarianceSetup) (calculation : framework.EquivarianceCalculation) : framework.actualTransportModel.NonvacuousBornEquivarianceAt framework.initialTime framework.selectedTime ∧ framework.ActualGuidedEnsemble := framework.the_actual_guided_ensemble_has_born_equivariance_and_current_velocity setup calculation noncomputable def collapsedCoordinates (initial : ℝ) : History 1 ℝ := { history 0 initial with position := fun time => initial + time coordinates := fun _ _ => 0 fromCoordinates := fun _ => initial } theorem constant_coordinate_readouts_do_not_certify_a_moving_configuration (initial : ℝ) : (∀ time, HasDerivAt (fun value => (collapsedCoordinates initial).coordinates ((collapsedCoordinates initial).position value) 0) 0 time) ∧ ¬(collapsedCoordinates initial).SolvesOn initial 0 1 := by refine ⟨fun time => hasDerivAt_const time 0, ?_⟩ intro solution have decoder := (solution.2.2.2 1 Set.right_mem_uIcc).2.1 change initial = initial + 1 at decoder linarith end PhysicsLogic.IndependentReuse.BohmGuidanceFlowClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.BohmGuidanceFlowClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 15 do throwError "Missing independent Bohm guidance-flow theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent Bohm guidance-flow client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent Bohm guidance-flow client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.BohmGuidanceFlowClient.library_recovers_actual_motion, ``PhysicsLogic.Papers.Bohm1952.GuidanceFlow.History.guidance_is_the_actual_trajectory_derivative), (``PhysicsLogic.IndependentReuse.BohmGuidanceFlowClient.library_rejects_the_wrong_intermediate_motion, ``PhysicsLogic.Papers.Bohm1952.GuidanceFlow.History.zero_guidance_keeps_the_actual_configuration_fixed), (``PhysicsLogic.IndependentReuse.BohmGuidanceFlowClient.library_transports_the_actual_born_law, ``PhysicsLogic.Framework.GuidedDensityTransport.Model.common_configuration_transport_preserves_quantum_equilibrium), (``PhysicsLogic.IndependentReuse.BohmGuidanceFlowClient.normalized, ``PhysicsLogic.Framework.GuidedDensityTransport.measurable_wave_with_normalized_born_measure_is_square_integrable), (``PhysicsLogic.IndependentReuse.BohmGuidanceFlowClient.source_certificate_requires_actual_guidance, ``PhysicsLogic.Papers.Bohm1952.SemanticContext.certified_equilibrium_route_retains_actual_guidance), (``PhysicsLogic.IndependentReuse.BohmGuidanceFlowClient.source_calculation_links_the_actual_flow_and_current, ``PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework.the_actual_guided_ensemble_has_born_equivariance_and_current_velocity), (``PhysicsLogic.IndependentReuse.BohmGuidanceFlowClient.source_calculation_links_the_actual_flow_and_current, ``PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework.actual_equivariance_uses_the_phase_gradient_guidance_flow)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Bohm guidance-flow client audit passed: {theoremCount} theorems, {uses.size} public API uses, normalized moving Gaussian, actual phase-derived trajectories, false endpoint-only flow rejected, no production diagnostics." namespace PhysicsLogic.IndependentReuse.BohmTrajectoryUniquenessClient open PhysicsLogic.Papers.Bohm1952.GuidanceFlow open PhysicsLogic.Paradigm.Dynamics /-- A mass-one kinematic phase; this test does not assert Schrodinger evolution. -/ def phase (position : ℝ) : ℝ := position ^ 2 theorem actual_phase_gradient (position : ℝ) : deriv phase position = 2 * position := by unfold phase convert (hasDerivAt_pow 2 position).deriv using 1 norm_num noncomputable def history (start initial : ℝ) : History 1 ℝ where position := fun time => initial * Real.exp (2 * (time - start)) coordinates := fun position _ => position fromCoordinates := fun coordinates => coordinates 0 velocity := fun _ position _ => deriv phase position admissible := fun _ position => 0 < position theorem the_history_has_actual_initial_value_and_velocity (start finish initial : ℝ) (distinct : start ≠ finish) (positive : 0 < initial) : (history start initial).SolvesOn initial start finish := by refine ⟨by norm_num, distinct, by simp [history], ?_⟩ intro time _inside refine ⟨mul_pos positive (Real.exp_pos _), rfl, ?_⟩ intro axis change HasDerivAt (fun value => initial * Real.exp (2 * (value - start))) (deriv phase (initial * Real.exp (2 * (time - start)))) time rw [actual_phase_gradient] have derivative := (((((hasDerivAt_id time).sub_const start).const_mul 2).exp).const_mul initial) simp only [id_eq, mul_one] at derivative convert derivative using 1 ring theorem actual_velocity_has_a_uniform_spatial_bound (start finish initial : ℝ) : (history start initial).LipschitzVelocityOn start finish := by refine ⟨2, ?_⟩ intro time _inside have vectorField : (history start initial).coordinateVelocity time = fun position : Fin 1 → ℝ => (2 : ℝ) • position := by funext position axis fin_cases axis simp [History.coordinateVelocity, history, actual_phase_gradient] rw [vectorField] convert (lipschitzWith_smul (β := Fin 1 → ℝ) (2 : ℝ)).lipschitzOnWith using 1 norm_num theorem library_identifies_every_admissible_trajectory (start finish initial : ℝ) (distinct : start ≠ finish) (positive : 0 < initial) (candidate : ℝ → ℝ) (solution : ((history start initial).withPosition candidate).SolvesOn initial start finish) : Set.EqOn candidate (history start initial).position (Set.uIcc start finish) := (history start initial).lipschitz_guidance_has_one_actual_history (the_history_has_actual_initial_value_and_velocity start finish initial distinct positive) (actual_velocity_has_a_uniform_spatial_bound start finish initial) candidate solution theorem library_retains_actual_continuity : ContinuousOn (fun time => (history 0 2).coordinates ((history 0 2).position time)) (Set.uIcc (0 : ℝ) 1) := (history 0 2).actual_coordinate_history_is_continuous (the_history_has_actual_initial_value_and_velocity 0 1 2 (by norm_num) (by norm_num)) theorem reverse_time_uses_the_same_uniqueness_theorem (candidate : ℝ → ℝ) (solution : ((history 1 2).withPosition candidate).SolvesOn 2 1 0) : candidate 0 = 2 * Real.exp (-2) := by have same := library_identifies_every_admissible_trajectory 1 0 2 (by norm_num) (by norm_num) candidate solution Set.right_mem_uIcc simpa [history] using same def encode (position : ℝ) : ℝ := 3 * position + 5 noncomputable def decode (state : ℝ) : ℝ := (state - 5) / 3 theorem physical_configuration_is_not_erased : Function.LeftInverse decode encode := by intro position unfold decode encode ring noncomputable def stateDynamics : Dynamics ℝ ℝ where evolves := fun time initial final => initial = encode 2 ∧ time = 1 ∧ ∃ candidate : ℝ → ℝ, ((history 0 2).withPosition candidate).SolvesOn 2 0 1 ∧ final = encode (candidate time) theorem selected_state_evolution_exists : stateDynamics.HasForwardEvolutionAt 1 (encode 2) := ⟨encode ((history 0 2).position 1), rfl, rfl, (history 0 2).position, the_history_has_actual_initial_value_and_velocity 0 1 2 (by norm_num) (by norm_num), rfl⟩ theorem state_determinism_uses_differential_uniqueness : stateDynamics.ForwardDeterministic := by rintro time initial first second ⟨_initial, timeIdentity, firstPath, firstSolution, firstIdentity⟩ ⟨_otherInitial, _otherTime, secondPath, secondSolution, secondIdentity⟩ subst time rw [firstIdentity, secondIdentity, library_identifies_every_admissible_trajectory 0 1 2 (by norm_num) (by norm_num) firstPath firstSolution Set.right_mem_uIcc, library_identifies_every_admissible_trajectory 0 1 2 (by norm_num) (by norm_num) secondPath secondSolution Set.right_mem_uIcc] theorem library_derives_a_nonvacuous_unique_physical_successor : stateDynamics.UniqueForwardEvolutionAt 1 (encode 2) := forward_existence_and_determinism_establish_unique_evolution_at stateDynamics 1 (encode 2) selected_state_evolution_exists state_determinism_uses_differential_uniqueness theorem library_recovers_the_actual_configuration_from_every_successor (final : ℝ) (evolution : stateDynamics.evolves 1 (encode 2) final) : decode final = (history 0 2).position 1 := by obtain ⟨_, _, candidate, solution, identity⟩ := evolution rw [identity, physical_configuration_is_not_erased] exact library_identifies_every_admissible_trajectory 0 1 2 (by norm_num) (by norm_num) candidate solution Set.right_mem_uIcc theorem selected_interval_existence_is_not_global_time_existence : stateDynamics.UniqueForwardEvolutionAt 1 (encode 2) ∧ ¬stateDynamics.HasForwardEvolutionAt 2 (encode 2) := by refine ⟨library_derives_a_nonvacuous_unique_physical_successor, ?_⟩ rintro ⟨final, _, impossible, _⟩ norm_num at impossible theorem matching_initial_position_does_not_supply_guidance : ¬((history 0 2).withPosition (fun _ => 2)).SolvesOn 2 0 1 := by intro solution have derivative := (solution.2.2.2 0 Set.left_mem_uIcc).2.2 0 have incompatible := derivative.unique (hasDerivAt_const (0 : ℝ) (2 : ℝ)) change deriv phase 2 = 0 at incompatible rw [actual_phase_gradient] at incompatible norm_num at incompatible theorem actual_source_calculation_uses_the_same_initial_value_problem {T : PhysicsLogic.Papers.Bohm1952.BohmTypes} (framework : PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework T) (inputs : framework.ActualTrajectoryCalculationInputs) : framework.ActualSelectedTrajectory := framework.trajectory_inputs_determine_the_actual_state inputs theorem source_certificate_retains_the_actual_particle_trajectory {T : PhysicsLogic.Papers.Bohm1952.BohmTypes} (context : PhysicsLogic.Papers.Bohm1952.SemanticContext T) (certificate : PhysicsLogic.Papers.Bohm1952.routedArgument.RouteSemanticallyCertified context.interpretAtom .configurationGuidance) : context.framework.ActualSelectedTrajectory := context.certified_guidance_route_retains_actual_selected_trajectory certificate end PhysicsLogic.IndependentReuse.BohmTrajectoryUniquenessClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.BohmTrajectoryUniquenessClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 15 do throwError "Missing independent Bohm trajectory-uniqueness theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent Bohm trajectory client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent Bohm trajectory client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.BohmTrajectoryUniquenessClient.library_identifies_every_admissible_trajectory, ``PhysicsLogic.Papers.Bohm1952.GuidanceFlow.History.lipschitz_guidance_has_one_actual_history), (``PhysicsLogic.IndependentReuse.BohmTrajectoryUniquenessClient.library_retains_actual_continuity, ``PhysicsLogic.Papers.Bohm1952.GuidanceFlow.History.actual_coordinate_history_is_continuous), (``PhysicsLogic.IndependentReuse.BohmTrajectoryUniquenessClient.library_derives_a_nonvacuous_unique_physical_successor, ``PhysicsLogic.Paradigm.Dynamics.forward_existence_and_determinism_establish_unique_evolution_at), (``PhysicsLogic.IndependentReuse.BohmTrajectoryUniquenessClient.reverse_time_uses_the_same_uniqueness_theorem, ``PhysicsLogic.Papers.Bohm1952.GuidanceFlow.History.lipschitz_guidance_has_one_actual_history), (``PhysicsLogic.IndependentReuse.BohmTrajectoryUniquenessClient.actual_source_calculation_uses_the_same_initial_value_problem, ``PhysicsLogic.Papers.Bohm1952.HiddenVariableFramework.trajectory_inputs_determine_the_actual_state), (``PhysicsLogic.IndependentReuse.BohmTrajectoryUniquenessClient.actual_source_calculation_uses_the_same_initial_value_problem, ``PhysicsLogic.Papers.Bohm1952.GuidanceFlow.History.lipschitz_guidance_has_one_actual_history), (``PhysicsLogic.IndependentReuse.BohmTrajectoryUniquenessClient.source_certificate_retains_the_actual_particle_trajectory, ``PhysicsLogic.Papers.Bohm1952.SemanticContext.certified_guidance_route_retains_actual_selected_trajectory)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Bohm trajectory-uniqueness client audit passed: {theoremCount} theorems, {uses.size} public API uses, nonconstant phase-gradient flow, both time orientations, invertible physical-state encoding and no global-time inference, no production diagnostics." namespace PhysicsLogic.IndependentReuse.FPUContinuousEnergyClient open PhysicsLogic.Papers.FermiPastaUlam1955 open HamiltonianEnergy SineCoordinateConvention open Finset noncomputable section def interaction : Interaction := ⟨.quadratic, 1 / 7, 0, 0, 0⟩ def frequency : Real := Real.sqrt 2 def position (time : Real) : Real := (3 / 2) * Real.cos (frequency * time) def speed (time : Real) : Real := -(3 / 2) * frequency * Real.sin (frequency * time) /-- A nonconstant one-interior-site diagnostic. The nonlinear terms of its two bond forces cancel in the net force, not separately in either force. -/ def chain : ContinuousChain where interaction := interaction endpointSpan := 2 displacement := fun site time => if site = 1 then position time else 0 velocity := fun site time => if site = 1 then speed time else 0 theorem frequency_squared : frequency ^ 2 = 2 := Real.sq_sqrt (by norm_num) theorem actual_position_and_velocity_derivatives (time : Real) : HasDerivAt position (speed time) time ∧ HasDerivAt speed (-2 * position time) time := by constructor · convert ((Real.hasDerivAt_cos (frequency * time)).comp time ((hasDerivAt_id time).const_mul frequency)).const_mul (3 / 2) using 1 dsimp [speed] ring · convert ((Real.hasDerivAt_sin (frequency * time)).comp time ((hasDerivAt_id time).const_mul frequency)).const_mul (-(3 / 2) * frequency) using 1 dsimp [position] calc -2 * (3 / 2 * Real.cos (frequency * time)) = -(3 / 2) * frequency ^ 2 * Real.cos (frequency * time) := by rw [frequency_squared]; ring _ = -(3 / 2) * frequency * (Real.cos (frequency * time) * (frequency * 1)) := by ring theorem the_selected_chain_has_actual_fixed_endpoints (time : Real) : (chain.phase time).FixedEndpointsAt := by norm_num [ContinuousChain.phase, chain, FixedEndPhase.FixedEndpointsAt] theorem the_nonlinear_bonds_give_the_actual_net_force (time : Real) : interaction.particleForce (chain.phase time) 1 = -2 * position time := by simp only [Interaction.particleForce, Interaction.bondStretch, ContinuousChain.phase, chain, interaction, Interaction.force] norm_num ring theorem actual_history_solves_the_source_force_equations (time : Real) : chain.NewtonAt time := by refine ⟨by norm_num [chain], ?_, ?_⟩ · intro site _ by_cases moving : site = 1 · subst site exact (actual_position_and_velocity_derivatives time).1 simpa [chain, moving] using hasDerivAt_const time (0 : Real) · intro site first last have selected : site = 1 := by change site < 2 at last; omega subst site change HasDerivAt speed (interaction.particleForce (chain.phase time) 1) time rw [the_nonlinear_bonds_give_the_actual_net_force] exact (actual_position_and_velocity_derivatives time).2 theorem library_derives_actual_zero_energy_derivative (time : Real) : HasDerivAt chain.energy 0 time := chain.fixed_end_newton_motion_has_zero_energy_derivative time (actual_history_solves_the_source_force_equations time) (the_selected_chain_has_actual_fixed_endpoints time) theorem library_conserves_energy_at_every_selected_time (start finish : Real) : chain.energy finish = chain.energy start := chain.fixed_end_newton_motion_conserves_actual_hamiltonian start finish (fun time _ => actual_history_solves_the_source_force_equations time) (fun time _ => the_selected_chain_has_actual_fixed_endpoints time) finish Set.right_mem_uIcc theorem the_actual_initial_energy_is_positive : chain.energy 0 = 9 / 4 := by norm_num [ContinuousChain.energy, ContinuousChain.phase, Interaction.hamiltonian, chain, FixedEndPhase.particleKineticEnergy, Interaction.potential, Interaction.nonlinearPotential, Interaction.bondStretch, interaction, position, speed, sum_range_succ] theorem the_library_retains_positive_energy_for_a_moving_solution (time : Real) : chain.NewtonAt time ∧ (chain.phase time).FixedEndpointsAt ∧ chain.energy time = 9 / 4 := by refine ⟨actual_history_solves_the_source_force_equations time, the_selected_chain_has_actual_fixed_endpoints time, ?_⟩ exact (library_conserves_energy_at_every_selected_time 0 time).trans the_actual_initial_energy_is_positive def halfPeriod : Real := Real.pi / frequency theorem energy_conservation_does_not_keep_configuration_fixed : chain.energy halfPeriod = chain.energy 0 ∧ chain.displacement 1 halfPeriod ≠ chain.displacement 1 0 := by refine ⟨library_conserves_energy_at_every_selected_time 0 halfPeriod, ?_⟩ have positive : 0 < frequency := Real.sqrt_pos.mpr (by norm_num) have phase : frequency * halfPeriod = Real.pi := by dsimp [halfPeriod] field_simp norm_num [chain, position, phase] theorem neither_initial_bond_force_is_its_linear_part : interaction.force (position 0) ≠ position 0 ∧ interaction.force (-position 0) ≠ -position 0 := by norm_num [interaction, Interaction.force, position] /-- The initial phase point at every displayed time, frozen instead of following the derivative law. The conserved-energy test cannot detect this failure. -/ def frozen : ContinuousChain := { chain with displacement := fun site _ => chain.displacement site 0 velocity := fun _ _ => 0 } theorem constant_positive_energy_does_not_validate_the_dynamics : (∀ time, (frozen.phase time).FixedEndpointsAt ∧ frozen.energy time = 9 / 4) ∧ ¬frozen.NewtonAt 0 := by have same : ∀ time, frozen.phase time = chain.phase 0 := by intro time simp [frozen, ContinuousChain.phase, chain, speed] constructor · intro time exact ⟨same time ▸ the_selected_chain_has_actual_fixed_endpoints 0, (congrArg (fun phase => interaction.hamiltonian phase) (same time)).trans the_actual_initial_energy_is_positive⟩ · intro newton have derivative := newton.2.2 1 (by norm_num) (by norm_num [frozen, chain]) have force : frozen.interaction.particleForce (frozen.phase 0) 1 = -3 := by change interaction.particleForce (frozen.phase 0) 1 = -3 rw [same, the_nonlinear_bonds_give_the_actual_net_force] norm_num [position] rw [force] at derivative have impossible := derivative.unique (hasDerivAt_const (0 : Real) (0 : Real)) norm_num at impossible theorem source_equation_uses_the_same_energy_balance {T : FPUTypes} (framework : NonlinearChainFramework T) (run : T.Run) (equation : framework.EquationOneQuadraticChainFormulaAt run) (start finish : Real) : (framework.continuousChain run).energy finish = (framework.continuousChain run).energy start := framework.equation_one_conserves_the_actual_nonlinear_hamiltonian run equation start finish end end PhysicsLogic.IndependentReuse.FPUContinuousEnergyClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.FPUContinuousEnergyClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 13 do throwError "Missing independent FPU energy theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent FPU energy client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent FPU energy client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.FPUContinuousEnergyClient.library_derives_actual_zero_energy_derivative, ``PhysicsLogic.Papers.FermiPastaUlam1955.HamiltonianEnergy.ContinuousChain.newton_equations_give_actual_energy_balance), (``PhysicsLogic.IndependentReuse.FPUContinuousEnergyClient.library_conserves_energy_at_every_selected_time, ``PhysicsLogic.Papers.FermiPastaUlam1955.HamiltonianEnergy.ContinuousChain.fixed_end_newton_motion_conserves_actual_hamiltonian), (``PhysicsLogic.IndependentReuse.FPUContinuousEnergyClient.source_equation_uses_the_same_energy_balance, ``PhysicsLogic.Papers.FermiPastaUlam1955.NonlinearChainFramework.equation_one_realizes_actual_newton_motion)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"FPU continuous-energy client audit passed: {theoremCount} theorems, {uses.size} public API uses, actual nonconstant fixed-end solution, positive energy and nonlinear bond forces, frozen-history rejection, no production diagnostics." namespace PhysicsLogic.IndependentReuse.JarzynskiLiouvilleClient open MeasureTheory ProbabilityTheory PhysicsLogic.Papers.Jarzynski1997 noncomputable section abbrev Phase := Real × Real /-- The time-one free-flight canonical shear, in position and momentum. -/ def shear : Phase ≃ᵐ Phase where toFun := fun point => (point.1 + point.2, point.2) invFun := fun point => (point.1 - point.2, point.2) left_inv := by rintro ⟨position, momentum⟩; simp right_inv := by rintro ⟨position, momentum⟩; simp measurable_toFun := (measurable_fst.add measurable_snd).prodMk measurable_snd measurable_invFun := (measurable_fst.sub measurable_snd).prodMk measurable_snd def transport : PhaseTransport Phase where phaseVolume := (volume : Measure Real).prod volume evolution := shear initialDensity := fun point => gaussianPDFReal 0 1 point.1 * gaussianPDFReal 0 1 point.2 def energy (point : Phase) : Real := (point.1 ^ 2 + point.2 ^ 2) / 2 def work (point : Phase) : Real := energy point - energy (shear.symm point) theorem the_initial_density_is_canonical_for_the_same_energy (point : Phase) : transport.initialDensity point = (2 * Real.pi)⁻¹ * Real.exp (-energy point) := by have square : Real.sqrt (2 * Real.pi) * Real.sqrt (2 * Real.pi) = 2 * Real.pi := Real.mul_self_sqrt (by positivity) change gaussianPDFReal 0 1 point.1 * gaussianPDFReal 0 1 point.2 = _ simp only [gaussianPDFReal, sub_zero, NNReal.coe_one, mul_one] calc _ = (Real.sqrt (2 * Real.pi) * Real.sqrt (2 * Real.pi))⁻¹ * (Real.exp (-point.1 ^ 2 / 2) * Real.exp (-point.2 ^ 2 / 2)) := by rw [mul_inv_rev] ring _ = (2 * Real.pi)⁻¹ * Real.exp (-energy point) := by rw [square, ← Real.exp_add] congr 2 dsimp [energy] ring theorem the_actual_shear_is_free_flight (initial : Phase) (time : Real) : HasDerivAt (fun value => initial.1 + value * initial.2) initial.2 time ∧ HasDerivAt (fun _ : Real => initial.2) 0 time ∧ shear initial = (initial.1 + 1 * initial.2, initial.2) := by refine ⟨?_, hasDerivAt_const time _, ?_⟩ · simpa using ((hasDerivAt_id time).mul_const initial.2).const_add initial.1 · simp [shear] theorem the_actual_shear_preserves_canonical_phase_volume : transport.LiouvilleAt := measurePreserving_add_prod (volume : Measure Real) volume theorem the_canonical_initial_phase_density_is_normalized : transport.ProbabilityDensityAt := by refine ⟨(measurable_gaussianPDFReal 0 1 |>.comp measurable_fst).mul (measurable_gaussianPDFReal 0 1 |>.comp measurable_snd), ?_, ?_, ?_⟩ · intro point exact mul_nonneg (gaussianPDFReal_nonneg 0 1 point.1) (gaussianPDFReal_nonneg 0 1 point.2) · exact (integrable_gaussianPDFReal 0 1).mul_prod (integrable_gaussianPDFReal 0 1) · change (∫ point, gaussianPDFReal 0 1 point.1 * gaussianPDFReal 0 1 point.2 ∂(volume : Measure Real).prod volume) = 1 have product := integral_prod_mul (μ := (volume : Measure Real)) (ν := (volume : Measure Real)) (gaussianPDFReal 0 1) (gaussianPDFReal 0 1) exact product.trans (by rw [integral_gaussianPDFReal_eq_one 0 (by norm_num)]; norm_num) theorem library_normalizes_the_actual_final_ensemble : IsProbabilityMeasure transport.finalLaw := transport.actual_pushforward_preserves_probability the_canonical_initial_phase_density_is_normalized theorem the_selected_work_uses_the_actual_inverse_trajectory (point : Phase) : work point = (point.1 ^ 2 - (point.1 - point.2) ^ 2) / 2 := by dsimp [work, energy, shear] ring theorem gaussian_weight_and_work_cancel_on_the_same_trajectory (point : Phase) : gaussianPDFReal 0 1 (point.1 - point.2) * Real.exp (-work point) = gaussianPDFReal 0 1 point.1 := by simp only [gaussianPDFReal, sub_zero, NNReal.coe_one, mul_one] rw [mul_assoc, ← Real.exp_add] congr 1 congr 1 rw [the_selected_work_uses_the_actual_inverse_trajectory] ring theorem actual_phase_density_work_cancellation (point : Phase) : transport.advectedDensity point * Real.exp (-work point) = transport.initialDensity point := by change (gaussianPDFReal 0 1 (point.1 - point.2) * gaussianPDFReal 0 1 point.2) * Real.exp (-work point) = gaussianPDFReal 0 1 point.1 * gaussianPDFReal 0 1 point.2 calc _ = (gaussianPDFReal 0 1 (point.1 - point.2) * Real.exp (-work point)) * gaussianPDFReal 0 1 point.2 := by ring _ = _ := by rw [gaussian_weight_and_work_cancel_on_the_same_trajectory] theorem library_derives_the_exact_exponential_work_average : (∫ point, Real.exp (-work point) ∂transport.finalLaw) = 1 := by rw [transport.liouville_transport_integrates_final_observables the_canonical_initial_phase_density_is_normalized the_actual_shear_preserves_canonical_phase_volume] have same := MeasureTheory.integral_congr_ae (μ := transport.phaseVolume) (Filter.Eventually.of_forall actual_phase_density_work_cancellation) exact same.trans the_canonical_initial_phase_density_is_normalized.2.2.2 theorem the_exponential_work_moment_is_genuinely_integrable : Integrable (fun point => Real.exp (-work point)) transport.finalLaw := by by_contra notIntegrable have zero := integral_undef notIntegrable rw [library_derives_the_exact_exponential_work_average] at zero exact one_ne_zero zero def ensemble : PhysicsLogic.Framework.ClassicalWorkEnsemble Phase where measurableSpace := inferInstance measure := transport.finalLaw theorem the_shared_work_ensemble_is_normalized_with_finite_exponential_moment : ensemble.ExponentialWorkDomainAt 1 work := by refine ⟨library_normalizes_the_actual_final_ensemble, ?_⟩ simpa [PhysicsLogic.Framework.ClassicalWorkEnsemble.IntegrableAt, ensemble] using the_exponential_work_moment_is_genuinely_integrable theorem work_can_have_either_sign_in_the_exact_identity : (∫ point, Real.exp (-work point) ∂transport.finalLaw) = 1 ∧ work (0, 1) = -1 / 2 ∧ work (1, 1) = 1 / 2 := by refine ⟨library_derives_the_exact_exponential_work_average, ?_, ?_⟩ <;> norm_num [the_selected_work_uses_the_actual_inverse_trajectory] theorem the_advected_density_is_not_the_final_canonical_density : transport.advectedDensity (1, 1) > transport.initialDensity (1, 1) := by change gaussianPDFReal 0 1 ((1 : Real) - 1) * gaussianPDFReal 0 1 1 > gaussianPDFReal 0 1 1 * gaussianPDFReal 0 1 1 rw [sub_self] have comparison : gaussianPDFReal 0 1 1 < gaussianPDFReal 0 1 0 := by have exponential : Real.exp (-(1 : Real) ^ 2 / 2) < Real.exp (-(0 : Real) ^ 2 / 2) := Real.exp_lt_exp.mpr (by norm_num) have scale : 0 < (Real.sqrt (2 * Real.pi))⁻¹ := by positivity simp only [gaussianPDFReal, sub_zero, NNReal.coe_one, mul_one] nlinarith have positive := gaussianPDFReal_pos 0 1 1 (by norm_num) nlinarith theorem the_direct_paper_api_uses_actual_transport {Configuration Protocol Realization : Type} (framework : SwitchingFramework Configuration Protocol Realization) (protocol : Protocol) (canonical : framework.CanonicalInitialDensityEquation protocol) (liouville : framework.LiouvilleDensityTransportEquation protocol) (workEnergy : framework.WorkEnergyDifferenceEquation protocol) (finalIntegral : framework.FinalBoltzmannIntegralEquation protocol) (ratio : framework.SystemPartitionRatioDefinition protocol) (partition : framework.SystemPartitionRatioEncodesFreeEnergy protocol) : framework.JarzynskiEquality protocol := framework.actual_isolated_transport_yields_jarzynski protocol canonical liouville workEnergy finalIntegral ratio partition end end PhysicsLogic.IndependentReuse.JarzynskiLiouvilleClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.JarzynskiLiouvilleClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 14 do throwError "Missing independent Jarzynski phase-transport theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent Jarzynski phase client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent Jarzynski phase client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.JarzynskiLiouvilleClient.library_normalizes_the_actual_final_ensemble, ``PhysicsLogic.Papers.Jarzynski1997.PhaseTransport.actual_pushforward_preserves_probability), (``PhysicsLogic.IndependentReuse.JarzynskiLiouvilleClient.library_derives_the_exact_exponential_work_average, ``PhysicsLogic.Papers.Jarzynski1997.PhaseTransport.liouville_transport_integrates_final_observables), (``PhysicsLogic.IndependentReuse.JarzynskiLiouvilleClient.the_direct_paper_api_uses_actual_transport, ``PhysicsLogic.Papers.Jarzynski1997.SwitchingFramework.actual_liouville_transport_derives_equation_six), (``PhysicsLogic.IndependentReuse.JarzynskiLiouvilleClient.the_direct_paper_api_uses_actual_transport, ``PhysicsLogic.Papers.Jarzynski1997.SwitchingFramework.actual_phase_integral_scaling)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Jarzynski Liouville client audit passed: {theoremCount} theorems, {uses.size} public API uses, canonical Gaussian phase density, actual free-flight shear and finite fluctuating work moment, noncanonical final density, no production diagnostics." namespace PhysicsLogic.IndependentReuse.JarzynskiHamiltonianClient open MeasureTheory ProbabilityTheory PhysicsLogic.Papers.Jarzynski1997 open HamiltonianWork noncomputable section def system : Hamiltonian 1 where energy := fun control point => (1 + control) * ((point.1 0) ^ 2 + (point.2 0) ^ 2) / 2 controlGradient := fun _ point => ((point.1 0) ^ 2 + (point.2 0) ^ 2) / 2 positionGradient := fun control point i => (1 + control) * point.1 i momentumGradient := fun control point i => (1 + control) * point.2 i def positionProjection : (Real × Phase 1) →L[Real] Real := (ContinuousLinearMap.proj 0).comp ((ContinuousLinearMap.fst Real (Fin 1 → Real) (Fin 1 → Real)).comp (ContinuousLinearMap.snd Real Real (Phase 1))) def momentumProjection : (Real × Phase 1) →L[Real] Real := (ContinuousLinearMap.proj 0).comp ((ContinuousLinearMap.snd Real (Fin 1 → Real) (Fin 1 → Real)).comp (ContinuousLinearMap.snd Real Real (Phase 1))) theorem the_selected_gradients_differentiate_the_actual_hamiltonian (control : Real) (point : Phase 1) : system.DifferentialAt control point := by have differential := ((((ContinuousLinearMap.fst Real Real (Phase 1)).hasFDerivAt (x := (control, point))).const_add 1).mul ((positionProjection.hasFDerivAt.pow 2).add (momentumProjection.hasFDerivAt.pow 2))).mul_const (1 / 2 : Real) have calibrated := differential.congr_fderiv (g' := canonicalDifferential (system.controlGradient control point) (system.positionGradient control point) (system.momentumGradient control point)) (by apply ContinuousLinearMap.ext intro variation simp [canonicalDifferential, system, positionProjection, momentumProjection] ring) simpa [Hamiltonian.DifferentialAt, system, positionProjection, momentumProjection, div_eq_mul_inv] using calibrated def duration : Real := 4 * Real.pi / 3 def angle (time : Real) : Real := time + time ^ 2 / (2 * duration) def history : SwitchingHistory 1 where hamiltonian := system duration := duration evolution := MeasurableEquiv.refl (Phase 1) trajectory := fun point time => (fun _ => point.1 0 * Real.cos (angle time) + point.2 0 * Real.sin (angle time), fun _ => point.2 0 * Real.cos (angle time) - point.1 0 * Real.sin (angle time)) theorem the_switching_duration_is_positive : 0 < duration := by unfold duration; positivity theorem the_rotation_angle_has_the_controlled_rate (time : Real) : HasDerivAt angle (1 + time / duration) time := by convert (hasDerivAt_id time).add (((hasDerivAt_id time).pow 2).div_const (2 * duration)) using 1 simp only [id_eq, mul_one] field_simp ring theorem the_actual_trajectory_satisfies_hamilton_equations (point : Phase 1) (time : Real) : history.HamiltonEquationsAt point time := by refine ⟨the_selected_gradients_differentiate_the_actual_hamiltonian _ _, ?_⟩ have phase := the_rotation_angle_has_the_controlled_rate time have position := (phase.cos.const_mul (point.1 0)).add (phase.sin.const_mul (point.2 0)) have momentum := (phase.cos.const_mul (point.2 0)).sub (phase.sin.const_mul (point.1 0)) convert (hasDerivAt_pi.mpr (fun _ : Fin 1 => position)).prodMk (hasDerivAt_pi.mpr (fun _ : Fin 1 => momentum)) using 1 ext i <;> simp [history, system, Hamiltonian.velocity, SwitchingHistory.parameter] <;> ring theorem the_actual_trajectory_preserves_the_unscaled_quadratic_form (point : Phase 1) (time : Real) : system.controlGradient (history.parameter time) (history.trajectory point time) = ((point.1 0) ^ 2 + (point.2 0) ^ 2) / 2 := by calc _ = (Real.cos (angle time) ^ 2 + Real.sin (angle time) ^ 2) * ((point.1 0) ^ 2 + (point.2 0) ^ 2) / 2 := by dsimp [system, history]; ring _ = _ := by rw [Real.cos_sq_add_sin_sq]; ring theorem the_selected_duration_completes_one_actual_orbit : angle duration = 2 * Real.pi := by have nonzero := ne_of_gt the_switching_duration_is_positive dsimp [angle] field_simp dsimp [duration] ring theorem the_work_power_is_constant_along_each_actual_orbit (point : Phase 1) (time : Real) : history.power point time = duration⁻¹ * (((point.1 0) ^ 2 + (point.2 0) ^ 2) / 2) := by exact congrArg (fun value => duration⁻¹ * value) (the_actual_trajectory_preserves_the_unscaled_quadratic_form point time) theorem the_finite_time_switching_protocol_is_realized : history.RealizesSwitching := by refine ⟨the_switching_duration_is_positive, fun point => ⟨?_, ?_, ?_, ?_⟩⟩ · ext i <;> fin_cases i <;> simp [history, angle] · ext i <;> fin_cases i <;> simp [history, the_selected_duration_completes_one_actual_orbit] · intro time _ exact the_actual_trajectory_satisfies_hamilton_equations point time · have constant : history.power point = fun _ => duration⁻¹ * (((point.1 0) ^ 2 + (point.2 0) ^ 2) / 2) := funext (the_work_power_is_constant_along_each_actual_orbit point) rw [constant] exact intervalIntegrable_const theorem library_recovers_equation_three_for_the_actual_protocol : history.toTrajectoryWorkFormula.EquationThreeAt := history.actual_history_supplies_equation_three the_finite_time_switching_protocol_is_realized theorem library_derives_the_actual_controlled_work (point : Phase 1) : history.work point = ((point.1 0) ^ 2 + (point.2 0) ^ 2) / 2 := by rw [history.actual_trajectory_work_equals_endpoint_energy_change the_finite_time_switching_protocol_is_realized] simp [history, system] ring end end PhysicsLogic.IndependentReuse.JarzynskiHamiltonianClient namespace PhysicsLogic.IndependentReuse.JarzynskiHamiltonianClient open MeasureTheory ProbabilityTheory PhysicsLogic.Papers.Jarzynski1997 open HamiltonianWork noncomputable section def coordinates : (Real × Real) ≃ Phase 1 where toFun := fun point => (fun _ => point.1, fun _ => point.2) invFun := fun point => (point.1 0, point.2 0) left_inv := fun _ => rfl right_inv := by intro point; ext i <;> fin_cases i <;> rfl def scalarEnergy (point : Real × Real) : Real := (point.1 ^ 2 + point.2 ^ 2) / 2 def actualWork (point : Real × Real) : Real := history.work (coordinates point) def transport : PhaseTransport (Real × Real) where phaseVolume := (volume : Measure Real).prod volume evolution := MeasurableEquiv.refl (Real × Real) initialDensity := fun point => gaussianPDFReal 0 1 point.1 * gaussianPDFReal 0 1 point.2 theorem the_phase_transport_has_the_actual_history_endpoints (point : Real × Real) : history.trajectory (coordinates point) 0 = coordinates point ∧ history.trajectory (coordinates point) history.duration = coordinates (transport.evolution point) := by obtain ⟨initial, final, _equations, _integrable⟩ := the_finite_time_switching_protocol_is_realized.2 (coordinates point) exact ⟨initial, final⟩ theorem the_protocol_preserves_phase_volume : transport.LiouvilleAt := MeasurePreserving.id _ theorem the_initial_density_is_normalized : transport.ProbabilityDensityAt := by refine ⟨(measurable_gaussianPDFReal 0 1 |>.comp measurable_fst).mul (measurable_gaussianPDFReal 0 1 |>.comp measurable_snd), ?_, ?_, ?_⟩ · intro point exact mul_nonneg (gaussianPDFReal_nonneg 0 1 point.1) (gaussianPDFReal_nonneg 0 1 point.2) · exact (integrable_gaussianPDFReal 0 1).mul_prod (integrable_gaussianPDFReal 0 1) · have product := integral_prod_mul (μ := (volume : Measure Real)) (ν := (volume : Measure Real)) (gaussianPDFReal 0 1) (gaussianPDFReal 0 1) exact product.trans (by rw [integral_gaussianPDFReal_eq_one 0 (by norm_num)]; norm_num) theorem the_initial_density_is_canonical_for_the_actual_hamiltonian (point : Real × Real) : transport.initialDensity point = (2 * Real.pi)⁻¹ * Real.exp (-system.energy 0 (coordinates point)) := by have square : Real.sqrt (2 * Real.pi) * Real.sqrt (2 * Real.pi) = 2 * Real.pi := Real.mul_self_sqrt (by positivity) change gaussianPDFReal 0 1 point.1 * gaussianPDFReal 0 1 point.2 = _ simp only [gaussianPDFReal, sub_zero, NNReal.coe_one, mul_one] calc _ = (Real.sqrt (2 * Real.pi) * Real.sqrt (2 * Real.pi))⁻¹ * (Real.exp (-point.1 ^ 2 / 2) * Real.exp (-point.2 ^ 2 / 2)) := by rw [mul_inv_rev]; ring _ = _ := by rw [square, ← Real.exp_add] congr 2 dsimp [system, coordinates] ring theorem the_actual_work_is_the_initial_quadratic_energy (point : Real × Real) : actualWork point = scalarEnergy point := library_derives_the_actual_controlled_work (coordinates point) theorem the_actual_final_boltzmann_integral_is_pi : (∫ point, Real.exp (-system.energy 1 (coordinates point)) ∂transport.phaseVolume) = Real.pi := by have factorization : (fun point : Real × Real => Real.exp (-system.energy 1 (coordinates point))) = fun point => Real.exp (-(1 : Real) * point.1 ^ 2) * Real.exp (-(1 : Real) * point.2 ^ 2) := by funext point rw [← Real.exp_add] congr 1 dsimp [system, coordinates] ring rw [factorization] change (∫ point, Real.exp (-(1 : Real) * point.1 ^ 2) * Real.exp (-(1 : Real) * point.2 ^ 2) ∂(volume : Measure Real).prod volume) = Real.pi have product := integral_prod_mul (μ := (volume : Measure Real)) (ν := (volume : Measure Real)) (fun value : Real => Real.exp (-(1 : Real) * value ^ 2)) (fun value : Real => Real.exp (-(1 : Real) * value ^ 2)) exact product.trans (by rw [integral_gaussian, div_one]; exact Real.mul_self_sqrt Real.pi_pos.le) theorem actual_work_cancels_the_transported_density (point : Real × Real) : transport.advectedDensity point * Real.exp (-actualWork point) = (2 * Real.pi)⁻¹ * Real.exp (-system.energy 1 (coordinates point)) := by change transport.initialDensity point * Real.exp (-actualWork point) = _ rw [the_initial_density_is_canonical_for_the_actual_hamiltonian, the_actual_work_is_the_initial_quadratic_energy, mul_assoc, ← Real.exp_add] congr 2 dsimp [system, coordinates, scalarEnergy] ring theorem library_derives_the_exact_driven_oscillator_work_identity : (∫ point, Real.exp (-actualWork point) ∂transport.finalLaw) = 1 / 2 := by rw [transport.liouville_transport_integrates_final_observables the_initial_density_is_normalized the_protocol_preserves_phase_volume] have cancellation := integral_congr_ae (μ := transport.phaseVolume) (Filter.Eventually.of_forall actual_work_cancels_the_transported_density) rw [cancellation] have scale := integral_const_mul (μ := transport.phaseVolume) (2 * Real.pi)⁻¹ (fun point : Real × Real => Real.exp (-system.energy 1 (coordinates point))) exact scale.trans (by rw [the_actual_final_boltzmann_integral_is_pi]; field_simp) theorem the_driven_work_moment_is_finite : Integrable (fun point => Real.exp (-actualWork point)) transport.finalLaw := by by_contra notIntegrable have zero := integral_undef notIntegrable rw [library_derives_the_exact_driven_oscillator_work_identity] at zero norm_num at zero theorem the_exact_work_average_has_the_correct_free_energy_change : (∫ point, Real.exp (-actualWork point) ∂transport.finalLaw) = Real.exp (-Real.log 2) := by rw [library_derives_the_exact_driven_oscillator_work_identity, Real.exp_neg, Real.exp_log (by norm_num)] norm_num theorem work_is_not_a_constant_of_the_phase_point : actualWork (0, 0) = 0 ∧ actualWork (1, 0) = 1 / 2 := by simp [the_actual_work_is_the_initial_quadratic_energy, scalarEnergy] theorem an_actual_intermediate_phase_point_moves : duration / 2 ∈ Set.Icc 0 duration ∧ (history.trajectory (coordinates (0, 1)) (duration / 2)).1 0 = 1 / 2 := by have angleMid : angle (duration / 2) = Real.pi - Real.pi / 6 := by dsimp [angle] field_simp dsimp [duration] ring refine ⟨⟨by linarith [the_switching_duration_is_positive], by linarith [the_switching_duration_is_positive]⟩, ?_⟩ simp [history, coordinates, angleMid, Real.sin_sub, Real.sin_pi_div_six] theorem the_direct_history_api_retains_the_actual_work_mechanism {Configuration Protocol : Type} (framework : SwitchingFramework Configuration Protocol (Phase 1)) (protocol : Protocol) (represented : framework.RepresentsHamiltonianHistory protocol history) (selected : framework.trajectoryWorkFormulaRepresentsAt history.toTrajectoryWorkFormula protocol) (canonical : framework.CanonicalInitialDensityEquation protocol) (liouville : framework.LiouvilleDensityTransportEquation protocol) (finalIntegral : framework.FinalBoltzmannIntegralEquation protocol) (ratio : framework.SystemPartitionRatioDefinition protocol) (partition : framework.SystemPartitionRatioEncodesFreeEnergy protocol) : framework.workDefinedByGeneralizedForceIntegral protocol ∧ framework.JarzynskiEquality protocol := framework.actual_hamiltonian_history_yields_jarzynski protocol history represented the_finite_time_switching_protocol_is_realized selected canonical liouville finalIntegral ratio partition theorem scaled_boltzmann_integrals_have_their_actual_values (scale : Real) (positive : 0 < scale) : (∫ point, Real.exp (-scale * scalarEnergy point) ∂transport.phaseVolume) = 2 * Real.pi / scale := by have factorization : (fun point : Real × Real => Real.exp (-scale * scalarEnergy point)) = fun point => Real.exp (-(scale / 2) * point.1 ^ 2) * Real.exp (-(scale / 2) * point.2 ^ 2) := by funext point rw [← Real.exp_add] congr 1 dsimp [scalarEnergy] ring rw [factorization] have product := integral_prod_mul (μ := (volume : Measure Real)) (ν := (volume : Measure Real)) (fun value : Real => Real.exp (-(scale / 2) * value ^ 2)) (fun value : Real => Real.exp (-(scale / 2) * value ^ 2)) exact product.trans (by rw [integral_gaussian, Real.mul_self_sqrt (by positivity)] field_simp) theorem the_initial_partition_integral_is_two_pi : (∫ point, Real.exp (-system.energy 0 (coordinates point)) ∂transport.phaseVolume) = 2 * Real.pi := by simpa [system, coordinates, scalarEnergy] using scaled_boltzmann_integrals_have_their_actual_values 1 (by norm_num) theorem the_actual_partition_ratio_matches_the_work_average : (∫ point, Real.exp (-actualWork point) ∂transport.finalLaw) = (∫ point, Real.exp (-system.energy 1 (coordinates point)) ∂transport.phaseVolume) / (∫ point, Real.exp (-system.energy 0 (coordinates point)) ∂transport.phaseVolume) := by rw [library_derives_the_exact_driven_oscillator_work_identity, the_actual_final_boltzmann_integral_is_pi, the_initial_partition_integral_is_two_pi] field_simp /-- A separately prepared canonical reference for the final Hamiltonian. -/ def finalCanonical : PhaseTransport (Real × Real) where phaseVolume := transport.phaseVolume evolution := MeasurableEquiv.refl (Real × Real) initialDensity := fun point => Real.pi⁻¹ * Real.exp (-system.energy 1 (coordinates point)) theorem the_final_canonical_reference_is_a_normalized_density : finalCanonical.ProbabilityDensityAt := by have normalized : (∫ point, finalCanonical.initialDensity point ∂finalCanonical.phaseVolume) = 1 := by have scale := integral_const_mul (μ := transport.phaseVolume) Real.pi⁻¹ (fun point : Real × Real => Real.exp (-system.energy 1 (coordinates point))) exact scale.trans (by rw [the_actual_final_boltzmann_integral_is_pi]; field_simp) refine ⟨?_, ?_, ?_, normalized⟩ · dsimp [finalCanonical, system, coordinates] fun_prop · intro point exact mul_nonneg (inv_nonneg.mpr Real.pi_pos.le) (Real.exp_pos _).le · by_contra notIntegrable have zero := integral_undef notIntegrable rw [normalized] at zero exact one_ne_zero zero theorem the_final_canonical_reference_has_a_different_observable_average : (∫ point, Real.exp (-actualWork point) ∂finalCanonical.initialLaw) = 2 / 3 := by rw [finalCanonical.initial_density_integrates_observables the_final_canonical_reference_is_a_normalized_density.1 the_final_canonical_reference_is_a_normalized_density.2.1] have weighting : (fun point : Real × Real => finalCanonical.initialDensity point * Real.exp (-actualWork point)) = fun point => Real.pi⁻¹ * Real.exp (-3 * scalarEnergy point) := by funext point change (Real.pi⁻¹ * Real.exp (-system.energy 1 (coordinates point))) * Real.exp (-actualWork point) = _ rw [the_actual_work_is_the_initial_quadratic_energy, mul_assoc, ← Real.exp_add] congr 2 dsimp [system, coordinates, scalarEnergy] ring rw [weighting] have scale := integral_const_mul (μ := transport.phaseVolume) Real.pi⁻¹ (fun point : Real × Real => Real.exp (-3 * scalarEnergy point)) exact scale.trans (by rw [scaled_boltzmann_integrals_have_their_actual_values 3 (by norm_num)]; field_simp) theorem the_actual_final_probability_law_is_not_final_equilibrium : transport.finalLaw ≠ finalCanonical.initialLaw := by intro same have observed := library_derives_the_exact_driven_oscillator_work_identity rw [same, the_final_canonical_reference_has_a_different_observable_average] at observed norm_num at observed theorem the_actual_work_experiment_has_a_normalized_probability_law : IsProbabilityMeasure transport.finalLaw := transport.actual_pushforward_preserves_probability the_initial_density_is_normalized end end PhysicsLogic.IndependentReuse.JarzynskiHamiltonianClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.JarzynskiHamiltonianClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 30 do throwError "Missing independent Jarzynski Hamiltonian-history theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent Jarzynski history client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent Jarzynski history client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.JarzynskiHamiltonianClient.library_recovers_equation_three_for_the_actual_protocol, ``PhysicsLogic.Papers.Jarzynski1997.HamiltonianWork.SwitchingHistory.actual_history_supplies_equation_three), (``PhysicsLogic.IndependentReuse.JarzynskiHamiltonianClient.library_derives_the_actual_controlled_work, ``PhysicsLogic.Papers.Jarzynski1997.HamiltonianWork.SwitchingHistory.hamilton_equations_give_actual_energy_derivative), (``PhysicsLogic.IndependentReuse.JarzynskiHamiltonianClient.library_derives_the_exact_driven_oscillator_work_identity, ``PhysicsLogic.Papers.Jarzynski1997.HamiltonianWork.SwitchingHistory.actual_trajectory_work_equals_endpoint_energy_change), (``PhysicsLogic.IndependentReuse.JarzynskiHamiltonianClient.library_derives_the_exact_driven_oscillator_work_identity, ``PhysicsLogic.Papers.Jarzynski1997.PhaseTransport.liouville_transport_integrates_final_observables), (``PhysicsLogic.IndependentReuse.JarzynskiHamiltonianClient.the_direct_history_api_retains_the_actual_work_mechanism, ``PhysicsLogic.Papers.Jarzynski1997.SwitchingFramework.actual_hamiltonian_history_supplies_work_energy_balance)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Jarzynski Hamiltonian-history client audit passed: {theoremCount} theorems, {uses.size} public API uses, actual finite-time driven oscillator and equation-(3) work, canonical initial law and computed endpoint partitions, exact work average 1/2, final probability law not final equilibrium, no production diagnostics." namespace PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient open MeasureTheory PhysicsLogic.Papers.Jarzynski1997 HamiltonianWork noncomputable section def system : Hamiltonian 2 where energy := fun control point => control + ((point.1 0) ^ 2 + (point.1 1) ^ 2 + (point.2 0) ^ 2 + (point.2 1) ^ 2) / 2 controlGradient := fun _ _ => 1 positionGradient := fun _ point => point.1 momentumGradient := fun _ point => point.2 def positionProjection (i : Fin 2) : (Real × Phase 2) →L[Real] Real := (ContinuousLinearMap.proj i).comp ((ContinuousLinearMap.fst Real (Fin 2 → Real) (Fin 2 → Real)).comp (ContinuousLinearMap.snd Real Real (Phase 2))) def momentumProjection (i : Fin 2) : (Real × Phase 2) →L[Real] Real := (ContinuousLinearMap.proj i).comp ((ContinuousLinearMap.snd Real (Fin 2 → Real) (Fin 2 → Real)).comp (ContinuousLinearMap.snd Real Real (Phase 2))) theorem gradients_differentiate_the_full_energy (control : Real) (point : Phase 2) : system.DifferentialAt control point := by have quadratic := (((((positionProjection 0).hasFDerivAt (x := (control, point))).pow 2).add ((positionProjection 1).hasFDerivAt.pow 2)).add ((momentumProjection 0).hasFDerivAt.pow 2)).add ((momentumProjection 1).hasFDerivAt.pow 2) have differential := ((ContinuousLinearMap.fst Real Real (Phase 2)).hasFDerivAt (x := (control, point))).add (quadratic.mul_const (1 / 2 : Real)) have calibrated := differential.congr_fderiv (g' := canonicalDifferential (system.controlGradient control point) (system.positionGradient control point) (system.momentumGradient control point)) (by apply ContinuousLinearMap.ext intro variation simp [canonicalDifferential, system, positionProjection, momentumProjection, Fin.sum_univ_two] ring) simpa [Hamiltonian.DifferentialAt, system, positionProjection, momentumProjection, div_eq_mul_inv] using calibrated def coupling : SystemReservoirCoupling (Phase 2) where SystemPhase := Real × Real ReservoirPhase := Real × Real split := { toFun := fun point => ((point.1 0, point.2 0 + point.2 1), (point.1 1 - point.1 0, point.2 1)) invFun := fun point => (![point.1.1, point.1.1 + point.2.1], ![point.1.2 - point.2.2, point.2.2]) left_inv := fun point => by ext i <;> fin_cases i <;> simp right_inv := fun point => by rcases point with ⟨⟨qs, ps⟩, ⟨qb, pb⟩⟩; simp } systemEnergy := fun control point => control + point.1 ^ 2 + point.2 ^ 2 / 2 reservoirEnergy := fun point => point.1 ^ 2 / 2 + point.2 ^ 2 interactionEnergy := fun system bath => system.1 * bath.1 - system.2 * bath.2 theorem the_subsystem_split_preserves_canonical_pairing (left right : Phase 2) : (left.1 0 * right.2 0 - left.2 0 * right.1 0) + (left.1 1 * right.2 1 - left.2 1 * right.1 1) = ((coupling.split left).1.1 * (coupling.split right).1.2 - (coupling.split left).1.2 * (coupling.split right).1.1) + ((coupling.split left).2.1 * (coupling.split right).2.2 - (coupling.split left).2.2 * (coupling.split right).2.1) := by change _ = (left.1 0 * (right.2 0 + right.2 1) - (left.2 0 + left.2 1) * right.1 0) + ((left.1 1 - left.1 0) * right.2 1 - left.2 1 * (right.1 1 - right.1 0)) ring theorem the_interaction_is_part_of_the_actual_hamiltonian : coupling.DescribesHamiltonian system := by intro control point dsimp [system, coupling, SystemReservoirCoupling.totalEnergy] ring theorem the_initial_combined_energy_is_nonnegative (point : Phase 2) : 0 ≤ system.energy 0 point := by dsimp [system] positivity def evolution : Phase 2 ≃ᵐ Phase 2 where toFun := fun point => (point.2, -point.1) invFun := fun point => (-point.2, point.1) left_inv := fun point => by simp right_inv := fun point => by simp measurable_toFun := by fun_prop measurable_invFun := by fun_prop def duration : Real := Real.pi / 2 def history : SwitchingHistory 2 where hamiltonian := system duration := duration evolution := evolution trajectory := fun point time => (fun i => point.1 i * Real.cos (time - duration) + point.2 i * Real.sin (time - duration), fun i => point.2 i * Real.cos (time - duration) - point.1 i * Real.sin (time - duration)) theorem the_history_satisfies_full_hamilton_equations (point : Phase 2) (time : Real) : history.HamiltonEquationsAt point time := by refine ⟨gradients_differentiate_the_full_energy _ _, ?_⟩ have phase := (hasDerivAt_id time).sub_const duration have position := fun i : Fin 2 => (phase.cos.const_mul (point.1 i)).add (phase.sin.const_mul (point.2 i)) have momentum := fun i : Fin 2 => (phase.cos.const_mul (point.2 i)).sub (phase.sin.const_mul (point.1 i)) convert (hasDerivAt_pi.mpr position).prodMk (hasDerivAt_pi.mpr momentum) using 1 ext i <;> simp [history, system, Hamiltonian.velocity] <;> ring theorem the_switching_history_is_realized : history.RealizesSwitching := by refine ⟨by dsimp [history, duration]; positivity, fun point => ⟨?_, ?_, ?_, ?_⟩⟩ · ext i <;> simp [history, evolution, duration] · ext i <;> simp [history] · intro time _ exact the_history_satisfies_full_hamilton_equations point time · have power : history.power point = fun _ => duration⁻¹ := by funext time simp [SwitchingHistory.power, history, system] rw [power] exact intervalIntegrable_const theorem library_derives_system_equation_three : (coupling.toTrajectoryWorkFormula history).EquationThreeAt := coupling.actual_coupled_history_supplies_equation_three history the_interaction_is_part_of_the_actual_hamiltonian the_switching_history_is_realized theorem library_computes_system_controlled_work (point : Phase 2) : coupling.systemWork history point = 1 := by rw [coupling.controlled_system_work_equals_total_energy_change history the_interaction_is_part_of_the_actual_hamiltonian the_switching_history_is_realized] simp [system, history, evolution] ring def finalPoint : Phase 2 := (0, ![-1, -1]) theorem the_selected_trajectory_has_nonzero_initial_positions : history.evolution.symm finalPoint = (![1, 1], 0) := by ext i <;> fin_cases i <;> norm_num [history, evolution, finalPoint] <;> rfl theorem the_three_energy_changes_are_distinct : coupling.systemEnergyChange history finalPoint = 2 ∧ coupling.reservoirEnergyChange history finalPoint = 1 ∧ coupling.interactionEnergyChange history finalPoint = -2 := by unfold SystemReservoirCoupling.systemEnergyChange SystemReservoirCoupling.reservoirEnergyChange SystemReservoirCoupling.interactionEnergyChange rw [the_selected_trajectory_has_nonzero_initial_positions] change ((1 : Real) + 0 ^ 2 + (-1 + -1) ^ 2 / 2) - (0 + 1 ^ 2 + (0 + 0) ^ 2 / 2) = 2 ∧ ((0 - 0) ^ 2 / 2 + (-1) ^ 2) - ((1 - 1) ^ 2 / 2 + 0 ^ 2) = 1 ∧ (0 * (0 - 0) - (-1 + -1) * -1) - (1 * (1 - 1) - (0 + 0) * 0) = -2 norm_num exact ⟨sub_zero _, zero_add _⟩ theorem library_balance_keeps_the_interaction_energy : coupling.systemWork history finalPoint = coupling.systemEnergyChange history finalPoint + coupling.reservoirEnergyChange history finalPoint + coupling.interactionEnergyChange history finalPoint := coupling.controlled_work_resolves_all_three_energy_changes history the_interaction_is_part_of_the_actual_hamiltonian the_switching_history_is_realized finalPoint theorem actual_controlled_work_is_not_reduced_system_energy_change : coupling.systemWork history finalPoint ≠ coupling.systemEnergyChange history finalPoint := coupling.reservoir_and_interaction_exchange_obstruct_reduced_energy_work history the_interaction_is_part_of_the_actual_hamiltonian the_switching_history_is_realized finalPoint (by rw [the_three_energy_changes_are_distinct.2.1, the_three_energy_changes_are_distinct.2.2]; norm_num) theorem omitting_interaction_energy_breaks_the_balance : coupling.systemWork history finalPoint ≠ coupling.systemEnergyChange history finalPoint + coupling.reservoirEnergyChange history finalPoint := by rw [library_computes_system_controlled_work, the_three_energy_changes_are_distinct.1, the_three_energy_changes_are_distinct.2.1] norm_num theorem the_direct_paper_api_uses_system_work_and_full_dynamics {Configuration Protocol : Type} (framework : SwitchingFramework Configuration Protocol (Phase 2)) (protocol : Protocol) (represented : framework.RepresentsCoupledHamiltonianHistory protocol coupling history) (selected : framework.trajectoryWorkFormulaRepresentsAt (coupling.toTrajectoryWorkFormula history) protocol) (canonical : framework.FullCanonicalInitialDensityEquation protocol) (liouville : framework.FullLiouvilleDensityTransportEquation protocol) (finalIntegral : framework.FullFinalBoltzmannIntegralEquation protocol) (ratio : framework.FullPartitionRatioDefinition protocol) : framework.systemReservoirHamiltonianEmbedding protocol ∧ framework.workDefinedByGeneralizedForceIntegral protocol ∧ framework.workEqualsTotalEnergyChange protocol ∧ framework.ExponentialWorkAverage protocol = framework.fullPartitionRatio protocol := framework.actual_coupled_history_yields_full_partition_ratio protocol coupling history represented the_interaction_is_part_of_the_actual_hamiltonian the_switching_history_is_realized selected canonical liouville finalIntegral ratio abbrev Mode := Real × Real abbrev FullPhase := Mode × Mode def coordinates : FullPhase ≃ Phase 2 where toFun := fun point => (![point.1.1, point.2.1], ![point.1.2, point.2.2]) invFun := fun point => ((point.1 0, point.2 0), (point.1 1, point.2 1)) left_inv := fun _ => rfl right_inv := fun point => by ext i <;> fin_cases i <;> rfl abbrev modeVolume : Measure Mode := (volume : Measure Real).prod volume def modeEnergy (point : Mode) : Real := (point.1 ^ 2 + point.2 ^ 2) / 2 def modeEvolution : Mode ≃ᵐ Mode where toFun := fun point => (point.2, -point.1) invFun := fun point => (-point.2, point.1) left_inv := fun point => by simp right_inv := fun point => by simp measurable_toFun := by fun_prop measurable_invFun := by fun_prop def transport : PhaseTransport FullPhase where phaseVolume := modeVolume.prod modeVolume evolution := modeEvolution.prodCongr modeEvolution initialDensity := fun point => ((2 * Real.pi) ^ 2)⁻¹ * Real.exp (-system.energy 0 (coordinates point)) def actualWork (point : FullPhase) : Real := coupling.systemWork history (coordinates point) theorem scalar_coordinates_intertwine_the_actual_evolution (point : FullPhase) : coordinates (transport.evolution point) = history.evolution (coordinates point) ∧ coordinates (transport.evolution.symm point) = history.evolution.symm (coordinates point) := by constructor <;> ext i <;> fin_cases i <;> rfl theorem transported_endpoints_are_the_same_hamiltonian_history (point : FullPhase) : history.trajectory (coordinates point) 0 = coordinates (transport.evolution.symm point) ∧ history.trajectory (coordinates point) history.duration = coordinates point := by obtain ⟨initial, final, _⟩ := the_switching_history_is_realized.2 (coordinates point) exact ⟨initial.trans (scalar_coordinates_intertwine_the_actual_evolution point).2.symm, final⟩ theorem each_mode_rotation_preserves_actual_volume : MeasurePreserving modeEvolution modeVolume modeVolume := by exact ((MeasurePreserving.id (volume : Measure Real)).prod (Measure.measurePreserving_neg (volume : Measure Real))).comp Measure.measurePreserving_swap theorem the_combined_evolution_is_liouville : transport.LiouvilleAt := each_mode_rotation_preserves_actual_volume.prod each_mode_rotation_preserves_actual_volume theorem one_mode_has_partition_two_pi : (∫ point, Real.exp (-modeEnergy point) ∂modeVolume) = 2 * Real.pi := by have factorization : (fun point : Mode => Real.exp (-modeEnergy point)) = fun point => Real.exp (-(1 / 2 : Real) * point.1 ^ 2) * Real.exp (-(1 / 2 : Real) * point.2 ^ 2) := by funext point rw [← Real.exp_add] congr 1 dsimp [modeEnergy] ring rw [factorization] have product := integral_prod_mul (μ := (volume : Measure Real)) (ν := (volume : Measure Real)) (fun value : Real => Real.exp (-(1 / 2 : Real) * value ^ 2)) (fun value : Real => Real.exp (-(1 / 2 : Real) * value ^ 2)) exact product.trans (by rw [integral_gaussian, Real.mul_self_sqrt (by positivity)]; ring) theorem the_initial_full_partition_is_computed : (∫ point, Real.exp (-system.energy 0 (coordinates point)) ∂transport.phaseVolume) = (2 * Real.pi) ^ 2 := by have factorization : (fun point : FullPhase => Real.exp (-system.energy 0 (coordinates point))) = fun point => Real.exp (-modeEnergy point.1) * Real.exp (-modeEnergy point.2) := by funext point rw [← Real.exp_add] congr 1 dsimp [system, coordinates, modeEnergy] ring rw [factorization] have product := integral_prod_mul (μ := modeVolume) (ν := modeVolume) (fun point => Real.exp (-modeEnergy point)) (fun point => Real.exp (-modeEnergy point)) exact product.trans (by rw [one_mode_has_partition_two_pi]; ring) theorem the_final_full_partition_is_computed : (∫ point, Real.exp (-system.energy 1 (coordinates point)) ∂transport.phaseVolume) = Real.exp (-1) * (2 * Real.pi) ^ 2 := by have factorization : (fun point : FullPhase => Real.exp (-system.energy 1 (coordinates point))) = fun point => Real.exp (-1) * Real.exp (-system.energy 0 (coordinates point)) := by funext point rw [← Real.exp_add] congr 1 dsimp [system] ring rw [factorization] exact (integral_const_mul (μ := transport.phaseVolume) (Real.exp (-1)) (fun point => Real.exp (-system.energy 0 (coordinates point)))).trans (congrArg (Real.exp (-1) * ·) the_initial_full_partition_is_computed) theorem the_full_initial_canonical_density_is_normalized : transport.ProbabilityDensityAt := by have normalized : (∫ point, transport.initialDensity point ∂transport.phaseVolume) = 1 := by have scale := integral_const_mul (μ := transport.phaseVolume) ((2 * Real.pi) ^ 2)⁻¹ (fun point => Real.exp (-system.energy 0 (coordinates point))) exact scale.trans (by rw [the_initial_full_partition_is_computed]; field_simp) refine ⟨?_, ?_, ?_, normalized⟩ · dsimp [transport, system, coordinates] fun_prop · intro point exact mul_nonneg (inv_nonneg.mpr (sq_nonneg _)) (Real.exp_pos _).le · by_contra notIntegrable have zero := integral_undef notIntegrable rw [normalized] at zero exact one_ne_zero zero theorem actual_controlled_work_cancels_the_advected_full_density (point : FullPhase) : transport.advectedDensity point * Real.exp (-actualWork point) = ((2 * Real.pi) ^ 2)⁻¹ * Real.exp (-system.energy 1 (coordinates point)) := by change (((2 * Real.pi) ^ 2)⁻¹ * Real.exp (-system.energy 0 (coordinates (transport.evolution.symm point)))) * Real.exp (-coupling.systemWork history (coordinates point)) = _ rw [coupling.controlled_system_work_equals_total_energy_change history the_interaction_is_part_of_the_actual_hamiltonian the_switching_history_is_realized, (scalar_coordinates_intertwine_the_actual_evolution point).2, mul_assoc, ← Real.exp_add] congr 2 change -system.energy 0 (history.evolution.symm (coordinates point)) - (system.energy 1 (coordinates point) - system.energy 0 (history.evolution.symm (coordinates point))) = _ ring theorem library_derives_the_actual_combined_work_average : (∫ point, Real.exp (-actualWork point) ∂transport.finalLaw) = Real.exp (-1) := by rw [transport.liouville_transport_integrates_final_observables the_full_initial_canonical_density_is_normalized the_combined_evolution_is_liouville] rw [integral_congr_ae (μ := transport.phaseVolume) (Filter.Eventually.of_forall actual_controlled_work_cancels_the_advected_full_density)] have scale := integral_const_mul (μ := transport.phaseVolume) ((2 * Real.pi) ^ 2)⁻¹ (fun point => Real.exp (-system.energy 1 (coordinates point))) exact scale.trans (by rw [the_final_full_partition_is_computed]; field_simp) theorem the_actual_combined_average_is_the_full_partition_ratio : (∫ point, Real.exp (-actualWork point) ∂transport.finalLaw) = (∫ point, Real.exp (-system.energy 1 (coordinates point)) ∂transport.phaseVolume) / (∫ point, Real.exp (-system.energy 0 (coordinates point)) ∂transport.phaseVolume) := by rw [library_derives_the_actual_combined_work_average, the_initial_full_partition_is_computed, the_final_full_partition_is_computed] field_simp theorem the_full_work_experiment_has_an_actual_probability_law : IsProbabilityMeasure transport.finalLaw := transport.actual_pushforward_preserves_probability the_full_initial_canonical_density_is_normalized theorem the_actual_exponential_work_moment_is_finite : Integrable (fun point => Real.exp (-actualWork point)) transport.finalLaw := by by_contra notIntegrable have zero := integral_undef notIntegrable rw [library_derives_the_actual_combined_work_average] at zero exact Real.exp_ne_zero _ zero /-- A bounded position observable distinguishes actual subsystem measures. -/ def positionProbe (point : Mode) : Real := Real.exp (-point.1 ^ 2 / 2) theorem the_position_probe_is_positive_and_bounded (point : Mode) : 0 < positionProbe point ∧ positionProbe point ≤ 1 := by refine ⟨Real.exp_pos _, Real.exp_le_one_iff.mpr ?_⟩ exact div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (sq_nonneg _)) (by norm_num) def systemProjection (point : FullPhase) : Mode := (coupling.split (coordinates point)).1 def reducedInitialLaw : Measure Mode := Measure.map systemProjection transport.initialLaw def barePartition : Real := Real.sqrt Real.pi * Real.sqrt (2 * Real.pi) def bareSystem : PhaseTransport Mode where phaseVolume := modeVolume evolution := MeasurableEquiv.refl Mode initialDensity := fun point => barePartition⁻¹ * Real.exp (-coupling.systemEnergy 0 point) theorem gaussian_product_integral (positionScale momentumScale : Real) : (∫ point : Mode, Real.exp (-(positionScale * point.1 ^ 2 + momentumScale * point.2 ^ 2)) ∂modeVolume) = Real.sqrt (Real.pi / positionScale) * Real.sqrt (Real.pi / momentumScale) := by have factorization : (fun point : Mode => Real.exp (-(positionScale * point.1 ^ 2 + momentumScale * point.2 ^ 2))) = fun point => Real.exp (-positionScale * point.1 ^ 2) * Real.exp (-momentumScale * point.2 ^ 2) := by funext point rw [← Real.exp_add] congr 1 ring rw [factorization] exact (integral_prod_mul (μ := (volume : Measure Real)) (ν := (volume : Measure Real)) (fun value : Real => Real.exp (-positionScale * value ^ 2)) (fun value : Real => Real.exp (-momentumScale * value ^ 2))).trans (by rw [integral_gaussian, integral_gaussian]) theorem the_bare_system_partition_is_computed : (∫ point, Real.exp (-coupling.systemEnergy 0 point) ∂bareSystem.phaseVolume) = barePartition := by have integrands : (fun point : Mode => Real.exp (-coupling.systemEnergy 0 point)) = fun point => Real.exp (-((1 : Real) * point.1 ^ 2 + (1 / 2 : Real) * point.2 ^ 2)) := by funext point congr 1 change -(0 + point.1 ^ 2 + point.2 ^ 2 / 2) = _ ring rw [integrands] have half : Real.pi / (1 / 2) = 2 * Real.pi := by ring exact (gaussian_product_integral 1 (1 / 2)).trans (by rw [div_one, half]; rfl) theorem the_bare_system_canonical_reference_is_normalized : bareSystem.ProbabilityDensityAt := by have positive : 0 < barePartition := by dsimp [barePartition]; positivity have normalized : (∫ point, bareSystem.initialDensity point ∂bareSystem.phaseVolume) = 1 := by exact (integral_const_mul (μ := bareSystem.phaseVolume) barePartition⁻¹ (fun point => Real.exp (-coupling.systemEnergy 0 point))).trans (by rw [the_bare_system_partition_is_computed]; exact inv_mul_cancel₀ positive.ne') refine ⟨?_, ?_, ?_, normalized⟩ · change Measurable (fun point : Mode => barePartition⁻¹ * Real.exp (-(0 + point.1 ^ 2 + point.2 ^ 2 / 2))) fun_prop · intro point exact mul_nonneg (inv_nonneg.mpr positive.le) (Real.exp_pos _).le · by_contra notIntegrable have zero := integral_undef notIntegrable rw [normalized] at zero exact one_ne_zero zero theorem the_bare_final_partition_is_computed : (∫ point, Real.exp (-coupling.systemEnergy 1 point) ∂bareSystem.phaseVolume) = Real.exp (-1) * barePartition := by have integrands : (fun point : Mode => Real.exp (-coupling.systemEnergy 1 point)) = fun point => Real.exp (-1) * Real.exp (-coupling.systemEnergy 0 point) := by funext point rw [← Real.exp_add] congr 1 change -(1 + point.1 ^ 2 + point.2 ^ 2 / 2) = -1 + -(0 + point.1 ^ 2 + point.2 ^ 2 / 2) ring rw [integrands] exact (integral_const_mul (μ := bareSystem.phaseVolume) (Real.exp (-1)) (fun point => Real.exp (-coupling.systemEnergy 0 point))).trans (congrArg (Real.exp (-1) * ·) the_bare_system_partition_is_computed) theorem the_reduced_law_is_an_actual_probability_measure : IsProbabilityMeasure reducedInitialLaw := by letI := transport.initial_density_defines_a_probability_law the_full_initial_canonical_density_is_normalized have projection : Measurable systemProjection := by change Measurable (fun point : FullPhase => (point.1.1, point.1.2 + point.2.2)) fun_prop exact Measure.isProbabilityMeasure_map projection.aemeasurable theorem the_actual_reduced_probe_average_is_computed : (∫ point, positionProbe point ∂reducedInitialLaw) = barePartition / (2 * Real.pi) := by have projection : Measurable systemProjection := by change Measurable (fun point : FullPhase => (point.1.1, point.1.2 + point.2.2)) fun_prop have observable : Measurable positionProbe := by unfold positionProbe; fun_prop rw [reducedInitialLaw, integral_map projection.aemeasurable observable.aestronglyMeasurable] rw [transport.initial_density_integrates_observables the_full_initial_canonical_density_is_normalized.1 the_full_initial_canonical_density_is_normalized.2.1] have integrands : (fun point : FullPhase => transport.initialDensity point * positionProbe (systemProjection point)) = fun point => ((2 * Real.pi) ^ 2)⁻¹ * (Real.exp (-(point.1.1 ^ 2 + (1 / 2 : Real) * point.1.2 ^ 2)) * Real.exp (-modeEnergy point.2)) := by funext point change (((2 * Real.pi) ^ 2)⁻¹ * Real.exp (-system.energy 0 (coordinates point))) * Real.exp (-point.1.1 ^ 2 / 2) = _ rw [mul_assoc, ← Real.exp_add, ← Real.exp_add] congr 2 dsimp [system, coordinates, modeEnergy] ring rw [integrands] have factor := integral_prod_mul (μ := modeVolume) (ν := modeVolume) (fun point : Mode => Real.exp (-(point.1 ^ 2 + (1 / 2 : Real) * point.2 ^ 2))) (fun point => Real.exp (-modeEnergy point)) have first : (∫ point : Mode, Real.exp (-(point.1 ^ 2 + (1 / 2 : Real) * point.2 ^ 2)) ∂modeVolume) = barePartition := by have half : Real.pi / (1 / 2) = 2 * Real.pi := by ring simpa only [one_mul, div_one, half, barePartition] using gaussian_product_integral 1 (1 / 2) have scale := integral_const_mul (μ := transport.phaseVolume) ((2 * Real.pi) ^ 2)⁻¹ (fun point : FullPhase => Real.exp (-(point.1.1 ^ 2 + (1 / 2 : Real) * point.1.2 ^ 2)) * Real.exp (-modeEnergy point.2)) exact scale.trans ((congrArg (((2 * Real.pi) ^ 2)⁻¹ * ·) factor).trans (by rw [first, one_mode_has_partition_two_pi]; field_simp)) theorem the_bare_canonical_probe_average_is_computed : (∫ point, positionProbe point ∂bareSystem.initialLaw) = barePartition⁻¹ * (Real.sqrt (Real.pi / (3 / 2)) * Real.sqrt (2 * Real.pi)) := by rw [bareSystem.initial_density_integrates_observables the_bare_system_canonical_reference_is_normalized.1 the_bare_system_canonical_reference_is_normalized.2.1] have integrands : (fun point : Mode => bareSystem.initialDensity point * positionProbe point) = fun point => barePartition⁻¹ * Real.exp (-((3 / 2 : Real) * point.1 ^ 2 + (1 / 2 : Real) * point.2 ^ 2)) := by funext point change (barePartition⁻¹ * Real.exp (-(0 + point.1 ^ 2 + point.2 ^ 2 / 2))) * Real.exp (-point.1 ^ 2 / 2) = _ rw [mul_assoc, ← Real.exp_add] congr 2 ring rw [integrands] have half : Real.pi / (1 / 2) = 2 * Real.pi := by ring exact (integral_const_mul (μ := bareSystem.phaseVolume) barePartition⁻¹ (fun point : Mode => Real.exp (-((3 / 2 : Real) * point.1 ^ 2 + (1 / 2 : Real) * point.2 ^ 2)))).trans ((congrArg (barePartition⁻¹ * ·) (gaussian_product_integral (3 / 2) (1 / 2))).trans (by rw [half])) theorem the_actual_reduced_probe_average_has_square_one_half : (∫ point, positionProbe point ∂reducedInitialLaw) ^ 2 = 1 / 2 := by rw [the_actual_reduced_probe_average_is_computed, div_pow] simp only [barePartition, mul_pow, Real.sq_sqrt Real.pi_pos.le, Real.sq_sqrt (by positivity : 0 ≤ 2 * Real.pi)] field_simp theorem the_bare_canonical_probe_average_has_square_two_thirds : (∫ point, positionProbe point ∂bareSystem.initialLaw) ^ 2 = 2 / 3 := by rw [the_bare_canonical_probe_average_is_computed] simp only [barePartition, mul_pow, inv_pow, Real.sq_sqrt Real.pi_pos.le, Real.sq_sqrt (by positivity : 0 ≤ 2 * Real.pi), Real.sq_sqrt (by positivity : 0 ≤ Real.pi / (3 / 2))] field_simp theorem full_canonical_preparation_does_not_give_bare_system_equilibrium : transport.ProbabilityDensityAt ∧ IsProbabilityMeasure reducedInitialLaw ∧ bareSystem.ProbabilityDensityAt ∧ reducedInitialLaw ≠ bareSystem.initialLaw := by refine ⟨the_full_initial_canonical_density_is_normalized, the_reduced_law_is_an_actual_probability_measure, the_bare_system_canonical_reference_is_normalized, ?_⟩ intro equality have actual := the_actual_reduced_probe_average_has_square_one_half rw [equality, the_bare_canonical_probe_average_has_square_two_thirds] at actual norm_num at actual theorem coincident_partition_ratios_do_not_identify_the_reduced_state : (∫ point, Real.exp (-actualWork point) ∂transport.finalLaw) = (∫ point, Real.exp (-coupling.systemEnergy 1 point) ∂bareSystem.phaseVolume) / (∫ point, Real.exp (-coupling.systemEnergy 0 point) ∂bareSystem.phaseVolume) ∧ reducedInitialLaw ≠ bareSystem.initialLaw := by refine ⟨?_, full_canonical_preparation_does_not_give_bare_system_equilibrium.2.2.2⟩ rw [library_derives_the_actual_combined_work_average, the_bare_final_partition_is_computed, the_bare_system_partition_is_computed] have positive : 0 < barePartition := by dsimp [barePartition]; positivity field_simp [positive.ne'] end end PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 39 do throwError "Missing independent Jarzynski coupled-work and ensemble theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent Jarzynski coupled client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent Jarzynski coupled client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient.library_derives_system_equation_three, ``PhysicsLogic.Papers.Jarzynski1997.SystemReservoirCoupling.actual_system_control_derivative), (``PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient.library_computes_system_controlled_work, ``PhysicsLogic.Papers.Jarzynski1997.SystemReservoirCoupling.controlled_system_work_equals_total_energy_change), (``PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient.library_balance_keeps_the_interaction_energy, ``PhysicsLogic.Papers.Jarzynski1997.SystemReservoirCoupling.controlled_work_resolves_all_three_energy_changes), (``PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient.actual_controlled_work_is_not_reduced_system_energy_change, ``PhysicsLogic.Papers.Jarzynski1997.SystemReservoirCoupling.reservoir_and_interaction_exchange_obstruct_reduced_energy_work), (``PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient.the_direct_paper_api_uses_system_work_and_full_dynamics, ``PhysicsLogic.Papers.Jarzynski1997.SwitchingFramework.actual_coupled_history_supplies_full_work_energy_equation), (``PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient.the_direct_paper_api_uses_system_work_and_full_dynamics, ``PhysicsLogic.Papers.Jarzynski1997.SwitchingFramework.actual_full_liouville_transport_derives_the_work_integral), (``PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient.library_derives_the_actual_combined_work_average, ``PhysicsLogic.Papers.Jarzynski1997.SystemReservoirCoupling.controlled_system_work_equals_total_energy_change), (``PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient.library_derives_the_actual_combined_work_average, ``PhysicsLogic.Papers.Jarzynski1997.PhaseTransport.liouville_transport_integrates_final_observables), (``PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient.the_full_work_experiment_has_an_actual_probability_law, ``PhysicsLogic.Papers.Jarzynski1997.PhaseTransport.actual_pushforward_preserves_probability), (``PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient.full_canonical_preparation_does_not_give_bare_system_equilibrium, ``PhysicsLogic.Papers.Jarzynski1997.PhaseTransport.initial_density_integrates_observables), (``PhysicsLogic.IndependentReuse.JarzynskiCoupledWorkClient.full_canonical_preparation_does_not_give_bare_system_equilibrium, ``PhysicsLogic.Papers.Jarzynski1997.PhaseTransport.initial_density_defines_a_probability_law)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Jarzynski coupled-work client audit passed: {theoremCount} theorems, {uses.size} public API uses, actual coupled oscillator motion and canonical full ensemble, computed endpoint partitions and exact work average exp(-1), work distinct from reduced energy change, actual reduced law unequal to bare-system equilibrium by bounded probe, no production diagnostics." namespace PhysicsLogic.IndependentReuse.CHSHCountingClient open PhysicsLogic.Papers.CHSH1969 noncomputable section set_option backward.isDefEq.respectTransparency false def hiddenPair : Fin 3 -> Bool × Bool := ![(false, false), (true, false), (true, true)] def mean (readout : Fin 3 -> Real) : Real := (∑ hidden, readout hidden) / 3 def channel (setting : Option Bool) (bit : Bool) : Real := if setting.isNone then 1 else if bit then 1 else -1 def emergence (setting : Option Bool) (bit : Bool) : Real := (channel setting bit + 1) / 2 def rate (a b : Option Bool) : Real := 9 * mean (fun hidden => emergence a (hiddenPair hidden).1 * emergence b (hiddenPair hidden).2) def correlation (a b : Option Bool) : Real := mean (fun hidden => channel a (hiddenPair hidden).1 * channel b (hiddenPair hidden).2) def data : CountingRateData (Option Bool) where removed := none correlation := correlation coincidence := rate reference := rate none none leftConstant := 6 rightConstant := 3 theorem the_finite_source_average_is_normalized : mean (fun _ => 1) = 1 := by norm_num [mean, Fin.sum_univ_succ] theorem the_actual_marginals_are_unequal : data.reference = 9 ∧ ∀ orientation : Bool, data.coincidence (some orientation) none = 6 ∧ data.coincidence none (some orientation) = 3 := by norm_num [data, rate, mean, hiddenPair, emergence, channel, Fin.sum_univ_succ] theorem the_actual_rates_obey_the_marginal_identity : data.CorrelationRateIdentity := by refine ⟨rfl, ?_⟩ intro a b cases a <;> cases b <;> norm_num [data, correlation, rate, mean, hiddenPair, emergence, channel, Fin.sum_univ_succ] theorem the_selected_single_rates_are_calibrated : data.SelectedSingleRates (some false) (some false) (some true) (some true) := by norm_num [CountingRateData.SelectedSingleRates, data, rate, mean, hiddenPair, emergence, channel, Fin.sum_univ_succ] theorem the_actual_correlations_obey_the_generalized_bound : |data.correlation (some false) (some false) - data.correlation (some false) (some true)| ≤ 2 - data.correlation (some true) (some false) - data.correlation (some true) (some true) := by norm_num [data, correlation, mean, hiddenPair, channel, Fin.sum_univ_succ] theorem the_library_derives_the_nonunit_counting_bound : data.CountingInequality (some false) (some false) (some true) (some true) := by apply CountingRateData.selected_marginals_translate_the_correlation_bound data (some false) (some false) (some true) (some true) · rw [the_actual_marginals_are_unequal.1] norm_num · exact the_actual_rates_obey_the_marginal_identity · exact the_selected_single_rates_are_calibrated · exact the_actual_correlations_obey_the_generalized_bound theorem the_removed_setting_rejects_the_constant_substitution : ¬(∀ a b, data.correlation a b = 4 * data.coincidence a b / data.reference - 2 * data.leftConstant / data.reference - 2 * data.rightConstant / data.reference + 1) := by intro oldIdentity have removed := oldIdentity none none norm_num [data, correlation, rate, mean, hiddenPair, emergence, channel, Fin.sum_univ_succ] at removed def varyingChannel : Option Bool -> Bool -> Real | none, _ => 1 | some false, _ => -1 | some true, bit => if bit then -1 else 1 def varyingEmergence (setting : Option Bool) (bit : Bool) : Real := (varyingChannel setting bit + 1) / 2 def varyingRate (a b : Option Bool) : Real := 9 * mean (fun hidden => varyingEmergence a (hiddenPair hidden).1 * varyingEmergence b (hiddenPair hidden).2) def varyingCorrelation (a b : Option Bool) : Real := mean (fun hidden => varyingChannel a (hiddenPair hidden).1 * varyingChannel b (hiddenPair hidden).2) def varyingData : CountingRateData (Option Bool) where removed := none correlation := varyingCorrelation coincidence := varyingRate reference := varyingRate none none leftConstant := varyingRate (some false) none rightConstant := varyingRate none (some false) theorem the_varying_marginal_rates_are_computed : varyingData.reference = 9 ∧ varyingData.coincidence (some false) none = 0 ∧ varyingData.coincidence (some true) none = 3 ∧ varyingData.coincidence none (some false) = 0 ∧ varyingData.coincidence none (some true) = 6 ∧ varyingData.coincidence (some true) (some true) = 3 := by norm_num [varyingData, varyingRate, mean, hiddenPair, varyingEmergence, varyingChannel, Fin.sum_univ_succ] theorem the_varying_rates_retain_the_probability_identity : varyingData.CorrelationRateIdentity := by refine ⟨rfl, ?_⟩ intro a b fin_cases a <;> fin_cases b <;> norm_num [varyingData, varyingRate, varyingCorrelation, mean, hiddenPair, varyingEmergence, varyingChannel, Fin.sum_univ_succ] theorem the_varying_reference_has_allowed_imperfection : varyingData.correlation (some true) (some false) = 1 - (2 / 3 : Real) ∧ 0 ≤ (2 / 3 : Real) ∧ (2 / 3 : Real) ≤ 1 := by norm_num [varyingData, varyingCorrelation, mean, hiddenPair, varyingChannel, Fin.sum_univ_succ] theorem the_varying_correlations_obey_bells_bound : |varyingData.correlation (some false) (some false) - varyingData.correlation (some false) (some true)| ≤ 2 - varyingData.correlation (some true) (some false) - varyingData.correlation (some true) (some true) := by norm_num [varyingData, varyingCorrelation, mean, hiddenPair, varyingChannel, Fin.sum_univ_succ] theorem the_library_keeps_varying_marginal_terms : varyingData.MarginalCountingInequality (some false) (some false) (some true) (some true) := by apply CountingRateData.varying_marginals_translate_the_correlation_bound varyingData (some false) (some false) (some true) (some true) · rw [the_varying_marginal_rates_are_computed.1] norm_num · exact the_varying_rates_retain_the_probability_identity · exact the_varying_correlations_obey_bells_bound theorem uncalibrated_constants_can_fail_with_valid_bookkeeping : varyingData.CorrelationRateIdentity ∧ varyingData.MarginalCountingInequality (some false) (some false) (some true) (some true) ∧ ¬varyingData.SelectedSingleRates (some false) (some false) (some true) (some true) ∧ ¬varyingData.CountingInequality (some false) (some false) (some true) (some true) := by refine ⟨the_varying_rates_retain_the_probability_identity, the_library_keeps_varying_marginal_terms, ?_, ?_⟩ <;> norm_num [CountingRateData.SelectedSingleRates, CountingRateData.CountingInequality, varyingData, varyingRate, mean, hiddenPair, varyingEmergence, varyingChannel, Fin.sum_univ_succ] end end PhysicsLogic.IndependentReuse.CHSHCountingClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.CHSHCountingClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 13 do throwError "Missing independent CHSH counting-rate theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent CHSH counting client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent CHSH counting client depends on production diagnostics {name}" let declaration := ``PhysicsLogic.IndependentReuse.CHSHCountingClient.the_library_derives_the_nonunit_counting_bound let required := ``PhysicsLogic.Papers.CHSH1969.CountingRateData.selected_marginals_translate_the_correlation_bound unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" let varyingDeclaration := ``PhysicsLogic.IndependentReuse.CHSHCountingClient.the_library_keeps_varying_marginal_terms let varyingRequired := ``PhysicsLogic.Papers.CHSH1969.CountingRateData.varying_marginals_translate_the_correlation_bound unless DownstreamDependencyAudit.usesConstant environment varyingDeclaration varyingRequired do throwError "{varyingDeclaration} does not use {varyingRequired}" Lean.logInfo m!"CHSH counting client audit passed: {theoremCount} theorems, normalized three-state average, reference 9, unequal and varying marginal rates, reference imperfection 2/3 in the source regime, two public counting-bound deductions, uncalibrated equation-(2a) rejection, foundational axioms only, no production diagnostics." namespace PhysicsLogic.IndependentReuse.BlattInterpolationClient open PhysicsLogic.Papers.BlattJackson1949 open WellDepthInterpolation noncomputable section set_option backward.isDefEq.respectTransparency false /-- Numerical contract validation, not a reconstructed nuclear potential. -/ def parameters : Parameters := ⟨1 / 30000, 2, 3, 1 / 30000, 1⟩ theorem the_dimensionless_inputs_use_inverse_length_and_the_same_range : parameters.coordinateValue .inverseLengthEffectiveRange = 1 / 15000 ∧ parameters.coordinateValue .inverseLengthIntrinsicRange = 1 / 10000 ∧ parameters.coordinateValue .bindingMomentumIntrinsicRange = 1 / 10000 := by norm_num [parameters, Parameters.coordinateValue] theorem all_three_formulas_have_controlled_errors : parameters.ThreeCoordinateAccuracy .square := by intro coordinate cases coordinate <;> norm_num [Parameters.ThreeCoordinateAccuracy, AccurateAt, InReportedInterval, DefinedAt, printedApproximation, effectiveRangeNumerator, effectiveRangeDenominator, intrinsicRangePolynomial, bindingMomentumPolynomial, reportedRelativeError, parameters, Parameters.coordinateValue] theorem the_library_bounds_the_same_selected_depth (coordinate : Coordinate) : printedApproximation .square coordinate (parameters.coordinateValue coordinate) / (1 + reportedRelativeError) ≤ parameters.depth ∧ parameters.depth ≤ printedApproximation .square coordinate (parameters.coordinateValue coordinate) / (1 - reportedRelativeError) := three_coordinates_bound_one_physical_depth parameters .square all_three_formulas_have_controlled_errors coordinate theorem the_approximate_depth_is_not_exact : parameters.ThreeCoordinateAccuracy .square ∧ printedApproximation .square .inverseLengthIntrinsicRange (parameters.coordinateValue .inverseLengthIntrinsicRange) ≠ parameters.depth := by refine ⟨all_three_formulas_have_controlled_errors, ?_⟩ norm_num [printedApproximation, intrinsicRangePolynomial, parameters, Parameters.coordinateValue] def rescaledParameters : Parameters := ⟨1 / 60000, 4, 6, 1 / 60000, 1⟩ theorem dimensionless_interpolations_do_not_fix_the_length_scale : (∀ coordinate, parameters.coordinateValue coordinate = rescaledParameters.coordinateValue coordinate) ∧ parameters.intrinsicRange ≠ rescaledParameters.intrinsicRange := by constructor · intro coordinate cases coordinate <;> norm_num [parameters, rescaledParameters, Parameters.coordinateValue] · norm_num [parameters, rescaledParameters] theorem another_interior_yukawa_input_rejects_positive_depth_accuracy : InReportedInterval .inverseLengthEffectiveRange (359 / 1000) ∧ DefinedAt .yukawa .inverseLengthEffectiveRange (359 / 1000) ∧ ¬(∃ depth, AccurateAt .yukawa .inverseLengthEffectiveRange (359 / 1000) depth) := by refine ⟨by norm_num [InReportedInterval], by norm_num [DefinedAt, effectiveRangeDenominator], ?_⟩ rintro ⟨depth, accurate⟩ have positive := accurate_positive_depth_requires_positive_interpolation _ _ _ _ accurate norm_num [printedApproximation, effectiveRangeNumerator, effectiveRangeDenominator] at positive theorem the_actual_paper_extraction_supplies_the_selected_square_depth_bounds {T : BlattJacksonTypes} (framework : ScatteringFramework T) (extraction : framework.ConventionalShapeParameterExtraction) (coordinate : Coordinate) : printedApproximation .square coordinate ((framework.wellDepthParameters framework.squareWellShape).coordinateValue coordinate) / (1 + reportedRelativeError) ≤ framework.shapeConditionedWellDepthValue framework.squareWellShape ∧ framework.shapeConditionedWellDepthValue framework.squareWellShape ≤ printedApproximation .square coordinate ((framework.wellDepthParameters framework.squareWellShape).coordinateValue coordinate) / (1 - reportedRelativeError) := three_coordinates_bound_one_physical_depth _ .square extraction.2.2.1 coordinate end end PhysicsLogic.IndependentReuse.BlattInterpolationClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.BlattInterpolationClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 7 do throwError "Missing independent Blatt interpolation theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent Blatt interpolation client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent Blatt interpolation client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.BlattInterpolationClient.the_library_bounds_the_same_selected_depth, ``PhysicsLogic.Papers.BlattJackson1949.WellDepthInterpolation.three_coordinates_bound_one_physical_depth), (``PhysicsLogic.IndependentReuse.BlattInterpolationClient.another_interior_yukawa_input_rejects_positive_depth_accuracy, ``PhysicsLogic.Papers.BlattJackson1949.WellDepthInterpolation.accurate_positive_depth_requires_positive_interpolation), (``PhysicsLogic.IndependentReuse.BlattInterpolationClient.the_actual_paper_extraction_supplies_the_selected_square_depth_bounds, ``PhysicsLogic.Papers.BlattJackson1949.WellDepthInterpolation.three_coordinates_bound_one_physical_depth)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Blatt interpolation client audit passed: {theoremCount} theorems, {uses.size} public API uses, three actual dimensionless inputs, nonexact positive-depth bounds, separate length scale, independent interior Yukawa discrepancy, actual paper-extraction handoff, foundational axioms only, no production diagnostics." namespace PhysicsLogic.IndependentReuse.RiessSelectedDistanceClient open PhysicsLogic.Papers.RiessEtAl1998 open DistancePrediction open PhysicsLogic.Framework.StandardCandleCalibration noncomputable section set_option backward.isDefEq.respectTransparency false /-- A two-observation API diagnostic, not a reconstruction of the historical fit. -/ def distances : Data Bool Bool where selectedModels := {false} fitData := Finset.univ distance := fun model observation => if model then -3 else if observation then 40 else 20 distanceUnit := fun _ _ => .megaparsec modulus := fun model observation => distanceModulus (if model then -3 else if observation then 40 else 20) def observed (observation : Bool) : Real := distances.modulus false observation - if observation then 2 else 1 def variance (observation : Bool) : Real := if observation then 4 else 1 theorem the_selected_sample_is_nonempty_and_physically_interpreted : distances.HasSelectedInput ∧ distances.SelectedPredictions := by refine ⟨⟨⟨false, rfl⟩, ⟨false, Finset.mem_univ _⟩⟩, ?_⟩ intro model selected observation member have same : model = false := selected subst model cases observation <;> norm_num [distances] theorem the_library_fixes_the_physical_coordinate (observation : Bool) : 0 < distances.physicalDistanceMpc false observation ∧ distances.modulus false observation = distanceModulus (distances.physicalDistanceMpc false observation) := distances.selected_predictions_fix_the_actual_unit_and_distance the_selected_sample_is_nonempty_and_physically_interpreted.2 false rfl observation (Finset.mem_univ _) theorem the_library_determines_the_geometric_residual_sum : distances.geometricResidualSum false observed variance = 2 := by have geometric := distances.selected_predictions_determine_the_nonnegative_geometric_residual_sum the_selected_sample_is_nonempty_and_physically_interpreted.2 false rfl observed variance (by intro observation member; cases observation <;> norm_num [variance]) rw [← geometric.2] norm_num [Data.residualSum, distances, observed, variance, Fintype.sum_bool] theorem an_excluded_model_is_not_certified : distances.SelectedPredictions ∧ ¬(∀ model observation, 0 < distances.distance model observation) := by refine ⟨the_selected_sample_is_nonempty_and_physically_interpreted.2, ?_⟩ intro allModels have wrong := allModels true false norm_num [distances] at wrong def parsecCoordinates : Data Bool Bool := { distances with distanceUnit := fun _ _ => .parsec } theorem unit_conditionals_do_not_select_megaparsec_data : parsecCoordinates.UnitGuardedFormula ∧ ¬parsecCoordinates.SelectedPredictions := by constructor · intro model observation member unit cases unit · intro predictions have wrong := (predictions false rfl false (Finset.mem_univ _)).1 cases wrong theorem the_actual_paper_likelihood_uses_these_geometric_predictions {T : RiessTypes} (framework : SupernovaAccelerationFramework T) (inference : framework.EstablishedCosmologicalInference) (method : DistanceMethod) (parameters : PhysicsLogic.Framework.StandardCandlePosterior.Parameters × Real) (allowed : parameters ∈ framework.allowedPosteriorParameters) : 0 ≤ (framework.posteriorLikelihood method).chiSquare parameters ∧ (framework.posteriorLikelihood method).chiSquare parameters = framework.geometryBasedChiSquare method parameters := framework.established_inference_uses_selected_geometric_distances_in_the_likelihood inference method parameters allowed end end PhysicsLogic.IndependentReuse.RiessSelectedDistanceClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.RiessSelectedDistanceClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 6 do throwError "Missing independent Riess selected-distance theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent Riess selected-distance client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent Riess selected-distance client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.RiessSelectedDistanceClient.the_library_fixes_the_physical_coordinate, ``PhysicsLogic.Papers.RiessEtAl1998.DistancePrediction.Data.selected_predictions_fix_the_actual_unit_and_distance), (``PhysicsLogic.IndependentReuse.RiessSelectedDistanceClient.the_library_determines_the_geometric_residual_sum, ``PhysicsLogic.Papers.RiessEtAl1998.DistancePrediction.Data.selected_predictions_determine_the_nonnegative_geometric_residual_sum), (``PhysicsLogic.IndependentReuse.RiessSelectedDistanceClient.the_actual_paper_likelihood_uses_these_geometric_predictions, ``PhysicsLogic.Papers.RiessEtAl1998.SupernovaAccelerationFramework.established_inference_uses_selected_geometric_distances_in_the_likelihood)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Riess selected-distance client audit passed: {theoremCount} theorems, {uses.size} public API uses, two actual positive Mpc distances, unequal variances, geometric chi-square 2, excluded-model and unit-guard boundaries, actual paper likelihood handoff, foundational axioms only, no production diagnostics." namespace PhysicsLogic.IndependentReuse.SingularLayerClient open PhysicsLogic.Papers.Prandtl1904.SingularLayerLimit open PhysicsLogic.Framework.BoundaryLayerFieldReduction open Filter Set open scoped Topology noncomputable section set_option backward.isDefEq.respectTransparency false /-- Independent nonunit-speed, half-thickness API diagnostic, not a historical wake. -/ def physicalFields (epsilon : Real) : Data where density := 1 viscosity := epsilon ^ 2 / 4 tangential := fun _ _ y => 3 - 3 * Real.exp (-(2 * y) / epsilon) normal := fun _ _ _ => -epsilon / 2 pressure := fun _ _ _ => 0 outerPressure := fun _ _ => 0 outerTangential := fun _ _ => 3 theorem the_actual_fields_match_the_half_thickness_family (epsilon : Real) : physicalFields epsilon = suctionLayer 3 (epsilon / 2) := by have viscosity : (epsilon / 2) ^ 2 = epsilon ^ 2 / 4 := by ring have normal : -(epsilon / 2) = -epsilon / 2 := by ring unfold physicalFields suctionLayer rw [viscosity, normal] congr 1 funext time x y dsimp [profile] rw [div_div_eq_mul_div] ring_nf theorem the_library_verifies_both_momentum_balances_and_continuity {epsilon : Real} (positive : 0 < epsilon) (time x y : Real) : (physicalFields epsilon).FullPlanarFlowAt time x y := by rw [the_actual_fields_match_the_half_thickness_family] exact suction_layer_solves_full_steady_viscous_balances 3 (by positivity) time x y theorem the_viscosity_really_tends_to_zero : Tendsto (fun epsilon => (physicalFields epsilon).viscosity) (𝓝[>] 0) (𝓝 0) := by have identity : Tendsto (fun epsilon : Real => epsilon) (𝓝[>] 0) (𝓝 0) := tendsto_nhdsWithin_of_tendsto_nhds tendsto_id simpa [physicalFields] using (identity.pow 2).div_const 4 theorem the_library_keeps_the_distinct_shear_and_diffusion_scales {epsilon : Real} (positive : 0 < epsilon) : (physicalFields epsilon).viscosity * deriv ((physicalFields epsilon).tangential 0 0) 0 = 3 * epsilon / 2 ∧ (physicalFields epsilon).viscosity * deriv (deriv ((physicalFields epsilon).tangential 0 0)) (epsilon / 2) = -3 * Real.exp (-1) ∧ (physicalFields epsilon).normal 0 0 0 ≠ 0 := by rw [the_actual_fields_match_the_half_thickness_family] simpa [suctionLayer, mul_div_assoc] using actual_wall_shear_and_inner_diffusion_have_different_scales 3 (show 0 < epsilon / 2 by positivity) theorem the_library_bounds_the_actual_outer_velocity_defect {epsilon distance y : Real} (positive : 0 < epsilon) (away : distance ≤ y) : |(physicalFields epsilon).tangential 0 0 y - 3| ≤ 3 * Real.exp (-distance / (epsilon / 2)) := by rw [the_actual_fields_match_the_half_thickness_family] exact actual_velocity_defect_bound_away_from_wall (by norm_num) (by positivity) away theorem a_moving_positive_point_has_a_fixed_nonzero_velocity_defect {epsilon : Real} (positive : 0 < epsilon) : 0 < epsilon / 2 ∧ |(physicalFields epsilon).tangential 0 0 (epsilon / 2) - 3| = 3 * Real.exp (-1) := by refine ⟨by positivity, ?_⟩ rw [the_actual_fields_match_the_half_thickness_family] change |profile 3 (epsilon / 2) (epsilon / 2) - 3| = _ rw [moving_layer_coordinate_retains_a_nonzero_velocity_defect 3 (by positivity)] have identity : 3 * (1 - Real.exp (-1)) - 3 = -(3 * Real.exp (-1)) := by ring rw [identity, abs_neg, abs_of_pos (by positivity)] theorem uniform_outer_velocity_convergence_fails_even_without_the_wall : ¬TendstoUniformlyOn (fun epsilon y => (physicalFields epsilon).tangential 0 0 y) (fun _ => 3) (𝓝[>] 0) (Ioi 0) := by intro uniform have positive : 0 < 3 * Real.exp (-1) := by positivity have close := (Metric.tendstoUniformlyOn_iff.mp uniform) (3 * Real.exp (-1)) positive have positiveEpsilon : ∀ᶠ epsilon : Real in 𝓝[>] 0, 0 < epsilon := self_mem_nhdsWithin obtain ⟨epsilon, epsilonPositive, estimate⟩ := (positiveEpsilon.and close).exists have defect := a_moving_positive_point_has_a_fixed_nonzero_velocity_defect epsilonPositive have impossible := estimate (epsilon / 2) defect.1 rw [Real.dist_eq, abs_sub_comm, defect.2] at impossible exact lt_irrefl _ impossible end end PhysicsLogic.IndependentReuse.SingularLayerClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.SingularLayerClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 7 do throwError "Missing independent singular-layer theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent singular-layer client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent singular-layer client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.SingularLayerClient.the_library_verifies_both_momentum_balances_and_continuity, ``PhysicsLogic.Papers.Prandtl1904.SingularLayerLimit.suction_layer_solves_full_steady_viscous_balances), (``PhysicsLogic.IndependentReuse.SingularLayerClient.the_library_keeps_the_distinct_shear_and_diffusion_scales, ``PhysicsLogic.Papers.Prandtl1904.SingularLayerLimit.actual_wall_shear_and_inner_diffusion_have_different_scales), (``PhysicsLogic.IndependentReuse.SingularLayerClient.the_library_bounds_the_actual_outer_velocity_defect, ``PhysicsLogic.Papers.Prandtl1904.SingularLayerLimit.actual_velocity_defect_bound_away_from_wall), (``PhysicsLogic.IndependentReuse.SingularLayerClient.uniform_outer_velocity_convergence_fails_even_without_the_wall, ``PhysicsLogic.Papers.Prandtl1904.SingularLayerLimit.moving_layer_coordinate_retains_a_nonzero_velocity_defect)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Singular-layer client audit passed: {theoremCount} theorems, {uses.size} public API uses, independently written nonunit-speed half-thickness fields, full steady balances, actual shear and diffusion, moving interior-point defect, foundational axioms only, no production diagnostics." namespace PhysicsLogic.IndependentReuse.GamowRateComparisonClient open PhysicsLogic.Papers.Gamow1928 open LogRateComparison noncomputable section /-- Synthetic two-rate API test, not fitted nuclear data or a solved resonance. -/ def comparison : Data where lowerRate := Real.exp 2 higherRate := Real.exp (11 / 4) logBase := Real.exp 2 intercept := 1 slope := 1 / 2 energyIncrement := 2 lowerError := 1 / 10 higherError := 5 / 8 theorem the_two_rates_have_actual_log_calibrations : comparison.CalibratedComparison := by norm_num [comparison, Data.CalibratedComparison, Data.PositiveComparison, Data.ReferenceCalibration, Data.HigherApproximation, Data.logRate, Real.exp_pos, Real.one_lt_exp_iff] theorem the_library_bounds_the_actual_rate_ratio : |comparison.logRate (comparison.higherRate / comparison.lowerRate) - 1| ≤ 29 / 40 := by convert comparison.calibrated_comparison_bounds_the_actual_log_rate_ratio the_two_rates_have_actual_log_calibrations using 1 <;> norm_num [comparison] theorem a_resolved_margin_orders_the_actual_rates : Real.exp 2 < Real.exp (11 / 4) := by exact comparison.resolved_calibrated_comparison_has_faster_actual_decay the_two_rates_have_actual_log_calibrations (by norm_num [Data.ResolvesTrend, comparison]) def reversed : Data := { comparison with higherRate := Real.exp 1, higherError := 3 / 2 } theorem larger_allowed_error_admits_the_opposite_order : reversed.CalibratedComparison ∧ reversed.higherRate < reversed.lowerRate := by norm_num [reversed, comparison, Data.CalibratedComparison, Data.PositiveComparison, Data.ReferenceCalibration, Data.HigherApproximation, Data.logRate, Real.exp_pos, Real.one_lt_exp_iff, Real.exp_lt_exp] theorem the_paper_calculation_reaches_the_actual_rate_order {T : GamowTypes} (framework : AlphaDecayFramework T) (calculation : framework.EquationSevenCalculationAt) (referenceError : Real) (errorNonnegative : 0 ≤ referenceError) (lowerPositive : 0 < framework.decayRateValueAt framework.lowerDecayRate) (calibration : (framework.equationSevenRateComparison referenceError).ReferenceCalibration) (resolved : (framework.equationSevenRateComparison referenceError).ResolvesTrend) : framework.decayRateValueAt framework.lowerDecayRate < framework.decayRateValueAt framework.higherDecayRate := framework.equation_seven_with_a_resolved_margin_derives_faster_decay calculation referenceError errorNonnegative lowerPositive calibration resolved end end PhysicsLogic.IndependentReuse.GamowRateComparisonClient run_cmd do let environment ← Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.GamowRateComparisonClient let mut traversal : Lean.CollectAxioms.State := {} let mut theoremCount := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theoremCount := theoremCount + 1 let (_, next) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := next unless theoremCount >= 5 do throwError "Missing independent Gamow rate-comparison theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Independent Gamow rate-comparison client uses nonfoundational axiom {name}" for name in traversal.visited.toList do if name.toString.startsWith "PhysicsLogic." && (name.toString.splitOn "." |>.any fun part => part == "deprecated" || part.endsWith "Countermodels" || part.endsWith "Controls") then throwError "Independent Gamow rate-comparison client depends on production diagnostics {name}" let uses := #[ (``PhysicsLogic.IndependentReuse.GamowRateComparisonClient.the_library_bounds_the_actual_rate_ratio, ``PhysicsLogic.Papers.Gamow1928.LogRateComparison.Data.calibrated_comparison_bounds_the_actual_log_rate_ratio), (``PhysicsLogic.IndependentReuse.GamowRateComparisonClient.a_resolved_margin_orders_the_actual_rates, ``PhysicsLogic.Papers.Gamow1928.LogRateComparison.Data.resolved_calibrated_comparison_has_faster_actual_decay), (``PhysicsLogic.IndependentReuse.GamowRateComparisonClient.the_paper_calculation_reaches_the_actual_rate_order, ``PhysicsLogic.Papers.Gamow1928.AlphaDecayFramework.equation_seven_with_a_resolved_margin_derives_faster_decay)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Gamow rate-comparison client audit passed: {theoremCount} theorems, {uses.size} public API uses, actual log ratios in base exp(2), unequal endpoint errors, resolved and reversed orders, main-paper calculation handoff, foundational axioms only, no production diagnostics." namespace PhysicsLogic.IndependentReuse.CasimirHeatCurrentClient open PhysicsLogic.Framework.CoordinateCurrent open PhysicsLogic.Papers.Casimir1945.HeatCurrent open scoped BigOperators noncomputable section def temperature (point : Point 2) : Real := 9 + point 0 ^ 2 + 2 * point 0 * point 1 def original : Tensor 2 := fun _ => !![-2, 0; 0, -2] def correction : Tensor 2 := fun _ => !![0, 3; -3, 0] def combined : Tensor 2 := fun point row column => original point row column + correction point row column theorem independent_temperature_has_actual_gradient (point : Point 2) : (fun axis => coordinateDerivative temperature point axis) = ![2 * point 0 + 2 * point 1, 2 * point 0] := by funext axis fin_cases axis · change deriv (fun value : Real => 9 + value ^ 2 + 2 * value * point 1) (point 0) = _ simpa using ((((hasDerivAt_id (point 0)).pow 2).const_add 9).add (((hasDerivAt_id (point 0)).const_mul 2).mul_const (point 1))).deriv · change deriv (fun value : Real => 9 + point 0 ^ 2 + 2 * point 0 * value) (point 1) = _ simpa using (((hasDerivAt_id (point 1)).const_mul (2 * point 0)).const_add (9 + point 0 ^ 2)).deriv theorem the_independent_gradient_has_regular_symmetric_second_derivatives (point : Point 2) : GradientRegularAt temperature point ∧ MixedPartialsAgreeAt temperature point := by have scaled (a b : Real) : deriv (HMul.hMul a) b = a := by simpa using ((hasDerivAt_id b).const_mul a).deriv constructor · constructor · intro column fin_cases column all_goals simp [temperature, Function.update] all_goals fun_prop · intro row column simp_rw [independent_temperature_has_actual_gradient] fin_cases row <;> fin_cases column <;> simp [Function.update] <;> fun_prop · intro row column simp_rw [independent_temperature_has_actual_gradient] fin_cases row <;> fin_cases column <;> simp [coordinateDerivative, Function.update, scaled] theorem the_independent_correction_satisfies_both_source_conditions (point : Point 2) : TensorRegularAt correction point ∧ AntisymmetricAt correction point ∧ ColumnDivergenceZeroAt correction point := by refine ⟨fun _ _ => differentiableAt_const _, ?_, ?_⟩ · intro row column fin_cases row <;> fin_cases column <;> norm_num [correction] · intro column simp [coordinateDerivative, correction] theorem the_library_preserves_the_clients_actual_bulk_operator (point : Point 2) : divergence (current combined temperature) point = divergence (current original temperature) point := by have correctionInputs := the_independent_correction_satisfies_both_source_conditions point have thermal := the_independent_gradient_has_regular_symmetric_second_derivatives point exact equation_eighteen_preserves_the_bulk_current_divergence original correction temperature point (fun _ _ => differentiableAt_const _) correctionInputs.1 thermal.1 thermal.2 correctionInputs.2.1 correctionInputs.2.2 theorem the_library_balances_the_actual_added_current : NonvacuousLocalBalance (current correction temperature) Set.univ := by apply nonvacuous_balance_of_local_balance _ _ (by decide) Set.univ_nonempty intro point _ have correctionInputs := the_independent_correction_satisfies_both_source_conditions point have thermal := the_independent_gradient_has_regular_symmetric_second_derivatives point exact ⟨tensor_current_is_regular _ _ _ correctionInputs.1 thermal.1, equation_eighteen_gives_actual_zero_divergence _ _ _ correctionInputs.1 thermal.1 thermal.2 correctionInputs.2.1 correctionInputs.2.2⟩ theorem the_original_current_has_actual_nonzero_bulk_divergence (point : Point 2) : divergence (current original temperature) point = -4 := by have components (location : Point 2) : current original temperature location = ![-4 * location 0 - 4 * location 1, -4 * location 0] := by funext row simp only [current] simp_rw [independent_temperature_has_actual_gradient] fin_cases row <;> simp [original, Fin.sum_univ_two] <;> ring have scaled (a b : Real) : deriv (HMul.hMul a) b = a := by simpa using ((hasDerivAt_id b).const_mul a).deriv simp only [divergence] simp_rw [components] simp [coordinateDerivative, Function.update, Fin.sum_univ_two, scaled] theorem preserved_bulk_operator_has_distinct_local_normal_readouts : (∀ point, divergence (current combined temperature) point = -4 ∧ divergence (current original temperature) point = -4) ∧ temperature ![1, 0] = 10 ∧ current original temperature ![1, 0] 0 = -4 ∧ current combined temperature ![1, 0] 0 = 2 := by refine ⟨fun point => ⟨(the_library_preserves_the_clients_actual_bulk_operator point).trans (the_original_current_has_actual_nonzero_bulk_divergence point), the_original_current_has_actual_nonzero_bulk_divergence point⟩, by norm_num [temperature], ?_, ?_⟩ all_goals simp only [current] all_goals simp_rw [independent_temperature_has_actual_gradient] all_goals norm_num [original, correction, combined, Fin.sum_univ_two] all_goals rfl theorem the_independent_combined_coefficients_and_transpose_are_regular (point : Point 2) : TensorRegularAt combined point ∧ TensorRegularAt (transpose combined) point := by constructor all_goals intro row column all_goals dsimp [combined, original, correction, transpose] all_goals exact differentiableAt_const _ theorem the_independent_tensor_has_reciprocal_operator (point : Point 2) : ReciprocalOperatorAt combined point := by have regular := the_independent_combined_coefficients_and_transpose_are_regular point intro test thermal mixed rw [divergence_retains_both_product_rule_terms combined test point regular.1 thermal, divergence_retains_both_product_rule_terms (transpose combined) test point regular.2 thermal] have cross := mixed 0 1 simp [coordinateDerivative] at cross simp [combined, original, correction, transpose, coordinateDerivative, Fin.sum_univ_two] linarith theorem the_library_extracts_columns_from_an_independently_checked_operator (point : Point 2) : ColumnDivergenceZeroAt (antisymmetricPart combined) point := reciprocal_operator_forces_zero_antisymmetric_column_divergences combined point (the_independent_combined_coefficients_and_transpose_are_regular point).1 (the_independent_combined_coefficients_and_transpose_are_regular point).2 (the_independent_tensor_has_reciprocal_operator point) theorem the_independent_symmetric_representative_is_not_the_original_tensor : symmetricPart combined = original ∧ ¬SymmetricAt combined ![1, 0] ∧ (∀ point, divergence (current (symmetricPart combined) temperature) point = divergence (current combined temperature) point) := by refine ⟨?_, ?_, ?_⟩ · funext point row column fin_cases row <;> fin_cases column <;> norm_num [symmetricPart, combined, original, correction] · intro symmetric have cross := symmetric 0 1 norm_num [combined, original, correction] at cross · intro point have thermal := the_independent_gradient_has_regular_symmetric_second_derivatives point have regular := the_independent_combined_coefficients_and_transpose_are_regular point exact reciprocal_operator_preserves_the_actual_symmetric_representative combined temperature point regular.1 regular.2 thermal.1 thermal.2 (the_independent_tensor_has_reciprocal_operator point) theorem the_main_paper_derives_bulk_equality_from_its_admissible_current_correction {T : PhysicsLogic.Papers.Casimir1945.CasimirTypes} (analysis : PhysicsLogic.Papers.Casimir1945.MicroscopicReversibilityAnalysis T) (setup : analysis.HeatCurrentRepresentationSetup) (calculation : analysis.HeatCurrentAmbiguityCalculation) : Set.EqOn (divergence (analysis.heatCurrentField analysis.selectedHeatCurrent)) (divergence (analysis.heatCurrentField analysis.alternativeHeatCurrent)) analysis.heatDomain := analysis.heat_current_source_conditions_derive_actual_bulk_equivalence setup calculation theorem the_main_paper_assigns_symmetry_to_the_selected_representative {T : PhysicsLogic.Papers.Casimir1945.CasimirTypes} (analysis : PhysicsLogic.Papers.Casimir1945.MicroscopicReversibilityAnalysis T) (setup : analysis.HeatCurrentRepresentationSetup) (equivalence : analysis.DivergenceEquivalenceBridge) (calculation : analysis.HeatCurrentAmbiguityCalculation) (convention : analysis.conventionVanishesInVacuumAndIgnoresShape analysis.vacuumAndShapeConvention) (selection : analysis.HeatCurrentConventionCalculation) : ¬analysis.LocalHeatCurrentUniquelyDetermined ∧ (∀ point ∈ analysis.heatDomain, SymmetricAt (analysis.heatTensorField analysis.symmetricHeatRepresentative) point) := by have result := analysis.heat_current_symmetry_requires_a_representative_convention setup equivalence calculation convention selection exact ⟨result.1, result.2.2.2.2⟩ end end PhysicsLogic.IndependentReuse.CasimirHeatCurrentClient run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.CasimirHeatCurrentClient let mut traversal : Lean.CollectAxioms.State := {} let mut theorems := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theorems := theorems + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theorems >= 13 do throwError "Missing independent Casimir heat-current theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Nonfoundational Casimir heat-current client axiom {name}" for name in traversal.visited.toList do let parts := name.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls") || (`PhysicsLogic.Papers.Casimir1945.HeatCurrent.Examples).isPrefixOf name) then throwError "Independent heat-current client depends on production diagnostics {name}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.CasimirHeatCurrentClient.the_library_preserves_the_clients_actual_bulk_operator, ``PhysicsLogic.Papers.Casimir1945.HeatCurrent.equation_eighteen_preserves_the_bulk_current_divergence), (``PhysicsLogic.IndependentReuse.CasimirHeatCurrentClient.the_library_balances_the_actual_added_current, ``PhysicsLogic.Papers.Casimir1945.HeatCurrent.equation_eighteen_gives_actual_zero_divergence), (``PhysicsLogic.IndependentReuse.CasimirHeatCurrentClient.the_library_balances_the_actual_added_current, ``PhysicsLogic.Framework.CoordinateCurrent.nonvacuous_balance_of_local_balance), (``PhysicsLogic.IndependentReuse.CasimirHeatCurrentClient.the_library_extracts_columns_from_an_independently_checked_operator, ``PhysicsLogic.Papers.Casimir1945.HeatCurrent.reciprocal_operator_forces_zero_antisymmetric_column_divergences), (``PhysicsLogic.IndependentReuse.CasimirHeatCurrentClient.the_independent_symmetric_representative_is_not_the_original_tensor, ``PhysicsLogic.Papers.Casimir1945.HeatCurrent.reciprocal_operator_preserves_the_actual_symmetric_representative), (``PhysicsLogic.IndependentReuse.CasimirHeatCurrentClient.the_main_paper_derives_bulk_equality_from_its_admissible_current_correction, ``PhysicsLogic.Papers.Casimir1945.MicroscopicReversibilityAnalysis.heat_current_source_conditions_derive_actual_bulk_equivalence), (``PhysicsLogic.IndependentReuse.CasimirHeatCurrentClient.the_main_paper_assigns_symmetry_to_the_selected_representative, ``PhysicsLogic.Papers.Casimir1945.MicroscopicReversibilityAnalysis.heat_current_symmetry_requires_a_representative_convention)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Casimir heat-current client audit passed: {theorems} theorems, {uses.size} public API uses, actual independent temperature gradient, tensor addition and unchanged bulk operator, distinct normal components, no production diagnostics." namespace PhysicsLogic.IndependentReuse.CasimirHeatRouteClient open PhysicsLogic.Papers.Casimir1945 open PhysicsLogic.Papers.Casimir1945.HeatCurrent open PhysicsLogic.Framework.CoordinateCurrent open scoped BigOperators noncomputable section def temperature (point : Point 3) : Real := 17 + point 0 * point 1 + point 2 ^ 2 def energy (time : Real) (point : Point 3) : Real := 2 * (temperature point + 3 * time) def body : Set (Point 3) := {point | ∀ axis, 0 ≤ point axis ∧ point axis ≤ 2} def coefficient (amplitude : Real) : Tensor 3 := fun _ => !![3, amplitude, 0; -amplitude, 3, 0; 0, 0, 3] def correction : Tensor 3 := fun _ => !![0, 2, 0; -2, 0, 0; 0, 0, 0] def flow (amplitude : Real) : Current 3 := current (coefficient amplitude) temperature def candidate (amplitude : Real) (keepSkew : Bool) : Real := if keepSkew then amplitude else 0 def sampleTensor (amplitude : Real) (keepSkew : Bool) (shape : Set (Point 3)) : Tensor 3 := by classical exact fun point => if point ∈ shape then coefficient (candidate amplitude keepSkew) point else 0 def BareVacuumAndShapeConvention (amplitude : Real) (keepSkew : Bool) : Prop := (∀ shape point, point ∉ shape -> sampleTensor amplitude keepSkew shape point = 0) ∧ (∀ first second point, point ∈ first -> point ∈ second -> sampleTensor amplitude keepSkew first point = sampleTensor amplitude keepSkew second point) def SelectsActualSymmetricPart (amplitude : Real) (keepSkew : Bool) (selected : Real) : Prop := selected = candidate amplitude keepSkew ∧ Set.EqOn (coefficient selected) (symmetricPart (coefficient amplitude)) body def MacroscopicEnergyRate (amplitude : Real) : Prop := ∀ point ∈ body, HasDerivAt (fun time => energy time point) (divergence (flow amplitude) point) 0 def BulkIndistinguishable (first second : Real) : Prop := Set.EqOn (divergence (flow first)) (divergence (flow second)) body theorem temperature_has_its_actual_gradient (point : Point 3) : (fun axis => coordinateDerivative temperature point axis) = ![point 1, point 0, 2 * point 2] := by funext axis fin_cases axis · change deriv (fun value : Real => 17 + value * point 1 + point 2 ^ 2) (point 0) = _ simp · change deriv (fun value : Real => 17 + point 0 * value + point 2 ^ 2) (point 1) = _ simpa using ((((hasDerivAt_id (point 1)).const_mul (point 0)).const_add 17).add_const (point 2 ^ 2)).deriv · change deriv (fun value : Real => 17 + point 0 * point 1 + value ^ 2) (point 2) = _ simp theorem temperature_is_regular_with_commuting_mixed_partials (point : Point 3) : GradientRegularAt temperature point ∧ MixedPartialsAgreeAt temperature point := by have scaled (scale value : Real) : deriv (HMul.hMul scale) value = scale := by simpa using ((hasDerivAt_id value).const_mul scale).deriv constructor · constructor · intro axis fin_cases axis all_goals simp [temperature, Function.update] all_goals fun_prop · intro row column simp_rw [temperature_has_its_actual_gradient] fin_cases row <;> fin_cases column all_goals simp [Function.update] all_goals fun_prop · intro row column simp_rw [temperature_has_its_actual_gradient] fin_cases row <;> fin_cases column <;> simp [coordinateDerivative, Function.update, scaled] theorem temperature_is_positive_on_the_same_inhabited_body : (∀ point ∈ body, 0 < temperature point) ∧ ![2, 1, 1] ∈ body := by constructor · intro point inside have product := mul_nonneg (inside 0).1 (inside 1).1 have square := sq_nonneg (point 2) dsimp [temperature] linarith · intro axis fin_cases axis <;> norm_num theorem actual_flow_components (amplitude : Real) (point : Point 3) : flow amplitude point = ![3 * point 1 + amplitude * point 0, -amplitude * point 1 + 3 * point 0, 6 * point 2] := by funext row simp only [flow, current] simp_rw [temperature_has_its_actual_gradient] fin_cases row all_goals simp [coefficient, Fin.sum_univ_three] all_goals ring theorem actual_bulk_divergence_is_six (amplitude : Real) (point : Point 3) : divergence (flow amplitude) point = 6 := by have scaled (scale value : Real) : deriv (HMul.hMul scale) value = scale := by simpa using ((hasDerivAt_id value).const_mul scale).deriv simp only [divergence] simp_rw [actual_flow_components] simp [coordinateDerivative, Function.update, Fin.sum_univ_three, scaled] theorem bulk_divergence_is_the_actual_energy_rate (amplitude : Real) : MacroscopicEnergyRate amplitude := by intro point _ rw [actual_bulk_divergence_is_six] have derivative := (((hasDerivAt_id (0 : Real)).const_mul 3).const_add (temperature point)).const_mul 2 norm_num at derivative exact derivative theorem every_regular_test_has_the_reciprocal_bulk_operator (amplitude : Real) (point : Point 3) : ReciprocalOperatorAt (coefficient amplitude) point := by intro test thermal mixed rw [divergence_retains_both_product_rule_terms (coefficient amplitude) test point (fun _ _ => differentiableAt_const _) thermal, divergence_retains_both_product_rule_terms (transpose (coefficient amplitude)) test point (fun _ _ => differentiableAt_const _) thermal] have cross := mixed 0 1 simp [coordinateDerivative] at cross simp [coefficient, transpose, coordinateDerivative, Fin.sum_univ_three] rw [cross] ring theorem correction_has_actual_equation_eighteen_properties (point : Point 3) : TensorRegularAt correction point ∧ AntisymmetricAt correction point ∧ ColumnDivergenceZeroAt correction point := by refine ⟨fun _ _ => differentiableAt_const _, ?_, ?_⟩ · intro row column fin_cases row <;> fin_cases column <;> norm_num [correction] · intro column simp [coordinateDerivative, correction] theorem actual_tensor_addition_changes_the_current_parameter (amplitude : Real) : (fun point row column => coefficient amplitude point row column + correction point row column) = coefficient (amplitude + 2) := by funext point row column fin_cases row <;> fin_cases column all_goals simp [coefficient, correction] all_goals ring theorem the_actual_symmetric_part_is_positive_isotropic_conductivity (amplitude : Real) : symmetricPart (coefficient amplitude) = coefficient 0 ∧ (∀ point axis, coefficient 0 point axis axis = 3) := by constructor · funext point row column fin_cases row <;> fin_cases column <;> simp [symmetricPart, coefficient] · intro point axis fin_cases axis <;> norm_num [coefficient] theorem the_skew_part_does_not_change_the_positive_conductivity_quadratic_form (amplitude : Real) (point : Point 3) (gradient : Fin 3 -> Real) : (∑ row, ∑ column, gradient row * coefficient amplitude point row column * gradient column) = 3 * (gradient 0 ^ 2 + gradient 1 ^ 2 + gradient 2 ^ 2) := by simp [coefficient, Fin.sum_univ_three] ring theorem the_actual_temperature_gradient_has_strictly_positive_conductivity_response (amplitude : Real) : (∑ row, ∑ column, coordinateDerivative temperature ![2, 1, 1] row * coefficient amplitude ![2, 1, 1] row column * coordinateDerivative temperature ![2, 1, 1] column) = 27 := by rw [the_skew_part_does_not_change_the_positive_conductivity_quadratic_form] simp_rw [temperature_has_its_actual_gradient] norm_num [Matrix.cons_val_two] theorem both_selection_rules_have_bare_vacuum_and_shape_agreement (amplitude : Real) (keepSkew : Bool) : BareVacuumAndShapeConvention amplitude keepSkew := by constructor · intro shape point outside simp [sampleTensor, outside] · intro first second point insideFirst insideSecond simp [sampleTensor, insideFirst, insideSecond] theorem the_vacuum_extension_does_not_erase_material_conductivity (amplitude : Real) (keepSkew : Bool) : sampleTensor amplitude keepSkew body ![2, 1, 1] 0 0 = 3 ∧ sampleTensor amplitude keepSkew body ![3, 1, 1] = 0 := by have inside := temperature_is_positive_on_the_same_inhabited_body.2 have outside : ![(3 : Real), 1, 1] ∉ body := by intro assumed have forbidden := (assumed 0).2 norm_num at forbidden simp [sampleTensor, inside, outside, coefficient] def types : CasimirTypes where Sample := Unit sampleMeasurable := inferInstance Description := Unit State := Unit Process := Unit Variable := Unit Field := Unit Time := Unit Distribution := Unit Correlation := Unit Coefficient := Unit FourPole := Unit Tensor := Real Current := Real Convention := Bool /-- Only the heat route is asserted. Unrelated slots in the paper's common context have no asserted physical hypotheses and cannot certify other routes. -/ def model (amplitude : Real) (keepSkew : Bool) : MicroscopicReversibilityAnalysis types where equilibriumSystem := () selectedEquilibriumState := () selectedProcess := () thermomolecularProcess := () heatConduction := () electricConduction := () zeroField := () magneticField := () observationInterval := () microscopicLag := () macroscopicRelaxationTime := () selectedDistribution := () selectedCorrelation := () selectedEvenVariable₁ := () selectedEvenVariable₂ := () selectedOddVariable := () selectedOddVariable₂ := () selectedFourPole := () heatTransportTensor := amplitude symmetricHeatRepresentative := candidate amplitude keepSkew electricResistanceTensor := (0 : Real) hallTensor := (0 : Real) selectedHeatCurrent := amplitude alternativeHeatCurrent := amplitude + 2 vacuumAndShapeConvention := keepSkew heatTensorField := coefficient heatCurrentField := flow heatTemperature := temperature heatCurrentCorrection := correction heatDomain := body gaussianRegressionReadout := by constructor <;> exact 0 thermomolecularOrificeReadout := by constructor <;> exact 0 fieldReversedResistanceHallReadout := by constructor <;> exact 0 gaussianRegressionReadoutRepresentsAt := fun _ _ _ _ => False thermomolecularOrificeReadoutRepresentsAt := fun _ _ => False fieldReversedResistanceHallReadoutRepresentsAt := fun _ _ _ _ => False reverseField := id parity := fun _ => .even kineticCoefficient := fun _ _ _ => 0 resistanceCoefficient := fun _ _ _ => 0 equilibriumLaw := fun _ => MeasureTheory.Measure.dirac () microscopicEvolution := fun _ _ => MeasurableEquiv.refl Unit microscopicSampleReversal := MeasurableEquiv.refl Unit fluctuationObservable := fun _ _ _ => 0 lagValue := fun _ => 0 relativeLagTolerance := 0 adiabaticallyInsulated := fun _ => False equilibriumStateOf := fun _ _ => False entropyQuadraticNearEquilibrium := fun _ => False entropyQuadraticFormPositive := fun _ => False gaussianEquilibriumDistribution := fun _ _ => False conjugateVariablesDefined := fun _ => False conjugateCovarianceNormalized := fun _ => False nearEquilibrium := fun _ => False equilibriumFluctuations := fun _ => False microscopicReversalRelatesFields := fun _ _ _ => False fieldReversalAdmissible := fun _ _ _ => False averageRegressionObeysMacroscopicLaw := fun _ => False macroscopicLinearLaw := fun _ => False instantaneousDerivativeAverageZero := fun _ => False steadyFlowEstablishedAfterLag := fun _ => False regressionFollowsFromMicroscopicReversibility := fun _ => False thermomolecularLinearTransport := fun _ => False thermomolecularCrossCoefficientRelation := fun _ => False stationaryConcentrationDifference := fun _ => False molecularEnergyTransportCalculation := fun _ => False knudsenPressureLaw := fun _ => False pseudoThermostaticDisplacementRule := fun _ => False pseudoThermostaticEntropyStationary := fun _ => False pseudoThermostaticDerivationIsThermodynamic := fun _ => False divergenceHasDirectMacroscopicMeaning := MacroscopicEnergyRate currentsPhysicallyEquivalent := BulkIndistinguishable conventionVanishesInVacuumAndIgnoresShape := BareVacuumAndShapeConvention amplitude conventionSelectsSymmetricRepresentative := SelectsActualSymmetricPart amplitude fourPoleLinearResponse := fun _ => False capacitorsMakeTerminalCurrentsChargeRates := fun _ => False fourPoleReciprocal := fun _ => False everyFourPoleReciprocal := fun _ => False oneFourPoleSufficesForTensorSymmetry := fun _ => False localResistanceTensorSymmetric := fun _ => False symmetricResistanceEvenInField := fun _ => False antisymmetricHallResponseOddInField := fun _ => False theorem the_fixture_does_not_assert_unrelated_paper_routes (amplitude : Real) (keepSkew : Bool) : ¬(model amplitude keepSkew).FluctuationStatisticsSetup ∧ ¬(model amplitude keepSkew).ElectricalFourPoleSetup := ⟨fun setup => setup.1, fun setup => setup.1⟩ theorem all_actual_heat_setup_inputs_hold_together (amplitude : Real) (keepSkew : Bool) : (model amplitude keepSkew).HeatCurrentRepresentationSetup := by refine ⟨bulk_divergence_is_the_actual_energy_rate amplitude, rfl, ?_, ?_, ?_⟩ · change flow (amplitude + 2) = current (fun point row column => coefficient amplitude point row column + correction point row column) temperature rw [actual_tensor_addition_changes_the_current_parameter] rfl · intro point inside exact ⟨temperature_is_positive_on_the_same_inhabited_body.1 point inside, temperature_is_regular_with_commuting_mixed_partials point⟩ · refine ⟨![2, 1, 1], temperature_is_positive_on_the_same_inhabited_body.2, ?_⟩ intro same have normal := congrFun same 0 change flow amplitude ![2, 1, 1] 0 = flow (amplitude + 2) ![2, 1, 1] 0 at normal simp only [actual_flow_components] at normal norm_num at normal theorem all_actual_kernel_and_correction_inputs_hold_together (amplitude : Real) (keepSkew : Bool) : (model amplitude keepSkew).HeatCurrentAmbiguityCalculation := by intro point _ exact ⟨fun _ _ => differentiableAt_const _, fun _ _ => differentiableAt_const _, every_regular_test_has_the_reciprocal_bulk_operator amplitude point, correction_has_actual_equation_eighteen_properties point⟩ theorem the_bulk_observation_bridge_has_actual_observables (amplitude : Real) (keepSkew : Bool) : (model amplitude keepSkew).DivergenceEquivalenceBridge := fun _ _ _ equality => equality theorem the_symmetric_selection_input_is_realized (amplitude : Real) : (model amplitude false).HeatCurrentConventionCalculation := by intro _ _ have symmetricPart := (the_actual_symmetric_part_is_positive_isotropic_conductivity amplitude).1 have equality : Set.EqOn (coefficient 0) (HeatCurrent.symmetricPart (coefficient amplitude)) body := by rw [symmetricPart] exact Set.eqOn_refl _ _ exact ⟨⟨rfl, equality⟩, equality⟩ theorem a_complete_independent_heat_route_instance_exists (amplitude : Real) : (model amplitude false).HeatCurrentRepresentationSetup ∧ (model amplitude false).DivergenceEquivalenceBridge ∧ (model amplitude false).HeatCurrentAmbiguityCalculation ∧ (model amplitude false).conventionVanishesInVacuumAndIgnoresShape (model amplitude false).vacuumAndShapeConvention ∧ (model amplitude false).HeatCurrentConventionCalculation := ⟨all_actual_heat_setup_inputs_hold_together amplitude false, the_bulk_observation_bridge_has_actual_observables amplitude false, all_actual_kernel_and_correction_inputs_hold_together amplitude false, both_selection_rules_have_bare_vacuum_and_shape_agreement amplitude false, the_symmetric_selection_input_is_realized amplitude⟩ theorem the_main_paper_route_runs_on_the_complete_independent_instance (amplitude : Real) : (model amplitude false).HeatCurrentRepresentationResult := by have inputs := a_complete_independent_heat_route_instance_exists amplitude exact (model amplitude false).heat_current_symmetry_requires_a_representative_convention inputs.1 inputs.2.1 inputs.2.2.1 inputs.2.2.2.1 inputs.2.2.2.2 theorem the_completed_route_has_nonzero_evolution_and_different_normal_currents (amplitude : Real) : (model amplitude false).HeatCurrentRepresentationResult ∧ temperature ![2, 1, 1] = 20 ∧ energy 0 ![2, 1, 1] = 40 ∧ divergence (flow amplitude) ![2, 1, 1] = 6 ∧ flow (amplitude + 2) ![2, 1, 1] 0 - flow amplitude ![2, 1, 1] 0 = 4 := by refine ⟨the_main_paper_route_runs_on_the_complete_independent_instance amplitude, by norm_num [temperature, Matrix.cons_val_two], by norm_num [energy, temperature, Matrix.cons_val_two], actual_bulk_divergence_is_six _ _, ?_⟩ simp only [actual_flow_components] norm_num ring /-- Bare pointwise vacuum extension and common-interior shape agreement do not include a distributional interface condition. This does not contradict the paper. -/ theorem bare_vacuum_shape_and_local_reciprocity_do_not_supply_selection : (model 3 true).HeatCurrentRepresentationSetup ∧ (model 3 true).DivergenceEquivalenceBridge ∧ (model 3 true).HeatCurrentAmbiguityCalculation ∧ (model 3 true).conventionVanishesInVacuumAndIgnoresShape (model 3 true).vacuumAndShapeConvention ∧ ¬(model 3 true).HeatCurrentConventionCalculation ∧ ¬(model 3 true).HeatCurrentRepresentationResult := by have rejects : ¬(model 3 true).HeatCurrentRepresentationResult := by intro result have symmetric := result.2.2.2.2 ![2, 1, 1] temperature_is_positive_on_the_same_inhabited_body.2 0 1 change coefficient 3 ![2, 1, 1] 0 1 = coefficient 3 ![2, 1, 1] 1 0 at symmetric norm_num [coefficient] at symmetric refine ⟨all_actual_heat_setup_inputs_hold_together 3 true, the_bulk_observation_bridge_has_actual_observables 3 true, all_actual_kernel_and_correction_inputs_hold_together 3 true, both_selection_rules_have_bare_vacuum_and_shape_agreement 3 true, ?_, rejects⟩ intro selection exact rejects ((model 3 true).heat_current_symmetry_requires_a_representative_convention (all_actual_heat_setup_inputs_hold_together 3 true) (the_bulk_observation_bridge_has_actual_observables 3 true) (all_actual_kernel_and_correction_inputs_hold_together 3 true) (both_selection_rules_have_bare_vacuum_and_shape_agreement 3 true) selection) theorem the_rejected_extension_has_a_nonzero_normal_skew_jump_arbitrarily_near_the_face : ∀ distance : Real, 0 < distance -> antisymmetricPart (sampleTensor 3 true body) ![2, 1, 1] 0 1 = 3 ∧ antisymmetricPart (sampleTensor 3 true body) ![2 + distance, 1, 1] 0 1 = 0 := by intro distance positive have inside := temperature_is_positive_on_the_same_inhabited_body.2 have outside : ![2 + distance, (1 : Real), 1] ∉ body := by intro assumed have bound := (assumed 0).2 simp only [Matrix.cons_val_zero] at bound linarith norm_num [antisymmetricPart, sampleTensor, inside, outside, coefficient, candidate] rfl #check (the_main_paper_route_runs_on_the_complete_independent_instance : ∀ amplitude : Real, (model amplitude false).HeatCurrentRepresentationResult) #check (a_complete_independent_heat_route_instance_exists : ∀ amplitude : Real, (model amplitude false).HeatCurrentRepresentationSetup ∧ (model amplitude false).DivergenceEquivalenceBridge ∧ (model amplitude false).HeatCurrentAmbiguityCalculation ∧ (model amplitude false).conventionVanishesInVacuumAndIgnoresShape (model amplitude false).vacuumAndShapeConvention ∧ (model amplitude false).HeatCurrentConventionCalculation) end end PhysicsLogic.IndependentReuse.CasimirHeatRouteClient run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.CasimirHeatRouteClient let mut traversal : Lean.CollectAxioms.State := {} let mut theorems := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theorems := theorems + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theorems >= 24 do throwError "Missing complete independent Casimir heat-route theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Nonfoundational complete Casimir heat-route client axiom {name}" for name in traversal.visited.toList do let parts := name.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls") || (`PhysicsLogic.Papers.Casimir1945.HeatCurrent.Examples).isPrefixOf name) then throwError "Complete heat-route client depends on production diagnostics {name}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.CasimirHeatRouteClient.every_regular_test_has_the_reciprocal_bulk_operator, ``PhysicsLogic.Papers.Casimir1945.HeatCurrent.divergence_retains_both_product_rule_terms), (``PhysicsLogic.IndependentReuse.CasimirHeatRouteClient.the_main_paper_route_runs_on_the_complete_independent_instance, ``PhysicsLogic.Papers.Casimir1945.MicroscopicReversibilityAnalysis.heat_current_symmetry_requires_a_representative_convention), (``PhysicsLogic.IndependentReuse.CasimirHeatRouteClient.the_completed_route_has_nonzero_evolution_and_different_normal_currents, ``PhysicsLogic.Papers.Casimir1945.MicroscopicReversibilityAnalysis.heat_current_source_conditions_derive_actual_bulk_equivalence), (``PhysicsLogic.IndependentReuse.CasimirHeatRouteClient.the_completed_route_has_nonzero_evolution_and_different_normal_currents, ``PhysicsLogic.Papers.Casimir1945.HeatCurrent.reciprocal_operator_forces_zero_antisymmetric_column_divergences)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Complete Casimir heat-route client audit passed: {theorems} theorems, {uses.size} public API uses, all main-route premises jointly realized, positive conductivity, actual energy evolution, nonzero normal-current difference, explicit missing interface condition, no production diagnostics." namespace PhysicsLogic.IndependentReuse.BlattThresholdCentralClient open PhysicsLogic.Papers.BlattJackson1949 open ThresholdExtraction noncomputable section def lengths : Lengths := ⟨3, -15⟩ def readout : Readout := ⟨252 * Real.pi, -3, 2 / 3, 1 / 100⟩ theorem the_independent_source_equations_hold : PhysicalSignBranch lengths ∧ records.recordAt () lengths = readout.record := by refine ⟨by norm_num [PhysicalSignBranch, lengths], ?_⟩ apply Prod.ext · change crossSection lengths = readout.epithermalCentral norm_num [crossSection, lengths, readout] ring · change coherentAmplitude lengths = readout.coherentCentral norm_num [coherentAmplitude, lengths, readout] theorem all_central_reduction_inputs_hold_with_nonzero_rounding : readout.ArithmeticReductionAt (301 / 200) := by refine ⟨by norm_num [readout], by norm_num [readout], lengths, the_independent_source_equations_hold.1, congrArg Prod.fst the_independent_source_equations_hold.2, congrArg Prod.snd the_independent_source_equations_hold.2, ?_⟩ norm_num [PhysicsLogic.Framework.NeutralBoundState.extractedRange, lengths, readout] theorem the_public_reduction_identifies_the_independent_central_range : readout.ExtractedAt (301 / 200) := arithmetic_inputs_derive_a_unique_central_range _ _ all_central_reduction_inputs_hold_with_nonzero_rounding theorem the_central_parameter_is_not_the_rounded_report : readout.tripletCentral = 3 ∧ readout.rangeCentral = 3 / 2 ∧ |(301 / 200 : Real) - readout.rangeCentral| = 1 / 200 := by have calibrated := the_selected_branch_calibrates_the_triplet_decoder lengths the_independent_source_equations_hold.1 rw [the_independent_source_equations_hold.2] at calibrated have triplet : readout.tripletCentral = 3 := calibrated refine ⟨triplet, ?_⟩ simp only [Readout.rangeCentral, triplet] norm_num [PhysicsLogic.Framework.NeutralBoundState.extractedRange, readout] theorem the_main_paper_route_retains_the_calibrated_range_bound {T : BlattJacksonTypes} (framework : ScatteringFramework T) (setup : framework.TripletExtractionSetup) (relation : framework.EstablishedBoundStateRelation) (calculation : framework.TripletExtractionCalculation) : let data := framework.tripletThresholdReadout framework.neutronProtonExperiment framework.neutronProtonTriplet framework.nuclearRegime |framework.reportedTripletEffectiveRange - PhysicsLogic.Framework.NeutralBoundState.extractedRange (framework.reciprocalScatteringLength framework.neutronProtonTriplet) (framework.bindingMomentumValue framework.deuteronBoundEnergy)| ≤ data.rangeRoundingBudget := (framework.established_triplet_extraction_retains_the_measured_central_reduction (framework.scattering_and_binding_data_extract_triplet_effective_range setup relation calculation)).2 def firstComparison : CoherentComparison.Configuration := ⟨lengths, 0, 0, 0, 0⟩ def biasedComparison : CoherentComparison.Configuration := ⟨⟨342 / 103, -1524 / 103⟩, -60 / 103, 0, 0, 0⟩ def comparisonMomentum : Real := 7 / 20 theorem the_independent_comparison_inputs_are_realized : firstComparison.IdealComparisonAt comparisonMomentum ∧ biasedComparison.IdealComparisonAt comparisonMomentum ∧ (CoherentComparison.records comparisonMomentum).recordAt false firstComparison = (CoherentComparison.records comparisonMomentum).recordAt false biasedComparison ∧ firstComparison.rangeValue comparisonMomentum ≠ biasedComparison.rangeValue comparisonMomentum := by refine ⟨by norm_num [CoherentComparison.Configuration.IdealComparisonAt, PhysicalSignBranch, firstComparison, lengths, comparisonMomentum], by norm_num [CoherentComparison.Configuration.IdealComparisonAt, PhysicalSignBranch, biasedComparison, comparisonMomentum], ?_, ?_⟩ · change (⟨crossSection firstComparison.lengths, firstComparison.parahydrogen, firstComparison.crystal, 0⟩ : CoherentComparison.Readings) = ⟨crossSection biasedComparison.lengths, biasedComparison.parahydrogen, biasedComparison.crystal, 0⟩ congr 1 · norm_num [crossSection, firstComparison, lengths, biasedComparison] ring · norm_num [CoherentComparison.Configuration.parahydrogen, coherentAmplitude, firstComparison, lengths, biasedComparison] · norm_num [CoherentComparison.Configuration.crystal, coherentAmplitude, firstComparison, lengths, biasedComparison] · norm_num [CoherentComparison.Configuration.rangeValue, PhysicsLogic.Framework.NeutralBoundState.extractedRange, firstComparison, lengths, biasedComparison, comparisonMomentum] theorem matching_measurements_need_an_independent_calibration : (¬∃ value, (CoherentComparison.records comparisonMomentum).RecordDeterminesValueAt false ((CoherentComparison.records comparisonMomentum).recordAt false firstComparison) value) ∧ (CoherentComparison.records comparisonMomentum).RecordDeterminesValueAt true ((CoherentComparison.records comparisonMomentum).recordAt true biasedComparison) (biasedComparison.rangeValue comparisonMomentum) := by refine ⟨?_, CoherentComparison.reference_access_determines_the_selected_range _ _ the_independent_comparison_inputs_are_realized.2.1⟩ exact (CoherentComparison.records comparisonMomentum).ambiguous_record_does_not_determine_a_value false firstComparison biasedComparison the_independent_comparison_inputs_are_realized.1 the_independent_comparison_inputs_are_realized.2.1 the_independent_comparison_inputs_are_realized.2.2.1 the_independent_comparison_inputs_are_realized.2.2.2 def noisyComparison : CoherentComparison.Configuration := { firstComparison with commonOffset := 1 / 20 paraOnlyOffset := 1 / 10 paraError := 1 / 100 crystalError := -1 / 100 } theorem the_comparison_transfers_only_its_explicit_error_bounds : |noisyComparison.paraOnlyOffset| ≤ 7 / 50 ∧ |noisyComparison.parahydrogen - coherentAmplitude noisyComparison.lengths| ≤ 9 / 50 := by have agree : |noisyComparison.parahydrogen - noisyComparison.crystal| ≤ (3 / 25 : Real) := by norm_num [CoherentComparison.Configuration.parahydrogen, CoherentComparison.Configuration.crystal, coherentAmplitude, noisyComparison, firstComparison, lengths] constructor · have specific := CoherentComparison.agreement_and_error_bounds_control_the_method_specific_contamination noisyComparison (3 / 25) (1 / 100) (1 / 100) agree (by norm_num [noisyComparison]) (by norm_num [noisyComparison]) convert specific using 1 norm_num · have absolute := CoherentComparison.absolute_accuracy_additionally_requires_reference_control noisyComparison (3 / 25) (1 / 20) (1 / 100) agree (by norm_num [noisyComparison]) (by norm_num [noisyComparison]) convert absolute using 1 norm_num def errorReadout : Readout := { readout with bindingMomentumCentral := comparisonMomentum, rangeRoundingBudget := 1 / 2000 } theorem the_independent_error_readout_inputs_hold : errorReadout.ArithmeticReductionAt (40 / 147 + 1 / 2000) ∧ errorReadout.tripletCentral = 3 := by have same : records.recordAt () lengths = errorReadout.record := the_independent_source_equations_hold.2 have decoded := the_selected_branch_calibrates_the_triplet_decoder lengths the_independent_source_equations_hold.1 rw [same] at decoded refine ⟨⟨by norm_num [errorReadout, comparisonMomentum], by norm_num [errorReadout], lengths, the_independent_source_equations_hold.1, congrArg Prod.fst same, congrArg Prod.snd same, ?_⟩, decoded⟩ norm_num [PhysicsLogic.Framework.NeutralBoundState.extractedRange, lengths, errorReadout, comparisonMomentum] theorem the_independent_rounding_and_model_bounds_must_be_combined : |(40 / 147 + 1 / 2000 : Real) - (40 / 147 - 1 / 1000)| ≤ 3 / 2000 ∧ ¬|(40 / 147 + 1 / 2000 : Real) - (40 / 147 - 1 / 1000)| ≤ max (1 / 2000) (1 / 1000) := by have extracted := arithmetic_inputs_derive_a_unique_central_range _ _ the_independent_error_readout_inputs_hold.1 have residual : |PhysicsLogic.Framework.NeutralBoundState.effectiveRangeResidual errorReadout.tripletCentral⁻¹ (40 / 147 - 1 / 1000) errorReadout.bindingMomentumCentral| ≤ (49 / 800000 : Real) := by rw [the_independent_error_readout_inputs_hold.2] norm_num [PhysicsLogic.Framework.NeutralBoundState.effectiveRangeResidual, errorReadout, comparisonMomentum] have total := reported_range_error_combines_rounding_and_binding_residual errorReadout _ _ _ extracted residual refine ⟨?_, by norm_num⟩ convert total using 1 norm_num [errorReadout, comparisonMomentum] theorem the_main_route_retains_the_combined_reported_error {T : BlattJacksonTypes} (framework : ScatteringFramework T) (setup : framework.TripletExtractionSetup) (relation : framework.EstablishedBoundStateRelation) (calculation : framework.TripletExtractionCalculation) : |framework.reportedTripletEffectiveRange - framework.effectiveRangeValue framework.neutronProtonTriplet| ≤ (framework.tripletThresholdReadout framework.neutronProtonExperiment framework.neutronProtonTriplet framework.nuclearRegime).rangeRoundingBudget + 2 * framework.boundStateShapeCoefficientBound framework.neutronProtonTriplet * framework.effectiveRangeValue framework.neutronProtonTriplet ^ 3 * framework.bindingMomentumValue framework.deuteronBoundEnergy ^ 2 := framework.scattering_binding_and_rounding_control_bound_the_reported_range_error setup relation calculation #check (matching_measurements_need_an_independent_calibration : (¬∃ value, (CoherentComparison.records comparisonMomentum).RecordDeterminesValueAt false ((CoherentComparison.records comparisonMomentum).recordAt false firstComparison) value) ∧ (CoherentComparison.records comparisonMomentum).RecordDeterminesValueAt true ((CoherentComparison.records comparisonMomentum).recordAt true biasedComparison) (biasedComparison.rangeValue comparisonMomentum)) #check (the_independent_error_readout_inputs_hold : errorReadout.ArithmeticReductionAt (40 / 147 + 1 / 2000) ∧ errorReadout.tripletCentral = 3) #check (all_central_reduction_inputs_hold_with_nonzero_rounding : readout.ArithmeticReductionAt (301 / 200)) #check (the_public_reduction_identifies_the_independent_central_range : readout.ExtractedAt (301 / 200)) end end PhysicsLogic.IndependentReuse.BlattThresholdCentralClient run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.BlattThresholdCentralClient let mut traversal : Lean.CollectAxioms.State := {} let mut theorems := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theorems := theorems + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theorems >= 11 do throwError "Missing independent Blatt-Jackson threshold and calibration theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Nonfoundational Blatt-Jackson threshold client axiom {name}" for name in traversal.visited.toList do let parts := name.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Blatt-Jackson threshold client depends on production diagnostics {name}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.BlattThresholdCentralClient.the_public_reduction_identifies_the_independent_central_range, ``PhysicsLogic.Papers.BlattJackson1949.ThresholdExtraction.arithmetic_inputs_derive_a_unique_central_range), (``PhysicsLogic.IndependentReuse.BlattThresholdCentralClient.the_central_parameter_is_not_the_rounded_report, ``PhysicsLogic.Papers.BlattJackson1949.ThresholdExtraction.the_selected_branch_calibrates_the_triplet_decoder), (``PhysicsLogic.IndependentReuse.BlattThresholdCentralClient.the_main_paper_route_retains_the_calibrated_range_bound, ``PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework.scattering_and_binding_data_extract_triplet_effective_range), (``PhysicsLogic.IndependentReuse.BlattThresholdCentralClient.the_main_paper_route_retains_the_calibrated_range_bound, ``PhysicsLogic.Papers.BlattJackson1949.ThresholdExtraction.arithmetic_inputs_derive_a_unique_central_range), (``PhysicsLogic.IndependentReuse.BlattThresholdCentralClient.matching_measurements_need_an_independent_calibration, ``PhysicsLogic.Philosophy.Measurement.RecordAccessContext.ambiguous_record_does_not_determine_a_value), (``PhysicsLogic.IndependentReuse.BlattThresholdCentralClient.matching_measurements_need_an_independent_calibration, ``PhysicsLogic.Papers.BlattJackson1949.ThresholdExtraction.CoherentComparison.reference_access_determines_the_selected_range), (``PhysicsLogic.IndependentReuse.BlattThresholdCentralClient.the_comparison_transfers_only_its_explicit_error_bounds, ``PhysicsLogic.Papers.BlattJackson1949.ThresholdExtraction.CoherentComparison.agreement_and_error_bounds_control_the_method_specific_contamination), (``PhysicsLogic.IndependentReuse.BlattThresholdCentralClient.the_comparison_transfers_only_its_explicit_error_bounds, ``PhysicsLogic.Papers.BlattJackson1949.ThresholdExtraction.CoherentComparison.absolute_accuracy_additionally_requires_reference_control), (``PhysicsLogic.IndependentReuse.BlattThresholdCentralClient.the_independent_rounding_and_model_bounds_must_be_combined, ``PhysicsLogic.Papers.BlattJackson1949.ThresholdExtraction.reported_range_error_combines_rounding_and_binding_residual), (``PhysicsLogic.IndependentReuse.BlattThresholdCentralClient.the_main_route_retains_the_combined_reported_error, ``PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework.scattering_binding_and_rounding_control_bound_the_reported_range_error), (``PhysicsLogic.IndependentReuse.BlattThresholdCentralClient.the_main_route_retains_the_combined_reported_error, ``PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework.bound_state_route_quantifies_the_inverse_range_extraction_error)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Blatt-Jackson threshold client audit passed: {theorems} theorems, {uses.size} public API uses, actual two-channel inputs, independent common-offset ambiguity, nonzero rounding and model errors, conditional calibration, no production diagnostics." namespace PhysicsLogic.IndependentReuse.BlattFiniteAccuracyClient open PhysicsLogic.Papers.BlattJackson1949 open ThresholdExtraction open ThresholdExtraction.FiniteAccuracy noncomputable section def readout : Readout := ⟨12721 / 50 * Real.pi, -59 / 20, 7 / 20, 0⟩ def budgets : Budgets := ⟨121 / 50 * Real.pi, 1 / 20, 1 / 100⟩ def lowerLengths : Lengths := ⟨3, -15⟩ def upperLengths : Lengths := ⟨31 / 10, -151 / 10⟩ def first : Configuration := ⟨lowerLengths, 17 / 50, 121 / 50 * Real.pi, 1 / 20, 1 / 100⟩ def second : Configuration := ⟨upperLengths, 9 / 25, -121 / 50 * Real.pi, -1 / 20, -1 / 100⟩ theorem the_independent_interval_has_a_regular_physical_branch : (inputBox readout budgets).tripletLower = 3 ∧ (inputBox readout budgets).tripletUpper = 31 / 10 ∧ (inputBox readout budgets).Regular := by have lower := the_selected_branch_calibrates_the_triplet_decoder lowerLengths (by norm_num [PhysicalSignBranch, lowerLengths]) have upper := the_selected_branch_calibrates_the_triplet_decoder upperLengths (by norm_num [PhysicalSignBranch, upperLengths]) have lowerRecord : (inputBox readout budgets).rateLower = crossSection lowerLengths ∧ (inputBox readout budgets).coherentLower = coherentAmplitude lowerLengths := by norm_num [inputBox, readout, budgets, crossSection, coherentAmplitude, lowerLengths] ring have upperRecord : (inputBox readout budgets).rateUpper = crossSection upperLengths ∧ (inputBox readout budgets).coherentUpper = coherentAmplitude upperLengths := by norm_num [inputBox, readout, budgets, crossSection, coherentAmplitude, upperLengths] ring have low : (inputBox readout budgets).tripletLower = 3 := by unfold Box.tripletLower rw [lowerRecord.1, lowerRecord.2] exact lower have high : (inputBox readout budgets).tripletUpper = 31 / 10 := by unfold Box.tripletUpper rw [upperRecord.1, upperRecord.2] exact upper refine ⟨low, high, ?_⟩ unfold Box.Regular rw [low] norm_num [inputBox, readout, budgets] theorem the_independent_candidates_retain_all_three_nonzero_errors : (FiniteAccuracy.records budgets).RecordFiberAt () (reportedInputs readout) first ∧ (FiniteAccuracy.records budgets).RecordFiberAt () (reportedInputs readout) second ∧ first.rateError ≠ 0 ∧ first.coherentError ≠ 0 ∧ first.momentumError ≠ 0 ∧ second.rateError ≠ 0 ∧ second.coherentError ≠ 0 ∧ second.momentumError ≠ 0 := by refine ⟨?_, ?_, ?_⟩ · constructor · norm_num [FiniteAccuracy.records, PhysicalSignBranch, first, lowerLengths, budgets, abs_mul, abs_of_pos Real.pi_pos] · apply Prod.ext · change crossSection first.lengths + first.rateError = readout.epithermalCentral norm_num [crossSection, first, lowerLengths, readout] ring · apply Prod.ext <;> norm_num [FiniteAccuracy.records, reportedInputs, coherentAmplitude, first, lowerLengths, readout] · constructor · norm_num [FiniteAccuracy.records, PhysicalSignBranch, second, upperLengths, budgets, abs_mul, abs_of_pos Real.pi_pos] · apply Prod.ext · change crossSection second.lengths + second.rateError = readout.epithermalCentral norm_num [crossSection, second, upperLengths, readout] ring · apply Prod.ext <;> norm_num [FiniteAccuracy.records, reportedInputs, coherentAmplitude, second, upperLengths, readout] · norm_num [first, second, Real.pi_ne_zero] theorem the_public_api_gives_an_interval_without_an_exact_value : (FiniteAccuracy.records budgets).RecordConstrainsValueAt () (reportedInputs readout) (fun range => 25 / 243 ≤ range ∧ range ≤ 5800 / 8959) ∧ ¬∃ range, (FiniteAccuracy.records budgets).RecordDeterminesValueAt () (reportedInputs readout) range := by have lower : (inputBox readout budgets).rangeLower = 25 / 243 := by unfold Box.rangeLower rw [the_independent_interval_has_a_regular_physical_branch.1] norm_num [inputBox, readout, budgets] have upper : (inputBox readout budgets).rangeUpper = 5800 / 8959 := by unfold Box.rangeUpper rw [the_independent_interval_has_a_regular_physical_branch.2.1] norm_num [inputBox, readout, budgets] constructor · have interval := an_inhabited_finite_accuracy_record_constrains_the_range readout budgets the_independent_interval_has_a_regular_physical_branch.2.2 ⟨first, the_independent_candidates_retain_all_three_nonzero_errors.1⟩ simpa only [lower, upper] using interval · have ambiguity := (FiniteAccuracy.records budgets).ambiguous_record_does_not_determine_a_value () first second the_independent_candidates_retain_all_three_nonzero_errors.1.1 the_independent_candidates_retain_all_three_nonzero_errors.2.1.1 (the_independent_candidates_retain_all_three_nonzero_errors.1.2.trans the_independent_candidates_retain_all_three_nonzero_errors.2.1.2.symm) (by norm_num [FiniteAccuracy.records, PhysicsLogic.Framework.NeutralBoundState.extractedRange, first, second, lowerLengths, upperLengths]) rw [the_independent_candidates_retain_all_three_nonzero_errors.1.2] at ambiguity exact ambiguity theorem the_main_route_accepts_finite_accuracy_without_exact_central_calibration {T : BlattJacksonTypes} (framework : ScatteringFramework T) (budgets : ThresholdExtraction.FiniteAccuracy.Budgets) (candidate : ThresholdExtraction.FiniteAccuracy.Configuration) (regular : (ThresholdExtraction.FiniteAccuracy.inputBox (framework.tripletThresholdReadout framework.neutronProtonExperiment framework.neutronProtonTriplet framework.nuclearRegime) budgets).Regular) (compatible : (ThresholdExtraction.FiniteAccuracy.records budgets).RecordFiberAt () (ThresholdExtraction.FiniteAccuracy.reportedInputs (framework.tripletThresholdReadout framework.neutronProtonExperiment framework.neutronProtonTriplet framework.nuclearRegime)) candidate) (inverseLength : candidate.lengths.triplet⁻¹ = framework.reciprocalScatteringLength framework.neutronProtonTriplet) (bindingMomentum : candidate.momentum = framework.bindingMomentumValue framework.deuteronBoundEnergy) (boundRelation : framework.EstablishedBoundStateRelation) : let box := ThresholdExtraction.FiniteAccuracy.inputBox (framework.tripletThresholdReadout framework.neutronProtonExperiment framework.neutronProtonTriplet framework.nuclearRegime) budgets let modelError := 2 * framework.boundStateShapeCoefficientBound framework.neutronProtonTriplet * framework.effectiveRangeValue framework.neutronProtonTriplet ^ 3 * framework.bindingMomentumValue framework.deuteronBoundEnergy ^ 2 box.rangeLower - modelError ≤ framework.effectiveRangeValue framework.neutronProtonTriplet ∧ framework.effectiveRangeValue framework.neutronProtonTriplet ≤ box.rangeUpper + modelError := framework.finite_scattering_and_binding_records_enclose_the_effective_range budgets candidate regular compatible inverseLength bindingMomentum boundRelation #check (the_independent_candidates_retain_all_three_nonzero_errors : (FiniteAccuracy.records budgets).RecordFiberAt () (reportedInputs readout) first ∧ (FiniteAccuracy.records budgets).RecordFiberAt () (reportedInputs readout) second ∧ first.rateError ≠ 0 ∧ first.coherentError ≠ 0 ∧ first.momentumError ≠ 0 ∧ second.rateError ≠ 0 ∧ second.coherentError ≠ 0 ∧ second.momentumError ≠ 0) #check (the_public_api_gives_an_interval_without_an_exact_value : (FiniteAccuracy.records budgets).RecordConstrainsValueAt () (reportedInputs readout) (fun range => 25 / 243 ≤ range ∧ range ≤ 5800 / 8959) ∧ ¬∃ range, (FiniteAccuracy.records budgets).RecordDeterminesValueAt () (reportedInputs readout) range) end end PhysicsLogic.IndependentReuse.BlattFiniteAccuracyClient run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.BlattFiniteAccuracyClient let mut traversal : Lean.CollectAxioms.State := {} let mut theorems := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theorems := theorems + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theorems >= 4 do throwError "Missing independent finite-accuracy theorems" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Nonfoundational finite-accuracy client axiom {name}" for name in traversal.visited.toList do let parts := name.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Finite-accuracy client depends on production diagnostics {name}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.BlattFiniteAccuracyClient.the_independent_interval_has_a_regular_physical_branch, ``PhysicsLogic.Papers.BlattJackson1949.ThresholdExtraction.the_selected_branch_calibrates_the_triplet_decoder), (``PhysicsLogic.IndependentReuse.BlattFiniteAccuracyClient.the_public_api_gives_an_interval_without_an_exact_value, ``PhysicsLogic.Papers.BlattJackson1949.ThresholdExtraction.FiniteAccuracy.an_inhabited_finite_accuracy_record_constrains_the_range), (``PhysicsLogic.IndependentReuse.BlattFiniteAccuracyClient.the_public_api_gives_an_interval_without_an_exact_value, ``PhysicsLogic.Philosophy.Measurement.RecordAccessContext.ambiguous_record_does_not_determine_a_value), (``PhysicsLogic.IndependentReuse.BlattFiniteAccuracyClient.the_main_route_accepts_finite_accuracy_without_exact_central_calibration, ``PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework.finite_scattering_and_binding_records_enclose_the_effective_range), (``PhysicsLogic.IndependentReuse.BlattFiniteAccuracyClient.the_main_route_accepts_finite_accuracy_without_exact_central_calibration, ``PhysicsLogic.Papers.BlattJackson1949.ScatteringFramework.bound_state_route_quantifies_the_inverse_range_extraction_error)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Blatt-Jackson finite-accuracy client audit passed: {theorems} theorems, {uses.size} public API uses, nonzero errors in all three inputs, inhabited interval without exact identification, main model-error composition, no production diagnostics." namespace PhysicsLogic.IndependentReuse.BoundedParityRouteClient open PhysicsLogic.Framework open PhysicsLogic.Papers.WuEtAl1957 PhysicsLogic.Papers.GarwinLedermanWeinrich1957 -- Generic integration clients check the full contracts; these are not reconstructed experiments. theorem the_wu_route_accepts_bounded_records_before_selected_controls {T : Wu1957Types} (framework : WuExperimentFramework T) (encode : T.CountRate → Real) (budget : RateRecordInference.ErrorBudget) (configuration : RateRecordInference.Configuration) (calibration : framework.EstablishedPolarizationCalibration) (observation : framework.BoundedBetaObservationCalculation encode budget configuration) (controlObservations : framework.ControlObservationCalculation) (artifactExclusion : framework.ArtifactExclusionCalculation) (prediction : framework.ParityConservationPredictionCalculation) (interpretation : framework.ParityViolationInterpretationCalculation) : framework.EstablishedParityViolation := framework.bounded_records_and_selected_controls_establish_beta_decay_parity_violation encode budget configuration calibration observation controlObservations artifactExclusion prediction interpretation theorem the_muon_route_accepts_bounded_records_before_stopped_pion_control {T : Garwin1957Types} (framework : GarwinExperimentFramework T) (encode : T.CountRate → Real) (budget : RateRecordInference.ErrorBudget) (readout : Real × Real) (configuration : RateRecordInference.Configuration) (apparatus : framework.StoppedMuonApparatusResult) (scanSetup : framework.PrecessionScanSetup) (scan : framework.BoundedPrecessionScanCalculation encode budget readout configuration) (polarization : framework.PolarizationInterpretationCalculation) (controlSetup : framework.StoppedPionControlSetup) (controlCalculation : framework.StoppedPionControlCalculation) (controlInterpretation : framework.StoppedPionControlInterpretation) (controlScope : framework.StoppedPionControlScope) (paritySetup : framework.ParityInferenceSetup) (prediction : framework.ParityConservationPredictionCalculation) (interpretation : framework.ParityViolationInterpretationCalculation) : framework.ParityViolationResult := framework.bounded_scan_and_stopped_pion_control_establish_decay_parity_violation encode budget readout configuration apparatus scanSetup scan polarization controlSetup controlCalculation controlInterpretation controlScope paritySetup prediction interpretation #check (the_wu_route_accepts_bounded_records_before_selected_controls : ∀ {T : Wu1957Types} (framework : WuExperimentFramework T) (encode : T.CountRate → Real) (budget : RateRecordInference.ErrorBudget) (configuration : RateRecordInference.Configuration), framework.EstablishedPolarizationCalibration → framework.BoundedBetaObservationCalculation encode budget configuration → framework.ControlObservationCalculation → framework.ArtifactExclusionCalculation → framework.ParityConservationPredictionCalculation → framework.ParityViolationInterpretationCalculation → framework.EstablishedParityViolation) #check (the_muon_route_accepts_bounded_records_before_stopped_pion_control : ∀ {T : Garwin1957Types} (framework : GarwinExperimentFramework T) (encode : T.CountRate → Real) (budget : RateRecordInference.ErrorBudget) (readout : Real × Real) (configuration : RateRecordInference.Configuration), framework.StoppedMuonApparatusResult → framework.PrecessionScanSetup → framework.BoundedPrecessionScanCalculation encode budget readout configuration → framework.PolarizationInterpretationCalculation → framework.StoppedPionControlSetup → framework.StoppedPionControlCalculation → framework.StoppedPionControlInterpretation → framework.StoppedPionControlScope → framework.ParityInferenceSetup → framework.ParityConservationPredictionCalculation → framework.ParityViolationInterpretationCalculation → framework.ParityViolationResult) end PhysicsLogic.IndependentReuse.BoundedParityRouteClient run_cmd do let environment <- Lean.getEnv let clientNamespace := `PhysicsLogic.IndependentReuse.BoundedParityRouteClient let mut traversal : Lean.CollectAxioms.State := {} let mut theorems := 0 for (name, info) in environment.constants do unless clientNamespace.isPrefixOf name do continue if let .thmInfo _ := info then theorems := theorems + 1 let (_, updated) := ((Lean.CollectAxioms.collect name).run environment).run traversal traversal := updated unless theorems == 2 do throwError "Missing bounded-parity integration clients" for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Nonfoundational bounded-parity client axiom {name}" let isDiagnostic := fun (name : Lean.Name) => let parts := name.toString.splitOn "." parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.dropLast.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) if isDiagnostic ``PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.EstablishedArtifactControls then throwError "Physical artifact-control propositions are not diagnostic namespaces" for name in #[`PhysicsLogic.Framework.RateRecordInference.Countermodels.example, `PhysicsLogic.Framework.SampleControls.example, `PhysicsLogic.deprecated.Example.witness] do unless isDiagnostic name do throwError "Diagnostic namespace escaped guard: {name}" for name in traversal.visited.toList do if isDiagnostic name then throwError "Bounded-parity client depends on production diagnostics {name}" let uses : Array (Lean.Name × Lean.Name) := #[ (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_wu_route_accepts_bounded_records_before_selected_controls, ``PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.bounded_records_and_selected_controls_establish_beta_decay_parity_violation), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_wu_route_accepts_bounded_records_before_selected_controls, ``PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.bounded_observation_reconstructs_the_beta_calculation), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_wu_route_accepts_bounded_records_before_selected_controls, ``PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.beta_record_establishes_time_correlated_directional_asymmetry), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_wu_route_accepts_bounded_records_before_selected_controls, ``PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.reversal_and_unpolarized_specimens_exclude_selected_artifacts), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_wu_route_accepts_bounded_records_before_selected_controls, ``PhysicsLogic.Papers.WuEtAl1957.WuExperimentFramework.controlled_asymmetry_establishes_beta_decay_parity_violation), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_muon_route_accepts_bounded_records_before_stopped_pion_control, ``PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.bounded_scan_and_stopped_pion_control_establish_decay_parity_violation), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_muon_route_accepts_bounded_records_before_stopped_pion_control, ``PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.bounded_scan_reconstructs_the_precession_calculation), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_muon_route_accepts_bounded_records_before_stopped_pion_control, ``PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.polarization_preservation_and_precession_scan_establish_strong_muon_polarization), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_muon_route_accepts_bounded_records_before_stopped_pion_control, ``PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.stopped_pion_null_scan_excludes_only_the_selected_field_alternative), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_muon_route_accepts_bounded_records_before_stopped_pion_control, ``PhysicsLogic.Papers.GarwinLedermanWeinrich1957.GarwinExperimentFramework.controlled_precession_asymmetry_refutes_parity_invariance), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_wu_route_accepts_bounded_records_before_selected_controls, ``PhysicsLogic.Framework.RateRecordInference.a_separated_record_distinguishes_the_bound_physical_rates), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_wu_route_accepts_bounded_records_before_selected_controls, ``PhysicsLogic.Philosophy.Measurement.RecordAccessContext.record_constraint_holds_on_compatible_configuration), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_muon_route_accepts_bounded_records_before_stopped_pion_control, ``PhysicsLogic.Framework.RateRecordInference.a_separated_record_distinguishes_the_bound_physical_rates), (``PhysicsLogic.IndependentReuse.BoundedParityRouteClient.the_muon_route_accepts_bounded_records_before_stopped_pion_control, ``PhysicsLogic.Philosophy.Measurement.RecordAccessContext.record_constraint_holds_on_compatible_configuration)] for (declaration, required) in uses do unless DownstreamDependencyAudit.usesConstant environment declaration required do throwError "{declaration} does not use {required}" Lean.logInfo m!"Bounded-parity integration audit passed: {theorems} generic clients, {uses.size} actual proof dependencies, measurement constraints through selected controls to parity conclusions, no production diagnostics." run_cmd do let environment <- Lean.getEnv let cases : Array (Lean.Name × Array Lean.Name) := #[ (``PhysicsLogic.Papers.Mori1965.CanonicalPreparation.sampled_projection_derives_the_zero_linear_preparation_force, #[``PhysicsLogic.Papers.Mori1965.ClassicalEnsemble.source_orthogonality_is_an_actual_classical_zero_cross_moment, ``PhysicsLogic.Framework.FiniteResidualCovariance.LinearResidualModel.actual_centered_initial_sampling_gives_zero_mean_force, ``PhysicsLogic.Framework.FiniteCanonicalData.the_canonical_mean_is_the_actual_ensemble_average, ``PhysicsLogic.Framework.FiniteCanonicalData.canonical_mean_response_is_the_actual_covariance, ``PhysicsLogic.Framework.FiniteCanonicalData.normalized_canonical_preparation_realizes_the_field_response]), (``PhysicsLogic.Papers.Kubo1957.FiniteCanonicalPreparation.isothermal_admittance_is_the_actual_normalized_preparation_derivative, #[``PhysicsLogic.Papers.Kubo1957.FiniteCanonicalPreparation.a_positive_temperature_finite_preparation_is_admissible, ``PhysicsLogic.Papers.Kubo1957.FiniteCanonicalPreparation.diagonal_isothermal_response_is_the_actual_canonical_covariance, ``PhysicsLogic.Papers.Kubo1957.FiniteCanonicalPreparation.diagonal_isolated_static_response_vanishes_without_changing_populations, ``PhysicsLogic.Framework.FiniteCanonicalData.canonical_mean_response_is_the_actual_covariance, ``PhysicsLogic.Framework.FiniteCanonicalData.normalized_canonical_preparation_realizes_the_field_response])] let mut dependencies := 0 for (declaration, required) in cases do let (_, traversal) := ((Lean.CollectAxioms.collect declaration).run environment).run {} for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Nonfoundational canonical-preparation axiom {name}" for name in traversal.visited.toList do let parts := name.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.dropLast.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Generic canonical-preparation proof depends on diagnostic {name}" for target in required do unless DownstreamDependencyAudit.usesConstant environment declaration target do throwError "Canonical preparation {declaration} does not use {target}" dependencies := dependencies + 1 unless dependencies == 10 do throwError "Missing canonical-preparation dependency tests" Lean.logInfo m!"Canonical-preparation integration audit passed: two generic paper endpoints, {dependencies} actual dependencies, no diagnostic witnesses." run_cmd do let environment <- Lean.getEnv let cases : Array (Lean.Name × Array Lean.Name) := #[ (``PhysicsLogic.Papers.Mori1965.CanonicalPreparation.orthogonality_controls_the_prepared_force_on_a_local_field_window, #[``PhysicsLogic.Framework.FiniteCanonicalData.every_positive_tolerance_has_a_model_dependent_preparation_window, ``PhysicsLogic.Framework.FiniteCanonicalData.canonical_mean_response_is_the_actual_covariance, ``PhysicsLogic.Framework.FiniteCanonicalData.normalized_canonical_preparation_realizes_the_field_response]), (``PhysicsLogic.Papers.Kubo1957.FiniteCanonicalPreparation.the_isothermal_coefficient_controls_a_local_preparation_prediction, #[``PhysicsLogic.Framework.FiniteCanonicalData.every_positive_tolerance_has_a_model_dependent_preparation_window, ``PhysicsLogic.Framework.FiniteCanonicalData.canonical_mean_response_is_the_actual_covariance, ``PhysicsLogic.Framework.FiniteCanonicalData.a_preparation_window_controls_actual_predictions, ``PhysicsLogic.Papers.Kubo1957.FiniteCanonicalPreparation.diagonal_isothermal_response_is_the_actual_canonical_covariance, ``PhysicsLogic.Framework.FiniteCanonicalData.normalized_canonical_preparation_realizes_the_field_response])] let mut dependencies := 0 for (declaration, required) in cases do let (_, traversal) := ((Lean.CollectAxioms.collect declaration).run environment).run {} for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Nonfoundational preparation-window axiom {name}" for name in traversal.visited.toList do let parts := name.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.dropLast.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Generic preparation-window proof depends on diagnostic {name}" for target in required do unless DownstreamDependencyAudit.usesConstant environment declaration target do throwError "Preparation window {declaration} does not use {target}" dependencies := dependencies + 1 unless dependencies == 8 do throwError "Missing preparation-window dependency tests" Lean.logInfo m!"Preparation-window integration audit passed: two generic paper endpoints, {dependencies} actual dependencies, no diagnostic witnesses." namespace PhysicsLogic.IndependentReuse.RadiationHorizonClient open PhysicsLogic.Papers.Guth1981 open RadiationHorizon MeasureTheory noncomputable section set_option backward.isDefEq.respectTransparency false -- Only the radiation geometry is asserted. The other historical calibration -- fields are not claimed to satisfy Guth's full thermal or numerical model. def radiationReadout : StandardHorizonFormulaReadout where planckMassValue := 1 gammaValue := 1 effectiveRelativisticDegreesValue := 1 presentEntropyDensityValue := 1 presentPhotonTemperatureValue := 1 presentObservedRadiusValue := 1 temperatureValueAt := fun _ => 1 scaleFactorValueAt := scaleFactor 3 entropyDensityValueAtTemperature := fun _ => 1 observedRegionRadiusValueAt := fun _ => 6 theorem the_nonunit_radiation_readout_derives_its_integrable_horizon (time : Real) (positive : 0 < time) : radiationReadout.EquationTwoEighteenHorizonDistanceAt time := radiationReadout.radiation_scaling_derives_the_actual_integral_and_horizon_formula ⟨3, by norm_num, fun _ _ => rfl⟩ time positive theorem the_age_four_horizon_has_physical_radius_eight : radiationReadout.horizonDistanceValueAt 4 = 8 := by have horizon := (the_nonunit_radiation_readout_derives_its_integrable_horizon 4 (by norm_num)).2.2 norm_num at horizon ⊢ exact horizon theorem the_same_epoch_has_an_accessible_inner_source : Reachable 3 4 1 := by rw [the_actual_physical_horizon_bounds_all_regular_radial_signals 3 4 1 (by norm_num) (by norm_num)] norm_num [scaleFactor] theorem the_same_epoch_has_an_inaccessible_outer_source : ¬Reachable 3 4 2 := by rw [the_actual_physical_horizon_bounds_all_regular_radial_signals 3 4 2 (by norm_num) (by norm_num)] norm_num [scaleFactor] theorem the_outer_source_can_send_a_new_signal_at_age_four : SignalFrom 3 4 25 2 := by have early : conformalTime 3 4 = 4 / 3 := by rw [the_actual_conformal_integral_is_two_sqrt_time_over_normalization 3 4 (by norm_num)] norm_num have late : conformalTime 3 25 = 10 / 3 := by rw [the_actual_conformal_integral_is_two_sqrt_time_over_normalization 3 25 (by norm_num)] norm_num refine ⟨by norm_num, by norm_num, ?_⟩ rw [early, late] exact an_affine_radial_signal_realizes_every_allowed_separation _ _ _ (by norm_num) (by norm_num) theorem the_fixed_emission_null_boundary_is_closed : SignalFrom 3 4 25 2 ∧ ¬SignalFrom 3 4 25 3 := by refine ⟨the_outer_source_can_send_a_new_signal_at_age_four, ?_⟩ rw [a_regular_signal_exists_iff_its_endpoints_are_causally_separated 3 4 25 3 (by norm_num) (by norm_num) (by norm_num)] norm_num def coastingReadout : StandardHorizonFormulaReadout := { radiationReadout with scaleFactorValueAt := fun time => time } theorem a_totalized_zero_integral_does_not_supply_a_finite_particle_horizon : coastingReadout.horizonDistanceValueAt 1 = 0 ∧ ¬IntervalIntegrable (fun time => (coastingReadout.scaleFactorValueAt time)⁻¹) volume 0 1 := by have divergent : ¬IntervalIntegrable (fun time : Real => time⁻¹) volume 0 1 := by rw [intervalIntegrable_inv_iff] norm_num [Set.uIcc_of_le (by norm_num : (0 : Real) ≤ 1)] refine ⟨?_, divergent⟩ change 1 * (∫ time in (0 : Real)..1, time⁻¹) = 0 rw [intervalIntegral.integral_undef divergent, mul_zero] #check (the_outer_source_can_send_a_new_signal_at_age_four : SignalFrom 3 4 25 2) #check (the_same_epoch_has_an_inaccessible_outer_source : ¬Reachable 3 4 2) #check (a_regular_signal_exists_iff_its_endpoints_are_causally_separated : ∀ normalization emission reception coordinate : Real, 0 < normalization → 0 < emission → emission < reception → (SignalFrom normalization emission reception coordinate ↔ |coordinate| ≤ 2 * (Real.sqrt reception - Real.sqrt emission) / normalization)) #check (the_actual_physical_horizon_bounds_all_regular_radial_signals : ∀ normalization reception coordinate : Real, 0 < normalization → 0 < reception → (Reachable normalization reception coordinate ↔ scaleFactor normalization reception * |coordinate| < 2 * reception)) end end PhysicsLogic.IndependentReuse.RadiationHorizonClient run_cmd do let environment <- Lean.getEnv let cases : Array (Lean.Name × Array Lean.Name) := #[ (``PhysicsLogic.Papers.Guth1981.InflationFramework.the_standard_cosmology_calculation_exposes_actual_causal_reachability, #[``PhysicsLogic.Papers.Guth1981.StandardHorizonFormulaReadout.source_formulas_determine_the_actual_radial_causal_region, ``PhysicsLogic.Papers.Guth1981.RadiationHorizon.the_origin_light_cone_integral_is_finite, ``PhysicsLogic.Papers.Guth1981.RadiationHorizon.regular_signals_reach_exactly_the_open_particle_horizon, ``PhysicsLogic.Papers.Guth1981.RadiationHorizon.an_affine_radial_signal_realizes_every_allowed_separation]), (``PhysicsLogic.IndependentReuse.RadiationHorizonClient.the_nonunit_radiation_readout_derives_its_integrable_horizon, #[``PhysicsLogic.Papers.Guth1981.StandardHorizonFormulaReadout.radiation_scaling_derives_the_actual_integral_and_horizon_formula, ``PhysicsLogic.Papers.Guth1981.RadiationHorizon.the_origin_light_cone_integral_is_finite]), (``PhysicsLogic.IndependentReuse.RadiationHorizonClient.the_age_four_horizon_has_physical_radius_eight, #[``PhysicsLogic.Papers.Guth1981.StandardHorizonFormulaReadout.radiation_scaling_derives_the_actual_integral_and_horizon_formula]), (``PhysicsLogic.IndependentReuse.RadiationHorizonClient.the_same_epoch_has_an_accessible_inner_source, #[``PhysicsLogic.Papers.Guth1981.RadiationHorizon.the_actual_physical_horizon_bounds_all_regular_radial_signals]), (``PhysicsLogic.IndependentReuse.RadiationHorizonClient.the_same_epoch_has_an_inaccessible_outer_source, #[``PhysicsLogic.Papers.Guth1981.RadiationHorizon.the_actual_physical_horizon_bounds_all_regular_radial_signals]), (``PhysicsLogic.IndependentReuse.RadiationHorizonClient.the_outer_source_can_send_a_new_signal_at_age_four, #[``PhysicsLogic.Papers.Guth1981.RadiationHorizon.an_affine_radial_signal_realizes_every_allowed_separation]), (``PhysicsLogic.IndependentReuse.RadiationHorizonClient.a_totalized_zero_integral_does_not_supply_a_finite_particle_horizon, #[``intervalIntegral.integral_undef, ``intervalIntegrable_inv_iff]), (``PhysicsLogic.IndependentReuse.RadiationHorizonClient.the_fixed_emission_null_boundary_is_closed, #[``PhysicsLogic.Papers.Guth1981.RadiationHorizon.a_regular_signal_exists_iff_its_endpoints_are_causally_separated, ``PhysicsLogic.Papers.Guth1981.RadiationHorizon.an_affine_radial_signal_realizes_every_allowed_separation])] let mut dependencies := 0 for (declaration, required) in cases do let (_, traversal) := ((Lean.CollectAxioms.collect declaration).run environment).run {} for name in traversal.axioms do unless name == ``propext || name == ``Classical.choice || name == ``Quot.sound do throwError "Nonfoundational causal-horizon axiom {name}" for name in traversal.visited.toList do let parts := name.toString.splitOn "." if parts.contains "PhysicsLogic" && (parts.contains "deprecated" || parts.dropLast.any (fun part => part.endsWith "Countermodels" || part.endsWith "Controls")) then throwError "Independent causal-horizon proof depends on diagnostic {name}" for target in required do unless DownstreamDependencyAudit.usesConstant environment declaration target do throwError "Causal horizon {declaration} does not use {target}" dependencies := dependencies + 1 unless dependencies == 14 do throwError "Missing causal-horizon dependency tests" Lean.logInfo m!"Causal-horizon integration audit passed: one generic endpoint, seven independent clients, {dependencies} actual dependencies."