`ComparatorChallenges/SphericalField.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

abbrev OAI.SphericalPerceptron.Time line 11
Also declared: OAI/Probability/SphericalField/Model.lean:11 (same declaration text).
abbrev OAI.SphericalPerceptron.Spin line 13
Also declared: OAI/Probability/SphericalField/Model.lean:13 (same declaration text).
def OAI.SphericalPerceptron.unitSphereLaw line 15
Also declared: OAI/Probability/SphericalField/Model.lean:15 (same declaration text).
structure OAI.SphericalPerceptron.Trial line 19
Also declared: OAI/Probability/SphericalField/Model.lean:19 (same declaration text).
instance OAI.SphericalPerceptron.(instance at line 26) line 26
def OAI.SphericalPerceptron.timeLaw line 28
Also declared: OAI/Probability/SphericalField/Model.lean:28 (same declaration text).
def OAI.SphericalPerceptron.tailIntegral line 30
Also declared: OAI/Probability/SphericalField/Model.lean:30 (same declaration text).
def OAI.SphericalPerceptron.entropy line 33
Also declared: OAI/Probability/SphericalField/Model.lean:33 (same declaration text).
theorem OAI.SphericalPerceptron.timeLaw_eq_volume line 37
Also declared: OAI/Probability/SphericalField/Model.lean:37 (same declaration text).
instance OAI.SphericalPerceptron.(instance at line 39) line 39
def OAI.SphericalPerceptron.finitePointMeasure line 43
Also declared: OAI/Probability/SphericalField/Model.lean:43 (same declaration text).
lemma OAI.SphericalPerceptron.finitePointMeasure_measurable line 46
Also declared: OAI/Probability/SphericalField/Model.lean:46 (same declaration text).
def OAI.SphericalPerceptron.finitePoissonLaw line 51
Also declared: OAI/Probability/SphericalField/Model.lean:51 (same declaration text).
instance OAI.SphericalPerceptron.finitePoissonLaw_probability line 56
Also declared: OAI/Probability/SphericalField/Model.lean:56 (same declaration text).
lemma OAI.SphericalPerceptron.measureSum_measurable line 70
Also declared: OAI/Probability/SphericalField/Model.lean:70 (same declaration text).
def OAI.SphericalPerceptron.countablePoissonLaw line 77
Also declared: OAI/Probability/SphericalField/Model.lean:77 (same declaration text).
instance OAI.SphericalPerceptron.countablePoissonLaw_probability line 81
Also declared: OAI/Probability/SphericalField/Model.lean:81 (same declaration text).
def OAI.SphericalPerceptron.finiteIntensityMarks line 87
Also declared: OAI/Probability/SphericalField/Model.lean:87 (same declaration text).
instance OAI.SphericalPerceptron.finiteIntensityMarks_probability line 90
Also declared: OAI/Probability/SphericalField/Model.lean:90 (same declaration text).
def OAI.SphericalPerceptron.poissonRandomMeasureLaw line 98
Also declared: OAI/Probability/SphericalField/Model.lean:98 (same declaration text).
instance OAI.SphericalPerceptron.poissonRandomMeasureLaw_probability line 103
Also declared: OAI/Probability/SphericalField/Model.lean:103 (same declaration text).
def OAI.SphericalPerceptron.stableLogIntensity line 108
Also declared: OAI/Probability/SphericalField/Model.lean:108 (same declaration text).
instance OAI.SphericalPerceptron.stableLogIntensity_sigmaFinite line 111
Also declared: OAI/Probability/SphericalField/Model.lean:111 (same declaration text).
def OAI.SphericalPerceptron.normalizedMeasure line 115
Also declared: OAI/Probability/SphericalField/Model.lean:115 (same declaration text).
lemma OAI.SphericalPerceptron.measurable_measure_withDensity_fixed line 118
Also declared: OAI/Probability/SphericalField/Model.lean:118 (same declaration text).
lemma OAI.SphericalPerceptron.normalizedMeasure_measurable line 124
Also declared: OAI/Probability/SphericalField/Model.lean:124 (same declaration text).
def OAI.SphericalPerceptron.markedStableTotalE line 133
Also declared: OAI/Probability/SphericalField/Model.lean:133 (same declaration text).
lemma OAI.SphericalPerceptron.markedStableTotalE_measurable line 136
Also declared: OAI/Probability/SphericalField/Model.lean:136 (same declaration text).
def OAI.SphericalPerceptron.markedStableTotal line 139
Also declared: OAI/Probability/SphericalField/Model.lean:139 (same declaration text).
lemma OAI.SphericalPerceptron.markedStableTotal_measurable line 141
Also declared: OAI/Probability/SphericalField/Model.lean:141 (same declaration text).
def OAI.SphericalPerceptron.markedStableMassKernel line 144
Also declared: OAI/Probability/SphericalField/Model.lean:144 (same declaration text).
instance OAI.SphericalPerceptron.markedStableMassKernel_finite line 148
Also declared: OAI/Probability/SphericalField/Model.lean:148 (same declaration text).
def OAI.SphericalPerceptron.markedStableCountKernel line 155
Also declared: OAI/Probability/SphericalField/Model.lean:155 (same declaration text).
def OAI.SphericalPerceptron.DecoratedCascade line 160
Also declared: OAI/Probability/SphericalField/Model.lean:160 (same declaration text).
instance OAI.SphericalPerceptron.decoratedCascadeNonempty line 164
Also declared: OAI/Probability/SphericalField/Model.lean:164 (same declaration text).
def OAI.SphericalPerceptron.decoratedCascadeLaw line 170
Also declared: OAI/Probability/SphericalField/Model.lean:170 (same declaration text).
def OAI.SphericalPerceptron.decoratedTerminalTotalE line 177
Also declared: OAI/Probability/SphericalField/Model.lean:177 (same declaration text).
def OAI.SphericalPerceptron.decoratedTerminalTotal line 183
Also declared: OAI/Probability/SphericalField/Model.lean:183 (same declaration text).
def OAI.SphericalPerceptron.gaussianShiftStep line 187
Also declared: OAI/Probability/SphericalField/Model.lean:187 (same declaration text).
def OAI.SphericalPerceptron.standardGaussianMark line 190
Also declared: OAI/Probability/SphericalField/Model.lean:190 (same declaration text).
def OAI.SphericalPerceptron.piMarkLaw line 192
Also declared: OAI/Probability/SphericalField/Model.lean:192 (same declaration text).
def OAI.SphericalPerceptron.canonicalCoordinateStep line 196
Also declared: OAI/Probability/SphericalField/Model.lean:196 (same declaration text).
def OAI.SphericalPerceptron.decoratedLeafIntegral line 200
Also declared: OAI/Probability/SphericalField/Model.lean:200 (same declaration text).
def OAI.SphericalPerceptron.decoratedFieldFactor line 209
Also declared: OAI/Probability/SphericalField/Model.lean:209 (same declaration text).
def OAI.SphericalPerceptron.decoratedAngularFactor line 214
Also declared: OAI/Probability/SphericalField/Model.lean:214 (same declaration text).
def OAI.SphericalPerceptron.coordinateLeafField line 220
Also declared: OAI/Probability/SphericalField/Model.lean:220 (same declaration text).
def OAI.SphericalPerceptron.coordinateCascadeLaw line 223
Also declared: OAI/Probability/SphericalField/Model.lean:223 (same declaration text).
def OAI.SphericalPerceptron.coordinateAngularLog line 228
Also declared: OAI/Probability/SphericalField/Model.lean:228 (same declaration text).
def OAI.SphericalPerceptron.normalizedCoordinateSphereLog line 234
Also declared: OAI/Probability/SphericalField/Model.lean:234 (same declaration text).
def OAI.SphericalPerceptron.expectedCoordinateSphereLog line 238
Also declared: OAI/Probability/SphericalField/Model.lean:238 (same declaration text).
def OAI.SphericalPerceptron.weightedStepTrial line 241
Also declared: OAI/Probability/SphericalField/Model.lean:241 (same declaration text).
def OAI.SphericalPerceptron.stepCumulative line 268
Also declared: OAI/Probability/SphericalField/Model.lean:268 (same declaration text).
def OAI.SphericalPerceptron.stepFieldIncrement line 271
Also declared: OAI/Probability/SphericalField/Model.lean:271 (same declaration text).
def OAI.SphericalPerceptron.finiteSphericalFieldValue line 275
Also declared: OAI/Probability/SphericalField/Model.lean:275 (same declaration text).
def OAI.SphericalPerceptron.finiteSphericalDualObjective line 279
Also declared: OAI/Probability/SphericalField/Model.lean:279 (same declaration text).
def OAI.SphericalPerceptron.finiteSphericalDualValues line 283
Also declared: OAI/Probability/SphericalField/Model.lean:283 (same declaration text).
structure OAI.SphericalPerceptron.FiniteFieldModel line 288
Also declared: OAI/Probability/SphericalField/Model.lean:288 (same declaration text).
def OAI.SphericalPerceptron.FiniteFieldModel.weight line 297
Also declared: OAI/Probability/SphericalField/Model.lean:297 (same declaration text).
def OAI.SphericalPerceptron.FiniteFieldModel.pressure line 300
Also declared: OAI/Probability/SphericalField/Model.lean:300 (same declaration text).
theorem OAI.SphericalPerceptron.exists_finiteFieldModel line 303
Also declared: OAI/Probability/SphericalField/Model.lean:303 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
structure OAI.SphericalPerceptron.BoundedField line 354
Also declared: OAI/Probability/SphericalField/Model.lean:354 (same declaration text).
instance OAI.SphericalPerceptron.(instance at line 360) line 360
lemma OAI.SphericalPerceptron.BoundedField.measurable line 362
Also declared: OAI/Probability/SphericalField/Model.lean:362 (same declaration text).
def OAI.SphericalPerceptron.roundLower line 364
Also declared: OAI/Probability/SphericalField/Model.lean:364 (same declaration text).
lemma OAI.SphericalPerceptron.roundLower_nonneg line 366
Also declared: OAI/Probability/SphericalField/Model.lean:366 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
lemma OAI.SphericalPerceptron.roundLower_monotone line 368
Also declared: OAI/Probability/SphericalField/Model.lean:368 (same declaration text).
lemma OAI.SphericalPerceptron.roundLower_le line 373
Also declared: OAI/Probability/SphericalField/Model.lean:373 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
def OAI.SphericalPerceptron.BoundedField.round line 378
Also declared: OAI/Probability/SphericalField/Model.lean:378 (same declaration text).
lemma OAI.SphericalPerceptron.BoundedField.round_finite line 384
Also declared: OAI/Probability/SphericalField/Model.lean:384 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
def OAI.SphericalPerceptron.BoundedField.roundModel line 394
Also declared: OAI/Probability/SphericalField/Model.lean:394 (same declaration text).
def OAI.SphericalPerceptron.quantileTrial line 397
Also declared: OAI/Probability/SphericalField/Model.lean:397 (same declaration text).
def OAI.SphericalPerceptron.quantileTail line 411
Also declared: OAI/Probability/SphericalField/Model.lean:411 (same declaration text).
lemma OAI.SphericalPerceptron.quantileTail_integrable line 414
Also declared: OAI/Probability/SphericalField/Model.lean:414 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
lemma OAI.SphericalPerceptron.quantileTail_lower line 431
Also declared: OAI/Probability/SphericalField/Model.lean:431 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
lemma OAI.SphericalPerceptron.quantileTail_lipschitz line 438
Also declared: OAI/Probability/SphericalField/Model.lean:438 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
def OAI.SphericalPerceptron.boundedSphericalFieldValue line 457
Also declared: OAI/Probability/SphericalField/Model.lean:457 (same declaration text).
def OAI.SphericalPerceptron.BoundedField.toQuantile line 460
Also declared: OAI/Probability/SphericalField/Model.lean:460 (same declaration text).
lemma OAI.SphericalPerceptron.BoundedField.toQuantile_measurable line 463
Also declared: OAI/Probability/SphericalField/Model.lean:463 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
6 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
def OAI.SphericalPerceptron.quantileA line 466
Also declared: OAI/Probability/SphericalField/Model.lean:466 (same declaration text).
lemma OAI.SphericalPerceptron.quantileA_integrand_continuous line 469
Also declared: OAI/Probability/SphericalField/Model.lean:469 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
lemma OAI.SphericalPerceptron.quantileA_integrand_bound line 476
Also declared: OAI/Probability/SphericalField/Model.lean:476 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
lemma OAI.SphericalPerceptron.quantileA_nonneg line 483
Also declared: OAI/Probability/SphericalField/Model.lean:483 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
lemma OAI.SphericalPerceptron.quantileA_mono line 486
Also declared: OAI/Probability/SphericalField/Model.lean:486 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
lemma OAI.SphericalPerceptron.quantileA_bound line 494
Also declared: OAI/Probability/SphericalField/Model.lean:494 (same declaration text).

Checked implications (found by the pipeline and checked by Lean's kernel when it ran; a recorded result, not checked again in this build: what these are)
1 more checked entries are routine (the same statement, a restatement, a generated lemma, one challenge file, a field of a structure, a route the released proof already takes, a duplicate): in the catalogue.
def OAI.SphericalPerceptron.BoundedField.stationary line 505
Also declared: OAI/Probability/SphericalField/Model.lean:505 (same declaration text).
def OAI.SphericalPerceptron.clippedQuantile line 520
Also declared: OAI/Probability/SphericalField/Model.lean:520 (same declaration text).
def OAI.SphericalPerceptron.stationaryQuantileField line 526
Also declared: OAI/Probability/SphericalField/Model.lean:526 (same declaration text).
def OAI.SphericalPerceptron.SphericalLinearFieldFormula line 529
Also declared: OAI/Probability/SphericalField/Model.lean:529 (same declaration text).
theorem OAI.SphericalPerceptron.spherical_linear_field_formula Challenge statement: Spherical linear-field dual formula and approximation sorry line 553
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 3 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
2. The repository's comparator configuration selects OAI.Probability.SphericalField.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Probability/SphericalField/Main.lean:26 (same statement text; different proof or body). Qualify references as OAI.SphericalPerceptron.spherical_linear_field_formula@path.
```
theorem spherical_linear_field_formula : SphericalLinearFieldFormula := by
…
```
Mentions: OAI.SphericalPerceptron.SphericalLinearFieldFormula