structureProof-free data for one selected anomaly-reduction calculation. The model retains the exact upstream current model and all six downstream relation families used by the route. No field carries a proof of anomaly, coefficient calculation, sector selection, phase test, exact symmetry, theory use, or physical representation.
inductiveThe shared claims transferred by the Witten and Seiberg-Witten adapters. The candidate-subgroup phase test and the obstruction of larger candidates are distinct claims. Condensates, mass deformations, spontaneous breaking, and vacuum multiplicities are downstream and are not constructors.
structureProof-free constraint selecting one exact anomaly-restriction model. It stores no route applicability, calculation result, bridge, or established endpoint.
structureTechnical data for identifying the exact finite quantum subgroup left by a current anomaly in a selected physical system. The current-level anomaly route is supplied by the independently earned `QuantumCurrentAnomalyContext`. The coefficient, anomaly-sector test, measure-phase tests, residual order, and exact finite-law predicate remain technical, paper-local data.
structureThe selected model constraint together with one typed paper-local derivation. The proposition-valued source relation stores no proof and keeps distinct anomaly-phase arguments from collapsing into one anonymous calculation.
structurePaper-local interpretation of an anomaly-reduction model. The embedded current-anomaly interpretation requires the upstream calculation to transfer to the same theory and be sound for the same represented world.
structureThe situated system, symmetry, and residual order explained by an anomaly-restriction route. Its fields are selected physical data, not proofs.
structureThe containing theory is distinct from the anomaly-reduction model.
structureThe represented world is distinct from model and theory.
structureThe situated anomaly-selected finite-vacuum phenomenon. It retains the two systems, regime, symmetry label, and four finite orders/counts, but no proofs.
structureThe composite calculation stores only the two selected systems, regime, symmetry label, and the four finite integers. It contains no proof of anomaly, selection, breaking, coset realization, or vacuum multiplicity.
inductive
structureProof-free constraint selecting one composite anomaly/vacuum calculation. It stores no anomaly, parent-selection, breaking, coset, route, or endpoint proof.
structureComposition data linking an anomaly-restricted symmetry context to a finite symmetry-breaking vacuum context. `parentSymmetrySelection` is a paper-local physical calculation or regime restriction. It relates the anomaly-surviving order to the exact finite parent order that actually undergoes spontaneous breaking.
structureThe selected composite constraint together with one typed paper-local anomaly-to-vacuum derivation. Its realization relation remains proposition valued rather than storing proof evidence in the constraint.
structurePaper-local interpretation of the composite route. The two embedded interpretations retain the potentially different anomaly and vacuum theories and worlds.
structureThe anomaly source theory and vacuum-regime theory are separate fields. Witten may choose the same paper-local theory for both; Seiberg-Witten does not.
structureThe represented anomaly and vacuum-regime worlds are likewise separate.
inductiveThe seven propositions exposed by one exact functional assignment route.
structureA technical context owns the paper-selected charge map, carrier, and regime in which physical carriers may be assigned charges. Admissibility and charge assignment remain relations: existence, graph realization, and uniqueness remain propositions to be established by a paper or calculation route.
structurePaper-local semantics relating a functional charge model to its containing theory and represented physical carrier. The exact assignment route does not determine these relations or the physical meaning of a charge claim.
structureA proof-free charge-assignment model retains the selected carrier and regime, the paper-local charge map, and both relation families used by the exact assignment route. No field carries admissibility, functional realization, sector occupancy, or well-definedness as a proof.
structureA paper-local theory object paired with the selected mathematical charge model. Its relation to that model is supplied by a `ChargeInterpretation`.
structureA paper-local world object paired with the selected mathematical charge model. Representation remains an explicit interpretation proposition.
structureProof-free data for one selected finite-symmetry vacuum calculation. The model retains the five relation families used by the counting route. No field carries a proof of exact symmetry, breaking, an order calculation, coset realization, or vacuum multiplicity.
inductiveClaims transferred by both paper adapters. An independent vacuum calculation is intentionally absent because Witten does not provide one in this route and Seiberg-Witten treats it as a separate later calculation.
structureProof-free constraint selecting one exact finite-symmetry vacuum model. It stores no symmetry, breaking, order, coset, count, route, or endpoint proof.
structureTechnical data for a finite exact symmetry broken to a residual subgroup in a selected physical regime. `independentlyCalculatedVacuumCount` records a route other than the symmetry-breaking count. It is relational so a paper must state which calculation establishes which number.
structureThe selected constraint together with one typed paper-local counting derivation. The source relation stores no proof and prevents Witten's chiral count and Seiberg-Witten's deformed two-vacuum count from becoming anonymous.
structurePaper-local interpretation of a finite-symmetry vacuum model.
structureThe containing theory is distinct from the finite-vacuum model.
structureThe represented world is distinct from model and theory.
structureThe situated finite-vacuum multiplicity explained by one complete counting route. Its fields retain selected physical and finite-order data, not proofs.
structureProof-free constraint selecting one exact functional charge-assignment model. It stores no route applicability, occupied sector, uniqueness proof, or established assignment.
structureThe selected charge-model constraint together with one typed paper-local derivation. The proposition-valued source relation stores no proof and keeps physically different charge computations from collapsing into one map.
structureThe situated carrier, regime, and selected charge value explained by a functional assignment route. Its fields are selected physical data, not assignment proofs.
structureThe mathematical quantity "charge of this carrier".
structureThe corresponding physical charge property of a carrier.
structureProof-free data selected from one protected-index calculation. It retains all seven selected objects and all nine relation families used by the shared route. It stores no proof that either description is valid, that the source is calculable, that the regimes are connected, or that the indexed value transfers.
inductive
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structureProof-free data identifying one exact source-calculation and protected- transport constraint. The source and target descriptions remain distinct roles even when a paper selects the same description at both ends.
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structureThe target phenomenon explained by protected transport: one indexed observable has the selected value in one valid target description and regime.
structureProof-free data for one selected current-anomaly calculation. No field carries a proof of setup, a completed quantum calculation, an anomaly, theory use, or physical representation.
inductiveExactly the current-level claims shared by all three paper adapters.
structureSelected system and current on the source side of an anomaly explanation.
structureTechnical data for applying the neutral current-anomaly semantics to a selected physical system and a paper-local quantum calculation.
structureA common anomaly constraint paired with one paper-local derivation.
structurePaper-local grounding of the selected technical model.
structureThe selected classically conserved current with its quantum anomaly.
structureThe containing theory is distinct from the selected calculation.
structureThe represented physical world is distinct from model and theory.
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structureA common invariant inner-product domain, not an assertion of essential self-adjointness.
structureOne normalized Hermitian charge and an actual fermion grading.
structureThe ordinary vacuum constraint plus one typed paper-local derivation. The derivation relation remains proposition-valued and independent of the source description, so common zero-energy output cannot identify two paper routes.
structureThe selected vacuum-energy calculation retains its physical regime, the two energy values being compared, and exactly the three relation families used by the shared routes. It stores no proof of a zero-energy state, spectral bridge, supersymmetry status, theory use, or physical representation.
inductiveOnly the repeated vacuum-energy claims transfer. Supercharge algebras, indices, Goldstini, instantons, and gauge-theory identities are not constructors.
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structurePaper-local meanings for a selected vacuum-energy calculation, the theory using it, and the physical system it represents.
structureThe containing supersymmetric theory is distinct from the calculation.
structureThe represented physical system is distinct from model and theory.
structureProof-free physical indices selected by the repeated zero-state route. The constraint stores no zero-state witness, spectral bridge, breaking criterion, or unbroken conclusion.
structureThe selected zero-energy, supersymmetry-unbroken phenomenon. Its occurrence still requires the route; the record itself stores no proof.
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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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structureKinematic data for the source Gauss-law argument, pp. 292-295. The quantum representation is not a constructed SYM Hilbert space. The topology, physical subspace and restriction to Cartan profiles must be physically identified separately. Weyl representatives are a function, not an assumed splitting of the Weyl quotient inside SU(N).
inductive
structureA vacuum selected by the proposed gluino-bilinear condensate.
inductiveThe residual chiral-symmetry order read from the selected state.
structureThe quantum discrete chiral law for pure `SU(N)` theory.
inductiveThe anomaly-free generator acting on the gluino condensate.
structureSection 2, pp. 255-256: a selected finite rotation representation, with no infinitesimal J_z.
structureLegacy extensions retain their own data. The periodic fields have one canonical declaration, inherited here rather than independently specified.
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structureThe physical regime in which the chiral-vacuum count is interpreted.
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inductiveSpatial-volume data separating finite calculations from vacuum limits.
inductiveTyped identity of Witten's full instanton-phase restriction of the classical chiral U(1) to its exact finite 2N subgroup.
inductiveTyped identity of Witten's full anomaly-selected finite-vacuum route: the instanton phase restricts the chiral symmetry to Z_(2N), identity selection uses that group as the vacuum parent, the gluino condensate leaves Z_2, and cosets count N vacua.
inductiveSection 8 joins the established instanton restriction to a separately hypothesized infinite-volume gluino condensate. Identifying the resulting vacua with the independently counted finite-volume states is an additional author expectation, not part of the index calculation.
inductiveTyped identity of Witten's section-8 finite-vacuum route: the instanton anomaly leaves Z_(2N), the gluino condensate breaks it to Z_2, and the cosets count N vacua. The token stores no proof of any of those steps.
inductiveTyped identity of Witten's finite-volume nonzero-index vacuum-energy route. The token stores no proof of index realization, nonvanishing, or the spectral bridge.
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inductiveTyped identity of Witten's instanton/ABJ chiral-current derivation.
structureThe `U`-generated zero-mode tower in section 8. Level `p` represents `U^p |Ω⟩`. It is the physical invariant tower only in the true-invariant vacuum branch; before that branch is selected it is a candidate construction.
inductiveTyped identity of Witten's Cartan-pair tower calculation used to assign the selected level its chiral charge.
structureSection 2, p. 255: actual operators on a common invariant domain and a joint spectral representation.
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structureSection 2, p. 255: four actual rest-domain charges and the one selected for index pairing.
inductiveElectric-flux sectors used in Witten's ordinary-periodic discussion: the zero vector and every nonzero flux vector.
structureAn ordinary electric-flux vector with at least one nonzero component. The paper's phrase "other sectors" ranges over these vectors, not over one undifferentiated nonzero tag.
structureA candidate complete energy spectrum with finitely many zero modes and paired positive levels. A physical spectral-basis representation, positive energies, and trace convergence are separate requirements.
structureThe actual coordinate, translated coordinate, box size, and wavefunction used in the periodic gauge argument on pp. 291-292.
structureThe data of the complete ordinary periodic argument. No twisted-sector, electric-flux, or chiral-vacuum extension is required to state this argument. The physical interpretation of these readouts and predicates is still an external obligation; this record stores no proofs or completed index answer.
structureA spatial momentum label and its excitation energy. Identification with the entire physical discarded sector is part of the imported reduction.
structureData only. The selected discarded spectrum may not be empty. Its completeness and the physical reading of cartanEnergy are external to this interface and remain in the low-energy quantization/trace identification.
structureThe alternative physical construction when `|Ω⟩` is pseudo-invariant. Witten's pseudo-invariant operator `V` has `r+1` spin components and spin `r/2`. Its reported common grading is separate data: the printed general formula and the rank-one discussion disagree, so geometry must not fix it.
inductiveThe sign of the empty Cartan Fock state under the residual Weyl action. The source does not establish which choice the full gauge theory realizes.
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structureData for one selected regulated trace, with no convergence or physical spectral-basis proof stored. The regulator has inverse-energy units.
structureThe reduced common domain is isometrically identified with the actual momentum kernel.
inductiveBoundary conditions compared in Witten's infinite-volume argument.
structureDimensionless spin factors for the three spatial cycles (p. 254). Common gauge/internal transitions are separate physical data, not these scalar factors. No quantum charge or compatibility proof is stored.
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inductiveElectric-flux sectors in the presence of the selected magnetic flux. The paper distinguishes every nonparallel flux vector from each of the `N` possible parallel `T_z` eigenvalues; undifferentiated direction tags would lose both sector-by-sector conclusions.
structureThe fixed-theta magnetic-flux orbit retains the winding of `(T_z)^N` and the separate contribution of every parallel electric-flux sector.
structureAn electric-flux value with at least one nonzero component transverse to the selected magnetic flux in the `z` direction. The `z` entry is retained because Witten's zero-index statement ranges over every nonparallel sector, not just over one transverse direction.
inductiveTyped token for Witten's periodic-index route: supersymmetric spectral pairing, the small-volume zero-mode calculation, and a deformation that preserves the large-field asymptotics.
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structurePrediction data together with paper-local relations recording admissible regime connection and protection of a selected observable.
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structurePhysical transformation and observable data need neither a selected vacuum family nor a variational principle.
structurePhysical breaking requires both interpretation and selected-family data, sharing exactly the same underlying law and state action.
structureLaw symmetry and admissible state action, without interpretation, vacuum selection, or variational claims.
structureA law-specific selected family adds no physical or descriptive classification of its representative action.
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Definitions and domain choices
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definitionDirect content of one completed anomaly-restriction calculation. The nested current-anomaly model must concern the same selected system and symmetry, and its current-level result must be nonvacuous. The coefficient, sector, order, positivity, both phase tests, and exact finite-law output then hold together directly on one proof-free model, without using context equality or counting an implication-valued residual bridge by itself.
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definitionThe proof-free constraint selected by one calculation model.
definitionThe selected constraint plus one typed paper-local derivation relation.
definitionThe proof-free situated phenomenon selected by one calculation model.
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definitionApplicability keeps the current-anomaly route, coefficient calculation, selected nontrivial sector, positive calculated order, both anomaly-phase tests, and exact-law interpretation bridge separate.
definitionThe selected paper system has a complete applicable current-anomaly route.
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definitionSource-sensitive structural explanation of one selected anomaly restriction. The source relation is independent of the completed finite-symmetry endpoint.
definitionOrdinary structural explanation of one selected anomaly restriction.
definitionThe output-led established result keeps exact context selection and the direct content of the completed anomaly-restriction calculation. The implication-valued residual-symmetry bridge and both technical routes are reconstructed from these outputs rather than retained as evidence.
definitionThe physical interpretation bridge from the established anomaly, coefficient and order calculations, selected sector, phase-invariance test, and maximality test to an exact finite quantum law.
definitionExact selected semantics is equality with the context-owned model, not a model reconstructed from a candidate's own three indices.
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definitionThe complete context-induced model at the three selected indices.
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definitionThe unique anomaly-restriction model fixed by the upstream context's selected system and current, this context's selected order, the exact upstream current model, and all six downstream relation families.
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definitionSource-sensitive explanation additionally requires one realized typed paper-local derivation while retaining the same ordinary technical route.
definitionOrdinary structural explanation treats the raw anomaly-reduction route as the necessitating constraint for the selected finite-symmetry phenomenon.
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definitionThe proof-free situated phenomenon selected by one composite model.
definitionThe proof-free exact-calculation constraint selected by one model.
definitionThe selected composite constraint together with one typed paper source.
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definitionApplicability exposes every logical stage. In particular, anomaly reduction, parent-symmetry selection, spontaneous breaking, subgroup arithmetic, and the coset-to-vacuum bridge are separate inputs. The final premise certifies that the finite-vacuum model is the one selected by its child context; supplying a different system, regime, or counting integer cannot select itself merely by being passed to the composite route.
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definitionSource-sensitive explanation of one anomaly-selected vacuum count.
definitionOrdinary structural explanation of one anomaly-selected vacuum count.
definitionThe output-led established result retains exact child selections together with the direct completed composite endpoint. Both child routes and the implication-valued parent-selection bridge are reconstructed from these facts.
definitionDirect physical content of the completed composite inference. Unlike the implication-valued parent-selection bridge used by the route, this endpoint requires the anomaly-restricted symmetry, the paper-local selection of the vacuum parent, and the proper-subgroup multiple-vacuum prediction to hold together directly. It therefore cannot be obtained from a correct vacuum count alone, nor from an anomaly result whose selected parent is never connected to the vacuum regime.
definitionInterpretation bridge from the established anomaly result and the paper-local selection calculation to the exact finite law and parent order used by the vacuum-counting context.
definitionThe paper-local calculation selecting the finite parent symmetry used in the vacuum regime from the anomaly-surviving quantum symmetry.
definitionThe exact selected composite model before anomaly reduction, parent selection, breaking, arithmetic, coset realization, or vacuum counting is supplied.
definitionExact selection of both child calculations used by the composite route.
definitionSource-sensitive composition retains independently selected anomaly and vacuum models, their exact identity with the models used by the composite calculation, the directly inhabited composite result, three separately sourced explanations, and a further paper-local joining derivation. The extra derivation is essential: an instanton-restricted chiral symmetry and a finite vacuum count do not determine whether the parent symmetry is reused unchanged under a conjectural gluino condensate, or restricted by a mass deformation whose isolated vacua arise at monopole and dyon singularities.
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definitionThe exact anomaly-restricted-symmetry model induced by the embedded anomaly context at the selected anomaly system, symmetry, and finite order.
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definitionThe exact finite-vacuum parent model induced by the embedded vacuum context at the selected system, regime, and three counting integers.
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definitionEvery shared technical assignment claim is grounded when model conclusions transfer to the paper-local theory and are sound for the represented carrier.
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definitionThe selected numerical charge is grounded when equality with the paper-local charge map transfers to the corresponding theory carrier and represented physical carrier.
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definitionThe proposition-valued semantics of every shared assignment claim.
definitionSource-sensitive structural explanation of one selected charge assignment. The source relation is independent of the completed assignment endpoint.
definitionOrdinary structural explanation of one selected charge assignment.
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definitionThe established output retains exact paper-local semantics together with the direct functional, occupied, and well-defined charge assignment. The route can be reconstructed from these outputs, but is not part of the definition.
definitionThe exact functional charge-assignment route retains the paper-local charge map whose graph realizes the relation.
definitionThe weaker existential statement that some paper-local charge function realizes the relational context.
definitionThe relational context is realized by paper-local charge data exactly when its assignment relation is the graph of one charge function on admissible carriers. This is an additional proposition, not part of the base context.
definitionAn admissible carrier occupies a charge sector when the charge-assignment relation assigns that charge in the selected regime.
definitionTwo carriers occupy the same charge sector when one charge is assigned to both admissible carriers in the same regime. This does not identify the carriers themselves.
definitionThe paper-local setup needed before the selected charge model can be used in a theory/world bridge: exact model selection and admissibility of the selected carrier. It does not supply functional realization, occupancy, or uniqueness.
definitionExact selected semantics is equality with the context-owned model. Functional realization of the selected map remains a separate route premise.
definitionThe grounded result retains the exact functional assignment route, selected sector occupancy, charge well-definedness, and both philosophy-layer transfer bridges.
definitionComplete grounding retains both the selected numerical charge and all seven technical assignment claims.
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definitionThe complete context-induced model for one selected assignment.
definitionOptional realization of relational charge assignment by paper-local charge data. The base framework remains relational; this constructor applies only when a paper independently supplies a charge function on admissible carriers.
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definitionThe unique charge model induced by the context-owned paper-local charge map, carrier, and regime together with both retained relation families.
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definitionInterpret every shared technical charge-assignment claim without identifying the selected charge model with its containing theory or represented carrier.
definitionRepresent one grounded charge assignment as a mathematical quantity and a physical quantity without identifying those roles. The adapter is anchored to the selected model and represented world. Mathematical assignment is the retained charge relation; physical possession is the paper-local `physicalCarriesCharge` relation.
definitionSource-sensitive explanation additionally requires one realized typed paper-local derivation while retaining the same ordinary assignment route.
definitionOrdinary structural explanation treats the raw exact functional-assignment route as the necessitating constraint for the selected charge phenomenon.
definitionFunctional assignment is interpreted through distinct model, theory, and world roles. A model entails a selected charge only when its carrier is admissible in the selected relational context and the charge equals the value returned by the paper-local map. Paper-local interpretation predicates determine model use, representation, theory consequence, and physical truth.
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definitionThe retained relation is exactly the graph of the retained charge map.
definitionThe direct charge-assignment output says that the retained relation is functional, the selected carrier actually occupies its selected sector, and the selected charge exists uniquely. This is intentionally stronger than merely having an applicable derivation route.
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definitionThe proof-free constraint selected by one charge model.
definitionThe selected constraint plus one typed paper-local derivation relation.
definitionThe proof-free situated phenomenon selected by one charge model.
definitionThe finite breaking route predicts more than one vacuum.
definitionDirect physical content of a finite broken-symmetry vacuum prediction. Unlike the context-selected established proposition, this endpoint drops exact record equality and requires the proper residual symmetry, the coset bridge, the exact count, and a genuinely multiple-vacuum conclusion on one model.
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definitionThe residual symmetry is a proper subgroup at the level of finite orders. Without this condition, a nominal breaking label could retain the full parent symmetry and predict only one vacuum.
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definitionThe proof-free exact-model constraint selected by one counting model.
definitionThe selected vacuum constraint together with one typed paper source.
definitionThe proof-free situated finite-vacuum phenomenon selected by one model.
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definitionThe paper-local physical bridge saying that the distinct vacua exhaust the cosets of the residual subgroup in the parent finite symmetry.
definitionSource-sensitive structural explanation of one selected finite-vacuum count.
definitionOrdinary structural explanation of one selected finite-vacuum count.
definitionThe semantic content of each claim for one selected calculation.
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definitionThe output-led established prediction keeps exact context selection together with the direct proper-subgroup, exact-count, multiple-vacuum result. The counting route is reconstructed from those physical outputs rather than retained as evidence.
definitionThe finite breaking pattern keeps exact law symmetry, spontaneous breaking, and the two group orders visible as distinct ingredients.
definitionConnection to the philosophy layer: a paper-local technical breaking predicate realizes spontaneous breaking only when it agrees with an explicit physical-symmetry context. Exact law symmetry is related separately so it cannot be inferred from a non-invariant state alone.
definitionThe selected model is exactly induced by the paper-local context, retaining all five route-used relation families.
definitionThe finite-symmetry counting route is applicable when the physical breaking pattern, nonzero residual subgroup, subgroup-index arithmetic, and coset-to-vacuum bridge are all available. This proposition contains neither the predicted count as an assumed conclusion nor an independent calculation.
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definitionThe complete context-induced counting model at the selected five indices.
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definitionThe unique finite-vacuum calculation selected by this context.
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definitionThe complete inhabited protected-index transport route, stated directly at one model instead of through the implication-valued transfer bridge. The proposition keeps both valid descriptions, source calculability, target selection, observable availability, the source prediction, admissible connection, protection, and target prediction together at the same physical indices. It does not identify Witten's index with Strominger-Vafa's elliptic genus growth, and it does not require source and target regimes to be definitionally distinct when a paper does not state that extra condition.
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definitionThe exact protected-transport constraint selected by this model.
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definitionThe exact situated target-index phenomenon selected by this model.
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definitionThe complete selected route structurally explains its target prediction.
definitionExact model selection plus direct, inhabited source-to-target transport.
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definitionThe framework route needs only the transfer calculation at the selected source, target, observable, and value. A paper may prove this calculation by establishing the stronger uniform `ProtectedTransferBridge`, but the shared route does not store that universal rule as if it were an output.
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definitionThe direct physical content of a completed current-anomaly calculation, without the context-equality certificate used at paper boundaries. The implication-valued bridge and criterion are not retained here: their selected outputs are the actual failed quantum conservation and identified anomaly.
definitionThe selected anomaly constraint paired with one paper-local derivation.
definitionThe exact anomalous-current phenomenon selected by this model.
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definitionSource-sensitive explanation of one selected quantum current anomaly.
definitionStructural explanation of one selected quantum current anomaly.
definitionThe output-led established result retains exact selected setup, the performed quantum calculation, failed quantum conservation, and anomaly identification. The implication-valued bridge, criterion, and complete route are reconstructed from these direct facts.
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definitionThe exact pre-quantum-calculation boundary: context-owned semantics and the selected physical system, current, and classical conservation law.
definitionExact selected semantics is equality with the context-owned model, not a model reconstructed from a candidate's own two indices.
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definitionThe complete context-induced proof-free model at two selected indices.
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definitionThe unique current-anomaly model fixed by the context's two selected physical values and six route-used relation families.
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definitionDirect content of an unbroken zero-energy vacuum. The selected ground-energy coordinate, actual zero-energy state, one-way spectral bridge, breaking criterion, zero-energy output, and unbroken conclusion all hold together on one proof-free model.
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definitionThe selected route constraint with one paper-local derivation.
definitionThe proof-free route constraint selected by one vacuum-energy model.
definitionThe proof-free unbroken-vacuum phenomenon selected by one model.
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definitionThe global-supersymmetry criterion identifying spontaneous breaking with nonzero ground-state energy.
definitionSource-sensitive structural explanation of the selected unbroken vacuum.
definitionOrdinary structural explanation of the selected unbroken vacuum.
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definitionThe selected model carries exactly the context's energy values and relation families. This certificate asserts no route premise or conclusion.
definitionExact selection of the proof-free vacuum-energy model at this regime.
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definitionThe paper-local spectral bridge from existence of a zero-energy state to the value of the ground-state energy.
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definitionThe unique vacuum-energy model selected by this paper-local context.
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definitionSource-sensitive structural semantics. The selected model, ordinary route, and zero-energy phenomenon stay fixed; satisfaction additionally requires one independently realized paper-local derivation.
definitionStructural semantics for the repeated zero-state route. Constraint satisfaction requires exact selected semantics and the complete route, while the phenomenon retains only the selected zero-energy unbroken-vacuum output.
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definitionConstruct the exact proof-free vacuum-energy model at one regime.
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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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definitionThe three real Cartan coordinates per direction, not spatial positions.
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definitionThe global invariance form of Gauss' law is an imported quantum constraint, not a derivation from infinitesimal generators. Only transformations joined to the identity are constrained; no theta sector is inferred.
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definitionBoth the actual identity-connected gauge transformations and their coordinate pullback are required. The minus sign is the inverse pullback for the active positive cocharacter shift.
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definitionThe explicit quantization/restriction bridge to the same exterior states and coordinate action used by the trace. This does not follow from abstract Gauss invariance or from the scalar legacy readout.
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definitionA nonzero state has creation number n precisely when every mode scaling acts with weight n. The nonzero clause excludes the zero vector, which otherwise satisfies every degree equation. Physical occupation-number identification still belongs to the fermion quantization interpretation.
definitionA real orthonormal frame spanning precisely the traceless diagonal Cartan space. It is an explicit mathematical condition, not an assigned flag.
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definitionExistence of a basis with these exact vectors expresses independence, nonzero candidates and completeness of the whole gauge-fixed space. It does not follow from the U/V support readouts in the legacy interface.
definitionThe complete gauge-fixed-space classification still needs equality with the full span. Only the separately requested linear independence is replaced by concrete nonzero U/V vectors, using their proved distinct weights.
definitionThe exact algebraic outputs used on p. 296, in the actual exterior algebra. The V-product coefficient depends on both spin tensors and may be zero; neither a component square nor coefficient one is asserted. General-rank nonvanishing, products and cubic exclusion remain explicit mathematical inputs, distinct from full Weyl-fixed-space completeness.
definitionThe imported classification is two explicit statements about the actual U/V vectors: independence and equality of their span with the full Weyl-fixed space. A number of components alone supplies neither statement.
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definitionThe two adjacent exchanges generate symmetry of the Cartan tensor. No invariance of a coefficient is inferred merely from calling it d.
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definitionAn inhabited incompatible restriction boundary, not a SYM model. Its trivial quantum action satisfies global Gauss invariance, but its nonzero constant profile is a nonsinglet under the actual SU(3) Weyl action.
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definitionEvaluation at any coordinate reverses the constant-profile embedding. This exact equivalence introduces no additional physical quantization input.
definitionThe existing exterior grading transported to the joint constant sector. This is not a new independently chosen parity operator.
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definitionThe selected constant-boson sector, inside the full joint constraint. Constancy is not inferred from Gauss' law; the source Hamiltonian and its physical zero-mode interpretation supply that reduction separately.
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definitionWitten pp. 292-296: periodicity and Weyl invariance constrain the total coordinate-dependent fermionic state. A nonconstant coordinate factor can compensate a fermionic Weyl character. No measure, L2 domain, energy equation, or full SYM gauge representation is supplied by this kinematic subspace.
definitionThe polynomial extension of the creation-number phase action. Unit scalars give phase rotations; arbitrary complex scalars are only an algebraic grading test, not a claim of physical unitarity.
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definitionScale just the second Pauli-spin component. Arbitrary complex scalars are an algebraic weight test, not a claim of physical unitarity or a new quantum spin observable. The one-particle labels are the same actual modes.
definitionA cocharacter-lattice shift of all Cartan coordinates. The basis is the same one used in the source Fourier Hamiltonian; its physical identification and nonsingularity remain separate inputs.
definitionThe literal antisymmetric spin contraction. Its normalization is not silently replaced by a chosen pair-count coefficient.
definitionThe Cartan epsilon contraction, with every spin index retained.
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definitionCoordinates of the actual permutation of N diagonal eigenvalues in the selected Cartan frame.
definitionThe real Cartan-coordinate action of the actual eigenvalue permutation. The source frame condition ensures its entries are real.
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definitionEight quarter turns are the identity; four need not be the identity on spinorial states.
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definitionWitten's caveat excluding states that enter from infinity in field space.
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definitionThe paper's physical bridge: twisting by a commuting spontaneously broken global symmetry costs nonzero energy in finite volume.
definitionThe imported section-8 anomaly calculation contains the current obstruction, the coefficient `2N`, a nontrivial instanton-sector phase test, invariance of the candidate `2N` subgroup, obstruction of larger candidates, the residual order computation, and the physical exactness bridge. It does not contain the later condensate hypothesis.
definitionThe paper-local bridge from the anomalous current, coefficient and order calculations, selected instanton sector, subgroup measure invariance, and phase obstruction of larger candidates to the exact residual quantum law.
definitionWitten's true-invariant branch choice is an author-qualified response to the unresolved Weyl-vacuum sign, not a consequence of choosing names for the states.
definitionThe additional zero-mode evidence Witten gives for the interpretation.
definitionThe current-level output of Witten's section-8 anomaly calculation, before the residual discrete subgroup is calculated.
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definitionThe Fourier-vacuum construction consumes the selected true-invariant branch, the computed zero-mode charge sequence, and the conditional vacuum count.
definitionThe large-volume identification remains an explicit author expectation that consumes both the finite-volume multiplicity and the conditional chiral count.
definitionThe checked chiral-vacuum count consumes the routed scope, anomaly, and residual subgroup separately before returning Witten's conditional N-vacuum count.
definitionThe source-facing chiral route keeps the anomaly and condensate bridges in the order Witten uses them. Pure N = 1 SU(N) scope identifies the classical chiral symmetry; that selected symmetry feeds the anomaly calculation; and the selected anomaly plus the conjectural nonzero condensate feed the residual twofold subgroup. These are imported physics bridges, not proofs of the anomaly or condensate dynamics.
definitionOne selected parallel electric-flux representative can have unit index while another does not. This finite model blocks replacing Witten's `every possible eigenvalue` conclusion by a single undifferentiated parallel tag.
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definitionBroken discrete symmetry gives one vacuum for each coset.
definitionThe source-facing section-8 flux contract follows the route's displayed dependencies instead of storing only their conclusions. Pure SU(N) scope supplies the two supersymmetry-commutation facts; the periodic gauge shift and bosonic zero-mode result feed the ordinary flux analysis; and the calculated twisted index and fixed-theta count feed the transverse and parallel twisted-flux facts. These implication bridges are imported physics from Witten's argument.
definitionSection 8's inputs for refining the index by electric-flux sector. Ordinary zero-mode wavefunctions are independent of the Cartan coordinates shifted by the flux operators. In the magnetically twisted calculation, the zero-energy states are transverse-flux invariant while the `T_z` orbit supplies every parallel-flux eigenvalue.
definitionThe deferred section-8 electric-flux calculation consumes both already calculated finite-volume indices and the paper-local flux data before returning its ordinary and twisted sector refinements.
definitionEvery periodic finite volume is connected to the calculable regime.
definitionEvery twisted finite volume is connected to its calculable regime.
definitionFinite-volume indices can differ between periodic and global twists.
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definitionThe twisting global symmetry commutes with supersymmetry.
definitionThe twisting global symmetry is spontaneously broken.
definitionThe global-symmetry twist has nonzero ground energy at every finite volume.
definitionThe global-symmetry-twisted finite-volume index vanishes at every volume.
definitionThe global-symmetry-twisted finite-volume index is realized by zero modes.
definitionWitten's QCD-inspired condensate hypothesis leaves the twofold subgroup.
definitionAt least the stated number of zero-energy states are present.
definitionAt least one bosonic or fermionic zero-energy state exists.
definitionThe index is constant along an admissible deformation.
definitionThe finite-volume index is literally the zero-state integer difference.
definitionWitten's expectation that the two boundary conditions have the same infinite-volume vacuum energy.
definitionThe flux-sector outputs and every premise needed to interpret them are present at one paper-local system. This keeps the sector refinement from being used as an implication with absent ordinary or twisted calculations.
definitionThe nonvacuous paper-local endpoint of Witten's ordinary-periodic calculation. The deferred zero-mode evaluation is consumed only after the pure N = 1, SU(N) scope, weak-coupling reduction, and full Cartan/Weyl calculation data are present at the same selected system and color rank. This does not prove the imported zero-mode quantization. It records the physical setup and its two signed index branches together, rather than letting the implication-valued evaluation count when its premises are absent.
definitionThe nonvacuous paper-local endpoint of Witten's twisted-sector calculation. The sector sum is meaningful here only together with the pure-theory scope and the twisted setup, zero-mode elimination, pure-gauge classification, one-state-per-sector result, and fixed-theta sector count that feed it. The proposition keeps the imported sector evaluation visible as a consumed hypothesis of the constructor below; it does not replace that calculation by an unexplained equality.
definitionA nonzero ground-state energy excludes both bosonic and fermionic zero-energy states. This is the imported spectral step used in Witten's example.
definitionThe checked ordinary-flux evaluation consumes each routed section-8 premise separately before returning the ordinary sector refinement.
definitionThe ordinary-periodic flux refinement keeps Witten's unresolved Weyl branch: the zero-flux sector carries the complete periodic index, while every nonzero electric-flux sector has vanishing index.
definitionOrdinary periodic boundary conditions can retain a nonzero finite index.
definitionSection 8's gauge constraint on the periodic Cartan zero modes. The large periodic gauge transformations identify the Cartan coordinates, while the Weyl transformations are identity-connected gauge transformations. Gauss' law then forces the two conclusions that the zero-mode coordinates are periodic variables and physical states are Weyl invariant. The two implications are derived from the typed action. Gauge geometry, global Gauss invariance and quantum/Cartan covariance remain physical inputs.
definitionThe source-facing section-8 evaluation contract consumes the routed weak-coupling, gauge, and zero-mode conclusions separately. This keeps the deferred Cartan-Weyl calculation from bypassing the displayed reduction, periodicity, Weyl-invariance, tower, parity, pseudo-branch, and unresolved sign steps when coverage certifies the ordinary periodic route.
definitionSection 2 inputs for every ordinary periodic finite-volume theory.
definitionBeta-independent heat-kernel control for every periodic finite volume.
definitionSection 8's candidate constructions with a retained reported grading. The magnitude conclusion is independent of the printed sign discrepancy.
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definitionThe section 8 approximation is asserted only at the weak-coupling calculation point, not at every finite volume.
definitionWitten's direct periodic finite-to-infinite-volume limit principle.
definitionExact outputs of the Cartan/Weyl zero-mode calculation that Witten performs. The technical quantization may be imported, but these outputs cannot be hidden.
definitionQuantizing the zero modes and imposing Weyl invariance selects either the `U^p|Ω⟩` candidate tower or the separate `V|Ω⟩` multiplet. This is the exact logical output of the deferred section 8 calculation before specializing `r+1=N`. The index-to-basis identification retains the unresolved proper gauge-measure obligation noted on p. 293; the expected measure independence is not derived by evaluating these finite sums.
definitionPositive-energy Bose-Fermi pairs first make the heat-kernel trace beta-independent and then cancel from that regulated trace.
definitionThe target theory is pure four-dimensional `N = 1` `SU(N)` SYM.
definitionThe reported ground energy is an infimum of actual normalized-state energies. Unlike the 1981 broken-vacuum criterion, this nonzero-index route need not assume that the infimum is already attained.
definitionSelects the signed section-8 branch transported by one use of the paper's argument. Witten leaves the true-invariant versus pseudo-invariant vacuum choice unresolved, so source-sensitive transport must expose this bridge rather than silently replacing the branch-valued calculation by one integer.
definitionThe selected energy is the full-domain infimum; attainment is not assumed.
definitionThe selected paper counts and integer readout refer to this computed spectral trace.
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definitionConditions for a supersymmetry-preserving finite-volume regulator.
definitionSupersymmetry is broken when the ground-state energy is nonzero.
definitionAn energetic diagnostic for p. 255, not a claimed SYM spectrum or a construction of Appendix A's theory. Every finite energy is positive.
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definitionThe former Boolean-only fixture now evaluates actual real energies and uses actual cofinal convergence. The Boolean parameter asks whether the energy is positive or zero; it does not assign either physical property.
definitionAn actual unit cubic box and periodic constant wavefunction at every stage. A changing label does not change this selected spatial geometry.
definitionA positive cofinal volume family in fixed reference units.
definitionThe checked section-7 sector-sum contract consumes the conclusions displayed by the source route separately. In particular, it cannot turn the imported sector evaluation into I = N without the pure-theory scope, both twisted finite-volume setups, zero-mode elimination, pure-gauge classification, one-state result, and fixed-theta sector count.
definitionThe checked twisted-flux evaluation separately consumes the calculated twisted index, supersymmetry commutation, transverse invariance, and the full parallel-flux eigenstate family before returning the twisted refinement.
definitionThe twisted magnetic-flux refinement: every nonparallel electric-flux sector has zero index, while each parallel electric-flux sector has one zero-energy contribution and zero ground-state energy.
definitionThe corresponding section 2 inputs for twisted boundary conditions.
definitionBeta-independent heat-kernel control for every twisted finite volume.
definitionSection 7 gives `Tr(-1)^F = N`.
definitionSection 7's classification and sector-counting outputs.
definitionSumming one bosonic contribution from each of the `N` fixed-theta sectors is the exact output of the deferred twisted-sector calculation.
definitionThe source-facing section-7 calculation contract keeps Witten's logical order visible. The magnetic twist first removes the gauge zero mode; that classification then supports the pure-gauge description, one bosonic state per perturbative sector, and the fixed-theta sector count. These implications are imported physics from the paper, not mathematical proofs of the classification.
definitionThe corresponding limit for the twisted finite-volume regulator.
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definitionTotal number of zero-energy states in a regime.
definitionA zero-energy state forces zero ground-state energy.
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definitionThe same approximate bosonic mode multiplicity, not new legacy data.
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definitionThe exact shared anomaly-restriction model selected by Witten's pure four-dimensional `N = 1`, `SU(N)` calculation.
definitionThe exact composite calculation selected by Witten's section-8 route.
definitionPaper-local current semantics for the classical chiral `U(1)`.
definitionThe exact shared model selected by Witten's section-8 chiral-vacuum route. It retains the paper's color rank, vacuum regime, `Z_(2N)` parent order, unbroken `Z_2` order, predicted `N` vacua, and every route-used relation.
definitionThe exact shared current-anomaly model selected at rank `colors`.
definitionPaper-local physical realization of the foundational distinction between an exact law symmetry and a condensate-selected non-invariant state. The anomaly-free generator advances the gluino-bilinear condensate phase by one around the `N`-element orbit. Admissibility fixes the regime and orbit; vacuum selection additionally requires the same color number as the law and the nonzero-condensate hypothesis. The residual order remains observable.
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definitionShared finite-symmetry view of Witten's conditional chiral-vacuum argument. The finite-volume index is not registered as an independent exact vacuum calculation because the paper proves only a lower bound before making the large-volume identification.
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definitionCompatibility view of the same positive cubic-box domain.
definitionCompatibility views of the single explicit regulated spectral readout.
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definitionWitten's exact proof-free vacuum-energy model at one selected regime.
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definitionThe paper-local index as a protected prediction. A connection from a calculable source to a target is represented by Witten's admissible deformation in the reverse argument direction, from the target theory to the calculation point.
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definitionCompatibility view, not a second assignable U-tower family.
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definitionThe shared protected-index route is interpreted as a calculation in four-dimensional `N = 1` SYM and as a prediction for one selected periodic finite-volume system. Physical representation requires the actual supersymmetry-preserving finite-volume setup; it is not supplied by the index equality itself.
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definitionThe shared technical-framework view retains the periodic or twisted description, the selected spatial volume, the calculable source regime, and the finite-volume target. Its validity and description predicates contain no index conclusion.
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definitionThe candidate `V` contribution using its reported common grading. This does not hardcode the inconsistent printed exponent or identify the readout with a physical rotation; `HasSpinParity` is the explicit latter test.
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definitionCompatibility view of the single canonical SU(N) rank definition.
definitionShared technical-framework view of the paper's vacuum-energy criterion.
definitionShared technical view of Witten's anomaly-restricted chiral symmetry.
definitionThe residual-order calculation remains in the same pure `N = 1`, `SU(N)` theory and represented gauge system as the grounded current anomaly.
definitionShared composite view of Witten's section-8 chiral-vacuum route. The finite parent symmetry used for vacuum counting is the anomaly-surviving symmetry itself, so the paper-local selection relates the color label to the chiral symmetry label and preserves the calculated order.
definitionWitten's anomaly source and vacuum regime remain descriptions of the same pure `N = 1`, `SU(N)` theory and world. The composite interpretation states that identity explicitly rather than building it into the shared framework.
definitionThe finite chiral-vacuum calculation is interpreted in the same pure four-dimensional `N = 1`, `SU(N)` theory and represented gauge system as the anomaly calculation, at one fixed color rank.
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definitionWitten's actual periodic finite-volume family and its infinite-volume energy. The finite family, supersymmetric regulator, uniform zero-energy property, physical convergence relation, and preservation bridge remain separate.
definitionShared current-level view of Witten's chiral anomaly calculation.
definitionWitten's anomaly calculation is interpreted only as a calculation in the selected pure four-dimensional `N = 1`, `SU(N)` theory and its corresponding physical gauge system.
definitionThe vacuum-energy model is used for Witten's selected pure four-dimensional `N = 1`, `SU(N)` theory. Exact model identity, theory scope, and represented-world identity are independent of the supersymmetry-preserving regulator, index realization, nonzero-index calculation, zero-state inference, and spectral bridge.
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definitionCompatibility view of the quantitative ordinary-periodic scale data.
definitionThe source relation retains the complete section-8 anomaly-restriction calculation rather than only its already-established finite-symmetry endpoint.
definitionThe source relation retains the full raw section-8 anomaly-to-vacuum route rather than only the completed composite endpoint.
definitionThe joining relation retains the three independent source derivations, the assumed nonzero gluino condensate, its residual Z2, the conditional N-vacuum count, and only an implication requiring a separate large-volume bridge.
definitionThe source relation retains the full raw section-8 route rather than only the already-established N-vacuum endpoint.
definitionThe 1982 source relation retains the realized finite-volume index, its nonvanishing, and the separate zero-state-to-ground-energy bridge. The zero-energy state itself remains a derived output of the index route.
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definitionCompatibility views of the canonical selected-spectrum multiplicities.
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definitionThe represented levels are p = 0,...,r. Successors, creation grading and the p. 297 normalized axial readout are derived. Actual nonzero U powers and top-power nilpotence belong to the separate operator-algebra input.
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definitionWitten's p. 297 convention: a creator has axial charge +1 and the empty true-invariant state has charge zero. This is a normalized candidate readout, not an exact quantum U(1) symmetry or an infinite-volume vacuum identification. Outside the selected tower it is only a label formula.
definitionShared technical-framework view of a tower's chiral charge sectors.
definitionThe exact charge map as a model of the selected invariant tower and one represented tower level.
definitionExact proof-free charge model selected by one tower level.
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definitionLevel p is U^p applied to the empty state. Its exterior creation number is derived from the actual two-creator U expression below, not stored as an arbitrary readout. Outside the selected tower this is only a label formula.
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definitionThe next label in the selected finite U-tower. None at or beyond its last label is a totalization convention, not a proof of U nilpotence.
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definitionThe source relation retains the raw pair-power calculation, SU rank relation, and selected in-tower level rather than the completed shared assignment.
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definitionAn additional symmetry input for rotated vacua, not a premise of the nonzero-index criterion.
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definitionSection 2, p. 255, with c = 1. This bounds spectral momentum, not only its expectation.
definitionOne finite bound for the entire selected path, not a different bound for every field or a pointwise-continuity claim. The coefficient is dimensionless, the offset has energy units, and the radius has field-size units. No numerical tolerance or unit energy is assigned by definition.
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definitionA rest-domain normalization comparison, not a statement about individual global charge squares.
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definitionGauss-law invariance identifies the entire translated coordinate orbit.
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definitionThe paper's periodic gauge transformation shifts `c` by `2 pi / L`.
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definitionThe three equal spatial periods in p. 284, equation (45), specify a cubic torus. This volume uses the same dimensionful L as the gauge shift.
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definitionExact geometric output needed before quantizing the Cartan modes. It quantifies over every represented periodic flat connection, not an arbitrarily selected nonempty subset or an assigned validity flag.
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definitionFields on the trivial untwisted SU(N) bundle in the faithful N-dimensional representation. Positive box length is required by the classification input.
definitionStrict periodicity is not periodicity up to the center. The classical classification does not assert that every such transformation is connected to the identity, or settle the quantum theta-sector interpretation.
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definitionThe action on Hermitian potentials corresponding to D = d - i A. For a unitary gauge, its conjugate transpose is its inverse.
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definitionThe p. 255 energetic limit with its mathematical meaning exposed: the reported volume is the cube of the same box length used in the calculation; positive physical volumes grow without bound along a proper filter, and the actual finite ground energies converge to the selected periodic energy. The SUSY energy origin is zero. This does not assert a limit of vacuum vectors or identify an energy-density limit with a total-energy limit.
definitionCanonical cofinal limit for the very boxes used by the finite-volume argument. Positivity is supplied by PeriodicFiniteVolumeIndexSetup, not requested again here. Alone this condition does not assert a physical setup.
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definitionThe two inference implications formerly requested here are now derived from the typed gauge action. Geometry, Gauss' law and the scalar compatibility readout remain assumptions; the latter is not the full joint constraint.
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definitionThe selected approximate Cartan Hamiltonian and box are the same on both sides of the reduction. This is not equality of the full quantum spectra: the physical quantization flag remains an external obligation. Proper measure, completeness and approximation validity remain in the imported quantization and physical trace obligations, not in scale separation alone.
definitionThe selected physical ground energy is the infimum of this same spectrum at every periodic volume. This identifies a readout; it neither stores zero energy nor constructs a physical SYM basis or self-adjoint Hamiltonian. The source's nonnegative spectrum and zero-mode argument are pp. 255-257.
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definitionAn explicit energetic limit premise, not a theorem about converging normalizable vacuum vectors or about a chosen thermodynamic topology.
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definitionThe input to the represented-state primary: the bosonic Fourier count, rank relation and U-label count are derived. This retains the actual fermionic, V, U/V operator and unresolved-character obligations.
definitionThe imported physical basis/trace identification, not a completed answer. The proper gauge measure and physical grading are not established by the sums.
definitionPhysical trace and period-lattice identification in the Cartan Fock frame. Rank identification comes from the calculation data, so its quantified transport is not vacuous under the premises. Proper measure, constant-boson reduction, completeness and regulated-index interpretation remain imported. The source state-classification premise is separate. The explicit restriction covariance relates quantum gauge transformations to the same joint action used by the trace; it is not inferred from abstract Gauss invariance.
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definitionThis bridge refers to zero-state counts with a physical interpretation; it does not establish that interpretation or ground-state attainment.
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definitionMode multiplicity of the selected approximate Cartan Hamiltonian, not an independent count family or a claim about the exact SYM zero-state counts.
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definitionThe source's large-field order criterion for the selected energy readout and path. Quantum spectral continuity and state exclusion do not follow from this classical/readout-level condition.
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definitionA positive real side length specifies the selected finite cubic torus. This is not an independently assignable finite-volume predicate.
definitionFinite multiplicity below every energy cutoff, not finite Hilbert-space dimension. Identifying this candidate spectrum with SYM remains physical input.
definitionThe same retained Cartan mode count determines the classical SU(N) matrix size. The calculation input identifies it with the source's N-1; the box is the one already used by the periodic and spectral data.
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definitionThe source's arbitrary-positive-beta regularization requires convergence at every positive beta. Discreteness and pair cancellation do not supply it. Zero-state counts are already views of this spectrum; the reported integer index still needs trace identification.
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definitionOriginal p. 258, Eq. (4), defines the integer index by the difference of the same zero-state multiplicities. It is not another reported family. Physical realization and convergence of the full trace remain separate.
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definitionThe candidate domain in pp. 296-297 is p=0,...,r. This view counts labels, not physical states: nonvanishing, independence and completeness still belong to the actual exterior-state classification.
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definitionReindex the already retained period basis by the source's Cartan-rank identification. This transports indices only; it chooses no new lattice.
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definitionSource gauge shifts on the same calculation lattice and box. The rank transport is inhabited for the selected SU(N) calculation by its fermionic-rank input. No gauge assertion is made for unrelated theories.
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definitionThe SU(N) Cartan rank in the source's diagonal traceless construction (p. 295) is N-1, not an independently selected readout. Natural subtraction is totalized at N=0; the physical theory scope separately requires N>=2. Identifying the retained modes with this Cartan algebra remains an input.
definitionQuantitative source approximation data, not three freely assignable scale propositions. The tolerance is a chosen interpretation budget, not a numerical constant supplied by Witten or an approximation-error theorem.
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definitionThe zero-state multiplicities belong to the same selected spectrum whose trace is used below (pp. 258-259), not independent reported families. Physical SYM identification still requires an operator representation.
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definitionOne retained physical nonzero-momentum excitation, not a complete SYM spectrum. It suffices to test the scale distinction without empty domains.
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definitionThe zero-coupling limit is a singular Fourier model, not a SYM theory.
definitionIn a traceless SU(N) frame, periodic gauge transformations have integer diagonal generators. The basis must generate the entire cocharacter lattice, not a proper sublattice or a rectangular guess. This does not refer to a center-quotient gauge group or identify all topological sectors.
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definitionA chosen quantitative perturbative budget. A tolerance below one is not by itself a theorem that Born-Oppenheimer errors are controlled.
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definitionFourier-mode multiplicity in the selected approximate bosonic Hamiltonian. It is not the cardinality of all scalar multiples of a wavefunction, nor an exact physical SYM ground-state count. Completeness and the measure remain part of the physical quantization/trace identification.
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definitionPeriod vectors in an orthonormal Cartan frame are (2 pi/L) B. Fourier momenta are L B^(-T) n, not spatial momenta 2 pi n/L. The period basis need not be orthogonal or the identity.
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definitionReference threshold printed on p. 291, not an independently stored gap.
definitionLiteral normalization of the printed Hamiltonian g^2/V sum pi_i^2, with V=L^3. Its relation to the full theory and measure is not proved here.
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definitionEquality of the reported fermion grading and the stated full-rotation sign. This is a separate physical interpretation constraint, not part of the printed component/spin calculation.
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definitionInvariance is imposed on the total bosonic-coordinate/fermionic state. It is not invariance of the bosonic scalar factor by itself.
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definitionThe second creator acquires the sign required by fermionic ordering.
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definitionA gauge-invariant profile need not have an even fermionic factor: its odd bosonic coordinate dependence can compensate the fermionic Weyl sign. This excited profile is not asserted to have zero energy.
definitionThe rank-one U contraction, with the source's antisymmetric spin normalization retained. Rank-one V components are the creators themselves.
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definitionThe source's G reverses both fermion modes. Its overall sign on the empty state is not selected here.
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definitionThe fixed counterexample is a countable paired spectrum with energy log(n+2), one bosonic zero mode, and no fermionic zero modes. Its heat trace exists exactly for beta > 1. No trace value is independently assigned.
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definitionThe grading's symmetry and involution are derived from physical rotations, not supplied here.
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definitionBoundary law for an ordinary untwisted scalar field component. This is a spatial shift by L, not a Cartan gauge-coordinate shift by 2 pi/L. Colored magnetically twisted fields also need their common gauge action.
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definitionThe reported minimum is the infimum of actual unit-norm state energies, not only of the labels. Attainment is not an input (journal pp. 255-257).
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definitionA different finite paired spectrum with no zero modes. This and the three-state spectrum are diagnostics, not identified SYM realizations.
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definitionAlternative (A) in section 2: zero index and no zero-energy states.
definitionThe value zero alone is compatible with both alternatives in section 2. This is a statement about the information carried by the integer index, not an assertion that both alternatives occur in one physical system.
definitionAlternative (B) in section 2: zero index with paired zero-energy states.
definitionAt least one zero-energy state is present for these multiplicities.
definitionThe index determined by bosonic and fermionic zero-energy multiplicities.
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definitionThe literal printed pseudo-invariant formula on p. 296, not a derivation from the stated spin `r/2`. With `rank(SU(N))+1=N` it is `(-1)^N N`. At rank one this disagrees with the alternative `-2` on p. 294. The spin-parity comparison below keeps this source discrepancy explicit.
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definitionThe bridge is the substantive preservation rule. Connection and protection labels alone do not manufacture a prediction in the target regime.
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definitionUniform finite-volume control does not itself prove that the property is closed under the selected thermodynamic limit. That closure is this bridge.
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definitionA concrete mathematical assignment whose state, quantity, and value all have the selected representation relations. This proposition deliberately stops before asserting that the physical quantity possesses the represented value.
definitionThe universal soundness law for assignments made through the selected quantity representation. Every application consumes the complete state and quantity representation setup. The law may still hold vacuously when no complete represented assignment exists; `FaithfullyRepresentsAt` separately requires a concrete witness.
definitionA selected mathematical quantity faithfully represents a physical quantity when it has a concrete value-bearing representation instance and every such assignment is physically sound in the stated system and regime.
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definitionThe selected mathematical and physical quantities have at least one complete state-and-value representation instance. Unlike an established represented value, this witness contains no physical-value conclusion.
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definitionThese local action requirements do not assert an entire group representation. They exclude missing or conflicting images and require preservation of the selected family for the transformation in question.
definitionSelection specifies the relevant vacuum, equilibrium, or other physical state family for this law. It is not arbitrary kinematic admissibility and does not pick just one member of a degenerate family.
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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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inductiveEach row blocks a tempting but invalid strengthening of Witten's argument: zero index is not a breaking diagnosis, spectral data do not manufacture regularization, supersymmetry alone does not control large-field deformations, gauge translations require quotienting, the actual `U,V` operator algebra cannot be weakened, the periodic Weyl branches retain their parity-dependent signs, fixed-theta magnetic sectors require unit winding, and finite-volume or one-sector statements cannot be silently universalized.
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definitionThe chiral-vacuum interpretation remains downstream of the ordinary calculation and retains its conjectural and author-qualified ingredients.
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checked statementGiven the thirty-two raw paper-local inputs, the Witten source map, independent ledgers, complete semantic coverage, and typed boundary matrix form one checked fidelity certificate. The route graph is reconstructed from those inputs here; no completed route certificate is assumed.
definitionThe general finite-volume boundary-condition warning is intentionally independent of the specific magnetic-flux computation.
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definitionThe ordinary electric-flux refinement inherits the periodic calculation.
definitionThe ordinary periodic route to the paper's main no-breaking conclusion. This hand-written ledger is intentionally independent of rawSteps.
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definitionThe twisted electric-flux refinement inherits the sector calculation.
definitionThe independent twisted-boundary route retains Witten's explicit boundary-condition-independence expectation at the final infinite-volume step, rather than smuggling it into the finite-volume calculation.
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checked statementRaw paper-local inputs construct every Witten 1982 route in order.
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checked statementNo completed route certificate is needed to cover a route identifier.
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checked statementSection 8's anomaly, branch-selection, Fourier, and large-volume bridges enter as explicit source hypotheses rather than consequences of route certificates.
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checked statementSection 8's general boundary-condition warning is not a consequence of the particular SU(N) periodic or magnetic-flux index calculations.
checked statementThe finite-boundary warning keeps the broken-symmetry twist-energy bridge as an explicit source contract and applies every later step to its declared premises.
checked statementThe section-8 flux bridges and the two branch-specific sector evaluations enter as explicit source hypotheses.
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checked statementThe foundational route introduces paper-local setup, imported principles, and scope qualifications as independent inputs. Only the later regularization, index-definition, transport, and zero-energy steps apply those inputs.
checked statementBoth index-definition routes consume the beta-independent regulated trace rather than treating regularization as an unused neighboring claim.
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checked statementThe section-8 flux refinement and qualified chiral interpretation are downstream of the ordinary index result; neither feeds the main no-breaking conclusion back upstream.
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checked statementThe section-8 weak-coupling, gauge, zero-mode, and branch-evaluation packages are explicit source hypotheses rather than hidden theorem parameters.
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checked statementThe beta-independent trace is derived through the positive-energy pairing principle in both finite-volume branches.
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checked statementThe all-volume regulator, section-7 classification package, and checked sector-sum evaluation are explicit source hypotheses rather than hidden coverage parameters.
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Checked statements: parameters, hypotheses, and interpretation
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checked statementA genuine fixed exterior-state vector can be given duplicate component labels. Nonzero labels alone do not justify a state count; this is a Fock diagnostic, not a Yang-Mills spectrum or a classification theorem.
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checked statementAlready at rank one, a nonzero V component has zero self-product. This does not refute the source's spin-resolved proportionality statement.
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checked statementFor these explicitly constructed U/V vectors, not arbitrary labels, linear independence is exactly the remaining pointwise nonzero condition.
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checked statementNonvanishing already belongs to the actual source-algebra input. The only further classification input is equality with the full fixed space; containment alone would not exclude additional physical states.
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checked statementBoth actual source branches have distinct second-spin occupations. The V components have the same total fermion number but different weights under this component scaling, so total fermion number alone is insufficient.
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checked statementRemoving the quartic term is source-order bounded at large fields. This directional property is not preservation of the asymptotic degree.
checked statementThe exact source energy obeys the directional growth criterion on the reverse path, while the previously computed normalizable kernel index changes. No self-adjoint closure, regulated trace, or full SYM realization is inferred.
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checked statementIn units with m=1, switching on g changes the large-field degree. Pointwise continuity of this exact source potential is not an order bound.
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checked statementWithout the physical `N ≥ 2` scope, the quotient orbit can collapse: for `N = 1`, the modular generator fixes the only phase. This is why the shared spontaneous-breaking realization requires the paper's `SU(N)` scope.
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checked statementThe converse fails on a nonempty cofinal family with pointwise positive energies, not a volume-independent positive lower bound.
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checked statementSection 8: the anomaly restriction, identity selection of the surviving Z_(2N) parent, and proper Z_(2N) -> Z_2 vacuum splitting are jointly active in the shared composite endpoint.
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checked statementWitten's typed anomaly-condensate-coset route source-sensitively explains the selected composite multiplicity without consuming the ordinary explanation or established endpoint.
checked statementThe raw section-8 anomaly, identity selection, condensate, and coset-counting inputs supply the ordinary composite explanation without consuming the established composite endpoint.
checked statementThe typed Witten source relation reconstructs the raw composite route without using its established endpoint.
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checked statementInteger arithmetic turns `|I| = N` into a zero-state multiplicity bound.
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checked statementThe raw section-8 anomaly calculation realizes the ordinary shared structural explanation before any paper-source tag is retained.
checked statementThe named model retains every relation supplied by the paper context.
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checked statementThe typed Witten source relation reconstructs the raw shared route without using the completed residual-symmetry endpoint.
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checked statementWitten 1982, p. 269: the minimal supersymmetric gauge theory's naively conserved chiral current has an anomalous nonzero divergence.
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checked statementFor physical `SU(N)` with `N ≥ 2`, the anomaly-free generator sends the zero-phase condensate vacuum to the distinct phase-one vacuum.
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checked statementSection 8: the `Z_(2N)` to `Z_2` breaking route gives a proper residual symmetry and therefore a genuinely multiple-vacuum prediction for `N ≥ 2`.
checked statementWitten's typed anomaly-condensate-coset derivation source-sensitively explains the selected finite-vacuum multiplicity without consuming the ordinary explanation or established endpoint.
checked statementThe raw section-8 anomaly, condensate, and coset-counting inputs supply the ordinary shared structural explanation without consuming the packaged N-vacuum endpoint.
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checked statementThe typed Witten source relation reconstructs the raw finite-vacuum route without using the completed multiplicity endpoint.
checked statementThe technical chiral-breaking predicates are exactly the selected realization of foundational spontaneous breaking.
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checked statementWitten's original source yields the fully sourced anomaly-to-vacuum composition only conditional on the separately supplied gluino-condensate hypothesis. It neither establishes that hypothesis nor asserts the distinct finite-volume-to-infinite-volume identification.
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checked statementThe ordinary zero-energy states are in the zero electric-flux sector because the flux operators shift Cartan coordinates while those wavefunctions are independent of the shifted coordinates.
checked statementThe section-8 flux refinement is nonvacuous only after both ordinary and twisted index calculations have already been supplied at the same pure `N = 1`, `SU(N)` system.
checked statementThe physical `SU(N)` scope makes the ordinary nonzero, twisted nonparallel, and twisted parallel families inhabited. The universal sector conclusions therefore exhibit actual representatives rather than passing vacuously.
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checked statementWitten's chiral-vacuum reading is a checked conditional interpretation: the index supplies the finite-volume multiplicity, while the anomaly, condensate, symmetry-breaking count, charge evidence, and large-volume identification remain separately visible.
checked statementSelect the exact finite-volume regime before the zero-state route runs.
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checked statementEquality of the protected index values is the exact transfer rule; connection and protection remain separate premises.
checked statementThe uniform equality rule realizes the pointwise transfer calculation used by the shared technical framework. The established target result will retain the direct source and target index predictions, not this implication.
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checked statementThe checked conclusion of Witten's ordinary-periodic route.
checked statementThe twisted conclusion with its sector calculation fully exposed.
checked statementThe ordinary conclusion with the complete section 8 calculation exposed.
checked statementThe realized nonzero finite-volume index and spectral bridge realize the source-sensitive shared explanation without identifying this route with Witten's 1981 Hamiltonian-square derivation.
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checked statementEvery nonzero ordinary electric-flux vector has vanishing index.
checked statementAll ordinary zero-energy states lie in the zero electric-flux sector, so that sector retains the same unresolved `N` versus reported common-grade branch as the full periodic calculation. Every nonzero electric-flux sector remains at index zero.
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checked statementThe two gauge-theoretic consequences used by the zero-mode calculation are actual deductions from the explicit section-8 constraint data.
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checked statementThe same routed gauge constraint now retains the numerical `2 pi / L` translation and invariance of the physical wavefunction on its gauge orbit.
checked statementNormalize the imported graded sums after their actual counts are supplied. The U-tower even grading is derived rather than independently supplied.
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checked statementPairing plus periodic finite-volume data gives Witten's beta-independent heat-kernel regulator at every volume.
checked statementThe complete section-8 setup supplies graded sums, not completed signed answers. The disputed printed exponent is unnecessary for the subsequent magnitude, state-existence, and no-breaking conclusions.
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checked statementWitten's original journal sec. 2, pp. 257-260, defines the finite-volume index and preserves it under gentle parameter changes that leave the large-field asymptotics fixed and exclude states entering from infinity. The paper's later small-volume analysis supplies the calculable source. This adapter records only the shared direct transport conclusion after those paper-local inputs have been consumed.
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checked statementWitten's source calculation and large-field-asymptotics-qualified invariance route structurally explain the selected finite-volume index value. The phenomenon is only the situated target index prediction; no supersymmetry- breaking conclusion is added by this shared projection.
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checked statementEvery periodic regulator is zero-energy, and the family really has a member.
checked statementZero energy passes to the physical limit only through Witten's limit premise.
checked statementWitten's selected finite-to-infinite zero-energy inference in shared logic.
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checked statementThe routed calculation retains actual products with every spin slot, not the legacy support table. General-rank algebraic results are imported.
checked statementThe complete section-8 setup and its branch-valued answer hold together at one paper-local system. This is the direct logical endpoint used downstream; the imported evaluation remains a premise rather than a stored conclusion.
checked statementThe same `U^p|Ω⟩` tower used in the index calculation has exactly `N` levels, all of even relative fermion number, with chiral charges `2p`.
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checked statementThe pseudo-invariant candidate has `N` components and spin `(N-1)/2`. Its contribution uses the reported grading rather than a hardcoded sign; physical spin grading and the printed sign remain separate readings.
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checked statementWitten's typed instanton-phase derivation realizes the source-sensitive structural explanation without consuming the established shared endpoint or the ordinary explanation.
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checked statementMatching finite-volume states to condensate vacua consumes its own bridge.
checked statementThe composition exposes the conjectural condensate rather than proving it.
checked statementSection 8, p. 297: the selected condensate orbit uses the same color number as its law. Closure is a property of the modular map, not a proof of the condensate conjecture.
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checked statementRecovering the literal p. 296 formula requires its extra grading assignment. It is not inferred from geometry or assumed by the ordinary no-breaking route.
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checked statementThe magnetically twisted sector calculation keeps Witten's full directional and value-by-value distinction visible: transverse sectors have zero index, while every one of the `N` parallel sectors has unit index and zero ground-state energy.
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checked statementPairing plus twisted finite-volume data gives the corresponding beta-independent heat-kernel regulator at every volume.
checked statementThe section 7 sector sum gives the exact positive value `N`.
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checked statementThe complete section-7 twisted setup and its exact I = N answer hold together at one paper-local system. In particular, the sector evaluation cannot be used as a free-standing equality without its classification data.
checked statementSection 7's fixed-theta orbit carries unit winding after `N` applications of `T_z`; this is explicitly not the trivial gauge transformation.
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checked statementThe source chain really performs the zero-mode evaluation before selecting the signed value used by protected transport.
checked statementThe typed source exposes both parts that an endpoint equality would erase: spectral cancellation realizes the regulated finite-volume index, and the selected deformation separately satisfies all three of Witten's asymptotic safety conditions.
checked statementWitten's raw source chain constructs the shared protected-index route.
checked statementWitten's pairing, zero-mode evaluation, and asymptotically gentle deformation realize a source-sensitive explanation of the transported finite-volume index.
checked statementWitten's source-sensitive protected transport, a nonzero transported value, and the separate zero-state spectral bridge together retain the full index-to-vacuum chain. The result keeps both source explanations, rather than forgetting the zero-mode/deformation provenance when entering the vacuum-energy layer.
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checked statementThe raw Cartan-pair tower derivation realizes the source-sensitive selected charge explanation without consuming the established assignment or ordinary explanation.
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checked statementThe canonical readout is backed by the actual nonzero U states in the imported complete physical classification, not just a label count.
checked statementThe p. 297 normalized axial readout is realized by the same nonzero U states and exterior phase weights as the creation number. This does not upgrade the anomalous classical U(1) to an exact quantum symmetry.
checked statementThe true-invariant branch uses the derived U-power creation numbers. Even parity is no longer a separate input.
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checked statementComparison with freely reported legacy functions is restricted to the source's represented levels and endpoint. No equality of unrestricted old records, or of readouts beyond the tower, is asserted.
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checked statementExact readback of the former source constraints using the derived functions. The physical meaning of the chosen axial normalization and the later chiral-vacuum interpretation are not proved by these label equations.
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checked statementThe derived label agrees with actual multiplication by U at every nonterminal source level. Top-power nilpotence remains in the independent operator-support contract, not in this label theorem.
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checked statementOriginal source: section 8, p. 297. The states `|Omega>, U|Omega>, ..., U^(N-1)|Omega>` carry chiralities `0, 2, ..., 2N-2`. For any selected level that lies in the tower, the paper-local charge map therefore has an actual occupied sector.
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checked statementThe grounded chiral-charge map faithfully represents the selected tower level's physical chiral-charge quantity. The conclusion uses the explicit paper-local world transfer and is not obtained from the numerical map alone.
checked statementThe raw exact functional-assignment route explains the selected tower-level chiral charge before the established assignment is consumed.
checked statementWitten's typed Cartan-pair source reconstructs the raw shared assignment route without using the completed assignment endpoint.
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checked statementPositivity of these existential witnesses is a presentation choice, not three additional physical assumptions on the energy family.
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checked statementA convergent positive-beta trace already excludes infinitely many states below a fixed energy cutoff. This is a consequence of the selected trace domain, not an additional finite-volume physics input (p. 258).
checked statementOriginal p. 258, Eq. (4): the convergent, unnormalized parity trace admits precisely the integer difference of the same zero-state counts. This compares readouts; it does not identify a candidate spectrum with SYM.
checked statementThe p. 257 nonzero-index inference supplies a genuine zero-energy label. Its being a physical eigenstate still requires spectral identification.
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checked statementSpatially constant, periodic Hermitian traceless potentials can carry nonzero magnetic curvature. Commuting constants, not constantness alone, are needed for the classical zero-energy sector.
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checked statementThe same SU(N) counting magnitude, now derived from an explicit U/V state classification and an actual gauge-fixed parity trace. The printed pseudo-invariant exponent is not silently used as exterior grading.
checked statementThe full general-N route now exhibits at least N orthonormal normalized ground states at every finite box (p. 297), not just one vacuum. The single Hermitian charge suffices for the source's index argument (pp. 255-257); identifying it with physical SYM remains an input. The cofinal conclusion is energetic, not convergence of vacuum vectors.
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checked statementThe trace equation is now derived from the canonical spectral index. Positive regulator and convergence for every positive beta are retained.
checked statementThe classical matrix representation and quantized Cartan modes refer to the same SU(N), not two independently selected color counts.
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checked statementThe number of labels agrees with physical-space dimension only under the complete classification of the actual U/V vectors (pp. 296-297). This does not identify the approximate exterior space with the SYM Hilbert space.
checked statementThe old closure premise is derived from actual proper convergence. The reverse implication is not asserted: an arbitrary closure predicate need not describe any positive, cofinal family of physical volumes.
checked statementA kernel-checked readable characterization: neither "large volume" nor "convergence" is supplied by an uninterpreted relation in this refinement.
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checked statementExact consolidation under the existing physical box setup. The older unmatched volume predicate is not asserted equivalent: matching is a correction.
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checked statementThe complete ordinary periodic argument with its generic U/V states, actual SU(N) Weyl-fixed parity trace and proper cofinal energy limit exposed. Every original scope, gauge, calculation, transport and energy input remains; the old bare evaluation is replaced by frame, state-classification and trace identification inputs. No state-classification or proper-measure proof is claimed, and the conclusion is energetic rather than a vacuum-vector limit.
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checked statementThe complete SU(2) periodic argument with the source projector in place of the abstract U/V graded-sum evaluation. Every transport, cancellation, energy, and infinite-volume obligation is retained. This does not construct the physical SYM Hilbert space or settle the unresolved Weyl character.
checked statementThe complete periodic no-breaking implication with all eleven original premises, without any twisted, flux, chiral, or convergence-record dependency.
checked statementThe same complete ordinary SU(N) argument with the p. 255 limit replaced by its normalized proper, cofinal real-energy interpretation. No reduction, gauge, U/V, regulated-counting, or deformation premise is omitted.
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checked statementBoth possible common grades yield the same nonzero magnitude.
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checked statementThe common inference after any correctly identified nonzero calculation. Transport, the count/trace relation, and the energy bridge remain explicit.
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checked statementThe same transported index and regulated spectrum now determine the identified ground energy. No separately supplied count-to-energy implication is used. The conclusion is attained in the candidate label spectrum only; physical state realization remains part of the interpretation obligation.
checked statementThe full finite-volume route, independent of all legacy extension data. The calculation supplies finite graded sums, cancellation identifies their transported index with counts, and the explicit count/energy bridge gives zero.
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checked statementThe source's SU(2) count is obtained from an actual four-state projector. The physical trace identification includes the low-energy approximation, constant bosonic zero state, gauge quotient, and proper measure; it is not established by the finite matrices. No printed general-N sign is substituted.
checked statementThe source's one constant bosonic Fourier mode is derived from (70), positive coupling and an invertible period lattice. Physical state-space completeness, proper measure and approximation validity are not proved.
checked statementLiteral readable specification of the complete reduction input. The classical geometric classification and quantization remain explicit external inputs; the numerical scale content is not supplied by predicate names.
checked statementThe actual source scale argument at a selected finite Fourier cutoff. Not every spatially constant Cartan level is below the hard threshold.
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checked statementTrace cancellation and protected transport identify the full integer index at every periodic volume (pp. 257-260), not only its nonvanishing. The same-spectrum and convergence requirements remain in the setup.
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checked statementThe products are equations between the source's actual U/V operators, not freely assigned support labels. The spin-resolved V coefficient is quantified independently for every pair and may vanish.
checked statementWithin the actual reduction, the rank, bosonic mode and U-label count clauses are derived. Every remaining quantization, operator and character input is retained; in particular this does not identify chiral charge with spin by definition.
checked statementExact readable trace contract including its quantum/Cartan covariance. That relation exposes content formerly implicit in physical identification; this is not equivalence with arbitrary former gauge-predicate assignments. The calculation input supplies a rank identification, preventing a vacuous universal quantifier. The joint constant profiles use the same box, lattice, Weyl character and exterior grading.
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checked statementExact comparison with separately reported counts at a selected regime. Only reports consistent with that spectrum are retained. This is not an equivalence with unrestricted records that permit inconsistent counts.
checked statementExact preservation of the older separately reported index contract on consistent reports only, not equivalence with arbitrary old records.
checked statementOn a physically represented spectrum, neither a second integer readout nor a trace-value equation is an independent input. Boundary, representation and all-positive-beta convergence still require scrutiny.
checked statementOn the actual represented spectra, the older four-part setup is equivalent to boundary conditions and trace identification alone. Positive pairs come from representation; finite sublevels follow from convergence. No converse from discreteness to heat-trace convergence is claimed.
checked statementSpatial periodicity is explicit in addition to, not a replacement for, the unresolved common gauge/internal boundary and quantum SUSY obligations. This corrects the older setup that had no formal spin-phase condition.
checked statementExact comparison at the selected color count only. An arbitrary reported tower is admitted precisely when its length matches this domain; no equality of unrestricted legacy families or physics models is asserted.
checked statementExact comparison with the formerly arbitrary rank readout, on its positive-color domain. No equality of unrestricted records is asserted.
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checked statementThe p. 297 multiplicity consequence uses |b-f| <= b+f and the same represented Hilbert basis. The chosen states are orthonormal zero modes, not repeated labels. The bound allows additional cancelling vacua and does not identify finite-volume vectors with infinite-volume chiral vacua.
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Axioms and trust boundaries
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Constructors, projections, and other meaning dependencies
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Every physical statement, equation,
dependency, qualification, and locator in this edition is taken only
from WittenIndex1982.lean,
WittenIndex1982/SourceMap.lean, and
WittenIndex1982/Coverage.lean. In the foundational
finite-volume route, raw setup and imported contracts are
premise-free source inputs; beta-independent regularization, index
realization, deformation transport, and zero-energy consequences are
rebuilt from checked premise maps. The ordinary-periodic,
twisted-boundary, finite-boundary, electric-flux, and chiral-vacuum
routes now expose their imported calculation packages or physical
bridges as explicit source contracts and rebuild each displayed
consequence from checked premise maps.
01
The index argument has a foundation, two computations, and one interpretation
The formalization first establishes what a finite-volume
supersymmetric index can prove. It then branches into an ordinary
periodic zero-mode calculation and an independent twisted-sector
calculation. Both finite-volume routes produce a nonzero index, but
they enter the infinite-volume conclusion through different
premises. Only the periodic calculation feeds the later
chiral-vacuum interpretation. A separate route records why
finite-volume boundary-condition equality cannot be inferred from
the large-volume expectation.
Finite boundary-condition scopebroken global-symmetry twist · nonzero twist energy · zero index
Interpretive extension from the periodic routeat least N finite-volume ground states · anomaly · condensate conjecture · N chiral vacua
Figure 1. The six source-mapped routes. The two computations share
the finite-volume index logic but not each other’s calculational
premises. The boundary-scope route is independent of both
calculations; the chiral interpretation depends on the periodic
route.
Shared physical contract
The theory is pure four-dimensional 𝒩 = 1, SU(N) Yang–Mills with color number N ≥ 2.
The finite-volume regulator preserves supersymmetry and gives bosons and fermions matching spatial boundary conditions.
The finite spectrum is discrete, the graded trace defines the index, its heat-kernel regularization is β-independent at every relevant volume, and positive-energy states occur in Bose–Fermi pairs.
Deformations used to reach a calculable regime preserve supersymmetry, obey the selected source-relative large-field bound, and exclude quantum states arriving or disappearing at infinity along that same path.
Lean checks the deductions among explicitly named propositions. It
proves the integer consequences of the index and the dependency
structure of the routes. Zero-mode quantization, twisted-sector
classification, anomaly calculation, large-volume bridges, and
interpretive physics remain visible as theorem hypotheses with
precisely stated outputs.
02
Physical system, finite-volume regimes, and notation
The paper-local system separates physical regimes from claims about
them. It stores an integer index, bosonic and fermionic zero-energy
multiplicities, finite- and infinite-volume ground energies, and the
data produced by the two calculations. Statements such as
deformation invariance, weak coupling, or vanishing ground energy
remain propositions rather than proof fields in the data structure.
Boundary conditions
Lean case
Physical role
periodic
Ordinary spatially periodic regulator used for the direct physical limit
twisted
Magnetic-flux-twisted regulator used for the independent sector calculation
globalSymmetryTwisted
Separate section-8 counterexample: periodic only up to a broken global symmetry
Selected theory
Datum
Lean meaning
Gauge theory
Pure 4D 𝒩 = 1 SU(N) Yang–Mills
Color scope
N ≥ 2
Breaking criterion
Ground-state energy differs from the designated zero energy
Regimes kept distinct by the source map
Stage
Purpose in the argument
Infinite-volume theory
The physical conclusion about dynamical supersymmetry breaking
Finite periodic theory
An arbitrary ordinary supersymmetric finite volume
Weak-coupling periodic theory
The calculable periodic source regime
Periodic zero-mode theory
The effective low-energy Cartan/Weyl quantization
Finite twisted theory
An arbitrary twisted supersymmetric finite volume
Calculable twisted theory
The source regime for the sector-counting calculation
03
The finite-volume index detects an unpaired zero-energy state
In a supersymmetric finite volume, every positive-energy bosonic
state is paired with a fermionic state of the same energy. Their
cancellation, when the heat trace converges, makes the regulated graded trace independent of its
heat-kernel parameter and leaves an integer determined entirely by
zero-energy states.
I(β) = Tr[(-1)Fe−βH]
= nB(0) − nF(0).
the regulated trace is β-independent under the finite-volume setup(1)
Lean checks the decisive arithmetic step. If I is nonzero,
the two zero-energy counts cannot both vanish. Thus at least one
zero-energy state exists. A separate physical bridge identifies the
existence of such a state with zero ground-state energy and hence
unbroken supersymmetry at that finite volume.
The periodic and twisted index definitions each consume a named
volume-universal beta-independence proposition derived through the
pairing principle. Neither the zero-energy difference nor a bare
pairing label is allowed to stand in for that bridge.
pairingPositive-energy contributions cancel
The cancellation first gives β-independence, then leaves only zero-energy multiplicities.
→
integer obstructionI ≠ 0 forces nB(0) + nF(0) > 0
This implication is proved directly in Lean.
→
physical bridgeA zero-energy state fixes E0 = 0
The ground-energy interpretation remains an explicit premise.
I ≠ 0
⟹
nB(0) + nF(0) > 0
⟹
E0 = 0.
(2)
Finite-volume index theoremLean checked
Once the index is realized as the difference of zero-energy
multiplicities, any nonzero index produces a zero-energy state.
Together with the explicit vacuum-energy bridge, the selected
finite-volume gauge system has an unbroken zero-energy vacuum and
directly exhibits the selected zero-state, bridge, criterion, and
unbroken output on one proof-free model.
The converse fails. The value zero is compatible both with no
zero-energy states and with equal nonzero bosonic and fermionic
multiplicities. The index alone therefore cannot distinguish broken
supersymmetry from a paired zero-energy spectrum. A separate finite
countermodel also keeps the zero-energy difference while withholding
beta-independent regularization.
I = 0:
[nB(0)=nF(0)=0]
or
[nB(0)=nF(0)>0].
(3)
Formal auditFinite-volume assumptions and exact logical outputs
Complete input contract
The finite spatial volume uses matching bosonic and fermionic boundary conditions that preserve supersymmetry.
The finite spectrum is discrete and the graded trace defines the index.
Positive-energy states are Bose–Fermi paired, supplying the two-stage cancellation principle.
The pairing principle derives a β-independent regulator, which is then passed explicitly into index realization.
The separate bridge ZeroEnergyStateSetsGroundStateEnergy maps a zero-energy state to vanishing ground-state energy.
Finite-volume logical ledger
Transition
Status
Lean realization
Derive β-independence at every periodic or twisted volume
sec. 2, pp. 254–260 · equations (1)–(4) and alternatives (A)–(B)
03a
Actual pairing, a convergent trace, and a normalizable state
The section-2 refinement computes the normalized partner using an actual odd charge and actual grading. Witten's one-charge convention here is distinct from the four-charge sum in the 1981 paper. A complete paired energy basis and its common invariant operator domain are explicit mathematical and physical inputs.
H = Q^2; Q and Gamma are symmetric; Gamma^2 = I; Gamma Q = -Q Gamma. For ||b|| = 1, H b = E b, Gamma b = b, and E > 0: f = Q b/sqrt(E). ||f|| = 1; H f = E f; Gamma f = -f; Q f = sqrt(E) b; <b,f> = 0. At E = 0, Q psi = 0; positive-energy normalization does not pair zero modes.
PairedSpectralData computes the exponential heat weights and signed weights; HeatTraceDefinedAt requires positive beta and actual summability. SpectralOperatorData binds a complete Hilbert basis to the charge domain, actual Hamiltonian eigenvalues, and grading. The total Lean sum is not interpreted as a physical trace outside its convergence domain.
For beta > 0 with sum_s exp(-beta E_s) < infinity: I(beta) = sum_s Gamma_s exp(-beta E_s) = n_B(0) - n_F(0). I(beta1) = I(beta2) requires both traces to be defined. Complete spectral-basis representation and a common invariant operator domain are explicit inputs. A nonzero represented index gives an actual normalized zero state and an attained zero ground energy.
regulated_trace_and_actual_operators_establish_the_selected_unbroken_vacuum derives a normalized zero state and an attained ground energy, then supplies the shared unbroken-vacuum result. Its SelectedGroundEnergyInfimumAt input only identifies the selected energy as the infimum of actual normalized-state expectations; it does not assume attainment in advance.
E_n = log(n+2): positive pairs and finitely many levels below every finite energy cutoff. sum_n exp(-beta E_n) converges iff beta > 1. I(beta) = 1 on that convergence domain; beta = 1 is excluded. Pair-by-pair cancellation does not make the unregulated parity sum summable. This is a constructed spectral boundary, not a density-of-states claim for N=1 SYM.
The counterexample has a genuine convergent domain, with index one at beta two, but excludes beta one. It replaces independently assigned trace readouts. The fidelity tests also reject treating pairwise cancellation of an infinite parity sum as convergence.
For beta > 0, one bosonic zero mode and one pair at E = 1: I(beta) = 1; Z(beta) = 1 + 2 exp(-beta); <Gamma>_beta = I(beta)/Z(beta). <Gamma>_log(2) = 1/2; <Gamma>_(2 log(2)) = 2/3. The complete three-state Hilbert basis and its actual Q and Gamma operators are realized explicitly. The index is an unnormalized supertrace, not the Gibbs expectation of parity.
The finite witness verifies the new complete-basis representation contract rather than only postulating that it can be met. The three new fidelity boundaries are pairwiseCancellationNotUnregulatedTrace, oneConvergentRegulatorNotAllPositiveRegulators, and betaIndependentIndexNotNormalizedThermalParity. None asserts Yang-Mills trace convergence, admissible large-field deformation, or transport to infinite volume.
03b
The physical rest sector and fermion grading
The original section 2, journal pp. 255-256, justifies restricting the search for zero-energy states to zero momentum before writing the normalized single-charge algebra. JointEnergyMomentumData represents actual Hamiltonian and momentum operators on one invariant domain. The joint spectral bound implies an operator-kernel statement, not just a vanishing mean momentum.
Joint basis: H e_a = E_a e_a; P_j e_a = p_(a,j) e_a; ||p_a|| <= E_a. H psi = 0 implies P_j psi = 0 for every spatial component j. Zero expected momentum alone does not imply membership in the joint momentum kernel. The complete joint basis and common invariant domain are explicit representation inputs.
FiniteVolumeRotationData supplies the selected quarter-turn operator. Its fourth power defines rotationGrading; the spin-cover condition derives symmetry and involution. The covariance equations give the spatial meaning of the chosen rotation. This is neither a proof of the spin-statistics theorem nor an assumption that particle number is conserved.
R is the selected finite z-axis quarter turn; R^8 = I; Gamma = R^4. P_x R = -R P_y; P_y R = R P_x; P_z R = R P_z. Gamma^2 = I and Gamma is symmetric; bosonic and fermionic states have eigenvalues +1 and -1. A spinorial quarter turn can have R^4 = -I; no infinitesimal J_z is assumed. The chosen spatial geometry and boundary conditions must admit this rotation.
RestSectorData identifies the reduced common domain with the actual joint momentum kernel and intertwines both the Hamiltonian and physical grading. SourceRestSpectrumAt supplies the charge, basis, and eigenvalue inputs without independently assuming the grading algebra. The physical rotations derive that algebra, and the existing spectral-index route consumes it.
U is a norm-preserving linear bijection onto the actual joint momentum kernel. H U = U Q^2; Gamma U = U Gamma_rest; the reduced charge is odd under the physical grading. The restriction preserves the full zero-mode space and each finite bosonic or fermionic zero-mode multiplicity. A nonzero convergent represented index yields a normalized full-space zero-energy ground state. The selected full-space energy infimum is then attained; attainment is not a premise.
physical_momentum_rotation_and_regulated_index_establish_the_full_vacuum retains the full-space zero mode, its physical parity, the rest-sector charge vacuum, and the shared unbroken-vacuum conclusion. Separate linear isometries identify all zero modes and their parity subspaces; finite multiplicities refer to dimensions, not counts of normalized vectors.
Constructed five-state system: rest energies (0,1,1), moving energies (2,2). Z_full(beta) = Z_rest(beta) + 2 exp(-2 beta). At beta = log(2): Z_full = 5/2 while Z_rest = 2. Both sectors retain the same zero modes, not the same heat trace. In this model Gamma and -Gamma obey the abstract algebra, but only Gamma matches R^4.
The constructed rest sector is proper: normalized moving states are excluded, while a vacuum and a positive-energy pair remain. Additional two-state controls distinguish vanishing mean momentum from the actual kernel and nonnegative energy from the relativistic spectral bound. Five independent fidelity boundaries protect these distinctions. These are finite operator tests, not Yang-Mills models or a realization of the full super-Poincare algebra. Physical domain adequacy, Yang-Mills trace convergence, deformation control, and infinite-volume interpretation remain explicit obligations.
03c
Four charges, one index, and dynamical rotations
Section 2, pp. 255-256, gives four normalized Hermitian rest-frame charges for four-dimensional N = 1 supersymmetry, then selects one for index pairing. NOneRestChargeFamily retains the whole family: common Hamiltonian, distinct-charge anticommutation, oddness, and the selected charge. The half-rescaling below compares two energy conventions on that same rest domain, including the sum-square convention of Witten 1981; it is not a new assertion about charges at nonzero momentum.
On the common P = 0 domain: Q_i^2 = H for i = 1,2,3,4; Q_i Q_j + Q_j Q_i = 0 for i != j. Each Q_i is symmetric and Gamma Q_i = -Q_i Gamma; {Q_i,Q_j} = 2 delta_ij H. H = (1/4) sum_i Q_i^2 = sum_i (Q_i/2)^2. The normalization comparison is restricted to rest; it does not fix individual global charge squares away from rest.
Symmetry and the common-square relation identify the kernels of every charge with the zero-energy space. FourChargeSourceSpectrumAt connects that algebra to the existing paired basis without separately assuming its grading algebra. four_charge_source_index_establishes_a_full_supersymmetric_vacuum derives a normalized full-space vacuum annihilated by all four rest charges and supplies the shared unbroken-vacuum result.
ker H = ker Q_i = intersection_j ker Q_j for every rest charge i. A nonzero, convergent represented index gives ||psi|| = 1, H U psi = 0, and Q_i psi = 0 for all four i. Neither HR = RH nor momentum-rotation covariance is a premise of this vacuum-existence deduction. The common domains, complete spectral basis, and selected trace and energy interpretations remain explicit inputs.
The constructed witness contains a nontrivial four-state Clifford multiplet and one zero-energy bosonic state. A complete seven-state joint spectrum identifies the five rest states as the exact momentum kernel. Its four charges are explicit complex matrices, not four names for one operator. Separate controls retain valid pairing while rejecting duplicated charges and an incorrectly normalized sum.
Rest basis (v,b1,b2,f1,f2): H = diag(0,1,1,1,1); Gamma = diag(1,1,1,-1,-1). Four explicit Hermitian Clifford charges annihilate v and satisfy the full rest anticommutator. For beta > 0, I(beta) = 1; each charge gives a normalized positive-energy pair. Two additional moving states have energy 2, so the five-state rest sector is proper. Four copies of one charge preserve pairing but fail off-diagonal anticommutation; the unrescaled charge-square sum is 4H, not H.
HamiltonianRotationInvariantAt adds the actual commutation equation. The vacuum-orbit theorem uses it to preserve energy, then uses the spectral bound and rest-domain equivalence to recover an actual reduced four-charge vacuum. This is an extra dynamical conclusion, not a hidden premise of the index criterion. One counterexample swaps a vacuum with an excited boson while keeping the spin cover, physical grading, and momentum covariance. Another exchanges two degenerate zero-energy states.
With the additional input HR = RH: H psi = 0 and ||psi|| = 1 imply H R^n psi = 0 and ||R^n psi|| = 1. Every finite rotation power preserves the four-charge vacuum sector and commutes with Gamma = R^4. Spin cover, correct grading, and momentum covariance do not imply HR = RH. A kinematic-only candidate maps the normalized vacuum v to b1 with energy 1. Even with HR = RH, a rotation may mix degenerate vacua: R v != v can hold.
Four independent fidelity boundaries protect these distinctions. The finite examples test the represented algebra and inference dependencies; they do not construct a Yang-Mills Hilbert space, the full super-Poincare representation, or a spin-statistics theorem. Physical identification, domain adequacy, actual Yang-Mills trace convergence, admissible deformation, and infinite-volume transport remain explicit obligations.
03d
Actual normalizability and the large-field limit
The large-field caveat is tested with a one-dimensional differential model using the actual potential printed in section 2. Its primitive fixes the complex zero-mode profiles. The differential charge reverses grading for every coupling, but this algebraic fact and pointwise coefficient continuity do not protect the normalizable kernel index.
For m > 0: U_g(x) = m x - g x^2; V_g(x) = U_g(x)^2; S_g(x) = m x^2/2 - g x^3/3. For each epsilon > 0 and field bound R, there are 0 < g < epsilon and x > R with V_g(x) = 0. The additional classical minimum is x = m/g; pointwise continuity in g does not give uniform low-energy field containment. An escaping classical minimum is not itself a constructed normalizable quantum state.
With hbar = 1, K_s(g) is the complex space of square-integrable differentiable solutions of psi' = s U_g psi. Delta_ker(g) = dim_C K_-(g) - dim_C K_+(g). At g = 0: (dim_C K_-, dim_C K_+) = (1,0), so Delta_ker = 1. For g > 0: (dim_C K_-, dim_C K_+) = (0,0), so Delta_ker = 0. The kernel dimensions are calculated from actual solution spaces, not independently assigned. This one-dimensional diagnostic does not construct a self-adjoint Hilbert-space closure, regulated trace, or four-dimensional SYM spectrum.
This is a diagnostic of the deformation inference, not Witten's four-dimensional index calculation. The grading places the negative-exponent component first. First-order factorization, classical minima, actual square integrability, and the analytic realization of a self-adjoint quantum theory are separate claims.
The calculation is useful only because the index is protected
Neither finite-volume calculation is performed at every volume.
Instead, the argument connects each target theory to a convenient
calculable regime and transports the integer index along that
connection. Lean keeps the connection and the protection law as
separate hypotheses.
1
Supersymmetry
The deformation preserves supersymmetry between the target and calculable regimes.
2
Large-field behavior
The deformation preserves the asymptotic behavior of the theory at large field values.
3
No spectral influx
No states are allowed to enter the spectrum from infinity in field space.
4
Protected integer
Along such an admissible deformation, the source and target index values are equal.
admissible connection from Rtarget to Rcalc
⟹
I(Rtarget) = I(Rcalc).
(4)
calculateEvaluate I at a controlled source regime
The periodic and twisted routes use different source calculations.
→
connectSupply an admissible deformation to each target volume
Connectivity is a named route premise, not inferred from the value.
→
transportCarry the same integer to the target theory
Index equality is the exact protected-transfer rule.
Protected-index transportLean checked
If every target volume is admissibly connected to its calculable
source and the index is invariant along admissible deformations,
the calculated source value is also the index of every target
finite-volume theory.
Formal auditProtection contract and shared-framework realization
AdmissibleDeformation is the conjunction of
supersymmetry preservation, source-relative large-field
order control, and state exclusion along the selected path
and its derived reversal. The quantum conditions remain
explicit schematic obligations.
IndexInvariantUnderAdmissibleDeformation supplies
equality of the integer index for every such pair of regimes.
EveryPeriodicVolumeConnectsToCalculation and
EveryTwistedVolumeConnectsToCalculation are distinct
route premises.
The shared ProtectedIndexTransportContext records the
selected boundary-condition description, volume, calculable
source, target regime, connection, and protected prediction. Its
realization theorem transports only the index value; it does not
import the later zero-mode or sector calculations.
periodic_inputs_realize_structural_protected_index_explanation
retains the complete calculation-and-protection route as a typed
constraint and explains only the valid selected finite-volume
target index prediction. It adds no supersymmetry-breaking or
vacuum-count conclusion.
The stronger source-sensitive theorem retains the original
derivation. wittenProtectedIndexTransportDerivationObtains
combines regulated Bose-Fermi cancellation, the weak-coupling
Cartan/Weyl zero-mode calculation, an explicit selector for the
unresolved signed branch, and deformation conditions controlling
large-field energy growth while excluding states arriving or
disappearing at infinity along the same path. Lean first derives the selected calculable index value
and only then constructs the shared protected route.
The checked stages are
witten_source_derivation_establishes_selected_periodic_index_value,
witten_source_derivation_exposes_pairing_regularization_and_gentle_deformation,
witten_source_derivation_makes_shared_protected_index_route_applicable,
and
witten_source_derivation_realizes_source_structural_protected_index_explanation.
When that transported value is nonzero and the separate
zero-state spectral bridge is supplied,
witten_source_transport_and_nonzero_value_realize_protected_index_and_unbroken_vacuum_explanations
returns both the protected-index explanation and the
source-sensitive unbroken-vacuum explanation. The passage to
vacuum energy therefore does not erase the zero-mode and
deformation provenance.
sec. 2, pp. 257–260 · pairing, index invariance, and the large-field caveat
05
Ordinary periodic route: the zero modes determine |I|
At the calculable periodic point, weak coupling separates nonzero
momentum excitations from the much softer zero modes. Flat
connections reduce to commuting Cartan constants, and the low-energy
problem becomes a quantization of the bosonic and gluino zero modes.
Periodic gauge shifts identify the Cartan coordinates, while Gauss'
law forces invariance under identity-connected gauge
transformations.
Enonzero momentum ∼ 1/L,
Ezero mode ∼ g2/L,
at the weak-coupling calculation point g2 ≪ 1(5)
For SU(N), the rank satisfies r + 1 = N.
The gauge constraints first make the Cartan zero modes periodic
variables and make physical states Weyl invariant. The deferred
Cartan/Weyl calculation then produces one bosonic zero-mode ground
state and r fermionic Cartan pairs, leaving two possible
representations of the reference vacuum |Ω⟩.
True-invariant Weyl branch
The pair operator U generates |Ω⟩, U|Ω⟩, …, Ur|Ω⟩.
Fermi statistics terminates the chain at Ur+1|Ω⟩ = 0.
The tower has r + 1 = N states, all with even relative fermion number.
The chiral charges are 0, 2, …, 2N − 2.
Index branch: Iper = N.
Pseudo-invariant Weyl branch
A separate operator V produces the true-invariant multiplet V|Ω⟩.
The multiplet has r + 1 = N components.
Its spin is r/2 = (N − 1)/2.
All components carry a reported common fermion grade f; geometry alone does not fix it.
Reported index branch: Iper = (−1)fN.
Iper,calc = N
or
Iper,calc = (−1)fN,
|Iper,calc| = N.
(6)
The Weyl representation sign is explicitly unresolved. Lean does not
choose between the two branches. The nonbreaking argument needs only
their common magnitude, so the open sign does not obstruct the
conclusion.
Original-source sign discrepancy: p. 294 reports
−2 for the alternative SU(2) branch, but p. 296 prints
(−1)r+1(r+1), which gives +2 at rank one. The literal print
is retained separately. Equating grading with the stated spin
r/2 would instead give (−1)r; that physical
identification is an explicit constraint, not assumed by component
counting. The full no-breaking route needs neither disputed exponent:
it evaluates finite graded sums and uses their common magnitude.
This is not an author-approved erratum or a claim that the deferred
quantization and measure obligations have been proved.
Complete argument, smaller interface:PeriodicIndexData owns 26 data fields and preserves all
eleven input positions of the compatibility no-breaking theorem,
with the explicit spatial-spin setup correction described below. The full
IndexSystem inherits this record and retains its other
36 fields for twisted, flux, chiral and separate convergence
extensions. Legacy predicates and quantitative scale definitions delegate to the same
canonical definitions; the existing public theorem applies the
smaller interface's theorem. The gauge wavefunction, chiral-charge
tower, actual U/V operator equations and unresolved grading are all
retained. This reduces this argument's inherited inspection scope,
not the number of independent physical choices in the library.
Physical interpretation and human semantic review remain pending.
A low-energy window, not all Cartan excitations:
Three freely assignable scale predicates have been replaced by a
data-only spectrum with a Cartan period basis, squared coupling, chosen relative tolerance,
nonempty discarded excitations and a Cartan energy map. Nonzero
spatial momentum has threshold 2 pi/L. Cartan period vectors
(2 pi/L) B instead give conjugate momenta L B inverse-transpose n.
General SU(N) does not assume a rectangular orthonormal lattice.
The basis is nonsingular and must match the entire integer-diagonal
SU(N) cocharacter lattice in the selected frame; that identification
remains an imported condition of the physical trace. The printed
approximate Hamiltonian gives energy g squared/L times the Fourier
level. Lean separates a bounded Fourier window from the selected
hard spectrum when tolerance times the cutoff is below 2 pi.
A constructed nonempty interface at squared coupling 1/100 also
contains a Cartan excitation of energy 10000, above its hard
threshold 2 pi. Small coupling does not put every Cartan level below
that threshold. The reduced and calculable theories must use the
same box. This refines the former schematic scale predicates, not
their unrestricted interpretation. Spectrum completeness, physical
identification, approximation validity and proper measure remain
external; no full SYM realization or source error bound is claimed.
The large-field input is a selected-path energy bound.
The complete periodic route no longer stores an arbitrary asymptotic-preservation
flag. It uses a configuration type, nonnegative unbounded field-size map,
energy readout and parameter path with the specified endpoints. A single finite
affine source-relative bound applies to every point of the path and every
sufficiently large field. Coefficient, offset and radius have dimensionless,
energy and field-size units; strictly positive and nonnegative existential
witnesses are kernel-equivalent. The source-potential differential model
rejects the bound from zero to nonzero coupling, but satisfies it in reverse
while its computed normalizable graded kernel index changes from zero to one.
Directional classical order control is not quantum spectral continuity,
asymptotic-degree preservation or an index-protection theorem. All deformation
conditions remain. Supersymmetry and quantum state exclusion refer
to the same selected path on the closed unit interval. State exclusion
is required forward and on the derived parameter reversal, correcting
the earlier omission of disappearing eigenvalues in the p. 259 proviso.
These quantum predicates and the imported invariance principle remain
open obligations. No self-adjoint SYM realization, complete-input
independence, manageable human review or semantic approval is inferred.
One geometric input, not a free regime flag:
The reduction quantifies over every differentiable, strictly
periodic Hermitian traceless SU(N) matrix potential with curvature
dA-i[A,A] equal to zero. It requires a strictly periodic,
differentiable determinant-one unitary gauge transformation to
real diagonal constant potentials. The gauge convention is D=d-iA,
with coupling absorbed into A. The same retained Cartan count plus
one fixes the classical matrix size, and the existing source rank
equation identifies it with N. No second color count or box length
is chosen. Lean checks that diagonal constants are flat, whereas
constant noncommuting Pauli potentials are periodic SU(2) fields
with nonzero curvature. The complete index route
still consumes the geometric classification as an imported
calculation. This does not construct (75)-(77), identify
dimensionless holonomy logarithms with dimensionful potentials,
allow center-twisted gauges, or settle a quantum theta sector.
Removing one freely stored predicate does not establish fewer
independent meanings or human semantic approval.
One fewer independently stored readout:
The SU(N) Cartan rank is now defined as N-1, following p. 295,
rather than selected by an arbitrary function. Lean checks the
equivalence with r+1=N for positive N and derives the rank clause
from the pure SU(N) scope used by the complete counting argument.
Every remaining quantization, gauge, operator, grading, deformation,
and large-volume input is retained. This is a normalization of the
SU(N) interface, not a proof of the rank of an abstract gauge algebra,
equivalence of unrestricted old records, or human semantic approval.
The physical scope remains N>=2; N=0 totalization is not physics.
The large-volume input has actual mathematical meaning:
The complete SU(N) real-energy refinement retains all eleven
physical inputs, replacing generic zero-energy closure with positive
cofinal volumes matched to the calculation's cubic box, a proper filter, zero SUSY energy origin and actual
convergence of ground energies. Lean checks the equivalent eventual
volume and epsilon energy bounds and derives the old closure law,
not its converse. The diagnostic V=n+1, E=1/V has positive energy
at every finite stage and zero limiting energy, with actual real
convergence rather than assigned Boolean answers. This is not a
SYM spectrum or Appendix A theory. The result concerns total energy,
not an identification with energy density or a limit of normalizable
vacuum vectors. Gauge, trace and human-review obligations remain.
The calculation and limit now use the same physical box:
Journal p. 284, equation (45), uses one length L in all three spatial
directions, so the selected cubic torus has volume L cubed. The primary
general-SU(N) theorem no longer accepts an independent volume function.
Finite volume is a positive real L, not a freely stored predicate;
the existing finite-volume setup derives positivity of L cubed. Under
that setup, Lean checks that the corrected compatibility limit is
exactly volume matching together with the canonical box limit. Matching
fixes a missing identification in the earlier specification; no
equivalence with its arbitrary unmatched readout is claimed. Proper
cofinal growth and convergence of actual total ground energy remain
required. A periodic constant wavefunction in a unit box stays at
volume one although an unrelated numeric label n+1 grows without bound.
This diagnostic rejects relabelling as a physical infinite-volume limit,
not a SYM model or the author's Appendix A construction. Thirty outer
fields do not establish a minimal independent count or manageable
complete review scope. Human semantic review remains pending.
The general SU(N) calculation now uses actual U/V states:
The exterior space has two fermion spin components per Cartan
direction. Its induced parity makes U even and V have sign
(-1)^(N-1); the source's inconsistent printed exponent is preserved
separately, not silently corrected. Weyl permutations act on all N
diagonal entries through an explicitly real orthonormal traceless
Cartan frame. Physical states are fixed by every such action.
The actual operator-algebra input includes nonzero candidates;
exact completeness of their span remains separate.
Their distinct spin-component weights derive independence;
together they construct a basis from which Lean computes the trace
of actual restricted parity, not an assigned expected index.
The trace now acts on constant coordinate-dependent exterior states
satisfying both cocharacter periodicity and the joint coordinate/fermion
Weyl action, with the calculation's same lattice and box length.
Constancy is selected separately, not inferred from Gauss' law.
A checked linear equivalence preserves the former fermionic trace,
and transported grading is pointwise exterior parity.
Identifying it with the physical SYM index, including proper gauge
measure and constant-boson zero-energy reduction, is a separate
assumption. The complete argument derives magnitude
N and retains protected transport, proved cancellation and real energetic
convergence. A genuine SU(3) Weyl transformation mixes the two
Cartan coordinates and excludes a nonzero one-fermion state from
the fixed space. This diagnostic does not prove general classification.
The refinement exposes hidden meaning; it does not establish fewer
independent definitions, manageable human scope or semantic approval.
U/V independence is derived from the actual candidates.
Scaling the second Pauli-spin slot gives candidate k weight scalar^k
in both source branches. At scalar 2 these weights are distinct.
Pointwise nonzero vectors are now part of the actual algebra input.
The primary requests exact equality of their span with the full
Weyl-fixed space; Lean derives independence. V components all have total fermion
number r, so total number alone cannot distinguish them.
This is an algebraic scaling test, not a new quantum spin observable
or a claim of physical unitarity at arbitrary complex scalars.
The exact classification equivalence preserves all completeness,
character, physical trace, measure and approximation obligations.
A nonzero fixed vacuum with duplicate labels is not independent;
the equivalence is specific to the actual U/V construction.
The independence-only step removed no field. No full SYM spectrum is constructed, and human
review and manageability of the complete scope remain unestablished.
The operator input now consists of actual spin-resolved equations.
The source algebra retains nonzero U powers and V components,
U^(r+1)=0, UV=VU=0 for every spin tensor, products of two V tensors
proportional to U^r, and zero cubic contractions of symmetric
Cartan tensors. Each pair of V spin tensors has its own complex
proportionality coefficient, which may vanish: a nonzero actual
rank-one V component has zero self-product. The independently
assigned support table is removed from both argument records;
the routed source interpretation uses these actual equations.
Four former label-only fidelity rows now use actual two-mode CAR
matrices: nonzero U but U squared zero; nonzero creators but zero
mixed products; vanishing three-creator words; and spin-resolved
cross products U/2 and -U/2 with zero component self-squares.
The old support-table declarations are preserved outside the active
package in deprecated/paper-controls/Witten1982OperatorSupport.lean.
These rank-one diagnostics do not prove general Weyl completeness,
a full SYM realization, or a matrix/exterior-algebra identification.
This repairs a weaker specification, not an equivalence with arbitrary
legacy labels. Under the actual algebra input, the further
classification premise is exactly equality with the full Weyl-fixed
space. General-rank algebraic identities, nonvanishing and complete
invariant-state classification remain inputs. No full SU(2) spin
representation, physical SYM identification, proper measure or
human approval is inferred from the repair.
The U-tower no longer stores an independent charge map.
Original p. 297 chooses creator chirality +1 and empty-state chirality
zero, assuming the true-invariant branch for the chiral interpretation.
The candidate readout chiralChargeAt p is now the derived
creation number 2p. The same classified nonzero U^p states realize
this normalized readout through their actual exterior scaling weights.
The general tower record retains only a reported length; the periodic
argument derives that length from its candidate-label domain
Fin (suRank colors + 1). A separate reported length is
admitted exactly when it agrees at the selected color count. The
primary calculation no longer asks for that derived count condition;
the older full predicate is an exact compatibility view. Labels alone
do not prove physical dimension: the dimension bridge still requires
independence and complete span of the actual U/V vectors. The
existing duplicate nonzero gauge-singlet diagnostic rejects counting
the same physical vector twice. The comparison with freely
reported historical successor, number and charge functions is exact
on represented levels and the terminal endpoint; it does not fix
arbitrary values outside that physical domain. Complex scaling tests
algebraic degree, not physical unitary evolution. This convention
does not prove an exact quantum U(1) symmetry, a physical charge
measurement, or convergence to chiral vacua. The source's anomalous
discrete symmetry and suggestive infinite-volume interpretation
retain their separate obligations. Human semantic review remains pending.
The complete argument now gives at least N orthonormal ground states:
The primary general-N theorem keeps the U/V calculation, protected
transport and same-box cofinal limit. At every finite volume it requires
the trace labels to represent a complete Hilbert basis in a common
invariant charge domain, with the explicit Q squared and grading
equations. Its ground-energy input is the infimum of actual unit-norm
state energies, not merely the candidate energy labels. The readable
form exposes both bounding inequalities. Trace cancellation and protected
transport identify the full integer index at every finite volume.
Its magnitude N obeys |b-f| ≤ b+f. From the same Hilbert basis Lean
selects a family indexed by Fin N, orthonormal in the
actual Hilbert space, each a normalized charge-annihilated zero-energy
ground state. Repeating a vector cannot meet this conclusion.
This is the at-least-N degeneracy on original p. 297, not exactly N:
additional bosonic and fermionic zero modes may cancel in the index.
The identified ground-energy infimum is zero and attained. The older candidate
infimum follows in this nonzero-index setting; no unrestricted
equivalence of the two infima is claimed. The legacy actual-energy
inference delegates to the same contract. This follows journal pp.
255-257, where one Hermitian charge suffices for the index argument.
Identification with physical SYM, self-adjoint closure and full
four-charge/spacetime interpretation remain separate obligations.
No infinite-volume vacuum vector or chiral-vacuum identification is
inferred from energetic convergence or this finite-volume degeneracy.
A second inhabited finite paired spectrum with only positive energies,
index zero and minimum one demonstrates why the first spectrum's index
cannot identify another spectrum's ground energy. Compatibility routes
retain their old count bridge; equivalence with its arbitrary interpretation
is not claimed. The primary now has thirteen physics inputs; the older
candidate-spectrum route has twelve. The core now has twenty-six fields:
the count consolidation removed two, and the spatial-spin correction
added the explicit previously missing boundary readout; the later
actual-operator repair removed the support-table family, and the
periodic U-tower is now a candidate-domain view. The approximate
bosonic mode count is now derived from the Cartan Hamiltonian rather
than separately assigned; the primary calculation input no longer
requests it. It still retains all fermionic and operator obligations.
Exposing this additional obligation is not evidence of lower independent
review burden or completed human validation.
Cancellation is now proved, not independently assumed.
Four freely assigned spectral predicates use one data-only regulated
spectrum at each regime: level type, paired energy data and selected
inverse-energy regulator. Bosonic and fermionic zero-state counts are
views of the same spectrum, not two separately assigned readout families.
The finite-volume setup explicitly matches the integer index to its
unnormalized exponential trace. The older compatibility setup explicitly
lists finite energy sublevels, positive pair energies, a positive selected regulator and heat-weight
summability at every positive beta. The existing spectral sum derives
beta independence and the count/trace equality; the complete general-N
candidate-spectrum route therefore needs twelve external inputs instead of
its former thirteen. The stronger represented-state primary exposes
thirteen inputs without reinstating cancellation as an assumption.
Under its already required operator representation, the older setup is
exactly equivalent to boundary setup, a positive regulator and
heat-weight summability at every positive beta. The integer index
is derived from the same zero-state counts; its unnormalized
trace equation is not a separate input.
Positive pairs are already a clause of representation. A convergent
positive-beta heat trace implies finite energy sublevels, because every
level below a cutoff contributes at least exp(-beta*cutoff)>0.
The main theorem now derives these two conditions instead of requesting
them again. This removes repeated input conditions, not record fields
or an independently established semantic count.
The physical spectral identification and all-positive-beta
convergence are retained, not supplied by discreteness or pairing. The
logarithmic spectrum demonstrates the latter distinction. The core has
21 outer stored fields after the canonical-index and typed gauge-action corrections and the box-volume, exact-count, spatial-spin,
approximate-bosonic-count, actual-operator and candidate-domain corrections,
but the whole readout remains in review
scope. No SYM spectral realization, minimal independent count, manageable
complete inspection or human semantic approval is inferred.
Derive gauge consequences; expose the restriction boundary:
Five freely assigned gauge proposition families are now definitions
on a typed gauge action. Its eleven data and instance fields,
including topology, physical state subspace and linear Cartan
restriction, all remain in review scope. Global Gauss invariance
under identity-connected transformations, the actual shift pullback
and connected Weyl representatives derive periodic profiles and
quantum Weyl invariance. The gauge hypothesis no longer imports
those two inference arrows. Weyl representatives are not assumed to
split the Weyl quotient homomorphically. Quantum-to-Cartan covariance
is a separate explicit trace premise, used to obtain the joint
coordinate/fermion constraint. A nonzero SU(3) one-fermion profile
with a trivial abstract quantum action satisfies the other gauge
conditions but fails that covariance. This is not a SYM model.
The readable trace equivalence retains covariance on both sides;
equivalence with the old unrestricted premise is not claimed.
Proper measure, physical quantization and full SYM trace
identification remain open. Raw field reduction is not an
established reduction of independent meaning; human review is pending.
One integer index, derived from the same spectrum:
Original p. 258, Eq. (4), defines the integer by the bosonic minus
fermionic zero-state counts and warns that the unregulated sum is
ill-defined. The periodic record no longer stores a second index
family. For a convergent positive-beta trace, Lean proves that a
reported integer is admitted exactly when it matches this canonical
index. The older selected-regime report contract is retained
precisely on those matching reports, not arbitrary inconsistent
records. The complete primary retains boundary and representation
inputs, a positive regulator and all-positive-beta heat convergence,
but derives the numerical trace equation. All finite-volume and
conditional energetic-limit outputs remain unchanged. The finite
three-state diagnostic still distinguishes index one from normalized
thermal parity one-half and from an incorrect integer report of two.
This removes a freely assigned readout, not the physical spectrum,
Cartan/SYM trace identification or the convergence obligation.
Minimal independent semantics, manageable complete review and
human semantic approval remain unestablished.
One source for zero-state multiplicities:
Original pp. 258-259 use the same bosonic and fermionic zero-energy
sectors for the count definition and trace. Their finite label sectors
remain, but the core no longer stores two additional count functions
or asks that they equal the spectrum again. The exact theorem
reported_count_trace_setup_iff_canonical_trace_and_matching_counts
compares the older trace conditions for any reported counts with the
canonical trace setup plus their matching equations. Only consistent
reports are retained; equivalence with arbitrary inconsistent records
is not claimed. Legacy count views delegate to this same spectrum.
A separately supplied four-charge spectrum still needs explicit
multiplicity identification. No physical SYM interpretation, minimal
semantic count or human approval follows from this consolidation.
Spatial field periodicity is now an explicit premise:
Original p. 254 requires both bosons and fermions to be periodic on
all three spatial cycles. The complete setup now fixes both spin
phases to +1 for every axis, in addition to positive box length and
the existing common gauge/internal boundary and quantum SUSY
compatibility obligations. For ordinary untwisted scalar components,
Lean characterizes the domains by x_i shifted by L. This is not the
Cartan gauge-coordinate shift by 2*pi/L. Common magnetic gauge
transitions remain separate; unit spin phases do not make them
trivial. Nor are these spatial cycles thermal Euclidean time.
A constant bosonic component and a nonzero fermionic exp(i*pi*x_0)
in the same positive unit box show why bosonic periodicity alone
is insufficient. This diagnostic is not a SYM spectrum. The setup
is a source-fidelity strengthening, not an equivalence with its
unrestricted earlier interpretation or a construction of a quantum
SUSY algebra. Thirteen primary physics-input positions and the complete
conclusion remain; human semantic review is pending.
The approximate bosonic mode count is derived, not assigned:
Original pp. 291-294 use the printed Cartan Hamiltonian (70). The
bosonic count is now the cardinality of its zero-energy Fourier-label
sector. With positive L and g squared and an invertible period lattice,
Lean proves that only the zero Fourier label has zero energy; its profile
is constant. The source and effective approximate Cartan-energy readouts
now explicitly match in the same box. This is not equality of full
quantum spectra. The complete primary uses the remaining fermionic,
U/V, operator and character data and derives the bosonic-count condition.
The full compatibility contract follows by a checked equivalence under
the actual reduction and SU(N) scope, not under an arbitrary old relation.
One independently assignable count family and one repeated condition
are removed, but no minimal independent review count is established.
The approximate count is not the exact SYM bosonic zero-state count:
pp. 257-258 allow bosonic and fermionic zero modes to lift together, preserving
their difference. The p. 293 footnote defers the proper measure and
expects index independence. Physical quantization, completeness and
trace identification remain external obligations.
At zero coupling the retained positive box and nonempty hard gap
coexist with two distinct zero-energy profiles, with values +1 and -1
at the selected coordinate. No finite physical count is inferred from
Nat.card outside the finite valid sector. Neither diagnostic nor
derivation constructs physical SYM; human review remains pending.
Concrete SU(2) gauge count:RankOneCartan uses two actual complex creation matrices,
a checked number operator with occupations (0,1,1,2), and the
joint reversal of all three bosonic coordinates and both fermions.
On constant profiles, the gauge projector is (1+W)/2. The
unprojected four-state supertrace is zero; the projected traces are
+2 and −2 for the two unresolved empty-state Weyl characters.
An odd coordinate profile can compensate an odd fermionic factor,
so gauge invariance alone does not isolate the constant bosonic
ground state. The complete rank-one no-breaking route keeps the
physical trace identification, reduction, transport, cancellation,
energy bridge, and infinite-volume principle as explicit inputs.
This is not a full SYM Hilbert-space construction, a proof of the
proper gauge measure, or a replacement for general SU(N) quantization.
Protected transport carries an explicitly calculated branch value
to every ordinary periodic finite volume. Its setup certificate
retains supersymmetry-preserving finite-volume conditions at both
the weak-coupling source and selected target; the source equality is
a genuine premise rather than reflexivity. Since N ≥ 2, the index is nonzero at
every such volume; its magnitude also bounds the total number of
zero-energy states from below.
|Iper(L)| = N,
nB(0;L) + nF(0;L) ≥ N,
E0,per(L) = 0.
for every admissibly connected ordinary periodic volume L(7)
Under the weak-coupling reduction, zero-mode calculation,
deformation protection, finite-volume pairing, and vacuum-energy
bridge, every ordinary periodic finite volume has a nonzero index,
at least N zero-energy states, and zero ground-state energy.
Formal auditComplete periodic calculation contract and branch outputs
Calculational inputs
PeriodicWeakCouplingReduction supplies the supersymmetric finite-volume setup, weak coupling, flat Cartan connections, the two energy scales, and the zero-mode effective theory.
PeriodicCartanGaugeConstraintData supplies periodic gauge-shift identification, the identity-connected Gauss-law constraint, and the checked implications to periodic Cartan variables and Weyl-invariant physical states.
PeriodicZeroModeCalculationData supplies r + 1 = N, one bosonic zero-mode ground state, r Cartan gluino pairs, the U-tower realization, the V-multiplet realization, and the unresolved Weyl sign.
PeriodicZeroModeIndexEvaluation is the deferred quantization output selecting one of the two index branches only after the gauge constraints are present.
NonvacuousPeriodicZeroModeIndexCalculationAt is the direct endpoint: the theory scope, reduction, gauge constraints, complete data, and branch result must all hold together before the downstream route can use the calculation.
The foundational route supplies positive-energy cancellation, arbitrary periodic-volume setup, deformation invariance, and an admissible connection to the calculation point.
Periodic-route premise ledger
Step
Status
Exact output
Scale separation and zero-mode reduction
controlled approximation
Low-energy zero-mode quantization
Gauge shifts, Gauss law, and Weyl invariance
checked deduction
Periodic Cartan variables and Weyl-invariant physical states
Bosonic ground state and Cartan-pair count
deferred calculation
One bosonic state and r fermion pairs
Weyl-invariant U and V constructions
deferred calculation
Two branches with N states or components
Weyl representation and reported grade
open obligation
I = N or (−1)fN; printed exponent retained separately
Absolute value of either branch
checked deduction
|I| = N
Transport to all periodic volumes
deformation argument
|I(L)| = N
sec. 8, pp. 290–300 · weak-coupling zero modes, Weyl branches, and the nonzero index
06
Twisted route: sector counting gives the positive index N
The twisted boundary condition changes the finite-volume
calculation rather than merely rephrasing the periodic one. It
removes the zero-momentum gauge mode, permits a classification of
the classical zero-energy configurations as pure gauge, and reduces
the index to a count of perturbative sectors at fixed θ.
twistThe zero-momentum gauge mode is eliminated
The twisted regulator has a different low-energy structure.
→
one sectorEach perturbative sector has one bosonic ground state
Classical zero-energy configurations are classified as pure gauge.
→
sector sumThere are N sectors at fixed θ
Summing one positive contribution from each sector gives the index.
Itw,calc
=
Σk=1N 1
= N.
(8)
Admissible-deformation invariance transports this exact positive
value to every twisted finite volume. The finite-volume index logic
then supplies a zero-energy state and, through the explicit
vacuum-energy bridge, zero ground-state energy.
Itw(L) = N ≠ 0,
E0,tw(L) = 0
for every admissibly connected twisted finite volume L(9)
Twisted finite-volume theoremLean checked
Given the twisted-sector classification and evaluation, the
supersymmetric finite-volume setup, protected transport, and the
zero-state energy bridge, every twisted finite volume has index
N, contains a zero-energy state, and has zero ground-state
energy.
This finite-volume conclusion does not use the later expectation
that periodic and twisted boundary conditions agree at infinite
volume. That expectation enters only when the twisted result is used
to infer the physical periodic infinite-volume conclusion.
Formal auditComplete twisted-sector calculation and finite-volume premises
TwistedSectorRouteCalculationData keeps the causal section-7 package explicit: zero-mode elimination feeds pure-gauge classification; those selected conclusions feed one bosonic ground state per perturbative sector; and pure SU(N) scope plus classification feed exactly N fixed-θ sectors.
TwistedCheckedRouteIndexEvaluation is the deferred summation contract. It consumes the all-volume twisted setup, calculable twisted setup, elimination, classification, one-sector result, and fixed-theta count separately before yielding Itw,calc = N.
TwistedSectorCalculationData and TwistedSectorIndexEvaluation remain the compact nonvacuous endpoint used by paper-local calculation theorems; coverage does not use their bundled sector theorem as a shortcut.
NonvacuousTwistedSectorIndexCalculationAt is the direct endpoint: the pure-theory scope, classification data, and exact sector-sum result must hold together before the final twisted route uses it.
TwistedFiniteVolumeIndexSetup and positive-energy cancellation derive TwistedHeatKernelIsBetaIndependentAtEveryVolume; that result and IndexInvariantUnderAdmissibleDeformation realize and protect the index.
EveryTwistedVolumeConnectsToCalculation supplies a separate admissible connection for every target twisted volume.
ZeroEnergyStateSetsGroundStateEnergy supplies the physical ground-energy bridge.
Twisted-route premise ledger
Step
Status
Output
Retain β-independent twisted regulated trace
paper derivation
TwistedHeatKernelIsBetaIndependentAtEveryVolume
Eliminate the twisted zero mode
paper derivation
No zero-momentum gauge mode
Classify zero-energy configurations
paper derivation
Pure-gauge configurations
Compute one sector and count sectors
deferred calculation
One bosonic state in each of N sectors
Sum the sectors
deferred calculation
I = N
Transport to arbitrary twisted volume
deformation argument
I(L) = N and E0(L) = 0
sec. 7, pp. 283–290 · twisted boundary conditions, pure-gauge sectors, and Tr(−1)F = N
Electric-flux sector refinement
Section 8 refines both calculations by electric flux. In the
ordinary zero-mode theory, electric-flux operators shift the
Cartan coordinates while the zero-energy wavefunctions are
independent of them, so all such states lie in the zero-flux
sector. That sector inherits the unresolved periodic branch;
every nonzero ordinary flux vector has zero index. With magnetic flux,
every nonparallel electric-flux vector has zero index, while
every one of the N possible parallel
Tz eigenvalue sectors has unit index and zero
ground-state energy.
Iper(0) = N or (−1)fN,
∀ e ≠ 0, Iper(e) = 0,
∀ e ∦ m, Itw(e) = 0,
∀ e∥ ∈ ℤN,
Itw(e∥) = 1 and E0(e∥) = 0.
The physical N ≥ 2 scope constructs an actual
ordinary nonzero vector, an actual nonparallel twisted vector,
and an actual parallel eigenvalue, so none of the three universal
statements is accepted only through an empty sector type. A
finite countermodel separately shows that one chosen parallel
sector with index one does not establish the result for every
parallel sector.
ElectricFluxRouteCalculationData keeps the route's
causal bridges explicit, while
OrdinaryElectricFluxCheckedRouteIndexEvaluation and
TwistedElectricFluxCheckedRouteIndexEvaluation
consume the two branch-local premise lists separately. The older
bundled ElectricFluxSectorIndexEvaluation remains a
compact paper-local endpoint, not a coverage shortcut.
Two finite-volume routes, two different infinite-volume contracts
Vanishing ground energy at every finite volume is not silently
identified with the infinite-volume theory. Each route contains an
explicit large-volume principle. The twisted route requires one
additional expectation because the physical breaking criterion is
defined using the periodic infinite-volume energy.
Ordinary periodic limit
checked conditional theorem
The periodic calculation gives
E0,per(L) = 0 for every finite volume. The explicit
periodic limit principle transfers that statement directly to
E0,per(∞) = 0.
No comparison with another boundary condition is used.
Twisted limit
finite-volume theoremboundary expectation
The twisted calculation gives E0,tw(L) = 0 for every
finite volume, and the twisted limit principle gives
E0,tw(∞) = 0.
To reach the periodic physical conclusion, the route separately
assumes E0,per(∞) = E0,tw(∞).
[E0,per(L)=0 for every L]
⟹
E0,per(∞)=0.
the direct periodic large-volume principle(10)
[E0,tw(L)=0 for every L]
⟹
E0,tw(∞)=0,
E0,per(∞)=E0,tw(∞).
the final equality is an author expectation(11)
Main periodic-route theoremLean checked
Under the complete ordinary-periodic calculation, protection,
vacuum-energy, and periodic large-volume premises, the periodic
infinite-volume ground energy is zero. Therefore dynamical
supersymmetry breaking does not occur.
Under the complete twisted-sector calculation, protection,
vacuum-energy, and twisted large-volume premises, plus
infinite-volume boundary-condition independence, the same
no-breaking conclusion follows.
The nonbreaking conclusion does not depend on the condensate conjecture
The source map formally proves that the main infinite-volume
conclusion is independent of the later gluino-condensate
conjecture. The conjecture belongs only to the chiral-vacuum
interpretation.
Formal auditLarge-volume hypotheses and route dependence
InfiniteVolumeSupersymmetryBroken means that the
periodic infinite-volume ground energy differs from the
designated zero energy. The periodic route uses
PeriodicZeroEnergyLimitPrinciple. The twisted route
instead uses TwistedZeroEnergyLimitPrinciple and then
InfiniteVolumeBoundaryConditionIndependence to reach
the periodic criterion.
Infinite-volume dependency comparison
Route
Finite-volume output
Additional infinite-volume input
Epistemic status
Periodic
E0,per(L)=0 for every L
Periodic zero-energy limit principle
paper derivation
Twisted limit
E0,tw(L)=0 for every L
Twisted zero-energy limit principle
paper derivation
Boundary comparison
E0,tw(∞)=0
E0,per(∞)=E0,tw(∞)
author expectation
The coverage theorem certifies that the twisted finite-volume
ground-energy claim is independent of the boundary expectation,
while the twisted infinite-volume conclusion records that exact
expectation as an origin.
08
Finite volume is not boundary-condition independent
The large-volume equality used by the twisted route is not promoted
to a finite-volume theorem. Witten’s separate scope example chooses
a boundary condition periodic only up to a global symmetry that
commutes with supersymmetry and is spontaneously broken.
commuting broken global symmetry
⟹
E0,global twist(L) ≠ 0
⟹
Iglobal twist(L) = 0.
the twist-energy bridge and zero-state step remain explicit hypotheses(12)
Finite-boundary scope theoremLean checked
If the broken global-symmetry twist has nonzero ground energy,
nonzero ground energy excludes bosonic and fermionic zero-energy
states, and the index is realized by their graded difference,
then the globally twisted finite-volume index is zero. If ordinary
periodic boundary conditions can retain a nonzero index, the two
finite-volume boundary conditions have different indices.
globalSymmetryTwisted is a separate paper-local regime.
The magnetic-flux-twisted SU(N) regulator still supplies
the positive index N; the global-symmetry example exists
to block the invalid inference that all finite-volume boundary
conditions must agree.
Formal auditComplete boundary-scope chain
The global symmetry commutes with supersymmetry.
The global symmetry is spontaneously broken.
A proposition-valued paper-local bridge turns those two facts into nonzero twist ground energy.
The globally twisted index is still realized from zero-energy boson and fermion counts.
A spectral bridge says nonzero ground energy excludes both zero-energy counts.
An ordinary periodic finite-volume index may remain nonzero.
Lean checks the zero-index conclusion from the first five items
and then checks finite boundary-index dependence from that
conclusion and the last item. A finite data countermodel also
keeps equal infinite-volume energies while assigning unequal
finite indices.
The N-vacuum picture is a conditional interpretation
The periodic index calculation supplies more than existence: its
magnitude gives a lower bound of N zero-energy states at every
periodic finite volume. Witten’s later chiral argument proposes how
these states may become N distinct infinite-volume vacua. Lean
keeps the numerical bound, anomaly calculation, condensate
conjecture, charge evidence, and large-volume identification as
separate logical ingredients.
Coverage keeps those ingredients premise-aware. A causal
ChiralVacuumRouteCalculationData contract carries the
ordered scope-to-classical-symmetry, anomaly, and condensate-to-
residual-subgroup bridges. Separate
ChiralVacuumCountCheckedRouteEvaluation,
ChiralBranchSelectionContract,
ChiralFourierVacuumCheckedRouteEvaluation, and
ChiralLargeVolumeIdentificationContract values are
supplied as source hypotheses and consumed only after their exact
routed inputs have been reconstructed. The condensate remains a
premise-free conjectural claim, not a consequence of a neighboring
calculation.
|Iper| = N
⟹
nB(0) + nF(0) ≥ N.
a lower bound on finite-volume zero-energy multiplicity, not an exact vacuum count(13)
anomaly calculationClassical U(1) leaves an exact group of order 2N
The coefficient is 2N; an instanton-phase test preserves this subgroup and obstructs larger ones.
→
condensate conjecture⟨Tr ψψ⟩ ≠ 0 leaves a subgroup of order 2
The nonzero gluino bilinear is a QCD-inspired hypothesis.
→
coset countThe broken symmetry predicts N vacua
One vacuum is assigned to each coset of the residual subgroup.
(# vacua) × 2 = 2N,
# vacua = N.
(14)
The exact 2N-fold symmetry is not encoded as a bare group-order
assertion. ChiralAnomalyCalculation exposes the anomalous
current, coefficient 2N, a contributing instanton sector,
invariance of the candidate subgroup, and the anomaly-phase
obstruction to any larger candidate. The separate
ChiralAnomalyPhaseExactnessBridge turns those tests into
the exact surviving quantum symmetry.
chiral_anomaly_calculation_exhibits_nonvacuous_shared_restricted_symmetry
additionally exposes the completed same-system/current calculation
as a direct nonvacuous shared endpoint.
At the upstream current level,
chiral_current_anomaly_calculation_realizes_structural_quantum_current_anomaly_explanation
records the ordinary complete-route explanation. The typed
pureNOneInstantonABJCurrentObstruction
derivation satisfies
wittenQuantumCurrentAnomalyDerivationObtains, and
chiral_instanton_anomaly_inputs_realize_source_structural_quantum_current_anomaly_explanation
retains the instanton/ABJ current calculation, coefficient 2N,
and nontrivial instanton sector while projecting only the current
anomaly as the shared phenomenon. The subgroup phase tests remain
downstream.
At the full residual-symmetry level,
chiral_anomaly_calculation_realizes_structural_anomaly_restricted_symmetry_explanation
realizes the ordinary explanation directly from the complete
section-8 route. The distinct typed
WittenAnomalyRestrictedSymmetryDerivation.pureNOneInstantonPhaseRestriction
relation is exactly ChiralAnomalyCalculation, and
pure_n1_instanton_phase_restriction_realizes_source_structural_anomaly_restricted_symmetry_explanation
retains the coefficient, instanton sector, subgroup-invariance and
maximality tests, residual order, and exactness bridge without
consuming the established endpoint or ordinary explanation.
The vacuum-counting model separately selects four spacetime
dimensions and infinite spatial volume, so this route cannot be
confused with the finite-volume index calculation.
The composite established result retains both exact child
selections together with direct anomaly, identity-selection, and
multiple-vacuum outputs; its complete route and parent bridge are
reconstructed from those facts.
The finite-vacuum route has a separate provenance boundary.
chiral_symmetry_counting_realizes_structural_finite_vacuum_multiplicity_explanation
executes the anomaly, condensate, and coset route without consuming
the packaged prediction. The typed
WittenFiniteVacuumMultiplicityDerivation.anomalySurvivingZ2NCondensateBreakingToZ2AndCosetCounting
relation is reconstructed by
chiral_symmetry_source_derivation_makes_shared_route_applicable,
and
chiral_symmetry_counting_realizes_source_structural_finite_vacuum_multiplicity_explanation
retains that origin without identifying it with another paper's
finite-vacuum argument.
The full anomaly-selected composition keeps a stronger provenance
boundary. anomaly_condensate_and_counting_realize_structural_anomaly_selected_vacua_explanation
executes anomaly restriction, identity parent selection, condensate
breaking, and coset counting. The typed
WittenAnomalySelectedVacuaDerivation.instantonPhaseRestrictionCondensateAndCosetCounting
relation is reconstructed by
anomaly_selected_source_derivation_makes_shared_route_applicable,
while
anomaly_condensate_and_counting_realize_source_structural_anomaly_selected_vacua_explanation
retains that full section-8 origin without consuming the composite
endpoint.
Why the periodic zero-mode tower is suggestive
If the true-invariant Weyl branch is selected, the N tower
states carry chiral charges 0, 2, …, 2N − 2. A separate
premise assigns the appropriate vacuum transformation law to their
Fourier combinations. A final large-volume identification then says
that the finite-volume ground states approach the proposed chiral
vacua.
A
Anomaly
The imported calculation leaves an exact discrete chiral group of order 2N.
imported resultC
Condensate
The gluino bilinear is assumed nonzero and leaves the twofold subgroup.
conjectureB
Branch and limit
The true-invariant branch is selected and finite-volume states are identified with large-volume vacua.
author expectationsI
Vacuum reading
The matching multiplicity, charges, Fourier candidates, and count support the chiral-vacuum picture.
Under the periodic index route, anomaly calculation, nonzero
condensate hypothesis, discrete-symmetry counting rule,
true-invariant branch selection, Fourier transformation law, and
large-volume identification, the formalization supports an
interpretation with N infinite-volume chiral vacua.
The index does not independently calculate exactly N chiral vacua
Its checked arithmetic output is only that at least N
zero-energy states occur at finite volume. Exact chiral-vacuum
counting uses the anomaly, condensate, and coset rule; identifying
those vacua with the finite-volume tower additionally uses branch
selection, Fourier evidence, and the large-volume expectation.
Formal auditComplete chiral interpretation contract and epistemic origins
PeriodicIndexCalculation and the foundational route give the finite-volume lower bound of N zero-energy states.
ChiralAnomalyCalculation supplies the anomalous current, coefficient 2N, nontrivial instanton-sector contribution, candidate-subgroup phase invariance, obstruction of larger subgroups, and the exactness bridge.
GluinoCondensateBreakingHypothesis conjectures a nonzero gluino bilinear and residual subgroup order 2.
DiscreteChiralBreakingVacuumCountRule supplies the coset-to-vacuum counting bridge.
trueInvariantBranchSelectedForChiralInterpretation selects the tower branch whose charges are 2p.
fourierCombinationsHaveVacuumTransformationLaw supplies the Fourier-candidate evidence.
finiteVolumeGroundStatesApproachChiralVacua supplies the large-volume identification.
Epistemic origin ledger
Ingredient
Source-map status
Used for main nonbreaking theorem?
Chiral anomaly, instanton-phase tests, and exact group order 2N
imported result
No
Nonzero gluino condensate
conjecture
No
True-invariant branch selection
author expectation
No
Finite-to-infinite-volume vacuum identification
author expectation
No
Final chiral-vacuum reading
interpretive inference
No
The source-map audit records all three origin classes explicitly:
conjecture, author expectation, and interpretive inference. It
separately proves that the main nonbreaking conclusion is
independent of the condensate conjecture.
sec. 3, p. 269 · anomalous divergence; sec. 8, pp. 290–300 · coefficient and instanton-phase calculation, condensate proposal, charge tower, Fourier candidates, and qualified vacuum interpretation
Appendices
Selected-State Scope
Admissibility retains the infinite-volume regime and the bounded phase label. Selection additionally matches the color number to the law and retains the condensate hypothesis.
The actual modular generator is proved functional and preserves every selected vacuum in that family. Selection does not prove the condensate or identify finite-volume zero modes with infinite-volume vacua.
the_chiral_generator_is_a_selected_vacuum_action
Symmetry Role Scope
The chiral adapter retains a physical transformation, the actual phase action, and a law-specific selected vacuum family.
Condensate existence and the infinite-volume interpretation remain inputs; no descriptive, variational, or conservation classification is filled in.
Linked terms elsewhere in the article lead to the equation, route,
or paper-local definition that controls their meaning. These entries
are extracted from the Witten Lean semantics rather than from an
external glossary. A “Return to reading” control appears after a
concept link is followed.
Supersymmetric index
The index is an integer attached to a regime. Under the
finite-volume pairing assumptions it equals the bosonic minus
fermionic zero-energy multiplicity.
indexFromZeroEnergyCounts · IndexRealizedAt
Supersymmetric finite-volume setup
The regime has finite spatial volume, matching boundary
conditions for bosons and fermions, and a boundary condition
that preserves supersymmetry.
SupersymmetricFiniteVolumeSetup
Positive-energy Bose–Fermi pairing
Paired positive-energy states cancel in the regulated graded
trace. This is the bridge that turns the trace into a statement
purely about zero-energy multiplicities.
PositiveEnergyCancellationPrinciple
Logical meaning of I = 0
A zero index may mean that no zero-energy states exist, or that
nonzero bosonic and fermionic zero-energy counts cancel. Its
value alone does not decide supersymmetry breaking.
ZeroIndexIsLogicallyInconclusive
Periodic, magnetic, and global-symmetry twists
Ordinary periodic and magnetic-flux-twisted regulators are the
two SU(N) calculations. A third, globally twisted
boundary condition is Witten’s separate broken-symmetry
counterexample: it can have zero finite-volume index even when
ordinary periodic boundary conditions retain a nonzero value.
Equality is assumed only for the infinite-volume ground
energies in the magnetic-twist conclusion.
An admissible connection preserves supersymmetry on its selected
parameter path, obeys a uniform source-relative large-field bound,
and excludes incoming states on that path and its derived reversal.
The quantum predicates remain schematic. Index equality is supplied
separately.
At weak coupling, nonzero momentum modes have scale 1/L
while zero modes have scale g2/L.
The low-energy periodic problem is represented by zero-mode
quantization.
PeriodicWeakCouplingReduction
Cartan zero modes and Weyl invariance
Flat periodic connections reduce to commuting Cartan constants.
Periodic gauge shifts and Gauss law make the Cartan coordinates
periodic; identity-connected Weyl transformations and Gauss law
make physical states Weyl invariant. Quantization then produces
the candidate U tower and V multiplet, leaving
the branch ambiguity retained by the formalization.
A zero-energy state is detected from the zero-state counts. A
separate bridge fixes the ground-state energy to zero; breaking
is defined as ground energy different from zero.
The periodic and twisted regulators have separate principles
carrying zero ground energy at every finite volume to their
corresponding infinite-volume energies.
The imported calculation starts from the classical U(1), records
nonconservation of the quantum current, fixes the coefficient at
2N, tests a nontrivial instanton sector, verifies that the
candidate subgroup preserves the instanton measure, and rules
out larger candidates through their anomaly phase. An explicit
bridge identifies the resulting order-2N subgroup as the
exact quantum symmetry.
The gluino bilinear is conjectured to be nonzero and to leave an
unbroken discrete chiral subgroup of order two. This hypothesis
is not used by the index nonbreaking theorem.
GluinoCondensateBreakingHypothesis
Discrete-symmetry vacuum counting
The paper-local count rule equates vacuum count times residual
subgroup order with parent symmetry order. Orders 2N and
2 therefore yield N vacua.
Under the selected true-invariant branch, the tower charges are
2p. Fourier combinations and their vacuum transformation
law are separate evidence for the proposed chiral-vacuum reading.
Shared charge grounding requires the selected level to lie in
the finite tower, and the output-led endpoint directly retains
functional realization, occupancy, and uniqueness.
The raw exact-map tower route also realizes an ordinary
structural explanation of the selected chiral assignment.
A typed Cartan-pair and SU charge-sequence source relation
realizes the stronger source-sensitive explanation without
consuming the established endpoint or selecting the unresolved
true-invariant branch.
Shared technical frameworks are related back to the selected
pure gauge theory and represented physical system by explicit
grounding theorems. Grounding is not hidden inside the index
arithmetic.
The source map declares six routes. The coverage module interprets
every atom in the paper-local IndexSystem semantics and
supplies a semantic certificate for every declared route output.
For all six routes, premise-free source hypotheses and
premise-consuming derived steps are now distinguished explicitly.
Complete route inventory
Route
Kind
Depends on
Principal outputs
Finite-volume index logic
Foundational logic
—
Index realization, deformation protection, nonzero-index criterion, zero-index limitation
Ordinary periodic computation
Primary computation
Finite-volume logic
Four explicit section-8 contracts, two Weyl branches, |I| = N, zero ground energy, no infinite-volume breaking
Twisted-boundary computation
Independent computation
Finite-volume logic
I = N, zero twisted finite-volume energy, conditional infinite-volume conclusion
Finite boundary-condition scope
Scope qualification
—
Explicit twist-energy bridge, zero global-twist index, finite boundary indices can differ
Electric-flux sector refinement
Primary computation
Finite-volume, ordinary periodic, and twisted calculations
Zero/nonzero ordinary sectors, every nonparallel twisted vector, and every parallel Tz eigenvalue sector
Shared technical frameworks and their paper-local meanings
Framework realization ledger
Shared framework
Witten paper-local realization
What it does not supply
ProtectedIndexTransport
Calculable periodic or twisted index transported to a selected finite volume
The calculation or admissible connection itself
SupersymmetricVacuumEnergy
Zero-energy state, zero ground energy, and unbroken finite-volume assessment
The nonzero index that produces the state
QuantumCurrentAnomaly and AnomalyRestrictedSymmetry
Classical chiral current, coefficient 2N, instanton-phase restriction, and exact subgroup of order 2N
The gluino condensate
DiscreteSymmetryVacua and AnomalySelectedVacua
Parent order 2N, proper residual order 2 < 2N, and genuinely multiple predicted vacuum count N
The large-volume identification with finite-volume states
Charge
Unique charge 2p for each selected true-invariant tower level
Selection of the true-invariant branch
Technical establishment and physical grounding are distinct
The primary file separately proves the shared technical results
and the theory/model/world grounding contracts. The periodic and
twisted vacuum results are grounded in the selected pure 4D
𝒩 = 1 SU(N) theory and finite-volume gauge system.
The anomaly and chiral-vacuum frameworks likewise retain their
theory and world interpretations instead of treating a generic
framework theorem as the physical conclusion by itself.
Accepts theory scope, periodic finite-volume setup,
positive-energy cancellation, beta-independent regularization,
connections to the calculation
point, deformation invariance, the zero-state energy bridge, and
the periodic large-volume principle. It certifies the index
definition, protected transport, zero-state criterion, and the
logical limitation of a zero index.
Consumes the foundational certificate and accepts the weak-coupling
reduction, Cartan gauge-constraint data, periodic zero-mode
calculation data, and zero-mode index evaluation. It certifies
periodic Cartan variables, Weyl-invariant physical states, both
Weyl branches, the U-tower structure, the unresolved
sign, |I| = N, transport to every periodic volume, and
the infinite-volume no-breaking conclusion inherited through the
foundational route. Those four section-8 inputs are now explicit
premise-free source contracts, and each result step reconstructs
its checked ClaimsHold premises rather than extracting
predecessor outputs from a route certificate.
Consumes the foundational certificate and accepts an explicit
all-volume twisted-regulator contract, a causal section-7
classification contract, a checked sector-sum contract,
protected connections, the zero-state energy bridge, twisted
large-volume limit, and boundary-condition independence. The
source graph makes zero-mode elimination feed pure-gauge
classification, then feeds the one-sector result and fixed-theta
count into the final sum. Coverage reconstructs every selected
ClaimsHold premise separately, certifying
I = N, finite-volume zero energy, and the conditional
infinite-volume conclusion without replaying a route certificate.
Accepts a commuting spontaneously broken global symmetry, the
twist-energy bridge, realization of the globally twisted index,
the spectral zero-state exclusion bridge, and the possibility of
a nonzero ordinary periodic index. It certifies zero index for the
global-symmetry twist and finite-volume boundary-index dependence
without using the magnetic-flux route. A checked independence
guard also prevents this general warning from borrowing either
the ordinary zero-mode evaluation or the magnetic-flux sector sum.
Consumes the certified ordinary and twisted calculations plus
an explicit causal flux-data contract and separate checked
ordinary/twisted sector-evaluation contracts. Pure scope,
periodic gauge shifts, the bosonic zero-mode result, the
calculated twisted index, and fixed-theta counting are consumed
through their selected ClaimsHold premises before
producing commutation, shift, support, transverse, and parallel
facts. The ordinary branch keeps the zero-flux sign unresolved;
the twisted branch separately certifies zero index for every
nonparallel vector and index one with zero ground energy for
every allowed parallel Tz eigenvalue.
Consumes the foundational and ordinary-periodic certificates and
accepts the anomaly calculation, condensate hypothesis, vacuum
counting rule, true-invariant branch selection, Fourier-candidate
law, and large-volume vacuum identification. It certifies the
finite-volume multiplicity, the N-vacuum count, and the
qualified chiral-vacuum interpretation.
Claims the formalization refuses to overstate
A zero index does not by itself imply supersymmetry breaking.
The finite-volume zero-state conclusion requires the index to be realized through positive-energy pairing and a beta-independent regulated trace.
zero_energy_difference_does_not_supply_beta_independent_regularization keeps the zero-energy multiplicity formula from supplying beta-independent regularization.
positive_energy_pairing_without_principle_does_not_supply_beta_independence keeps a bare pairing label from replacing the imported pairing-to-regularization bridge.
Index invariance requires admissibility; states entering from infinity are explicitly excluded.
The periodic zero-mode calculation does not resolve the Weyl representation sign.
The sign ambiguity does not affect the common magnitude |I| = N.
The twisted finite-volume theorem does not require infinite-volume boundary-condition independence.
The twisted infinite-volume theorem does require that author expectation.
Infinite-volume boundary-energy agreement does not imply equal finite-volume indices.
The broken-global-symmetry twist example is distinct from the magnetic-flux twist used in the SU(N) calculation.
The source graph names every inherited periodic, twisted, flux, and chiral output, and checks that the later flux, chiral, and boundary-scope routes do not feed back into the main nonbreaking conclusion.
The index proves at least N finite-volume zero-energy states, not exactly N infinite-volume chiral vacua.
The gluino condensate is a conjectural input and is irrelevant to the main nonbreaking theorem.
The true-invariant branch and finite-to-infinite-volume vacuum identification remain author expectations.
Lean-encoded bibliographic record
E. Witten, Constraints on Supersymmetry Breaking (1982).
The formalization parses sections 2, 7, and 8; all locators in this
edition are those stored in SourceMap.lean.
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.
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.