`ComparatorChallenges/ProjectionMoments.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

91 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.NoiselessRegression.Vector line 17
Also declared: ComparatorChallenges/MemoryPrecision.lean:133 (same declaration text), ComparatorChallenges/NoiselessRegression.lean:18 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:17 (same declaration text).
abbrev OAI.NoiselessRegression.UnitSphere line 19
Also declared: ComparatorChallenges/MemoryPrecision.lean:134 (same declaration text), ComparatorChallenges/NoiselessRegression.lean:19 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:19 (same declaration text).
abbrev OAI.NoiselessRegression.State line 21
Also declared: ComparatorChallenges/NoiselessRegression.lean:20 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:21 (same declaration text).
abbrev OAI.NoiselessRegression.Samples line 23
Also declared: ComparatorChallenges/NoiselessRegression.lean:21 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:23 (same declaration text).
def OAI.NoiselessRegression.uniformSphere line 25
Also declared: ComparatorChallenges/NoiselessRegression.lean:23 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:25 (same declaration text).
def OAI.NoiselessRegression.sampleLaw line 29
Also declared: ComparatorChallenges/NoiselessRegression.lean:27 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:29 (same declaration text).
def OAI.NoiselessRegression.observationReference line 32
Also declared: ComparatorChallenges/NoiselessRegression.lean:30 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:32 (same declaration text).
structure OAI.NoiselessRegression.Learner line 35
Also declared: ComparatorChallenges/NoiselessRegression.lean:33 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:35 (same declaration text).
abbrev OAI.NoiselessRegression.Learner.RunStatus line 42
Also declared: ComparatorChallenges/NoiselessRegression.lean:40 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:42 (same declaration text).
def OAI.NoiselessRegression.Learner.start line 44
Also declared: ComparatorChallenges/NoiselessRegression.lean:42 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:44 (same declaration text).
def OAI.NoiselessRegression.Learner.run line 49
Also declared: ComparatorChallenges/NoiselessRegression.lean:47 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:49 (same declaration text).
def OAI.NoiselessRegression.Learner.terminal line 62
Also declared: ComparatorChallenges/NoiselessRegression.lean:60 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:62 (same declaration text).
def OAI.NoiselessRegression.Learner.estimate line 68
Also declared: ComparatorChallenges/NoiselessRegression.lean:66 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:68 (same declaration text).
structure OAI.NoiselessRegression.Learner.Admissible line 73
Also declared: ComparatorChallenges/NoiselessRegression.lean:71 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:73 (same declaration text).
def OAI.NoiselessRegression.Learner.angularError line 85
Also declared: ComparatorChallenges/NoiselessRegression.lean:83 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:85 (same declaration text).
def OAI.NoiselessRegression.Learner.uniformSuccess line 88
Also declared: ComparatorChallenges/NoiselessRegression.lean:91 (same declaration text), OAI/Probability/ProjectionMoments/Model.lean:88 (same declaration text).
def OAI.NoiselessRegression.sphereLaw line 106
Also declared: OAI/Probability/ProjectionMoments/Model.lean:106 (same declaration text).
def OAI.NoiselessRegression.ambientSphereLaw line 117
Also declared: OAI/Probability/ProjectionMoments/Model.lean:117 (same declaration text).
abbrev OAI.NoiselessRegression.Learner.TaggedState line 127
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:361 (different declaration text). Qualify references as OAI.NoiselessRegression.Learner.TaggedState@path.
instance OAI.NoiselessRegression.Learner.taggedMeasurableSpace line 129
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:363 (different declaration text). Qualify references as OAI.NoiselessRegression.Learner.taggedMeasurableSpace@path.
def OAI.NoiselessRegression.Learner.taggedStart line 131
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:365 (different declaration text). Qualify references as OAI.NoiselessRegression.Learner.taggedStart@path.
structure OAI.NoiselessRegression.Learner.MeasurableRules line 142
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:374 (different declaration text). Qualify references as OAI.NoiselessRegression.Learner.MeasurableRules@path.
structure OAI.NoiselessRegression.FiniteKernelLearner line 153
Also declared: OAI/Probability/ProjectionMoments/Model.lean:381 (same declaration text).
structure OAI.NoiselessRegression.FiniteKernelLearner.CompletedRules line 165
Also declared: OAI/Probability/ProjectionMoments/Model.lean:393 (same declaration text).
def OAI.NoiselessRegression.kernelRun line 177
Also declared: OAI/Probability/ProjectionMoments/Model.lean:403 (same declaration text).
abbrev OAI.NoiselessRegression.FiniteKernelLearner.Tape line 191
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:417 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.Tape@path.
def OAI.NoiselessRegression.FiniteKernelLearner.destination line 193
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:419 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.destination@path.
def OAI.NoiselessRegression.FiniteKernelLearner.taggedKernel line 196
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:422 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.taggedKernel@path.
def OAI.NoiselessRegression.FiniteKernelLearner.terminalOutput line 202
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:428 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.terminalOutput@path.
abbrev OAI.NoiselessRegression.FiniteKernelLearner.Result line 216
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:438 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.Result@path.
def OAI.NoiselessRegression.FiniteKernelLearner.terminalTag line 218
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:440 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.terminalTag@path.
def OAI.NoiselessRegression.FiniteKernelLearner.terminalJointOutput line 222
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:444 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.terminalJointOutput@path.
def OAI.NoiselessRegression.FiniteKernelLearner.jointExperimentLaw line 225
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:447 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.jointExperimentLaw@path.
def OAI.NoiselessRegression.FiniteKernelLearner.uniformSuccess line 228
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:450 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.uniformSuccess@path.
def OAI.NoiselessRegression.FiniteKernelLearner.uniformFixedSuccess line 233
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:455 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.uniformFixedSuccess@path.
def OAI.NoiselessRegression.Learner.uniformFixedSuccess line 251
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:473 (different declaration text). Qualify references as OAI.NoiselessRegression.Learner.uniformFixedSuccess@path.
def OAI.NoiselessRegression.FiniteKernelLearner.seededSuccess line 262
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:490 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.seededSuccess@path.
def OAI.NoiselessRegression.FiniteKernelLearner.seededFixedSuccess line 266
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:494 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.seededFixedSuccess@path.
def OAI.NoiselessRegression.FiniteKernelLearner.FixedQuadraticMemory line 270
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:498 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.FixedQuadraticMemory@path.
def OAI.NoiselessRegression.FiniteKernelLearner.conditionalSuccess line 287
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:513 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.conditionalSuccess@path.
def OAI.NoiselessRegression.FiniteKernelLearner.seededPointwiseSuccess line 291
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:517 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.seededPointwiseSuccess@path.
def OAI.NoiselessRegression.FiniteKernelLearner.JointSuccessMeasurable line 295
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:521 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.JointSuccessMeasurable@path.
def OAI.NoiselessRegression.FiniteKernelLearner.SubquadraticMemoryWithPointwise line 300
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:526 (different declaration text). Qualify references as OAI.NoiselessRegression.FiniteKernelLearner.SubquadraticMemoryWithPointwise@path.
def OAI.ProjectionMoments.vectorSmoothing line 324
Also declared: OAI/Probability/ProjectionMoments/Model.lean:134 (same declaration text).
abbrev OAI.ProjectionMoments.Label line 328
Also declared: OAI/Probability/ProjectionMoments/Model.lean:142 (same declaration text).
def OAI.ProjectionMoments.ballKernel line 330
Also declared: OAI/Probability/ProjectionMoments/Model.lean:144 (same declaration text).
def OAI.ProjectionMoments.rowMap line 334
Also declared: OAI/Probability/ProjectionMoments/Model.lean:153 (same declaration text).
def OAI.ProjectionMoments.ballAverage line 338
Also declared: OAI/Probability/ProjectionMoments/Model.lean:157 (same declaration text).
def OAI.ProjectionMoments.lowerDensity line 342
Also declared: OAI/Probability/ProjectionMoments/Model.lean:161 (same declaration text).
def OAI.ProjectionMoments.AllBallMass line 345
Also declared: OAI/Probability/ProjectionMoments/Model.lean:169 (same declaration text).
abbrev OAI.ProjectionMoments.Rotation line 359
Also declared: OAI/Probability/ProjectionMoments/Model.lean:179 (same declaration text).
instance OAI.ProjectionMoments.rotationMeasurableSpace line 361
Also declared: OAI/Probability/ProjectionMoments/Model.lean:181 (same declaration text).
instance OAI.ProjectionMoments.rotationBorelSpace line 363
Also declared: OAI/Probability/ProjectionMoments/Model.lean:183 (same declaration text).
instance OAI.ProjectionMoments.rotationCompact line 365
Also declared: OAI/Probability/ProjectionMoments/Model.lean:185 (same declaration text).
def OAI.ProjectionMoments.rotationLaw line 377
Also declared: OAI/Probability/ProjectionMoments/Model.lean:197 (same declaration text).
def OAI.ProjectionMoments.rotateRows line 384
Also declared: OAI/Probability/ProjectionMoments/Model.lean:204 (same declaration text).
def OAI.ProjectionMoments.coordinateMap line 395
Also declared: OAI/Probability/ProjectionMoments/Model.lean:215 (same declaration text).
def OAI.ProjectionMoments.coordinateDensity line 398
Also declared: OAI/Probability/ProjectionMoments/Model.lean:226 (same declaration text).
def OAI.ProjectionMoments.frameLaw line 401
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:236 (same statement text; different proof or body). Qualify references as OAI.ProjectionMoments.frameLaw@path.
def OAI.ProjectionMoments.rieszKernel line 408
Also declared: OAI/Probability/ProjectionMoments/Model.lean:246 (same declaration text).
def OAI.ProjectionMoments.rieszPotential line 410
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:248 (same statement text; different proof or body). Qualify references as OAI.ProjectionMoments.rieszPotential@path.
def OAI.ProjectionMoments.mixRows line 417
Also declared: OAI/Probability/ProjectionMoments/Model.lean:258 (same declaration text).
def OAI.ProjectionMoments.polarRadial line 420
Also declared: OAI/Probability/ProjectionMoments/Model.lean:261 (same declaration text).
def OAI.ProjectionMoments.rawPolarRows line 423
Also declared: OAI/Probability/ProjectionMoments/Model.lean:264 (same declaration text).
def OAI.ProjectionMoments.polarRows line 426
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:267 (same statement text; different proof or body). Qualify references as OAI.ProjectionMoments.polarRows@path.
def OAI.ProjectionMoments.unitBallVolume line 434
Also declared: OAI/Probability/ProjectionMoments/Model.lean:278 (same declaration text).
def OAI.ProjectionMoments.unitBallLaw line 436
Also declared: OAI/Probability/ProjectionMoments/Model.lean:280 (same declaration text).
def OAI.ProjectionMoments.normalizedDensity line 439
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:283 (same statement text; different proof or body). Qualify references as OAI.ProjectionMoments.normalizedDensity@path.
def OAI.ProjectionMoments.sphereWeightedRestriction line 447
Also declared: OAI/Probability/ProjectionMoments/Model.lean:287 (same declaration text).
def OAI.ProjectionMoments.capMeasure line 461
Also declared: OAI/Probability/ProjectionMoments/Model.lean:299 (same declaration text).
structure OAI.ProjectionMoments.CapDictionary line 465
Also declared: OAI/Probability/ProjectionMoments/Model.lean:303 (same declaration text).
def OAI.ProjectionMoments.radialLabel line 491
Also declared: OAI/Probability/ProjectionMoments/Model.lean:325 (same declaration text).
def OAI.ProjectionMoments.capKernelValue line 494
Also declared: OAI/Probability/ProjectionMoments/Model.lean:328 (same declaration text).
def OAI.ProjectionMoments.capReferenceLabel line 497
Also declared: OAI/Probability/ProjectionMoments/Model.lean:331 (same declaration text).
def OAI.ProjectionMoments.capReference line 500
Also declared: OAI/Probability/ProjectionMoments/Model.lean:334 (same declaration text).
def OAI.ProjectionMoments.fractionalBlock line 508
Also declared: OAI/Probability/ProjectionMoments/Model.lean:342 (same declaration text).
def OAI.ProjectionMoments.IntegratedFrameMoment line 514
Also declared: OAI/Probability/ProjectionMoments/Model.lean:551 (same declaration text).
def OAI.ProjectionMoments.NormalizedUnitBallMoment line 534
Also declared: OAI/Probability/ProjectionMoments/Model.lean:571 (same declaration text).
def OAI.ProjectionMoments.AllRadiiSphereMoment line 554
Also declared: OAI/Probability/ProjectionMoments/Model.lean:591 (same declaration text).
def OAI.ProjectionMoments.PositiveCapDomination line 576
Also declared: OAI/Probability/ProjectionMoments/Model.lean:613 (same declaration text).
def OAI.ProjectionMoments.DeterministicOffsetDensity line 592
Also declared: OAI/Probability/ProjectionMoments/Model.lean:629 (same declaration text).
def OAI.ProjectionMoments.AllRadiiBlockBound line 610
Also declared: OAI/Probability/ProjectionMoments/Model.lean:647 (same declaration text).
def OAI.ProjectionMoments.HaarStreamingCost line 623
Also declared: OAI/Probability/ProjectionMoments/Model.lean:660 (same declaration text).
def OAI.ProjectionMoments.RieszStreamingCost line 633
Also declared: OAI/Probability/ProjectionMoments/Model.lean:670 (same declaration text).
def OAI.ProjectionMoments.CapFixedStoppingCost line 643
Also declared: OAI/Probability/ProjectionMoments/Model.lean:680 (same declaration text).
def OAI.ProjectionMoments.CapTotalStoppingCost line 654
Also declared: OAI/Probability/ProjectionMoments/Model.lean:691 (same declaration text).
def OAI.ProjectionMoments.CapFixedStoppingAsymptotic line 665
Also declared: OAI/Probability/ProjectionMoments/Model.lean:702 (same declaration text).
def OAI.ProjectionMoments.CapTotalStoppingAsymptotic line 678
Also declared: OAI/Probability/ProjectionMoments/Model.lean:715 (same declaration text).
def OAI.ProjectionMoments.KernelModelBridge line 688
Also declared: OAI/Probability/ProjectionMoments/Model.lean:725 (same declaration text).
def OAI.ProjectionMoments.MainStatement line 695
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Model.lean:732 (same statement text; different proof or body). Qualify references as OAI.ProjectionMoments.MainStatement@path.
theorem OAI.ProjectionMoments.main Challenge statement: Projection, cap, and Riesz estimates with regression bounds sorry line 707
L4DEL explanation
This declaration is an exact comparator-selected statement. Read the Lean statement below for its mathematical content.
1. The linked formalization note covers 8 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Probability.ProjectionMoments.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Probability/ProjectionMoments/Main.lean:14 (same statement text; different proof or body). Qualify references as OAI.ProjectionMoments.main@path.
```
theorem main : MainStatement.{u} := by
…
```