`ComparatorChallenges/VertexAlgebraNet.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 120 curated AI-written annotations and 1273 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 9 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. Those 9 annotations were revised on 2026-10-14 to bind all of them to exact page-one quotations from 6 pinned OpenAI manuscripts, the matching pinned formalization notes, and named historical primary sources; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. The 3 manuscript-grounded matrix-multiplication annotations were composed from pinned OpenAI manuscripts and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Six additional target-specific annotations for Catalan's constant, the irrationality exponent of pi, dyadic Erdos similarity, Unique Games, quintic Lienard systems, and Yau nodal growth were composed from pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for Seymour's second-neighborhood conjecture, Cannon's conjecture, Koebe circle-domain uniformization, and circle-domain rigidity were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for simple Lebesgue spectrum on the three-torus, Rokhlin multiple mixing, the nine-dimensional Borsuk counterexample, and Tingley's sphere-isometry problem were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the Abhyankar-Sathaye counterexample, the circulant Hadamard conjecture, the Eilenberg-Ganea counterexample, and short Egyptian fractions were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and quotations from Sathaye, Turyn with Storer, Eilenberg with Ganea, and Erdos in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the complete Crouzeix inequality, unrestricted Euclidean five-coloring, the general Mahler conjecture, and Thompson F nonamenability were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, the matching formalization notes, and linked historical sources by Crouzeix, de Grey, Mahler, Cannon, Floyd, and Parry; no human review is claimed. Four additional target-specific annotations for algorithmic strong thin trees, Barnette's Hamiltonian-cycle conjecture, the explicit Artin CAT(0) counterexample, and generalized amplitude-damping capacity were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, matching formalization notes, and linked historical sources by Anari with Goddyn, Arts with Hopman and Timmermans, Charney, and Khatri with Sharma and Wilde; no human review is claimed. Fourteen solution-level annotations for Seymour's second-neighborhood theorem were added on 2026-10-14 from the pinned 15-page manuscript, covering the exact-neighborhood model, minimal-counterexample reduction, subset deficit, transitive blowup, pruning, extremal contradiction, and proof-bearing final theorem. The subset and blowup annotations also bind exact quotations from Tyler Seacrest (2019) and Brantner, Brockman, Kay, and Snively (2009); no human review is claimed. Thirty-seven declaration-level annotations were added on 2026-10-14 for the rest of ComparatorChallenges/AlgorithmicThinTrees.lean, covering its multigraph semantics, stack-machine cost model, explicit and binary-multiplicity encodings, output certificate, and complete algorithmic proposition. They are bound to the pinned OpenAI manuscript and linked earlier thin-tree sources by Luis Goddyn, Nima Anari, and Shayan Oveis Gharan; the annotation explicitly preserves that the concluding comparator theorem still has body by sorry, and no human review is claimed. Thirty-one supporting declarations in ComparatorChallenges/UniqueGamesTheorem.lean were upgraded or added on 2026-10-14, completing curated coverage of all thirty-two declarations in that file. The annotations cover the finite game model, Boolean formula and parser, binary-vector translations, explicit output serialization, finite-alphabet machine contract, and gap theorem; they distinguish manuscript-backed mathematics from Lean-specific canonical parsing and unary game output encoding, bind exact OpenAI manuscript quotations, and credit Subhash Khot through his 2010 survey. The comparator theorem remains explicitly labeled as having body by sorry; no human review is claimed. The source-derived generator was hardened on 2026-10-14 so a formalization note covering multiple comparator results no longer copies its combined Scope prose onto each declaration; those targets now use declaration-specific source comments when available and otherwise remain explicitly bounded to the exact Lean statement. The remaining source-derived annotations were generated mechanically from pinned formalization notes and source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the source prose.

Credit: OpenAI's pinned repository says an unreleased internal model produced the manuscripts and supporting proof artifacts. The collection builds on the mathematics community's accumulated problems, results, language, and formal libraries; each manuscript's BibTeX and bibliography are the authority for individual scholarly credit.

Project sources: Pinned openai/math documentation, Pinned formalization catalogue

89 declarations, 1 contain sorry, 0 declared as axioms

The strongly rational unitary vertex operator algebra main result, clause (1).

abbrev OAI.StronglyRationalVOA.StateField line 11
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/FiniteSums.lean:135 (different declaration text). Qualify references as OAI.StronglyRationalVOA.StateField@path.
def OAI.StronglyRationalVOA.mode line 15
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/FiniteSums.lean:140 (different declaration text). Qualify references as OAI.StronglyRationalVOA.mode@path.
def OAI.StronglyRationalVOA.JacobiIdentity line 19
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/FiniteSums.lean:187 (different declaration text). Qualify references as OAI.StronglyRationalVOA.JacobiIdentity@path.
structure OAI.StronglyRationalVOA.VertexAlgebra line 27
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/FiniteSums.lean:196 (same statement text; different proof or body). Qualify references as OAI.StronglyRationalVOA.VertexAlgebra@path.
def OAI.StronglyRationalVOA.VertexAlgebra.translation line 41
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:35 (different declaration text). Qualify references as OAI.StronglyRationalVOA.VertexAlgebra.translation@path.
def OAI.StronglyRationalVOA.VertexAlgebra.c2 line 46
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:41 (different declaration text). Qualify references as OAI.StronglyRationalVOA.VertexAlgebra.c2@path.
def OAI.StronglyRationalVOA.VertexAlgebra.C2Cofinite line 49
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:49 (different declaration text). Qualify references as OAI.StronglyRationalVOA.VertexAlgebra.C2Cofinite@path.
def OAI.StronglyRationalVOA.VertexAlgebra.IsIdeal line 51
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:52 (different declaration text). Qualify references as OAI.StronglyRationalVOA.VertexAlgebra.IsIdeal@path.
def OAI.StronglyRationalVOA.VertexAlgebra.IsSimple line 55
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:57 (different declaration text). Qualify references as OAI.StronglyRationalVOA.VertexAlgebra.IsSimple@path.
def OAI.StronglyRationalVOA.ModuleJacobiIdentity line 63
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:64 (different declaration text). Qualify references as OAI.StronglyRationalVOA.ModuleJacobiIdentity@path.
structure OAI.StronglyRationalVOA.WeakModule line 72
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:75 (different declaration text). Qualify references as OAI.StronglyRationalVOA.WeakModule@path.
def OAI.StronglyRationalVOA.WeakModule.IsInvariant line 85
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:87 (different declaration text). Qualify references as OAI.StronglyRationalVOA.WeakModule.IsInvariant@path.
def OAI.StronglyRationalVOA.WeakModule.CompletelyReducible line 88
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:91 (different declaration text). Qualify references as OAI.StronglyRationalVOA.WeakModule.CompletelyReducible@path.
structure OAI.StronglyRationalVOA.CFTTypeVOA line 97
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:116 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA@path.
def OAI.StronglyRationalVOA.CFTTypeVOA.virasoroMode line 126
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:144 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.virasoroMode@path.
def OAI.StronglyRationalVOA.CFTTypeVOA.physicalMode line 129
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:148 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.physicalMode@path.
def OAI.StronglyRationalVOA.CFTTypeVOA.energyShift line 132
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:152 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.energyShift@path.
def OAI.StronglyRationalVOA.CFTTypeVOA.gradingEquiv line 134
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/VacuumMemNegative.lean:159 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.gradingEquiv@path.
def OAI.StronglyRationalVOA.CFTTypeVOA.IsAdmissible line 137
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/GradeComponentC.lean:316 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.IsAdmissible@path.
def OAI.StronglyRationalVOA.CFTTypeVOA.IsRational line 144
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/GradeComponentC.lean:325 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.IsRational@path.
def OAI.StronglyRationalVOA.CFTTypeVOA.PolynomialEnergyBound line 155
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/GradeComponentC.lean:386 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.PolynomialEnergyBound@path.
def OAI.StronglyRationalVOA.CFTTypeVOA.PolynomialEnergyBounds line 159
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/GradeComponentC.lean:392 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.PolynomialEnergyBounds@path.
structure OAI.StronglyRationalVOA.CFTTypeVOA.UnitaryStructure line 161
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/BlockGrowth.lean:416 (same statement text; different proof or body). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.UnitaryStructure@path.
def OAI.StronglyRationalVOA.HilbertCore._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.HilbertCore.space line 183
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:11 (same declaration text).
def OAI.StronglyRationalVOA.HilbertCore._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.HilbertCore.equiv line 187
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:15 (same declaration text).
def OAI.StronglyRationalVOA.HilbertCore._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.HilbertCore.operator line 192
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:20 (same declaration text).
def OAI.StronglyRationalVOA.RapidCoefficients line 201
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/HilbertCore.lean:63 (different declaration text). Qualify references as OAI.StronglyRationalVOA.RapidCoefficients@path.
def OAI.StronglyRationalVOA.SummableLinearFamily.tsumMap line 211
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/HilbertCore.lean:110 (different declaration text). Qualify references as OAI.StronglyRationalVOA.SummableLinearFamily.tsumMap@path.
structure OAI.StronglyRationalVOA.SmoothCircleFunction line 223
Also declared: OAI/RepresentationTheory/VertexAlgebra/HilbertCore.lean:243 (same declaration text).
instance OAI.StronglyRationalVOA.SmoothCircleFunction.(instance at line 234) line 234
def OAI.StronglyRationalVOA.SmoothCircleFunction._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleFunction.coefficient line 236
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:30 (same declaration text).
abbrev OAI.StronglyRationalVOA.ClosedGraphData._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.ClosedGraphData.Double line 249
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:43 (same declaration text).
def OAI.StronglyRationalVOA.ClosedGraphData._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.ClosedGraphData.graph line 251
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:45 (same declaration text).
def OAI.StronglyRationalVOA.ClosedGraphData._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.ClosedGraphData.projection line 254
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:48 (same declaration text).
def OAI.StronglyRationalVOA.ClosedGraphData._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.ClosedGraphData.inclusion line 262
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:56 (same declaration text).
def OAI.StronglyRationalVOA.ClosedGraphData._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.ClosedGraphData.coordinate line 266
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:60 (same declaration text).
def OAI.StronglyRationalVOA.ClosedGraphData._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.ClosedGraphData.entry line 270
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:64 (same declaration text).
def OAI.StronglyRationalVOA.LocalOperatorAlgebra._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.LocalOperatorAlgebra.generated line 281
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:75 (same declaration text).
def OAI.StronglyRationalVOA.SmoothCircleFunction._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleFunction.onCircle line 296
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:90 (same declaration text).
structure OAI.StronglyRationalVOA.ProperCircleArc line 305
Also declared: OAI/RepresentationTheory/VertexAlgebra/GraphCommutants.lean:127 (same declaration text).
def OAI.StronglyRationalVOA.ProperCircleArc._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.ProperCircleArc.carrier line 317
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:99 (same declaration text).
def OAI.StronglyRationalVOA.ProperCircleArc._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.ProperCircleArc.complement line 320
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:102 (same declaration text).
def OAI.StronglyRationalVOA.BoundedFlow._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.BoundedFlow.flow line 339
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:121 (same declaration text).
structure OAI.StronglyRationalVOA.SmoothCircleLift line 346
Also declared: OAI/RepresentationTheory/VertexAlgebra/RotationConjugateCommutant.lean:201 (same declaration text).
instance OAI.StronglyRationalVOA.SmoothCircleLift.(instance at line 357) line 357
def OAI.StronglyRationalVOA.SmoothCircleLift._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleLift.identity line 359
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:128 (same declaration text).
def OAI.StronglyRationalVOA.SmoothCircleLift._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleLift.compose line 365
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:134 (same declaration text).
def OAI.StronglyRationalVOA.SmoothCircleLift._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleLift.inverse line 371
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:140 (same declaration text).
instance OAI.StronglyRationalVOA.SmoothCircleLift._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleLift.instGroup line 380
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:149 (same declaration text).
def OAI.StronglyRationalVOA.SmoothCircleLift._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleLift.onCircle line 412
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:181 (same declaration text).
def OAI.StronglyRationalVOA.SmoothCircleLift._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleLift.circleHomeomorph line 425
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:194 (same declaration text).
def OAI.StronglyRationalVOA.SmoothCircleLift._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleLift.circleAction line 450
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:219 (same declaration text).
def OAI.StronglyRationalVOA._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleDiffeomorphism line 470
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:239 (same declaration text).
instance OAI.StronglyRationalVOA._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.instGroupSmoothCircleDiffeomorphism line 472
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:241 (same declaration text).
def OAI.StronglyRationalVOA.scalarPhaseSubgroup line 480
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/DiffeomorphismStressSmooth.lean:300 (different declaration text). Qualify references as OAI.StronglyRationalVOA.scalarPhaseSubgroup@path.
instance OAI.StronglyRationalVOA.scalarPhaseSubgroup_normal line 499
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/DiffeomorphismStressSmooth.lean:318 (different declaration text). Qualify references as OAI.StronglyRationalVOA.scalarPhaseSubgroup_normal@path.
abbrev OAI.StronglyRationalVOA.ProjectiveUnitaryGroup line 508
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/DiffeomorphismStressSmooth.lean:328 (different declaration text). Qualify references as OAI.StronglyRationalVOA.ProjectiveUnitaryGroup@path.
def OAI.StronglyRationalVOA.SmoothCircleLift.jet line 517
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/SmoothTopology.lean:99 (different declaration text). Qualify references as OAI.StronglyRationalVOA.SmoothCircleLift.jet@path.
instance OAI.StronglyRationalVOA.SmoothCircleLift.(instance at line 521) line 521
instance OAI.StronglyRationalVOA.unitaryStrongTopology line 530
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/ProjectiveTopology.lean:12 (different declaration text). Qualify references as OAI.StronglyRationalVOA.unitaryStrongTopology@path.
instance OAI.StronglyRationalVOA._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.instTopologicalSpaceSmoothCircleDiffeomorphism line 534
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:250 (same declaration text).
def OAI.StronglyRationalVOA.CircleMobiusGeometry._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.Rel line 543
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:259 (same declaration text).
def OAI.StronglyRationalVOA.CircleMobiusGeometry._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.Implements line 547
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:263 (same declaration text).
def OAI.StronglyRationalVOA.CircleMobiusGeometry._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.lifts line 549
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:265 (same declaration text).
def OAI.StronglyRationalVOA.CircleMobiusGeometry._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.rotationMatrix line 598
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:314 (same declaration text).
abbrev OAI.StronglyRationalVOA.CircleMobiusGeometry._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.Group line 607
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:323 (same declaration text).
def OAI.StronglyRationalVOA.CircleMobiusGeometry._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.CircleMobiusGeometry.inclusion line 609
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:325 (same declaration text).
def OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleDiffeomorphism._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.SmoothCircleDiffeomorphism.action line 621
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:337 (same declaration text).
def OAI.StronglyRationalVOA._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.projectiveConjugation line 631
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:347 (same declaration text).
def OAI.StronglyRationalVOA._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.globalObservableAlgebra line 654
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:370 (same declaration text).
def OAI.StronglyRationalVOA._root_.OAI.MinimalVertex.Code.StronglyRationalVOA.globalCyclicSpace line 659
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:375 (same declaration text).
abbrev OAI.StronglyRationalVOA.CFTTypeVOA.GradedDual line 668
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/SourceStrongRationality.lean:10 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.GradedDual@path.
def OAI.StronglyRationalVOA.CFTTypeVOA.gradedDualPairing line 670
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/SourceStrongRationality.lean:13 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.gradedDualPairing@path.
def OAI.StronglyRationalVOA.CFTTypeVOA.IsSelfContragredient line 674
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/SourceStrongRationality.lean:18 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.IsSelfContragredient@path.
def OAI.StronglyRationalVOA.CFTTypeVOA.IsStronglyRational line 682
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/SourceStrongRationality.lean:27 (different declaration text). Qualify references as OAI.StronglyRationalVOA.CFTTypeVOA.IsStronglyRational@path.
def OAI.MinimalVertex.smearedMap line 694
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:388 (same declaration text).
def OAI.MinimalVertex.smoothSmearedField line 704
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:398 (same declaration text).
def OAI.MinimalVertex.closedSmoothField line 713
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:407 (same declaration text).
def OAI.MinimalVertex.localGraphData line 717
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:411 (same declaration text).
def OAI.MinimalVertex.localAlgebra line 723
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:417 (same declaration text).
def OAI.MinimalVertex.intervalAlgebra line 727
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:421 (same declaration text).
def OAI.MinimalVertex.CKLWStrongLocal line 731
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:425 (same declaration text).
def OAI.MinimalVertex.resolvent line 744
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:438 (same declaration text).
def OAI.MinimalVertex.boundedGenerator line 750
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:444 (same declaration text).
def OAI.MinimalVertex.approximatingFlow line 756
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:450 (same declaration text).
def OAI.MinimalVertex.flow line 767
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:461 (same declaration text).
def OAI.MinimalVertex.projectiveStandardAction line 775
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:469 (same declaration text).
structure OAI.MinimalVertex.IrreducibleConformalNetStructure line 783
Also declared: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:477 (same declaration text).
theorem OAI.MinimalVertex.main Challenge statement: Unitary vertex operator algebras and conformal nets sorry line 822
L4DEL explanation
The linked formalization proves the first construction step relating strongly rational unitary vertex operator algebras to conformal nets. For a simple unitary strongly rational vertex operator algebra, it establishes polynomial energy bounds and strong locality and constructs an irreducible conformal net with the stated covariance, vacuum, and positive-energy properties. The selected statement does not include complete rationality of the net, unitarizability of all simple modules, the braided tensor equivalence, or the classification of local extensions described in the paper.
1. This is an exact comparator target linked by the pinned repository's formalization note for this result.
2. The repository's comparator configuration selects OAI.RepresentationTheory.VertexAlgebra.MinimalComparator as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/RepresentationTheory/VertexAlgebra/MinimalComparator.lean:676 (different declaration text). Qualify references as OAI.MinimalVertex.main@path.
```
theorem main (_hSimple : A.toVertexAlgebra.IsSimple)
    (hSR : A.IsStronglyRational.{u,v}) :
    A.PolynomialEnergyBounds ∧ OAI.MinimalVertex.CKLWStrongLocal U hSR.2.2 ∧
      Nonempty (OAI.MinimalVertex.IrreducibleConformalNetStructure (OAI.MinimalVertex.intervalAlgebra U hSR.2.2)) := by
…
```
Mentions: OAI.MinimalVertex.CKLWStrongLocal, OAI.MinimalVertex.IrreducibleConformalNetStructure, OAI.MinimalVertex.intervalAlgebra