Formal reconstruction · Inflation, relic dilution, and the graceful-exit obstruction
The Inflationary Universe
A possible solution to the horizon and flatness problems—and the precise logical reason the original false-vacuum scenario can solve those problems without being a viable cosmology
Definitions, structures, and assumptions
Structures, classes, and inductive alternatives
- structureThe ordinary bubble-completion constraint paired with one typed paper-local derivation. The proposition-valued derivation relation keeps Guth's percolation/thermalization route distinct from Albrecht-Steinhardt's standard-bubble failure calculation even when both reach the same output.
- structureProof-free physical indices selected by one bubble-completion route. The constraint stores no setup, competition calculation, interpretation bridge, or obstruction output.
- structureThe selected bubble-transition calculation is proof-free data. Besides its physical indices, it retains exactly the proposition-valued relations used by the shared route. It contains no proof that the route applies or that the calculation represents a physical cosmology.
- inductiveThe exact shared claims established by the Guth and Albrecht-Steinhardt routes.
- structureTechnical data for a bubble-mediated first-order transition in which accelerated expansion competes with connected phase completion. The side-effect semantics are inherited from the mature explanation foundation. Bubble physics and the selected expansion regime remain paper-local predicates.
- structurePaper-local interpretation of a selected bubble-transition calculation. Theory use and representation of a physical cosmology are independent bridges.
- structureThe containing cosmological theory remains distinct from the calculation.
- structureThe represented cosmological world remains distinct from model and theory.
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- structureNeutral proposition-valued vocabulary for the repeated defect-formation inference. Each client supplies the physical meaning of separation, uncorrelated local choice, collective assembly, topological obstruction, and defect production.
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- structureThe selected domain-assembly calculation carries eight physical indices and all ten proposition-valued relation families used by the shared route. The domain family and local order-parameter assignment are proof-free selected data. No field stores a proof of assignment selection, independence, assembly, obstruction, or defect formation.
- structureThe selected physical formation of one defect in one transition scope.
- structureProof-free source constraint for one paper-local causal-domain derivation. The typed token and its interpretation relation are retained without storing a proof that the source route obtains or any defect-formation output.
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- structureThe typed structural constraint selected by one domain-assembly argument. It preserves the paper-local domain family, local order-parameter assignment, assembly, and obstruction instead of replacing them with a generic topology label.
- structureThe situated incomplete-transition phenomenon selected by the shared route. It retains the six physical indices but no proof of expansion dominance, failed connected completion, or the interpreted side effect.
- inductiveWhether patch expansion uses the obstructed mechanism or replaces it.
- structureProof-free comparison of an actually expanded cosmological patch with an independently calculated bubble-completion obstruction. The source-specific expansion/transition relation is retained independently of both endpoints.
- structureThe selected inflationary-patch calculation is proof-free data. Besides its physical indices, it retains exactly the proposition-valued relations used by the shared route.
- inductiveThe retained claims expose the complete narrow route. Transition completion, reheating, defect dilution, and global viability are not constructors.
- structureProof-free constraint selecting one exact inflationary-patch calculation. It stores no applicability, target-distinctness, expansion, coverage, or produced-target proof.
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- structureSource-sensitive constraint pairing the selected patch calculation with one typed paper-local derivation and its proposition-valued realization relation. It contains no proof that the derivation or patch route obtains.
- structurePaper-local meanings for a patch calculation, its containing theory, and the physical cosmology it represents.
- structureThe situated expanded-patch phenomenon explained by the shared route. It retains the selected physical indices while keeping expansion calculations, coverage, and target-transfer derivations outside the phenomenon data.
- structureThe cosmological theory using the patch calculation remains distinct.
- structureThe represented physical cosmology remains distinct from model and theory.
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- structureA selected claim must depend on exactly these source-map steps.
- structureA selected claim must retain exactly these epistemic statuses.
- structureA selected route must name exactly these immediate predecessor routes.
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- structureA converted phase is a time-dependent spatial region. No fraction, cluster size, connectivity verdict, or completed transition is a data field.
- structurePaper-local carrier types for Guth's cosmological vocabulary.
- structureSelected real-valued content of Guth's false-vacuum Friedmann calculation. The readout keeps equations (3.5) and (3.8)-(3.13) paper-local: the shared inflation framework only receives the physical outputs after this source calculation has been retained. The relative-error witness preserves the printed approximate cooling law without choosing a universal meaning for Guth's `simeq`.
- inductiveTyped source token for Guth's slow-nucleation, percolation, and thermalization-failure route. It is separate from the local patch argument and from the imported Kibble bubble-coalescence mechanism.
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- inductiveTyped token for Guth's explicitly imported Kibble production mechanism.
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- inductiveGuth's separate comparison of patch success with the same mechanism's exit failure.
- inductiveTyped source token for Guth's local sub-horizon patch argument. The token is separate from the imported Kibble mechanism and from graceful-exit failure.
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- structurePaper-local physical semantics. Fields contain data and predicates, never proofs of unresolved claims or desired conclusions.
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- structureA named entropy-amplification output does not determine Guth's p. 10 equation (3.4) ratio between reheating and supercooling temperatures.
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- structureReal and auxiliary births share the same actual centers, birth times, and expansion background; their roles are data, not coverage verdicts.
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- structureSource pp. 15-16 uses a retained constant-rate, exponentially expanding regime. It is not an assertion that an arbitrary background is exactly this model at every finite time. The tunneling parameters are independent data until the imported equation (3.19) is supplied.
- structureThe actual and retained ensembles are distinct model data. A selected actual history supplies the paper's converted region; probabilities remain ensemble quantities and do not become claims about that selected history.
- structureSelected real-valued content of Guth's preprint p. 19 monopole calculation. Equations (4.1)-(4.4) retain the causal bubble-size bound, the explicitly order-of-magnitude monopole-density estimate, Preskill's independent observational abundance bound, and the resulting coalescence-temperature bound. The order-one estimate is an explicit paper-local relation rather than an exact identification of monopole density with inverse bubble volume.
- structureSelected real-valued content of Guth's standard-cosmology flatness calculation. The readout keeps preprint pp. 6-7 equations (2.10)-(2.16) paper-local: the temperature-form Friedmann equation, conserved curvature, current observational bounds, entropy estimate, and relative-density fine-tuning bound remain visible before the named flatness diagnosis is used.
- structureSelected real-valued content of the radiation-era horizon calculation in Guth's standard cosmology. The readout keeps preprint p. 8 equations (2.17)-(2.20) paper-local: temperature-time evolution, physical horizon distance, entropy-rescaled observed-region size, and the resulting horizon-volume ratio are retained before the named horizon diagnosis is used. The horizon radius and light-cone integral are computed from the actual scale factor. The selected radiation scaling and the finite integration domain remain explicit physical requirements.
- structureThe later Tye-Guth calculation replaces rather than proves the stronger coarse horizon estimate. Its separately selected real-valued data retain preprint p. 20 equation (4.5).
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- structureNaturalness-relative fine tuning of initial conditions for one model.
- structureThe exact scope shared by target-production and viability questions.
- structureA mechanism produces targets within a declared scope.
- structureA mechanism's side effects are judged against explicit acceptability.
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Definitions and domain choices
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- definitionThe model-native paper-local competition calculation. It remains an explicit implication, but the direct endpoint below also requires its antecedent and outputs, so it cannot certify a bubble obstruction vacuously.
- definitionDirect obstruction and interpretation outputs after the route is used.
- definitionDirect setup for one model-native bubble-transition calculation. The selected phase must really be bubble mediated and the selected initial/regime/phase triple must really supply the accelerated expansion used by the route.
- definitionThe model-native interpretation bridge from the calculated obstruction to the selected incomplete-transition side effect.
- definitionDirect physical content of one completed bubble-transition obstruction. Unlike the context-selected established proposition, this endpoint drops exact record equality while retaining an actual setup, the calculated expansion dominance and failed connected completion, and the interpreted incomplete-transition side effect on one model. It stores neither implication-valued paper-local contract.
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- definitionThe selected route constraint with one paper-local derivation.
- definitionThe proof-free route constraint selected by one calculation.
- definitionThe proof-free incomplete-transition phenomenon selected by one model.
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- definitionThe paper-local competition calculation. The setup alone does not determine whether expansion outpaces bubble formation or whether a connected completion network forms.
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- definitionThe paper has selected a bubble-mediated first-order transition and an expansion regime sufficient for the intended cosmological role.
- definitionPaper-independent semantics of every claim retained by the shared bubble-completion obstruction.
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- definitionSource-sensitive structural explanation of one selected obstruction.
- definitionOrdinary structural explanation of one selected completion obstruction.
- definitionThe established shared result stores exact selected semantics and direct physical obstruction outputs, not the implication-valued route.
- definitionThe physical interpretation bridge from failed connected bubble completion to the selected incomplete-transition side effect.
- definitionThe calculation is exactly the context-selected physical data at its declared initial condition, regime, and phase. This equality retains all six indices and all five proposition-valued relation families.
- definitionExact context selection plus an inhabited bubble-expansion setup.
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- definitionThe unique six-index bubble-transition calculation selected by context.
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- definitionInterpret a selected competition calculation without identifying it with its containing cosmological theory or represented physical universe.
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- definitionIdentifying a topological obstruction with a physical defect is a distinct interpretive bridge. The topological calculation does not supply it.
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- definitionThe output-led established result fixes the exact context-owned model and records every realized setup and formation output directly. The implication-valued topological calculation and interpretation remain reconstructible derivation objects rather than stored evidence.
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- definitionThe direct stopping point of the shared route. The independent domain setup, collective assembly, failed global extension, topological obstruction, and formed defect all hold at one model rather than only behind the two implication-valued paper-local bridges.
- definitionExact selected semantics is equality with the context-owned model, not a model reconstructed from the candidate's own indices or functions.
- definitionThe inhabited physical setup needed to ground the selected defect-formation model. It fixes the context-owned model and a nontrivial family of domains in the selected transition with the selected local order-parameter assignment and separated uncorrelated choices. Collective assembly, nonextendibility, topological obstruction, and defect production remain outside this setup.
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- definitionSource-sensitive explanation by one typed paper-local route.
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- definitionThe topology remains an explicit paper-local calculation. Independent local choices and collective assembly do not themselves prove an obstruction.
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- definitionThe unique defect-formation model fixed by the context's eight selected physical values and ten route-defining relation families.
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- definitionExact proof-free source constraint selected by a model and source token.
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- definitionBoth the expanded patch and the first-order obstruction really obtain.
- definitionModel and initial condition agree, while mechanism identity is explicit.
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- definitionShared selected data still require the paper-local comparison argument.
- definitionThe exact cosmological handoff retains independently selected patch and bubble contexts, both paper-local source explanations, and a third realized derivation for their actual same-mechanism or replacement comparison.
- definitionAn actually expanded patch and actual bubble obstruction, retaining their distinct paper-local derivations and the independently sourced comparison.
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- definitionDirect physical content of one inflationary-patch route. Unlike the context-selected established proposition, this endpoint drops exact record equality and requires one actual setup, both produced targets, and the fact that homogeneity and flatness are distinct targets. The implication-valued transfer bridges are reconstructed from these direct outputs.
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- definitionThe exact proof-free constraint selected by this calculation.
- definitionThe selected source-sensitive version of the patch constraint.
- definitionThe exact situated expanded patch selected by this calculation.
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- definitionThe selected model is exactly the context's proof-free physical data at its declared initial condition and region.
- definitionThe pre-output physical boundary used for theory/model/world grounding. It fixes the exact selected calculation, an admissible model and applicable mechanism, one initially homogeneous patch, and two distinct target roles. Expansion, observed-region coverage, transfer bridges, and produced targets remain outside this setup.
- definitionThe two paper-independent targets selected by the patch route.
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- definitionThe unique patch calculation selected by the context.
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- definitionInterpret a selected patch calculation without identifying it with its containing cosmological theory or represented physical region. Model and world membership retain the exact selected calculation and explicit paper-local bridges without assuming the expansion route.
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- definitionConstruct the exact proof-free calculation selected by a shared context at one initial condition, regime, and phase.
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- definitionImmediate predecessor routes. Claims established by their transitive predecessors are inherited by the dependency closure.
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- definitionTemporary reductio-introduction steps discharged by this inference. Discharge removes only those exact origins from the conclusion's epistemic basis; it does not assert the temporary hypotheses globally.
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- definitionA source-mapped claim holds when its declared formula evaluates under the given paper-local semantic interpretation.
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- definitionEvery claim in the supplied identifier list holds.
- definitionA source-mapped step is semantically sound when the interpreted premises imply the interpreted conclusion. Physical principles and imported calculations therefore remain explicit hypotheses of paper-specific certificates rather than acquiring truth from their source-map labels. A reductio hypothesis is different: it introduces a temporary assumption into a subargument and therefore is not required to hold globally. Its later uses remain ordinary premise-sensitive steps, and discharging it into a negated conclusion remains an ordinary semantic obligation.
- definitionEvery step in the supplied identifier list is semantically sound.
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- definitionThe formula declared for a claim identifier, when that claim is present.
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- definitionEvery source-mapped step whose justification contributes to the claim through declared premise-to-conclusion steps.
- definitionAll justification statuses contributed by the exact source-mapped origins of a claim. This records epistemic provenance; it does not assert any paper claim or discharge any physical assumption.
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- definitionThe declared argument step with the requested identifier, when present.
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- definitionThe premise list and conclusion attached to a declared step identifier.
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- definitionThe complete Lean-side fidelity certificate. The first conjunct is computed from source-map data; the second is supplied by checked route semantics; the third requires each declared independence boundary to have a countermodel.
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- definitionA typed countermodel matrix avoids string-keyed audit rows. Each index selects one proposition-valued separation obligation.
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- definitionEvery route actually declared in the source map has a semantic certificate under the paper-local interpretation.
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- definitionOne countermodel row retains a weaker proposition while rejecting a stronger upgrade. This is a proposition, not a structure carrying proof fields.
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- definitionIndependent dependency and epistemic ledgers must review the same actual source claims, including every public conclusion and the selected paper conclusion. Empty, duplicated, nonexistent, or one-sided entries are invalid.
- definitionEvery actually declared source route must have exactly one independently reviewed predecessor row. Correct rows for a selected subset are insufficient.
- definitionThe declared main conclusion must be exposed by at least one public route output, rather than existing only as an unexported source-map claim.
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- definitionTwo finite lists describe the same dependency set. Duplicate expected entries are rejected so a ledger cannot pass by repeating a label.
- definitionEvery paper-level, claim-level, and step-level source reference is present and has a nonempty work and locator. PaperArgument.auditable separately checks that every claim and step carries at least one source.
- definitionA source reference is usable for review only when both the work and its locator are nonempty.
- definitionStatic fidelity checks that can be evaluated from a paper's source map alone. They do not assert the truth of any physical premise.
- definitionEvery declared output of a named source route holds under the semantic interpretation. This is the proof obligation discharged by paper-specific Lean theorems.
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- definitionA route is fully certified only when both its individual steps and its declared outputs are semantically checked.
- definitionEvery source-mapped inference step in a named route is semantically sound.
- definitionEvery route named in the supplied identifier list is semantically certified.
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- definitionThe source-mapped route with the requested identifier, when present.
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- definitionThe public semantic interface inherited from predecessor routes. Descendants receive only declared route outputs, not every internal intermediate claim. Outputs still carrying a temporary reductio origin are excluded.
- definitionAll direct and inherited route dependencies. The route-order check separately ensures that the declared graph is acyclic and topologically ordered.
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- definitionInfinite spatial extent of a connected component, not infinite point cardinality or connectedness of the entire converted phase.
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- definitionSpatial density uses an expanding family of finite coordinate windows, not division of two infinite volumes or a one-point survival probability.
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- definitionThe product sup metric makes these coordinate cubes, not spherical bubbles. They provide a geometric inference control, not a nucleation model.
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- definitionPreprint p. 12, equation (3.8): the temperature-form Friedmann equation with the radiation, curvature, and false-vacuum terms kept separately visible.
- definitionPreprint p. 11, equation (3.5): after the transition supercools toward a metastable false vacuum, the radiation density is shifted by `rho_0`. The paper calls this a good approximation after ignoring mass thresholds, so the displayed formula is retained with a selected relative-error witness.
- definitionPreprint p. 12, equation (3.9): the critical curvature value.
- definitionPreprint p. 12, equation (3.10): in the `epsilon > epsilon_0` branch the expansion halts at the displayed minimum temperature and then contracts.
- definitionThe literal dominance condition preceding equation (3.11): the false-vacuum term on the right-hand side of (3.8) is positive and exceeds the magnitudes of the radiation and curvature terms.
- definitionThe positive real domain needed by the displayed square roots, ratios, and expansion branch. The approximation bound is selected by the paper-local readout rather than fixed globally.
- definitionThe source-level false-vacuum calculation retains the density law, the temperature Friedmann equation, both curvature branches, and a nonempty unchecked-expansion regime. The universal clauses keep equations (3.5) and (3.8) from collapsing to a single decorative numerical example.
- definitionPreprint pp. 12-13, equations (3.11)-(3.13): in the unchecked `epsilon < epsilon_0` branch, vacuum dominance gives approximately exponential cooling, `chi^2 = (8 pi / 3) G rho_0`, exponential scale-factor growth, and `H = Rdot / R = chi`. The cooling approximation is a bounded relative error, not an exact equality.
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- definitionSection III: GUT baryon violation permits post-reheating regeneration.
- definitionSection V, points (i), (iii), and (v): finite clusters containing few of the largest bubbles cannot provide the large-bubble wall collisions needed for thermalization.
- definitionSection III: exact dilution output for a strictly conserved quantity.
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- definitionSection III, equations (3.1)-(3.4): the entropy calculation removes the two standard-model problems but moves rather than abolishes the horizon.
- definitionThe source readout is attached to Guth's selected phase and initial condition, rather than floating as an untyped list of equations. Each function-valued ingredient has a paper-local representation bridge; in particular, the readout's `rho_0` is the phase's false-vacuum energy density.
- definitionThe phase-attached source formula for Guth's equations (3.5) and (3.8)-(3.13). It retains both the representation bridge and the literal paper-local equation logic.
- definitionSection III: the supercooling, reheating, false-vacuum, and tunneling calculations that realize the inflationary mechanism.
- definitionGuth's appendix makes the naturalness comparison explicit and interpretive: at least one admissible baseline is judged natural, while naturally judged initial conditions do not realize the observed-flatness target that the standard model reaches only by extreme initial-value tuning.
- definitionThe flatness problem is the initial-value tuning diagnosed in sections I-II: the initial Hubble value is extraordinarily fine tuned, equivalently leaving the conserved curvature parameter startlingly small.
- definitionSection V: random bubbles without sufficient wall collisions yield Guth's inhomogeneous-exit assessment. This remains short of an impossibility proof.
- definitionRaw source premises interpreting Guth's monopole-formation token. The selected grand-unified transition, uncorrelated bubble family, collective coalescence, and explicitly imported Kibble calculation remain separate contracts.
- definitionThe horizon problem is the conjunction Guth emphasizes: the observed thermal uniformity descends from many precursor regions that had no causal contact.
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- definitionThe original and retained backgrounds must be compared after integration, not just by the limiting values of the rate and scale-factor ratio.
- definitionGuth's magnetic monopole as a stable, magnetically charged Higgs knot.
- definitionKibble's imported production argument: uncorrelated Higgs orientations in the collectively coalescing bubble family obstruct uniform alignment and produce magnetic Higgs knots at a count comparable to the bubble count.
- definitionSection III, pp. 16-17: only a sub-horizon locally equilibrated patch is needed; inflation then expands it beyond the observed region.
- definitionSection VI: the sharp-nucleation modification is conditional and remains unestablished at the required supercooling; particle production is omitted. The imported assessment returns those physical and epistemic facts, not the downstream conclusion that a viable modification has been established.
- definitionSection IV: imported abundance-bound and supercooling calculations.
- definition"Comparable within a few orders of magnitude" is represented by symmetric multiplicative bounds using the paper-local order tolerance.
- definitionThe original monopole horizon calculation retains the selected causal bubble-radius bound, order-qualified density estimate, independent Preskill bound, printed temperature cutoff, and four-order supercooling assessment.
- definitionSection V: imported connectivity result in the presumed subcritical regime. Large spatial conversion is derived separately from survival and averaging.
- definitionSection V: Guth presumes that an inflationary nucleation rate lies below the unknown critical percolation threshold. This is not supplied by slowness alone.
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- definitionLimiting rate and scale-factor correspondence records which actual model the retained regime approximates. It is not a bound on the integrated intensity error and does not make equation (3.16) exact for that model.
- definitionThe paper-local input boundary for Guth's selected imported Kibble model: a grand-unified transition and an inhabited family of causally independent bubbles with uncorrelated Higgs orientations. Coalescence, failed uniform alignment, the Higgs knot, and monopole production remain outputs.
- definitionApplying ensemble predictions to the selected universe needs a spatial averaging/typicality input about that very history. An a.e. theorem alone does not select an arbitrary history as typical.
- definitionThe standard-cosmology flatness readout is attached to Guth's selected initial condition before equations (2.10)-(2.16) are used. The bridge remains paper-local because the historical entropy and current-curvature normalizations have not yet been independently reused by another source.
- definitionThe standard-cosmology horizon readout is attached to Guth's selected initial condition before its literal equations (2.17)-(2.20) are used. The representation bridge is paper-local because the present entropy density, photon temperature, and observed radius are part of Guth's historical normalization, not yet a shared cosmology interface.
- definitionPaper-local interpretation of Guth's two physical diagnoses as the targets named by the generic explanation vocabulary.
- definitionSections I-II and the Appendix: exact output of the standard-model calculation and the paper's model-relative naturalness assessment.
- definitionThe Tye-Guth strong-transition result is a separate revision of the coarse horizon estimate, not a consequence of the original equation-(4.4) bound.
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- definitionA nontrivial family of bubbles belongs to the selected transition. Distinct members had no prior causal contact and carry pairwise uncorrelated Higgs orientations. This is the collective setup used by Guth's Kibble argument.
- definitionAn established viable modification must establish the late trigger needed for the advertised supercooling and include gravitational particle production whenever that omitted effect may matter. This is an explicit interpretive requirement, not a calculation hidden inside `ModificationAssessment`.
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- definitionSource p. 22 compares large converted spatial density with connected cluster extent. It does not identify density with a one-point probability.
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- definitionPreprint p. 10, Eq. (3.4): reheating from the supercooled temperature `T_s` to `T_r` fixes the entropy scale factor `Z = T_r / T_s`. The strict temperature ordering records the selected supercooling/reheating regime; the coarser entropy-density conclusion remains a separate physical output below.
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- definitionGuth's selected false-vacuum energy density is positive.
- definitionPreprint p. 13: Lorentz invariance of the false vacuum gives the exact equation of state `p = -ρ₀` before negative pressure is used to drive the exponential solution.
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- definitionThe negative-pressure output retains both Guth's equation of state and its strict sign, rather than reducing the physical step to an opaque label.
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- definitionRaw source premises for Guth preprint pp. 20-22. The relation retains the separate first-order setup, slow-nucleation regime assessment, imported percolation calculation, finite-cluster thermalization calculation, and graceful-exit interpretation without storing a shared obstruction endpoint.
- definitionThe source-local comparison retains the actual local-patch calculation, slow-nucleation regime, percolation and thermalization steps, and graceful-exit assessment; neither child framework supplies this same-mechanism argument.
- definitionRaw source premises for Guth's preprint pp. 9-10 and 16-17. A locally equilibrated sub-horizon patch, accelerated expansion, coverage of the observed region, and separate horizon/flatness transfer remain explicit. The relation stores neither an established shared patch nor graceful-exit, reheating, or monopole conclusions.
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- definitionGuth's paper-local naturalness and target-realization semantics.
- definitionScope common to Guth's target-production and viability claims.
- definitionGuth's homogeneous-patch success and slow-nucleation graceful-exit failure refer to the same selected mechanism, model, and initial condition. The source-specific percolation and thermalization calculations remain explicit.
- definitionGuth's paper-local target-production semantics.
- definitionGuth's paper-local side-effect and viability semantics.
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- definitionSection V realization of the shared expansion-limited bubble-transition vocabulary. The unknown percolation threshold and all thermalization physics remain in Guth's paper-local predicates.
- definitionGuth's old-inflation interpretation keeps the selected percolation and thermalization calculation distinct from the theory using it and from the physical false-vacuum cosmology it represents.
- definitionGuth's monopole route realizes the neutral causal-domain defect-formation vocabulary with bubbles as domains, Higgs orientations as order-parameter values, phase-wide coalescence as the collective assembly, and the selected Higgs knot as both obstruction and produced defect. Magnetic charge, topological stability, count comparability, abundance bounds, and inflationary suppression remain Guth-local.
- definitionInterpret Guth's exact selected bubble-family model once the independent, uncorrelated Higgs-orientation setup is available. Coalescence, the imported Kibble calculation, and monopole production remain downstream.
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- definitionGuth's local patch calculation is distinct from the containing inflationary theory and the represented expanded cosmological patch. The interpretation transfers only the shared patch claims, not graceful exit, transition completion, reheating, monopole suppression, or global viability. Primary-source anchors: sections III-IV for accelerated expansion, horizon and flatness resolution, and the local-patch qualification.
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- definitionThe remaining measure-integration step is an explicit mathematical input about this actual past cone and this actual spacetime intensity.
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- definitionThe source rate is per physical volume per time. The R^3 Jacobian converts it to the intensity in comoving birth coordinates.
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- definitionDeliberately incorrect comparison: the source rate is applied directly to coordinate volume, omitting the physical-volume Jacobian.
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- definitionActual probability-law controls, not proposed realizations of Guth's spatial Poisson process. They isolate invalid general probability inferences.
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- definitionThis configuration deliberately violates local finiteness. It tests the natural-number count's totalization, not a realization of a Poisson process.
- definitionA single auxiliary birth without an enclosing real birth. Both regions are computed from this configuration and the same light-front background.
- definitionPositive parameters with unit prefactor and a large barrier, but a rate-to-expansion ratio too large to produce even one e-fold per time constant. This tests the tunneling formula alone, not Guth's sufficient-supercooling setup.
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- definitionThe spatial averaging calculation is an explicit mathematical input; countability permits one common full-measure set of histories.
- definitionSource p. 14 idealizes the extended births as a Poisson point process. Independence concerns disjoint birth cells, not generally overlapping engulfing cones or correlated point-survival events.
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- definitionA statement about a selected realization at countably many observation times. It is not inferred from its one-point probabilities or from an a.e. law.
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- definitionA rate limit alone does not control the integrated correction. This is an additional mathematical approximation obligation for the original background.
- definitionA reconstructed evaluation of the actual source integral. This technical integral calculation can be proved independently; it is not stored as a proof or replaced by a free assigned decay curve.
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- definitionPhysical light fronts are Euclidean spheres in the flat comoving coordinates of source p. 14, not the earlier sup-metric cube controls.
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- definitionPreprint p. 19, equation (4.4): Guth's original horizon estimate combines the photon-density normalization with Preskill's bound.
- definitionPreprint p. 19, equation (4.1): the bubble radius is horizon bounded.
- definitionPreprint p. 19, equation (4.3): Preskill's observational abundance bound.
- definitionPreprint p. 19, equation (4.2): an order-one Kibble estimate bounds the monopole density, while the inverse bubble volume strictly exceeds the printed horizon-derived density scale.
- definitionPositive physical domain for the selected horizon and abundance data.
- definitionThe paper's example requires approximately four orders of supercooling.
- definitionThe source-level horizon route retains each displayed calculation.
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- definitionThe speed bound holds between every pair of points, not only endpoints.
- definitionOne actual past event can send a signal to every point of a nonempty region. The emission time and place are chosen before the reception point. This is geometric access, not interaction, equilibration, or an observed common cause.
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- definitionA causal curve from one comoving worldline to another, with regular emission.
- definitionRegular cosmic emission time; the curve itself uses conformal time.
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- definitionGuth's normalized curvature sign, with the flat case retained as the `R -> infinity` limit discussed after equation (2.12).
- definitionPreprint p. 7, equation (2.11): the conserved curvature parameter is both `k/(R^2 T^2)` and the entropy-rewritten expression. The equation-(2.6) entropy density and `S = R^3 s` relation used in that rewrite are retained instead of being hidden in a name.
- definitionPreprint p. 7, equation (2.15): the conserved curvature parameter is bounded by the entropy estimate at every selected high temperature.
- definitionPreprint p. 7, equation (2.14): three massless neutrino species give `S_nu = (21/22) S_gamma`, and photon plus neutrino entropy contributes to the printed total bound `S > 10^86`.
- definitionPreprint p. 7, equation (2.16): the relative deviation from critical density is the curvature contribution rewritten with `M_p`, and it is bounded by the printed `3 x 10^(-59)` expression.
- definitionPreprint p. 6, equation (2.10): away from mass thresholds, the temperature-form Friedmann equation keeps the expansion and conserved curvature terms separate.
- definitionPreprint p. 7, equation (2.13): the photon contribution to the conserved entropy is bounded below by `3 x 10^85`.
- definitionPreprint p. 7, equation (2.12): present observations bound the current curvature term by `9 H^2`.
- definitionThe positive real domain needed by the temperature, entropy, curvature-radius, and density ratios. The selected curvature normalization retains Guth's `k = plus or minus 1` choice together with the separately mentioned flat limit `k = 0`.
- definitionThe Planck-mass convention used to rewrite equation (2.16).
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- definitionThe source-level flatness calculation retains equations (2.10)-(2.16), their present observational entropy inputs, and the universal high-temperature curvature and density-deviation bounds.
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- definitionPreprint p. 8, equation (2.18): the physical horizon is the scale factor times the actual, integrable light-cone integral and equals `2t` in the radiation approximation.
- definitionPreprint p. 8, equation (2.19): adiabatic entropy conservation rescales the present observed radius `L_p` back to time `t` by `[s_p / s(t)]^(1/3)`.
- definitionPreprint p. 8, equation (2.17): in the radiation-dominated era, `T^2 = M_p / (2 gamma t)` with `gamma^2 = (4 pi^3 / 45) N`.
- definitionPreprint p. 8, equation (2.20): the horizon-to-observed-region volume ratio is retained both in the source expression and in Guth's printed numerical normalization. The later choice `N approximately 10^2`, `T = 10^17 GeV` remains a separate numerical interpretation rather than being smuggled into the universal formula.
- definitionThe positive real domain needed by Guth's displayed fractional powers, inverse scale factor, entropy ratio, and volume ratio. Positivity is retained for every selected positive time rather than only at the numerical example.
- definitionThe flat radiation approximation on nonnegative times, with an arbitrary positive normalization of the comoving coordinates.
- definitionThe source-level standard-horizon calculation retains all four displayed equations and the scale-factor law for every selected positive radiation-era time. Thermal and numerical calibration inputs remain separate obligations.
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- definitionNonvacuous success at one selected admissible initial condition.
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- definitionA witnessed unacceptable side effect at one selected initial condition.
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- definitionA model-to-world interpretation law. It does not assert that the model has produced a conclusion; use `GroundedModelConclusionAt` for that.
- definitionA model-to-theory interpretation law. It does not assert that the model has produced a conclusion; use `GroundedModelConclusionAt` for that.
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- definitionBuild the epistemic dependency basis in paper order. A conclusion inherits the justification origin of its own step and every origin already attached to its premises.
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- definitionExpand a set of route identifiers by one dependency edge, iterated with enough fuel to traverse any acyclic graph on the declared routes.
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Selected instances and coercions
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Source mappings and interpretation choices
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Checked statements: parameters, hypotheses, and interpretation
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- checked statementAn actual increasing three-dimensional converted region approaches unit spatial density while every connected component remains bounded. The entire converted region is unbounded; that is not the same as percolation.
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- checked statementThe selected old-inflation inputs realize exact shared setup and no output.
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- checked statementPreprint pp. 21-22: the failed-collision route supplies the selected side effect; its classification as unacceptable remains an explicit premise.
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- checked statementPreprint p. 16: the expanded local patch realizes only a scoped solution.
- checked statementPreprint pp. 16 and 21-22: retain target success and graceful-exit failure for the same selected mechanism without globalizing the local-patch result.
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- checked statementThe completed false-vacuum calculation retains Guth's displayed reheating ratio, not merely a named entropy-density amplification claim.
- checked statementThe completed false-vacuum calculation retains the equations that make Guth's exponential-expansion output physical: the shifted density law, temperature Friedmann equation, curvature split, approximate cooling law, and scale-factor/Hubble relations.
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- checked statementGuth's typed source reconstructs the raw shared route; it does not store the established obstruction as source data.
- checked statementGround Guth's selected bubble-family model from exact retained semantics and the independent-domain setup, before collective coalescence, obstruction, or defect production is established.
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- checked statementGuth's typed local-patch source reconstructs the raw shared route rather than storing an established patch endpoint.
- checked statementThe exact Guth patch model, its selected admissible homogeneous setup, and its theory/world roles establish grounding without assuming expansion, coverage, either transfer bridge, or either produced target.
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- checked statementGuth's selected bubble-family realization of Kibble's argument structurally explains the selected transition defect only after coalescence, failed global Higgs alignment, the knot calculation, and the physical monopole bridge.
- checked statementThe actual expanded local patch and the actual incomplete bubble transition compose without promoting patch success to successful graceful exit.
- checked statementThe original same-mechanism comparison independently retains the successful subhorizon-patch derivation, slow-nucleation failure derivation, and paper-local argument that both concern the same false-vacuum mechanism.
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- checked statementPreprint pp. 8-9 and 16: Guth treats the horizon/homogeneity and flatness problems as distinct targets, and the local-patch route reaches both directly.
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- checked statementThe original local-subhorizon-patch argument realizes the source-sensitive shared explanation without consuming an established patch, bubble failure, or monopole result.
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- checked statementPreprint pp. 20-22: the slow-nucleation route has an actual bubble setup, expansion dominance, failed connected completion, and the selected incomplete- transition side effect, rather than only vacuous implication-valued bridges.
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- checked statementThe standard-model calculation retains Guth's pp. 6-7 flatness formula chain rather than allowing the fine-tuning output to stand in for it.
- checked statementThe standard-model calculation retains Guth's p. 8 horizon formula chain rather than allowing the causal-disconnection output to stand in for it.
- checked statementPreprint pp. 2-4 and appendix pp. 24-25: the numerical flatness diagnosis, target-identification bridge, and separately stated naturalness judgment realize the neutral nonvacuous fine-tuning proposition.
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- checked statementEquations (3.16)-(3.18) hold for the retained late regime. The correction is computed from the same intensity and converges to a constant; it is not assumed zero or promoted to a finite-time exact decay law.
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- checked statementPhysical interval width is twice its radius. The particle-horizon radius is still 2t; the common-past bound on the full width is 4t.
- checked statementA regular common event needs a strict margin even when the region is closed. This is a geometric consequence of the radiation approximation, not a thermalization criterion or an additional historical claim attributed to Guth.
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- checked statementA readable characterization of the curve-existence definition. The inequality permits a null signal at equality. Unlike Reachable, the emission time is fixed and strictly positive.
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- checked statementSpatial translation changes both endpoints, not the speed bound or epochs.
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- checked statementThis particular conformal trajectory, not merely its endpoint separation, obeys the local light-speed bound and realizes the incoming signal.
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- checked statementGuth p. 9 changes the inferred size of the selected early region. Fitting that region within the horizon does not remove the horizon outside it. This compares sizes at the selected radiation epoch, not a radiation law through the intervening inflationary era.
- checked statementStrengthens the selected-radius comparison to one shared regular past event for its entire closed radial interval. Entropy production supplies the radius rescaling as a separate input; no inflationary evolution is assumed here.
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- checked statementA finite particle horizon is not a future event horizon. In the radiation model continued to arbitrarily late times, every fixed comoving source can reach the observer later. This continuation is not an assertion about the intervening inflationary history in Guth's scenario.
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- checked statementThe inverse expansion rate happens to equal the particle horizon in this flat radiation model. The equality is derived from the scale factor, not used as the definition of causal reachability or of a future event horizon.
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Axioms and trust boundaries
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Constructors, projections, and other meaning dependencies
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External mathematical declarations
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- external type boundary
- external type boundary
Every physical statement, dependency, number, and qualifier below comes only from Guth1981.lean, SourceMap.lean, Coverage.lean, and Fidelity.lean. The Lean paper retains the standard-cosmology flatness formulas in equations (2.10)–(2.16), the horizon formulas in equations (2.17)–(2.20), p = -ρ0, preprint p. 10 equation (3.4), Z = Tr/Ts, and the false-vacuum density, Friedmann, curvature-branch, cooling, scale-factor, and Hubble formulas in equations (3.5) and (3.8)–(3.13) as value-level propositions. It deliberately treats the numerical choice T = 1017 GeV and N ≈ 102, entropy thresholds, tunneling dynamics, dilution law, Kibble/Preskill/Tye–Guth inputs, percolation result, collision analysis, and modification assessment as named assumptions or interpretations where indicated. The source map keeps those setup and calculation contracts premise-free until their result steps apply them; it also keeps the independent monopole pressure separate from the false-vacuum route. This edition narrates those exact logical outputs; it does not supply missing calculations.
The particle horizon is a causal boundary, not an assigned radius
Candidate for human semantic review. The six physical definitions are isolated in PhysicsLogic/Papers/Guth1981/RadiationHorizon/Definitions.lean, importing mathematics only. The accompanying proofs establish that, for A > 0 and 0 < e < t, a signal from x emitted at e can arrive at t exactly when |x| ≤ 2(sqrt(t)-sqrt(e))/A. Equality is allowed for this fixed-emission null boundary; the particle horizon below is open because emission at zero is excluded. This checked equivalence makes the curve definition inspectable, but does not constitute human approval of its interpretation. Thermal and flatness formula obligations remain separate.
In the flat radiation approximation with c = 1, the scale factor is a(t) = A sqrt(t), A > 0. Lean evaluates the actual integrable conformal-time integral as 2 sqrt(t)/A and derives the physical horizon radius 2t. A radial signal curve must obey the light-speed bound between every pair of its points. Constructing such curves proves that signals emitted at strictly positive times reach exactly the region a(t)|x| < 2t. The origin is a limiting boundary, not a regular emission event.
If entropy rescaling makes the selected early radius L/Z smaller than 2t, all sources in that region are reachable, while inaccessible sources still exist outside the horizon. This comparison is at a radiation epoch; it does not impose radiation evolution through the inflationary era. A separate diagnostic continues the radiation geometry indefinitely: a signal emitted now from any fixed finite comoving coordinate can arrive later. A finite particle horizon therefore does not imply a future event horizon. This distinction is a formal diagnostic, not an additional claim attributed to Guth.
An independent geometry with A = 3 and age 4 has physical horizon 8. Coordinate 1 is reachable; coordinate 2 is not yet reachable. A new signal from coordinate 2 at age 4 arrives by age 25. A coasting geometry a(t) = t tests the opposite integration behavior: its divergent origin integral cannot certify a finite particle horizon, even though Lean assigns a default integral value. These tests do not instantiate Guth's full thermal or numerical calibration, nor do causal curves alone define measurement or empirical inference.
The paper is a branching argument, not a single chain from problem to success
The standard-problem diagnosis, entropy criterion, and monopole constraint begin independently. The false-vacuum route realizes the entropy criterion. From it branch the local-patch argument, conserved-charge qualification, and graceful-exit obstruction. Only the last of these feeds the proposed modification.
| Transition | Status retained in Lean | What the status means here |
|---|---|---|
| Standard inputs to causal and curvature facts | deferred calculation | Equations (2.10)–(2.16) and (2.17)–(2.20) are retained as separate flatness and horizon formula endpoints; the interpretation of the curvature relation as “fine tuning” stays explicit. |
| Entropy amplification to both target resolutions | deferred calculation | The threshold calculation is supplied, then its outputs are checked. |
| False vacuum to expansion and reheating | deferred calculation | The false-vacuum and tunneling calculation is explicit; equations (3.4), (3.5), and (3.8)–(3.13) are retained while “huge expansion plausible” remains an author expectation. |
| Local smoothing | imported + expected | Anisotropy damping is imported; local thermalization and inhomogeneity damping are expectations. |
| Monopole formation and abundance | imported result | Collective Kibble coalescence and abundance bounds are named inputs, with model-rate uncertainty retained. |
| Slow rate to subcritical regime | author presumption | Slowness alone does not prove the rate is below the unknown percolation threshold. |
| Subcritical bubbles to finite clusters | imported result | The percolation calculation is a separate input. |
| Failed collisions to unacceptable exit | qualified assessment | The conclusion is an assessment of nonviability, explicitly not an impossibility proof. |
| Sharp nucleation to viable modification | open obligation | The bubble repair is conditional; trigger establishment and relevant particle production remain unresolved. |
Horizon and flatness are different diagnoses
The standard setup assumes sub-Planck initial conditions and a radiation-dominated, adiabatically expanding Robertson–Walker model with homogeneous, isotropic, thermal initial data. A supplied calculation retains separate pp. 6–7 flatness and p. 8 horizon formula chains and produces two pairs of diagnostic facts. A separate identification premise says which pair constitutes which named problem.
flatness problem = fine-tuned initial Hubble value + startlingly small conserved curvature parameter. physical calculation plus explicit target identification (1)
ε(T) = k/[R(T)2T2] = k[(2π2/45)N(T)/S]−2/3,
|k/Rp2| < 9Hp2, Sγ > 3 × 1085, Sν = (21/22)Sγ, S > 1086,
|ε| < 10−58N2/3,
|(ρ − ρcr)/ρ| = 45Mp2|ε|/(4π3NT2) < 3 × 10−59N−1/3(Mp/T)2.
StandardFlatnessSourceFormulaAt; equations (2.10)–(2.16); the later grand-unified numerical choice is not made exact
(1a)
l(t) = R(t)∫0tdt′/R(t′) = 2t,
L(t) = [sp/s(t)]1/3Lp,
l(t)3/L(t)3 = (11/43)(45/4π3)3/2N−1/2[Mp/(LpTγT)]3 = 4 × 10−89N−1/2(Mp/T)3.
StandardHorizonSourceFormulaAt; equations (2.17)–(2.20); the light-cone integral is an explicit selected value
(1b)
The source map records roughly 1083 disconnected precursor regions and one-part-in-1055 Hubble tuning as the paper’s quoted diagnoses. Lean keeps the flatness and horizon formulas as separate endpoints instead of treating “fine tuning” or “causal disconnection” as substitutes for their calculations. It does not identify the two problems with one another: a concrete countermodel has the horizon profile but not the flatness profile.
The standard setup, calculation, and target-identification premise establish causal disconnection, equal temperature, both fine-tuning facts, both named problems, and both target propositions.
standard_model_calculation_exposes_horizon_and_flatness_problems · original preprint pp. 2–8 and 25The full standard setup and calculation expose equations (2.10)–(2.16) at the selected initial condition. Equation (2.16) also yields the signed one-sided density-deviation bound from its absolute-value estimate. Coverage reconstructs the same endpoint from the exact route premises.
standard_model_calculation_retains_flatness_source_formula · equation_two_sixteen_bounds_signed_density_deviation · standard_problems_route_recovers_flatness_source_formula · original preprint pp. 6–7The full standard setup and calculation expose equations (2.17)–(2.20) at the selected initial condition. Equation (2.18) also yields positive horizon distance at every positive selected time. Coverage reconstructs the same source endpoint from the exact route premises rather than replaying a route certificate.
standard_model_calculation_retains_horizon_source_formula · equation_two_eighteen_makes_positive_time_horizon_positive · standard_problems_route_recovers_horizon_source_formula · original preprint p. 8The diagnosed flatness target realizes the shared nonvacuous fine-tuning proposition only when the separately stated naturalness judgment supplies an admissible baseline and excludes the target throughout that class. The numerical diagnosis does not choose its own naturalness standard.
standard_model_flatness_diagnosis_realizes_shared_fine_tuning · original preprint pp. 2–4 and 24–25Entropy gives the criterion; a false vacuum gives the mechanism
The entropy route does not depend on the standard-problem route. It directly assumes nonadiabatic entropy production and amplification sufficient for each target. The source map records an entropy multiplication by Z3, thresholds above 3 × 1027 and 5 × 1027, and the paper’s summary as 28 orders of supercooling.
while the horizon still exists and lies beyond the observed scale. the two amplification thresholds and their outputs are supplied by the entropy calculation (2)
The physical realization is a first-order transition with a low nucleation rate and sufficient supercooling. The phase is a metastable false vacuum with positive, dominant, Lorentz-invariant energy; bubble nucleation occurs by tunneling, and the zero-temperature rate is very small. The supplied calculation retains the exact false-vacuum equation of state p = -ρ0 together with p < 0. It also retains the shifted density law, temperature-form Friedmann equation, the halted ε > ε0 branch, and the unchecked ε < ε0 branch. The “good approximation” qualifier on equation (3.5) and the printed approximate cooling law are represented by selected relative-error bounds rather than rewritten as exact equalities; the RT = const bridge used for the scale-factor law is retained separately. The calculation then yields latent-heat release, reheating near the critical temperature, the value-level equation (3.4) Z = Tr/Ts with 0 < Ts < Tr, entropy-density amplification, a continuous scale factor through reheating, exponential cooling and expansion, and applicability of the inflationary mechanism.
(Ṫ/T)2 = 4π3GN(T)T4/45 − ε(T)T2 + 8πGρ0/3,
ε0 = 8π2√30 G√(Nρ0)/45; ε > ε0 → Tmin4 = 30ρ0[ε/ε0 + √((ε/ε0)2 − 1)]2/(π2N),
ε < ε0 + vacuum-term dominance → T(t) ≃ CTe−χt, χ2 = 8πGρ0/3, R(t) = CReχt, H = Ṙ/R = χ.
FalseVacuumExpansionSourceFormulaAt; equations (3.5), (3.8)–(3.13); equation (3.5) and the ≃ sign carry bounded relative approximations, and RT = const is retained
(3a)
latent heat → 0 < Ts < Tr, Z = Tr/Ts → reheating + nonadiabatic entropy production → realization of both inflationary targets.
false_vacuum_supercooling_retains_equation_three_four_entropy_scale_factor · false_vacuum_supercooling_retains_expansion_source_formula; enormous pre-transition expansion is assessed as plausible
(3)
The first theorem establishes both targets and the displaced-horizon qualification. The second establishes the false-vacuum outputs. The checked theorem positive_false_vacuum_energy_and_equation_of_state_yield_negative_pressure derives the strict sign from ρ0 > 0 and p = -ρ0; negative_pressure_output_retains_false_vacuum_equation_of_state prevents the output from forgetting that equation. false_vacuum_supercooling_retains_equation_three_four_entropy_scale_factor projects the reheating ratio, while false_vacuum_supercooling_retains_expansion_source_formula projects equations (3.5) and (3.8)–(3.13) instead of leaving them behind an exponential-expansion label. Coverage exposes that source formula as a separate endpoint and uses the certified entropy output when it certifies the mechanism route.
sufficient_entropy_production_solves_both_standard_problems · false_vacuum_supercooling_realizes_inflationary_mechanism · false_vacuum_supercooling_retains_equation_three_four_entropy_scale_factor · false_vacuum_supercooling_retains_expansion_source_formulaResolution does not mean disappearance
The horizon countermodel makes the wording exact: solving the observed horizon problem is compatible with a positive horizon radius beyond the observed radius. Likewise, exponential expansion alone does not imply that the phase transition completes.
From light-front births to survival probability
The paper-local LightFrontNucleation model uses actual Euclidean spheres in three comoving coordinates. Births have centers, birth times, occurrence, and real-versus-auxiliary roles. No count, converted region, probability, or time constant is an independently assigned conclusion. The actual ensemble and the retained late-regime ensemble are distinct probability models; the selected actual history supplies phaseConversionHistory.
Real and fictitious bubbles use the same light-front geometry.
A fictitious birth inside a real bubble remains inside that bubble at every later time. Euclidean sphere volume, not a sup-metric cube; pointlike births have zero initial interior. Equations (3.14)-(3.15). (3b)
The containment theorem uses the triangle inequality and additivity of the conformal travel integral. The union of real bubbles therefore equals the union of all bubbles under the interior-birth condition. Real-birth exclusion and fictitious containment are explicit hypotheses, not consequences of their labels.
p(x,t) = Prob(N(x,t) = 0) = exp(-mu(x,t)); 0 < p(x,t) <= 1. Rate per physical volume per time. A normalized Poisson law, measurable actual counts, local finiteness, and explicit cone-intensity integration are required. Independence is for disjoint birth cells, not overlapping point-survival events. (3c)
equation_three_fourteen_follows_from_actual_nucleation derives the printed survival formula from the zero-count event of the same birth configurations. ConeIntensityIntegrationAt states the remaining measure-integration calculation for the actual spacetime intensity and past cone; it is not a free mean parameter. The public source result retains the background regularity needed by later deductions.
tau = 3 chi^3/(4 pi lambda_0); Z_tau = exp(chi tau) = exp(3 chi^4/(4 pi lambda_0)).
p_ret(x,t) = exp(-t/tau + c_ret(x,t)).
c_ret(x,t) = t/tau - mu_ret(x,t); c_ret(x,t) tends to 22 pi lambda_0/(9 chi^4). The exact retained-model integral evaluation is an explicit input. The computed correction converges to a constant; this is not exact finite-time pure exponential decay. Equations (3.16)-(3.18), with the correction coefficient reconstructed in Lean. (3d)
IntensityIntegralCalculationAt equates the actual retained-model integral to its explicit elementary expression. The checked limit theorem then computes the correction limit. Limiting agreement of the original rate and scale-factor ratio does not itself control the integrated error or make the retained formula exact for the original background.
Z_tau = exp([3 chi^4/(4 pi A rho_0)] exp(B)). Imported tunneling formula (3.19). Double-exponential parameter dependence does not guarantee huge expansion for every positive parameter choice; enormous expansion remains an author plausibility assessment. (3e)
The tunneling density is the same selected phase's false-vacuum density, and the retained expansion rate comes from the existing expansion readout. A control with unit prefactor and barrier value ten satisfies the tunneling formula but gives only Z_tau = exp(1/4). It tests that formula alone, not the full sufficient-supercooling or Friedmann setup.
actual_nucleation_inputs_derive_the_source_survival_formula; retained_nucleation_inputs_derive_the_source_asymptotic_formula; imported_tunneling_law_derives_double_exponential_parameter_dependence. Original preprint pp. 14-16.
Inflation enlarges a local patch and dilutes charges, but neither branch supplies everything it needs
Local initial conditions
The global initial state may be chaotic. It is enough to select a hot local region, expect local thermalization and inhomogeneity damping, import anisotropy damping, and inherit exponential expansion from the false-vacuum route. The patch calculation then yields local homogeneity, isotropy, thermality, and subhorizon size before expansion, followed by enlargement beyond the observed region.
without requiring global initial homogeneity. thermalization and inhomogeneity damping expected; anisotropy damping imported (4)
Conserved abundance and baryon recovery
For a strictly conserved quantity, entropy-density amplification and a supplied dilution calculation imply an abundance below measurability. The source map describes the suppression as Z−3. Applied to baryon number, this would make exact conservation fatal. Recovery is a second argument: it assumes grand-unified baryon violation, post-reheating baryogenesis, and a baryon-recovery calculation.
baryon recovery = dilution + baryon-number violation + post-reheating baryogenesis. dilution does not itself generate a new baryon abundance (5)
The local-patch theorem establishes the enlarged prepared region and rejects global homogeneity as a requirement. The charge theorem establishes dilution; the baryon theorem separately establishes the exact-conservation failure and the conditional escape from that objection.
local_subhorizon_patch_can_generate_observed_homogeneity · inflation_dilutes_strictly_conserved_quantities · exact_baryon_conservation_would_fail_but_post_reheating_baryogenesis_can_avoid_itMonopole production is collective, and its abundance creates independent pressure for inflation
A magnetic monopole is not defined by stability alone. Lean requires a topological Higgs knot, magnetic charge, and topological stability. Production begins with a nontrivial family of distinct transition bubbles whose Higgs orientations are uncorrelated while the bubbles are out of contact. It additionally requires the family to coalesce and fill space.
with monopole count comparable to bubble count. collective Kibble production is an imported calculation; stability or independence alone is insufficient (6)
The abundance calculation combines a grand-unified transition, absence in the high-temperature phase, standard-transition overproduction, a horizon production bound, and sufficient supercooling. It concludes that inflation suppresses the abundance. If rapid nucleation avoids exponential growth, it overproduces monopoles; hence the monopole problem favors inflation. Unknown model parameters prevent a definitive nucleation-rate answer.
The checked theorem composes the production and abundance calculations, while preserving both count comparability and model-rate uncertainty. This route is source-map independent of the previously established false-vacuum mechanism: it is a separate pressure toward inflation, not a consequence of target resolution. Count comparability by itself does not satisfy the cosmological bound: Lean gives a witness with 100 bubbles, 100 monopoles, and an allowed count of 1.
uncorrelated_coalescing_higgs_bubble_family_produces_magnetic_monopole · monopole_production_problem_also_favors_inflationThe mechanism’s success and its graceful-exit failure coexist
The exit route assumes slow nucleation relative to expansion, random bubbles, latent heat initially stored in bubble walls, a need for many wall collisions, and an enormous range of bubble sizes. One logical hinge must remain visible: the paper presumes that the slow rate lies below an unknown critical percolation ratio. A formal countermodel proves that “slow” does not establish “subcritical.”
→ finite separated clusters with few of the largest bubbles in each. Large converted spatial fraction is derived separately from survival, integrated approximation control, and selected-history averaging. Converted volume and connected percolation are distinct observables. (7)
Controlled actual survival therefore tends to zero.
At t_n = n+1 for n in Nat, selected-history spatial density = 1 - p_actual(0,t_n).
For every epsilon > 0, some positive time has converted density > 1-epsilon. Integrated error and selected-history spatial averaging are separate inputs, not percolation conclusions. Almost-sure averaging at countably many times does not select an arbitrary history as typical. (7e)
source_survival_and_selected_spatial_averaging_give_large_converted_fraction now derives the volume conclusion from the nucleation source outputs. IntegratedNucleationApproximationAt supplies the actual-minus-retained integrated-error limit; SelectedSpatialAveragingAt supplies the correspondence for the selected actual history. These reconstructed obligations are explicit, not calculations claimed to have been completed by the paper. Independently, a checked theorem combines countably many almost-sure spatial averages into one almost-sure nearly-complete-conversion statement. It does not apply that statement to an arbitrary selected history.
Percolates(C_t) iff some x in C_t has an unbounded connected component.
FiniteClusters(C_t) iff every x in C_t has a bounded connected component. One actual region history; no assigned fraction, cluster size, or connectivity verdict. (7a)
f_N(C) = Vol(C intersect W_N)/Vol(W_N); 0 < Vol(W_N) < infinity.
SpatialDensity(C,f): C measurable, 0 <= f <= 1, and f_N(C) tends to f as N tends to infinity. Expanding finite coordinate-cell windows, not a point survival probability or a quotient of infinite volumes. (7b)
t > 0: f_N(C_t) = (2r(t))^3 = [t/(t+1)]^3 for every N.
For every epsilon > 0, some positive time has density > 1-epsilon.
Every connected component stays bounded, while C_t is unbounded. A deterministic geometric inference control, not a stochastic spherical-bubble nucleation model. (7c)
A bounded open cube has infinitely many points and does not percolate.
C = {x in R^3 | x_0 != 0} percolates but is not connected. Spatial extent of a connected component is distinct from total extent, point cardinality, and whole-phase connectedness. (7d)
The spatial definitions and controls are implemented in ConvertedRegionGeometry. Each finite window has positive finite volume and every coordinate cell is eventually included. The same converted set supplies both the volume integral and its actual connected components. The three paper-framework predicates now compute these meanings from phaseConversionHistory. Fidelity's certified_exit_route_retains_actual_spatial_density_and_connectivity and certified_exit_route_retains_bounded_actual_components_at_every_positive_time carry them through the certified exit route. The cube and split-region controls are independent Lean constructions, not examples attributed to Guth.
The source survival and conformal-volume formulas are now formalized, with precise integration, approximation, and averaging obligations retained as hypotheses. The unknown critical ratio, stochastic connectivity result, large-bubble collision physics, and final qualified assessment remain separate inputs. Guth's unconverted survival probability is not Albrecht-Steinhardt's converted-volume convention, and no condition from this route is silently imposed on that later mechanism.
| Retained data | Invalid strengthening rejected |
|---|---|
| Infinitely many actual engulfing births; totalized natural count zero | A zero count without local finiteness means survival |
| One auxiliary birth with no enclosing real bubble | The auxiliary label alone preserves the converted region |
| R = 2, lambda = 1: actual integrated intensity pi/3; coordinate-only value pi/24 | The physical-volume Jacobian can be omitted |
| Positive tunneling parameters, A = 1, B = 10, Z_tau = exp(1/4) | The tunneling form alone guarantees enormous expansion |
| A shifted exponential lifetime has p(t) = exp(-t+1) for t >= 1 | A convergent correction implies exact exp(-t) decay |
| A normalized whole-space/empty-space mixture has point probability 1-exp(-1) | Its realized density must equal that probability rather than zero or one |
| Unit spatial density almost surely, but zero density at the selected null-exception history | Almost-sure truth makes every selected history typical |
These are actual configuration, probability-measure, and spatial-region controls, not claimed realizations of the full Guth Poisson cosmology. They test which assumptions the deductions require.
Because latent heat begins in walls and thermalization requires many collisions, the huge size range matters. Finite clusters contain too few largest bubbles for sufficient large-bubble collisions. A separate graceful-exit assessment maps random formation plus failed collisions to gross inhomogeneity, absence of a smooth exit, and both inhomogeneity and incomplete-transition side effects.
unacceptable inhomogeneity side effect → original scenario not viable. qualified assessment of the original scenario, not an impossibility theorem for every inflationary model (8)
The full route certifies large converted volume without percolation, finite clusters, failed thermalization, an inhomogeneous incomplete exit, and nonviability of the original scenario. The volume conclusion separately consumes actual and retained survival outputs, integrated-error control, and selected-history spatial averaging. Its nonviability step consumes the accelerated-expansion output specifically, not the distinct target-resolution conclusion. A separate theorem explicitly conjoins target-solving success with nonviability.
slow_nucleation_prevents_a_homogeneous_graceful_exit · unacceptable_graceful_exit_effect_makes_original_scenario_nonviable · inflation_can_solve_targets_while_original_scenario_is_nonviableThe decisive separation
A mechanism solves its selected targets when it produces the horizon and flatness targets for admissible initial data. It is viable only if none of its side effects is unacceptable. Lean constructs a mechanism that satisfies the former and fails the latter, so “solves both problems” cannot be silently upgraded to “viable cosmology.”
A sharply rising nucleation rate repairs the bubble geometry only conditionally
The proposed modification assumes a sharply rising nucleation rate and a Coleman–Weinberg candidate phase. Given the original scenario’s nonviability and a supplied assessment, it yields approximately uniform bubble sizes, bubbles that fill space, and avoidance of bubble inhomogeneity.
but missing late trigger + omitted possibly relevant particle production → no established viable modification. the source map places the required trigger near 10−28 of the critical temperature (9)
Viability has explicit establishment requirements: the late trigger must be established, and gravitational particle production must be included whenever it may matter. The assessment reports the opposite—no established trigger, particle production omitted, and possible relevance. Each omission independently blocks established viability.
The theorem proves all three conditional repair outputs and all three unresolved outputs. A countermodel with uniform, space-filling bubbles but zero established supercooling orders and zero included particle-production terms proves that repair is not viability.
sharp_nucleation_is_only_an_unestablished_modification · missing_required_trigger_blocks_established_modification · omitted_relevant_particle_production_blocks_established_modificationConcept guide
Horizon problem
Many causally disconnected precursors nevertheless share equal temperature; it is distinct from flatness.
HorizonProblemFlatness problem
Fine tuning of the initial Hubble value together with a startlingly small conserved curvature parameter.
FlatnessProblemEntropy criterion
Nonadiabatic entropy production sufficient for both targets, with nearly critical density and a displaced—not abolished—horizon.
EntropyResolutionCalculationFalse-vacuum inflation
A slowly decaying metastable phase with ρ0 > 0 and p = -ρ0 < 0; its negative pressure drives exponential expansion before latent-heat reheating fixes Z = Tr/Ts.
falseVacuumEquationOfStateAt · equationThreeFourEntropyScaleFactorValueAt · FalseVacuumMechanismCalculationLocal patch
A hot, smoothed, subhorizon region enlarged beyond the observed region without global initial homogeneity.
LocalPatchCalculationConserved-quantity dilution
Entropy amplification drives a strictly conserved abundance below present measurability.
ConservedQuantityDilutionForPhaseBaryon recovery
A separate post-reheating regeneration argument requiring baryon violation; dilution alone is not baryogenesis.
BaryonRecoveryCalculationCollective monopole production
Uncorrelated Higgs orientations form a defect only through collective coalescence; magnetic charge and stability complete the monopole predicate.
KibbleMonopoleProductionCalculationPercolation distinction
Percolation requires an unbounded connected component of the actual converted region. Its normalized spatial volume can approach full density while every component remains bounded. Infinite point cardinality and whole-phase connectedness are different notions.
ConvertedRegionGeometry · PercolationCalculationGraceful-exit obstruction
Finite clusters and insufficient large-bubble collisions yield inhomogeneous, incomplete exit and nonviability of the original scenario.
GracefulExitAssessmentTarget solution versus viability
Producing horizon and flatness targets does not guarantee that all side effects are acceptable.
InflationarySolutionRuleUnestablished modification
A conditional uniform-bubble repair whose late trigger and particle-production obligations remain unsatisfied.
ViableModificationEstablishmentRequirementsComplete route inventory, separating witnesses, and formal provenance
The source map declares eight routes. Coverage interprets every source atom, certifies every route under explicit premises and transitive certificates, and proves the route realization exhaustive.
| Route | Direct dependency | Declared outputs | Epistemic profile |
|---|---|---|---|
| Standard problems | — | Equations (2.10)–(2.20); causal disconnection; fine tuning; both named problems | retained flatness/horizon formulas + deferred calculation + interpretive identification |
| Entropy resolution | — | Both targets; displaced horizon | deferred threshold calculation |
| False-vacuum mechanism | Entropy resolution | Accelerated expansion; reheating/entropy; target realization; plausibility | calculation + author expectation |
| Local patch | False-vacuum mechanism | Enlarged patch; no global homogeneity requirement | imported + expected smoothing |
| Conserved quantities | False-vacuum mechanism | Diluted charge; baryon qualification | two distinct calculations |
| Monopoles | — | Suppression/pressure for inflation; rate uncertainty | imported collective result + open uncertainty |
| Graceful exit | False-vacuum mechanism | Volume/connectivity distinction; cluster failure; inhomogeneous exit; original nonviability | expectation + imported result + assessment |
| Modification outlook | Graceful exit | Conditional repair; missing trigger; omitted physics; viability open | unresolved modification |
Standard problems · standard_problems_route_is_semantically_realized
Entropy resolution · entropy_resolution_route_is_semantically_realized
False-vacuum mechanism · false_vacuum_mechanism_route_is_semantically_realized
Local patch · local_patch_route_is_semantically_realized
Conserved quantities · conserved_quantities_route_is_semantically_realized
Monopoles · monopoles_route_is_semantically_realized
Graceful exit · graceful_exit_route_is_semantically_realized
Modification outlook · modification_outlook_route_is_semantically_realized
Twenty-one separated conclusions across eighteen countermodel profiles
| Available fact | Conclusion not supplied | Lean witness |
|---|---|---|
| Positive energy density and negative pressure | The false-vacuum equation of state p = -ρ0 | negative_pressure_alone_does_not_supply_false_vacuum_equation_of_state |
| Entropy-density amplification with the cubic temperature ratio | The equation-(3.4) scale-factor ratio Z = Tr/Ts | entropy_amplification_label_does_not_supply_equation_three_four_ratio |
| Vanishing relative-density deviation at a positive temperature | The source equations (2.10)–(2.16) | named_flatness_fine_tuning_does_not_supply_source_formulas |
| A positive-time horizon smaller than the observed region | The source equations (2.17)–(2.20) | named_standard_problem_calculation_does_not_supply_horizon_formulas |
| Positive scale factor and positive expansion derivative | The source equations (3.5) and (3.8)–(3.13) | named_exponential_expansion_does_not_supply_false_vacuum_source_formulas |
| Horizon-problem profile | Flatness-problem profile | horizon_problem_does_not_imply_flatness_problem |
| Nonzero Higgs winding and no unwinding channel | Magnetic monopole without nonzero magnetic flux | topological_stability_alone_does_not_define_magnetic_monopole |
| Distinct positive-radius bubbles with different Higgs orientations and separation greater than their combined radii | Defect production without collective coalescence | uncorrelated_bubble_family_without_collective_coalescence_does_not_produce_defect |
| Monopole count comparable to bubble count | Cosmologically acceptable abundance | bubble_comparable_production_can_violate_cosmological_bound |
| Expanded scale factor | Completed phase transition | exponential_expansion_does_not_imply_phase_transition_completion |
| Slow nucleation | Placement below the percolation threshold | slow_nucleation_does_not_establish_subcritical_percolation |
| Converted fraction arbitrarily close to unity | A spanning converted cluster | arbitrarily_large_converted_fraction_does_not_imply_percolation |
| An unbounded, monotonically growing converted region | An unbounded connected component | arbitrarily_large_spatial_fraction_can_coexist_with_no_percolation |
| Infinitely many points in a bounded open cube | Spatial percolation | a_bounded_cube_has_infinitely_many_points_without_percolating |
| A percolating converted phase | Connectedness of the whole phase | percolation_does_not_require_the_entire_phase_to_be_connected |
| Mechanism produces both selected targets | Mechanism is viable | solving_horizon_and_flatness_does_not_imply_viability |
| Observed horizon problem resolved | Horizon ceases to exist | resolving_observed_horizon_problem_does_not_remove_horizon |
| Abundance diluted by expansion | Abundance regenerated after reheating | entropy_dilution_does_not_supply_baryogenesis |
| Uniform bubbles and complete converted volume | Required late trigger established | conditional_bubble_repair_does_not_establish_modified_viability |
| Uniform bubbles and complete converted volume | Particle production included | conditional_bubble_repair_does_not_establish_modified_viability |
| Conditional bubble repair | Viable modification established | conditional_bubble_repair_does_not_establish_modified_viability |
The target-solving but nonviable witness now retains the achieved and required expansion orders, inflated and observed radii, curvature tolerance, and excessive bubble-diameter spread. No constant proposition or Unit carrier substitutes for the horizon, flatness, magnetic-flux, contact, or graceful-exit distinctions.
Machine-checked audit boundary
argument_wellFormed, argument_auditable, and routes_wellFormed certify source-graph structure. routes_have_exact_predecessor_graph, early_mechanism_and_patch_inputs_are_independent_hypotheses, monopole_exit_and_modification_inputs_are_independent_hypotheses, key_result_steps_apply_selected_inputs_and_contracts, and monopole_and_graceful_exit_dependencies_match_primary_theorems prevent hidden inputs and false dependency edges. every_source_claim_has_semantics covers every source claim, and route_semantic_realization_is_exhaustive covers all eight routes. Dedicated status theorems preserve the literal standard-flatness, standard-horizon, and false-vacuum formula endpoints, interpretive flatness judgment, calculation/expectation split, local smoothing split, percolation presumption, imported collective monopole result, qualified graceful-exit assessment, and open modification obligations.
Lean-encoded bibliographic record
A. H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems (1981), original SLAC-PUB-2576 preprint, pp. 1–32. Page anchors above follow the source map’s preprint pagination.
TraceComplete Lean semantic identifier ledger
Reading note. The article above is the human exposition. This collapsed cross-reference records every public theorem and every constructor of the five paper-local source-map vocabularies. It introduces no additional claim; its purpose is to make semantic omissions mechanically detectable.
Primary semantic theorems (196)
finite_clusters_iff_no_percolationbounded_regions_do_not_percolateobservation_centers_are_nonemptydistinct_lattice_centers_are_separateddistinct_cubes_are_disjointpreconnected_subsets_of_separated_cubes_are_boundedseparated_cubes_have_actual_finite_clustersconverted_region_intersect_cell_is_the_actual_cubecoordinate_cube_has_actual_volumefinite_cube_union_has_actual_volumeobservation_window_has_positive_finite_volumeperiodic_converted_fraction_is_computed_from_the_same_regionperiodic_cubes_have_their_actual_spatial_densityobservation_windows_eventually_include_every_coordinate_cellseparated_converted_cubes_are_spatially_unboundedfinite_clusters_at_every_time_exclude_percolationpositive_time_gives_positive_separated_cubesseparated_growing_regions_are_physicalseparated_growing_regions_have_actual_spatial_densityseparated_growing_regions_have_arbitrarily_large_converted_fractionseparated_growing_regions_have_bounded_connected_componentsarbitrarily_large_spatial_fraction_can_coexist_with_no_percolationa_bounded_cube_has_infinitely_many_points_without_percolatingnegative_diagonal_is_inside_the_split_converted_regionthe_negative_diagonal_is_spatially_unboundedpercolation_does_not_require_the_entire_phase_to_be_connectedspherical_region_is_measurablespherical_region_has_actual_coordinate_volumenonnegative_conformal_radiuscommon_light_front_radii_adda_birth_inside_a_light_front_remains_insidepointlike_birth_has_zero_initial_regioneach_light_front_region_is_measurablelight_front_regions_grow_with_timepossible_engulfing_centers_are_the_same_spherical_regionequation_three_fifteen_is_actual_coordinate_volumeintegrated_intensity_has_the_source_light_cone_formphysical_intensity_is_nonnegativeactual_births_are_exactly_the_unconverted_birthsinterior_fictitious_births_preserve_the_actual_converted_regionzero_finite_engulfing_count_iff_no_actual_bubbleactual_converted_regions_are_measurableactual_conversion_history_is_physicala_past_cone_counts_exactly_the_engulfing_birthspoisson_birth_cells_determine_the_actual_engulfing_count_lawactual_poisson_count_has_the_source_zero_probabilitysource_poisson_survival_is_the_actual_unconverted_eventequation_three_fourteen_follows_from_actual_nucleationfinite_poisson_survival_is_positive_and_at_most_oneretained_background_is_regularretained_rates_are_measurablesource_time_constant_is_positiveequation_three_eighteen_retains_the_fourth_powertime_constant_expansion_is_the_actual_scale_factor_ratiointegrated_source_remainder_is_computed_from_the_same_intensityinitial_source_exponent_correction_is_zeroretained_decay_factor_tends_to_zeroclosed_source_correction_converges_to_a_constantactual_source_correction_converges_after_the_integral_calculationretained_poisson_model_has_the_source_late_survival_formequation_three_nineteen_makes_the_expansion_double_exponentialintegrated_error_control_transfers_the_retained_late_survivalcontrolled_late_survival_tends_to_zerocountable_spatial_averages_give_almost_sure_joint_typicalitydecay_and_selected_spatial_averaging_give_nearly_complete_conversiondecay_and_spatial_averaging_give_almost_sure_nearly_complete_conversionunit_static_light_front_engulfs_its_originevery_coincident_birth_engulfs_the_same_pointzero_natural_count_without_local_finiteness_does_not_mean_survivalan_auxiliary_label_alone_does_not_preserve_the_converted_regionrescaled_static_background_is_regularrescaled_static_light_front_radiusunit_reverse_cubic_integralrescaled_background_has_its_actual_integrated_intensityomitted_jacobian_has_a_different_integrated_intensitycoordinate_volume_alone_does_not_supply_the_physical_nucleation_intensitylarge_barrier_control_has_positive_physical_parameterslarge_barrier_control_retains_the_imported_tunneling_formulalarge_barrier_control_has_a_short_expansion_timetunneling_form_and_a_large_barrier_do_not_guarantee_large_expansionexponential_control_is_normalizedexponential_control_cumulative_probabilityexponential_control_survival_probabilitydelayed_lifetime_has_actual_late_survivaldelayed_lifetime_has_nonzero_late_correctiondelayed_lifetime_correction_convergesa_convergent_exponent_correction_does_not_give_exact_exponential_decayglobal_switch_has_identical_nontrivial_point_probabilitiesglobal_switch_probability_is_strictly_between_zero_and_onefull_space_has_unit_spatial_densityempty_space_has_zero_spatial_densityevery_global_switch_realization_has_actual_spatial_densitypoint_probabilities_do_not_determine_a_realized_spatial_densityexponential_control_has_no_zero_atomnull_exceptional_region_has_almost_sure_unit_densityalmost_sure_spatial_density_does_not_select_a_typical_historynamed_flatness_fine_tuning_does_not_supply_source_formulasa_finite_particle_horizon_does_not_imply_a_future_event_horizonnamed_standard_problem_calculation_does_not_supply_horizon_formulassource_formula_retains_horizon_radius_and_density_estimatenamed_monopole_horizon_bound_does_not_supply_printed_formulasbubble_horizon_bound_does_not_supply_monopole_density_estimatehorizon_density_estimate_does_not_supply_preskill_observational_boundrevised_strong_transition_bound_does_not_imply_coarse_horizon_estimatenamed_exponential_expansion_does_not_supply_false_vacuum_source_formulasactual_nucleation_inputs_derive_the_source_survival_formularetained_nucleation_inputs_derive_the_source_asymptotic_formulaimported_tunneling_law_derives_double_exponential_parameter_dependencesource_survival_and_integrated_control_give_actual_late_decaysource_survival_and_selected_spatial_averaging_give_large_converted_fractionpositive_false_vacuum_energy_and_equation_of_state_yield_negative_pressurenegative_pressure_output_retains_false_vacuum_equation_of_stateinflationary_patch_model_retains_paper_selected_databubble_transition_model_retains_paper_selected_databubble_transition_inputs_realize_shared_selected_setupuncorrelated_higgs_bubble_family_exposes_distinct_independent_membersdefect_formation_model_retains_paper_semanticspaper_inputs_make_shared_selected_defect_formation_setupstandard_model_calculation_exposes_horizon_and_flatness_problemsstandard_model_calculation_retains_horizon_source_formulathe_standard_cosmology_calculation_exposes_actual_causal_reachabilitystandard_model_calculation_retains_flatness_source_formulastandard_model_flatness_diagnosis_realizes_shared_fine_tuningsufficient_entropy_production_solves_both_standard_problemsfalse_vacuum_supercooling_realizes_inflationary_mechanismfalse_vacuum_supercooling_retains_equation_three_four_entropy_scale_factorfalse_vacuum_supercooling_retains_expansion_source_formulalocal_subhorizon_patch_can_generate_observed_homogeneityinflation_dilutes_strictly_conserved_quantitiesexact_baryon_conservation_would_fail_but_post_reheating_baryogenesis_can_avoid_ituncorrelated_coalescing_higgs_bubble_family_produces_magnetic_monopolekibble_monopole_inputs_make_shared_defect_formation_applicablekibble_monopole_route_realizes_shared_defect_formationkibble_monopole_route_exhibits_nonvacuous_shared_defect_formationkibble_monopole_route_realizes_structural_defect_formation_explanationguth_imported_kibble_inputs_realize_source_structural_defect_formation_explanationestablished_guth_defect_formation_realizes_every_shared_claimguth_defect_model_inputs_establish_groundingkibble_monopole_inputs_establish_grounded_shared_defect_formationgrounded_guth_defect_formation_transfers_every_shared_claimmonopole_production_problem_also_favors_inflationpresumed_subcritical_rate_yields_finite_clusterspresumed_subcritical_rate_allows_large_fraction_without_percolationfinite_clusters_block_large_bubble_thermalizationfailed_large_bubble_thermalization_yields_inhomogeneous_exitslow_nucleation_realizes_shared_bubble_competition_calculationpercolation_thermalization_realizes_shared_incomplete_transition_bridgeslow_nucleation_inputs_make_shared_bubble_route_applicableguth_bubble_source_derivation_makes_shared_route_applicableslow_nucleation_inputs_realize_structural_bubble_completion_obstruction_explanationslow_nucleation_source_inputs_realize_source_structural_bubble_completion_obstruction_explanationslow_nucleation_realizes_shared_bubble_completion_obstructionslow_nucleation_exhibits_nonvacuous_shared_bubble_completion_obstructionestablished_guth_bubble_obstruction_realizes_every_shared_claimestablished_guth_bubble_obstruction_and_unacceptability_realize_scoped_viability_failurebubble_transition_model_inputs_establish_theory_model_groundingslow_nucleation_establishes_theory_model_grounded_bubble_obstructiongrounded_guth_bubble_obstruction_transfers_every_shared_claimslow_nucleation_prevents_a_homogeneous_graceful_exitinflationary_solution_rule_realizes_shared_solution_semanticsinflationary_solution_rule_realizes_shared_patch_homogeneity_bridgeinflationary_solution_rule_realizes_shared_patch_flatness_bridgelocal_patch_and_solution_rule_make_shared_patch_route_applicableguth_inflationary_patch_source_derivation_makes_shared_route_applicablelocal_subhorizon_patch_source_inputs_realize_source_structural_inflationary_patch_explanationlocal_patch_and_solution_rule_realize_shared_inflationary_patchlocal_patch_and_solution_rule_exhibit_nonvacuous_shared_inflationary_patchlocal_patch_and_slow_nucleation_realize_composed_inflation_bubble_failurelocal_patch_and_slow_nucleation_realize_source_sensitive_composed_inflation_bubble_failureestablished_local_patch_realizes_every_shared_claimestablished_local_patch_realizes_scoped_solutionestablished_patch_and_bubble_obstruction_retain_scoped_solution_failureguth_inputs_realize_selected_inflationary_patch_setupinflationary_patch_model_inputs_establish_theory_model_groundinglocal_patch_and_solution_rule_establish_theory_model_grounded_patchgrounded_local_patch_transfers_every_shared_claimunacceptable_graceful_exit_effect_makes_original_scenario_nonviableinflation_can_solve_targets_while_original_scenario_is_nonviablemissing_required_trigger_blocks_established_modificationomitted_relevant_particle_production_blocks_established_modificationsharp_nucleation_is_only_an_unestablished_modificationnegative_pressure_alone_does_not_supply_false_vacuum_equation_of_stateentropy_amplification_label_does_not_supply_equation_three_four_ratiohorizon_problem_does_not_imply_flatness_problemtopological_stability_alone_does_not_define_magnetic_monopoleuncorrelated_bubble_family_without_collective_coalescence_does_not_produce_defectbubble_comparable_production_can_violate_cosmological_boundexponential_expansion_does_not_imply_phase_transition_completionslow_nucleation_does_not_establish_subcritical_percolationarbitrarily_large_converted_fraction_does_not_imply_percolationcountermodel_solves_horizon_and_flatnesscountermodel_is_not_viablesolving_horizon_and_flatness_does_not_imply_viabilityresolving_observed_horizon_problem_does_not_remove_horizonentropy_dilution_does_not_supply_baryogenesisconditional_bubble_repair_does_not_establish_modified_viability
Coverage theorems (16)
every_source_claim_has_semanticsstandard_problems_route_is_semantically_realizedstandard_problems_route_recovers_flatness_source_formulastandard_problems_route_recovers_horizon_source_formulaentropy_resolution_route_is_semantically_realizedfalse_vacuum_mechanism_route_is_semantically_realizedfalse_vacuum_mechanism_route_exposes_source_formulalocal_patch_route_is_semantically_realizedconserved_quantities_route_is_semantically_realizedmonopoles_route_is_semantically_realizedmonopoles_route_recovers_horizon_source_formulamonopoles_route_recovers_strong_transition_revision_formulagraceful_exit_route_is_semantically_realizedmodification_outlook_route_is_semantically_realizedall_routes_are_semantically_realizableroute_semantic_realization_is_exhaustive
Source-map audit theorems (27)
argument_wellFormedargument_auditableroutes_wellFormedroutes_have_exact_predecessor_graphearly_mechanism_and_patch_inputs_are_independent_hypothesesmonopole_exit_and_modification_inputs_are_independent_hypotheseskey_result_steps_apply_selected_inputs_and_contractsmonopole_and_graceful_exit_dependencies_match_primary_theoremsflatness_problem_retains_interpretive_naturalness_statusstandard_problem_diagnosis_uses_both_physical_facts_and_target_identificationinflationary_mechanism_retains_calculation_and_expectationlocal_patch_route_retains_imported_and_expected_smoothingpercolation_outcome_retains_expectation_and_imported_resultvolume_connectivity_distinction_retains_expectation_and_imported_resultgraceful_exit_conclusion_is_assessment_not_impossibility_proofmodification_outlook_retains_open_obligationsmodification_nonestablishment_uses_explicit_requirements_and_obligationscharge_and_baryon_inputs_are_independent_hypothesesstandard_entropy_and_vacuum_result_steps_apply_exact_semantic_inputsstandard_route_exposes_flatness_source_formulastandard_route_exposes_horizon_source_formulafalse_vacuum_route_exposes_source_formulapatch_and_charge_result_steps_apply_exact_semantic_inputslate_result_steps_apply_exact_semantic_inputsmonopole_production_retains_imported_collective_kibble_resultspatial_conversion_steps_preserve_independent_approximation_and_averagingnucleation_steps_preserve_the_separate_source_inputs
Semantic atoms (66)
standardModelSetupstandardProblemsCalculationstandardFlatnessSourceFormulastandardHorizonSourceFormuladisconnectedPrecursorsflatnessFineTuningstandardProblemTargetIdentificationhorizonAndFlatnessProblemsentropyIncreaseSetupentropyResolutionCalculationentropyResolvesBothProblemshorizonMovedNotRemovedfalseVacuumSetupfalseVacuumMechanismCalculationfalseVacuumExpansionSourceFormulanegativePressureAndExponentialExpansionreheatingAndEntropyProductionactualNucleationSetupactualPoissonBirthProcessactualNucleationIntegrationactualNucleationSourceFormularetainedNucleationSetupretainedNucleationProcessretainedNucleationCalculationretainedNucleationSourceFormulatunnelingNucleationFormulatunnelingExpansionFormulainflationaryMechanismResolvesProblemshugeExpansionPlausibilitylocalPatchSetuplocalSmoothingInputslocalPatchCalculationlocalPatchBecomesObservedRegionglobalHomogeneityNotRequiredconservedQuantitySetupconservedQuantityDilutionCalculationconservedQuantityDilutedbaryonRecoverySetupbaryonRecoveryCalculationbaryonQualificationmonopoleSetupmonopoleHorizonSourceFormulastrongTransitionMonopoleRevisionSourceFormulamonopoleProductionCalculationmonopoleSuppressionAndPressureForInflationnucleationRateModelUncertaintygracefulExitSetupintegratedNucleationApproximationselectedSpatialAveragingnearlyCompleteConvertedVolumepercolationRegimeExpectationpercolationCalculationlargeConvertedFractionWithoutPercolationnoPercolationAndFiniteClustersclusterThermalizationCalculationgracefulExitCalculationnoPercolationAndInsufficientThermalizationgrossInhomogeneityAndNoSmoothExitoriginalScenarioNonviablemodificationSetupmodificationAssessmentmodificationEstablishmentRequirementsconditionalUniformBubbleExitrequiredTriggerUnestablishedomittedParticleProductionviableModificationOpen
Claim identifiers (66)
standardSetupstandardCalculationflatnessFormulastandardFormulacausalDisconnectionfineTuningproblemIdentificationstandardProblemsentropySetupentropyCalculationentropyResolutiondisplacedHorizonvacuumSetupvacuumCalculationvacuumSourceFormulaacceleratedExpansionreheatingEntropynucleationSetuppoissonBirthProcessnucleationIntegrationnucleationSourceFormularetainedNucleationSetupretainedNucleationProcessretainedNucleationCalculationretainedNucleationFormulatunnelingNucleationCalculationtunnelingExpansionFormulamechanismResolutionexpansionPlausibilitypatchSetupsmoothingInputspatchCalculationenlargedPatchnoGlobalHomogeneitychargeSetupdilutionCalculationdilutedChargebaryonSetupbaryonCalculationbaryonQualificationmonopoleSetupmonopoleHorizonFormulastrongTransitionMonopoleRevisionFormulamonopoleCalculationmonopoleConclusionrateUncertaintyexitSetupintegratedNucleationApproximationselectedSpatialAveragingnearlyCompleteConvertedVolumepercolationExpectationpercolationCalculationvolumeConnectivityDistinctionpercolationOutcomeclusterCalculationexitCalculationclusterFailureinhomogeneousExitnonviableOriginalScenariomodificationSetupmodificationCalculationmodificationRequirementsconditionalRepairtriggerOpenparticleProductionOpenmodificationOpen
Step identifiers (66)
stateStandardSetupstateStandardCalculationretainStandardFlatnessFormularetainStandardHorizonFormuladeriveCausalDisconnectionassessFineTuningstateProblemIdentificationidentifyStandardProblemsstateEntropySetupstateEntropyCalculationderiveEntropyResolutionretainDisplacedHorizonstateVacuumSetupstateVacuumCalculationretainVacuumSourceFormuladeriveAcceleratedExpansionderiveReheatingEntropystateNucleationSetupstatePoissonBirthProcessstateNucleationIntegrationderiveNucleationSourceFormulastateRetainedNucleationSetupstateRetainedNucleationProcessstateRetainedNucleationCalculationderiveRetainedNucleationFormulastateTunnelingNucleationCalculationderiveTunnelingExpansionFormularealizeProblemResolutionassessExpansionPlausibilitystatePatchSetupstateSmoothingInputsstatePatchCalculationderiveEnlargedPatchrejectGlobalHomogeneityRequirementstateChargeSetupstateDilutionCalculationderiveDilutedChargestateBaryonSetupstateBaryonCalculationderiveBaryonQualificationstateMonopoleSetupretainMonopoleHorizonFormularetainStrongTransitionMonopoleRevisionFormulastateMonopoleCalculationderiveMonopoleConclusionretainRateUncertaintystateExitSetupstateIntegratedNucleationApproximationstateSelectedSpatialAveragingderiveNearlyCompleteConvertedVolumestatePercolationExpectationstatePercolationCalculationderiveVolumeConnectivityDistinctionderivePercolationOutcomestateClusterCalculationstateExitCalculationderiveClusterFailurederiveInhomogeneousExitassessOriginalNonviabilitystateModificationSetupstateModificationCalculationstateModificationRequirementsderiveConditionalRepairretainTriggerObligationretainParticleProductionOmissionretainModificationObligation
Route identifiers (8)
standardProblemsentropyResolutionfalseVacuumMechanismlocalPatchconservedQuantitiesmonopolesgracefulExitmodificationOutlook
Route kinds (8)
standardModelDiagnosisentropyCriterionphysicalInflationMechanismlocalInitialConditionArgumentconservedChargeQualificationtopologicalDefectConstraintviabilityObstructionunresolvedModification
Checked source-route details
This ledger is generated from the elaborated Lean route records. It lists direct predecessors, ordered inference steps, and exported claims. The independent fidelity audit checks every field; this ledger does not replace the physical semantics or certify every explanatory diagram.
standardProblems
Logical kind: standardModelDiagnosis.
Required predecessor routes: none.
Ordered inference steps: stateStandardSetup, stateStandardCalculation, retainStandardFlatnessFormula, retainStandardHorizonFormula, deriveCausalDisconnection, assessFineTuning, stateProblemIdentification, identifyStandardProblems.
Exported conclusions: flatnessFormula, standardFormula, causalDisconnection, fineTuning, standardProblems.
entropyResolution
Logical kind: entropyCriterion.
Required predecessor routes: none.
Ordered inference steps: stateEntropySetup, stateEntropyCalculation, deriveEntropyResolution, retainDisplacedHorizon.
Exported conclusions: entropyResolution, displacedHorizon.
falseVacuumMechanism
Logical kind: physicalInflationMechanism.
Required predecessor routes: entropyResolution.
Ordered inference steps: stateVacuumSetup, stateVacuumCalculation, retainVacuumSourceFormula, deriveAcceleratedExpansion, deriveReheatingEntropy, stateNucleationSetup, statePoissonBirthProcess, stateNucleationIntegration, deriveNucleationSourceFormula, stateRetainedNucleationSetup, stateRetainedNucleationProcess, stateRetainedNucleationCalculation, deriveRetainedNucleationFormula, stateTunnelingNucleationCalculation, deriveTunnelingExpansionFormula, realizeProblemResolution, assessExpansionPlausibility.
Exported conclusions: vacuumSourceFormula, acceleratedExpansion, reheatingEntropy, nucleationSourceFormula, retainedNucleationFormula, tunnelingExpansionFormula, mechanismResolution, expansionPlausibility.
localPatch
Logical kind: localInitialConditionArgument.
Required predecessor routes: falseVacuumMechanism.
Ordered inference steps: statePatchSetup, stateSmoothingInputs, statePatchCalculation, deriveEnlargedPatch, rejectGlobalHomogeneityRequirement.
Exported conclusions: enlargedPatch, noGlobalHomogeneity.
conservedQuantities
Logical kind: conservedChargeQualification.
Required predecessor routes: falseVacuumMechanism.
Ordered inference steps: stateChargeSetup, stateDilutionCalculation, deriveDilutedCharge, stateBaryonSetup, stateBaryonCalculation, deriveBaryonQualification.
Exported conclusions: dilutedCharge, baryonQualification.
monopoles
Logical kind: topologicalDefectConstraint.
Required predecessor routes: none.
Ordered inference steps: stateMonopoleSetup, retainMonopoleHorizonFormula, retainStrongTransitionMonopoleRevisionFormula, stateMonopoleCalculation, deriveMonopoleConclusion, retainRateUncertainty.
Exported conclusions: monopoleHorizonFormula, strongTransitionMonopoleRevisionFormula, monopoleConclusion, rateUncertainty.
gracefulExit
Logical kind: viabilityObstruction.
Required predecessor routes: falseVacuumMechanism.
Ordered inference steps: stateExitSetup, stateIntegratedNucleationApproximation, stateSelectedSpatialAveraging, deriveNearlyCompleteConvertedVolume, statePercolationExpectation, statePercolationCalculation, deriveVolumeConnectivityDistinction, derivePercolationOutcome, stateClusterCalculation, stateExitCalculation, deriveClusterFailure, deriveInhomogeneousExit, assessOriginalNonviability.
Exported conclusions: nearlyCompleteConvertedVolume, volumeConnectivityDistinction, clusterFailure, inhomogeneousExit, nonviableOriginalScenario.
modificationOutlook
Logical kind: unresolvedModification.
Required predecessor routes: gracefulExit.
Ordered inference steps: stateModificationSetup, stateModificationCalculation, stateModificationRequirements, deriveConditionalRepair, retainTriggerObligation, retainParticleProductionOmission, retainModificationObligation.
Exported conclusions: conditionalRepair, triggerOpen, particleProductionOpen, modificationOpen.