`ComparatorChallenges/UniformGamma.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

68 declarations, 1 contain sorry, 0 declared as axioms

The final claim constructs the tracial quotients and states the original gamma assertion. Analysis.TracialSplitting.Challenge.main_iff_original verifies statement preservation.

structure OAI.Pureness.CStarAlgebraNorm line 18
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:18 (same declaration text).
structure OAI.Pureness.TracialState line 22
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:22 (same declaration text).
structure OAI.Pureness.ClosedStarIdeal line 27
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:27 (same declaration text).
abbrev OAI.ComparatorModel.Pureness.CStarAlgebraNorm line 42
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:44 (same declaration text).
def OAI.ComparatorModel.Pureness.IsNuclear line 44
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:46 (same declaration text).
abbrev OAI.ComparatorModel.Pureness.TracialState line 47
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:49 (same declaration text).
def OAI.ComparatorModel.CurrentMain.RealRankZero line 57
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:59 (same declaration text).
def OAI.ComparatorModel.CurrentMain.traceTwoNorm line 61
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:63 (same declaration text).
abbrev OAI.ComparatorModel.Pureness.ClosedStarIdeal line 73
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:75 (different declaration text). Qualify references as OAI.ComparatorModel.Pureness.ClosedStarIdeal@path.
abbrev OAI.ComparatorModel.Pureness.ClosedStarIdeal.Quotient line 81
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:81 (same declaration text).
def OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientMap line 83
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:83 (same declaration text).
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientStar line 85
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:85 (same declaration text).
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientStarRing line 93
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:93 (same declaration text).
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientStarModule line 98
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:98 (same declaration text).
abbrev OAI.ComparatorModel.Pureness.ClosedStarIdeal.idealSubmodule line 101
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:101 (same declaration text).
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.idealSubmodule_closed line 104
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:104 (same declaration text).
def OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientLinearEquiv line 112
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:112 (same declaration text).
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientNormedAddCommGroup line 116
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:116 (same declaration text).
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientNormedSpace line 120
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:120 (same declaration text).
def OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientNormEquiv line 123
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:123 (same declaration text).
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientCompleteSpace line 128
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:128 (same declaration text).
class OAI.ComparatorModel.Pureness.ClosedStarIdeal.NormConstruction line 131
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:131 (same declaration text).
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientNormedRing line 141
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:141 (same declaration text).
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientNormedAlgebra line 145
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:145 (same declaration text).
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientNormedStarGroup line 148
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:148 (same declaration text).
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientCStarAlgebra line 163
Also declared: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:167 (same declaration text).
def OAI.ComparatorModel.CurrentMain.familyTwoNorm line 182
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:184 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.familyTwoNorm@path.
instance OAI.ComparatorModel.CurrentMain.boundedFamilyProductCStar line 192
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:194 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.boundedFamilyProductCStar@path.
def OAI.ComparatorModel.CurrentMain.IsFamilyNull line 201
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:203 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.IsFamilyNull@path.
class OAI.ComparatorModel.CurrentMain.NullConstruction line 204
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:206 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.NullConstruction@path.
def OAI.ComparatorModel.CurrentMain.familyNullTwoSided line 227
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:229 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.familyNullTwoSided@path.
def OAI.ComparatorModel.CurrentMain.familyNullIdeal line 237
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:239 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.familyNullIdeal@path.
def OAI.ComparatorModel.CurrentMain.IsFamilyCauchy line 259
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:261 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.IsFamilyCauchy@path.
class OAI.ComparatorModel.CurrentMain.CauchyConstruction line 262
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:264 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.CauchyConstruction@path.
def OAI.ComparatorModel.CurrentMain.familyCauchyAlgebra line 283
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:285 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.familyCauchyAlgebra@path.
instance OAI.ComparatorModel.CurrentMain.familyCauchy_complete line 294
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:296 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.familyCauchy_complete@path.
instance OAI.ComparatorModel.CurrentMain.familyCauchy_cstar line 297
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:299 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.familyCauchy_cstar@path.
def OAI.ComparatorModel.CurrentMain.familyCauchyNullIdeal line 299
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:301 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.familyCauchyNullIdeal@path.
abbrev OAI.ComparatorModel.CurrentMain.UniformTracialCompletion line 309
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:311 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.UniformTracialCompletion@path.
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientOrder line 321
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:323 (different declaration text). Qualify references as OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientOrder@path.
instance OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientStarOrdered line 324
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:326 (different declaration text). Qualify references as OAI.ComparatorModel.Pureness.ClosedStarIdeal.quotientStarOrdered@path.
def OAI.ComparatorModel.Pureness.ClosedStarIdeal.descendedValue line 331
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:333 (different declaration text). Qualify references as OAI.ComparatorModel.Pureness.ClosedStarIdeal.descendedValue@path.
def OAI.ComparatorModel.Pureness.ClosedStarIdeal.descendedLinear line 339
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:341 (different declaration text). Qualify references as OAI.ComparatorModel.Pureness.ClosedStarIdeal.descendedLinear@path.
def OAI.ComparatorModel.Pureness.ClosedStarIdeal.descendedTrace line 344
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:346 (different declaration text). Qualify references as OAI.ComparatorModel.Pureness.ClosedStarIdeal.descendedTrace@path.
instance OAI.ComparatorModel.CurrentMain.boundedFamilyProductOrder line 374
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:378 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.boundedFamilyProductOrder@path.
instance OAI.ComparatorModel.CurrentMain.boundedFamilyProductStarOrdered line 377
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:381 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.boundedFamilyProductStarOrdered@path.
def OAI.ComparatorModel.CurrentMain.ultraTraceValue line 380
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:384 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.ultraTraceValue@path.
class OAI.ComparatorModel.CurrentMain.LimitConstruction line 383
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:387 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.LimitConstruction@path.
def OAI.ComparatorModel.CurrentMain.ultraTraceLinear line 390
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:394 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.ultraTraceLinear@path.
def OAI.ComparatorModel.CurrentMain.boundedFamilyEval line 401
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:405 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.boundedFamilyEval@path.
def OAI.ComparatorModel.CurrentMain.ultraTrace line 410
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:414 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.ultraTrace@path.
abbrev OAI.ComparatorModel.CurrentMain.FamilyUltrapower line 440
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:442 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.FamilyUltrapower@path.
class OAI.ComparatorModel.CurrentMain.TraceConstruction line 442
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:444 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.TraceConstruction@path.
def OAI.ComparatorModel.CurrentMain.familyLimitTrace line 450
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:452 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.familyLimitTrace@path.
def OAI.ComparatorModel.CurrentMain.pullbackTrace line 470
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:478 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.pullbackTrace@path.
instance OAI.ComparatorModel.CurrentMain.familyCauchyOrder line 489
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:495 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.familyCauchyOrder@path.
instance OAI.ComparatorModel.CurrentMain.familyCauchyStarOrdered line 492
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:498 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.familyCauchyStarOrdered@path.
def OAI.ComparatorModel.CurrentMain.cauchyTrace line 495
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:501 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.cauchyTrace@path.
def OAI.ComparatorModel.CurrentMain.completionTrace line 499
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:505 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.completionTrace@path.
def OAI.ComparatorModel.CurrentMain.boundedFamilyConstant line 509
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:515 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.boundedFamilyConstant@path.
instance OAI.ComparatorModel.CurrentMain.completionNontrivial line 512
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:532 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.completionNontrivial@path.
def OAI.ComparatorModel.CurrentMain.familyUltrapowerMap line 528
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:552 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.familyUltrapowerMap@path.
def OAI.ComparatorModel.Pureness.TopologicallySimple line 549
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:573 (different declaration text). Qualify references as OAI.ComparatorModel.Pureness.TopologicallySimple@path.
def OAI.ComparatorModel.CurrentMain.StablyFinite line 560
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:586 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.StablyFinite@path.
def OAI.ComparatorModel.CurrentMain.UniformPropertyGammaAt line 564
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:590 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.UniformPropertyGammaAt@path.
def OAI.ComparatorModel.CurrentMain.Statement line 580
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:604 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.Statement@path.
def OAI.ComparatorModel.CurrentMain.MainClaim line 592
Distinct definitions of this name: OAI/Analysis/TracialSplitting/ChallengeDefinitions.lean:620 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.MainClaim@path.
theorem OAI.ComparatorModel.CurrentMain.main Challenge statement: Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability sorry line 600
L4DEL explanation
The formalization proves that real rank zero of the uniform tracial ultrapower of the uniform tracial completion implies uniform property $\Gamma$ for a simple separable unital infinite-dimensional nuclear stably finite $C^*$-algebra with traces. The conclusion holds at every specified free ultrafilter under the corresponding real-rank-zero hypothesis. The linked comparison results also prove Jiang–Su absorption and uniform property $\Gamma$ for simple separable unital infinite-dimensional nuclear algebras with strict comparison, and for the stated stably projectionless nuclear algebras with bounded densely finite traces, a nonempty compact normalized trace base, and the prescribed finite-target-rank comparison condition.
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.Analysis.TracialSplitting.Challenge as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/TracialSplitting/Challenge.lean:75 (different declaration text). Qualify references as OAI.ComparatorModel.CurrentMain.main@path.
```
theorem CurrentMain.main : MainClaim.{uNuclear} := by
…
```