`ComparatorChallenges/RuledCovering.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

47 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.RuledApproximation.Plane line 10
Also declared: ComparatorChallenges/CylinderCovering.lean:231 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:12 (same declaration text).
abbrev OAI.RuledApproximation.Space line 11
Also declared: ComparatorChallenges/CylinderCovering.lean:232 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:13 (same declaration text).
def OAI.RuledApproximation.vec2 line 13
Also declared: ComparatorChallenges/CylinderCovering.lean:233 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:15 (same declaration text).
def OAI.RuledApproximation.vec3 line 14
Also declared: ComparatorChallenges/CylinderCovering.lean:234 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:16 (same declaration text).
def OAI.RuledApproximation.axial line 15
Also declared: ComparatorChallenges/CylinderCovering.lean:235 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:17 (same declaration text).
def OAI.RuledApproximation.IsParallelogram line 17
Also declared: ComparatorChallenges/CylinderCovering.lean:237 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:19 (same declaration text).
def OAI.RuledApproximation.IsTriangle line 22
Also declared: OAI/Geometry/RuledCovering/Model.lean:24 (same declaration text).
structure OAI.RuledApproximation.Cylinder line 27
Also declared: ComparatorChallenges/CylinderCovering.lean:241 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:29 (same declaration text).
def OAI.RuledApproximation.Cylinder.carrier line 34
Also declared: ComparatorChallenges/CylinderCovering.lean:248 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:36 (same declaration text).
def OAI.RuledApproximation.Cylinder.area line 37
Also declared: ComparatorChallenges/CylinderCovering.lean:251 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:39 (same declaration text).
def OAI.RuledApproximation.Cylinder.parallelogram line 39
Also declared: ComparatorChallenges/CylinderCovering.lean:253 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:41 (same declaration text).
def OAI.RuledApproximation.Cylinder.triangle line 41
Also declared: OAI/Geometry/RuledCovering/Model.lean:43 (same declaration text).
def OAI.RuledApproximation.Cylinder.openBounded line 43
Also declared: ComparatorChallenges/CylinderCovering.lean:255 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:45 (same declaration text).
def OAI.RuledApproximation.familySet line 47
Also declared: ComparatorChallenges/CylinderCovering.lean:259 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:49 (same declaration text).
def OAI.RuledApproximation.familyArea line 48
Also declared: ComparatorChallenges/CylinderCovering.lean:260 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:50 (same declaration text).
def OAI.RuledApproximation.HasFiniteCover line 50
Also declared: ComparatorChallenges/CylinderCovering.lean:262 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:52 (same declaration text).
def OAI.RuledApproximation.SmoothNear No boundary hypothesis is included in this neighborhood regularity predicate line 55
Source documentation
No boundary hypothesis is included in this neighborhood regularity predicate.
L4DEL explanation
No boundary hypothesis is included in this neighborhood regularity predicate.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: ComparatorChallenges/CylinderCovering.lean:266 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:57 (same declaration text).
```
def SmoothNear (n : WithTop ℕ∞) (D : Set Plane) (v : Plane → Plane) : Prop :=
…
```
Mentions: OAI.RuledApproximation.Plane
def OAI.RuledApproximation.HyperbolicAt In dimension two these two distinct nonzero eigenvalues are all the eigenvalues line 59
Source documentation
In dimension two these two distinct nonzero eigenvalues are all the eigenvalues.
L4DEL explanation
In dimension two these two distinct nonzero eigenvalues are all the eigenvalues.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: ComparatorChallenges/CylinderCovering.lean:269 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:61 (same declaration text).
```
def HyperbolicAt (L : ℝ) (v : Plane → Plane) (p : Plane) : Prop :=
…
```
Mentions: OAI.RuledApproximation.Plane
def OAI.RuledApproximation.weight line 64
Also declared: ComparatorChallenges/CylinderCovering.lean:274 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:66 (same declaration text).
def OAI.RuledApproximation.physicalWeight line 65
Also declared: ComparatorChallenges/CylinderCovering.lean:275 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:67 (same declaration text).
def OAI.RuledApproximation.ruled line 68
Also declared: ComparatorChallenges/CylinderCovering.lean:278 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:70 (same declaration text).
def OAI.RuledApproximation.physicalRuled line 72
Also declared: ComparatorChallenges/CylinderCovering.lean:282 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:74 (same declaration text).
def OAI.RuledApproximation.HyperbolicStatement line 76
Also declared: ComparatorChallenges/CylinderCovering.lean:286 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:78 (same declaration text).
def OAI.RuledApproximation.PhysicalHyperbolicStatement line 83
Also declared: OAI/Geometry/RuledCovering/Model.lean:85 (same declaration text).
def OAI.RuledApproximation.OpenRectangleC2 line 91
Also declared: OAI/Geometry/RuledCovering/Model.lean:93 (same declaration text).
def OAI.RuledApproximation.LogarithmicStatement line 96
Also declared: OAI/Geometry/RuledCovering/Model.lean:98 (same declaration text).
def OAI.RuledApproximation.ProductCellStatement line 107
Also declared: OAI/Geometry/RuledCovering/Model.lean:109 (same declaration text).
def OAI.RuledApproximation.OrthogonalGridStatement line 114
Also declared: OAI/Geometry/RuledCovering/Model.lean:116 (same declaration text).
def OAI.RuledApproximation.unitSquare line 124
Also declared: ComparatorChallenges/CylinderCovering.lean:293 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:126 (same declaration text).
def OAI.RuledApproximation.IsSquareTile line 126
Also declared: ComparatorChallenges/CylinderCovering.lean:295 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:128 (same declaration text).
def OAI.RuledApproximation.auxiliaryCylinder line 130
Also declared: ComparatorChallenges/CylinderCovering.lean:299 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:132 (same declaration text).
def OAI.RuledApproximation.firstPhysicalMap line 133
Also declared: ComparatorChallenges/CylinderCovering.lean:302 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:135 (same declaration text).
def OAI.RuledApproximation.secondPhysicalMap line 134
Also declared: ComparatorChallenges/CylinderCovering.lean:303 (same declaration text), OAI/Geometry/RuledCovering/Model.lean:136 (same declaration text).
def OAI.RuledApproximation.NilpotentStatement line 137
Also declared: OAI/Geometry/RuledCovering/Model.lean:139 (same declaration text).
def OAI.RuledApproximation.tetrahedron The regular edge-2 tetrahedron in physical coordinates line 154
Source documentation
The regular edge-2 tetrahedron in physical coordinates.
L4DEL explanation
The regular edge-2 tetrahedron in physical coordinates.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Geometry/RuledCovering/Model.lean:156 (same declaration text).
```
def tetrahedron : Set Space :=
…
```
Mentions: OAI.RuledApproximation.Space
def OAI.RuledApproximation.singularField line 158
Also declared: OAI/Geometry/RuledCovering/Model.lean:160 (same declaration text).
def OAI.RuledApproximation.ceilingPolynomial line 163
Also declared: OAI/Geometry/RuledCovering/Model.lean:165 (same declaration text).
def OAI.RuledApproximation.singularDomain line 166
Also declared: OAI/Geometry/RuledCovering/Model.lean:168 (same declaration text).
def OAI.RuledApproximation.singularLower line 170
Also declared: OAI/Geometry/RuledCovering/Model.lean:172 (same declaration text).
def OAI.RuledApproximation.singularUpper line 174
Also declared: OAI/Geometry/RuledCovering/Model.lean:176 (same declaration text).
def OAI.RuledApproximation.lowerTip line 179
Also declared: OAI/Geometry/RuledCovering/Model.lean:181 (same declaration text).
def OAI.RuledApproximation.upperTip line 182
Also declared: OAI/Geometry/RuledCovering/Model.lean:184 (same declaration text).
def OAI.RuledApproximation.singularCost line 185
Also declared: OAI/Geometry/RuledCovering/Model.lean:187 (same declaration text).
def OAI.RuledApproximation.singularCoefficient line 189
Also declared: OAI/Geometry/RuledCovering/Model.lean:191 (same declaration text).
def OAI.RuledApproximation.SingularStatement line 192
Also declared: OAI/Geometry/RuledCovering/Model.lean:194 (same declaration text).
def OAI.RuledApproximation.FullMain line 211
Also declared: OAI/Geometry/RuledCovering/Model.lean:213 (same declaration text).
theorem OAI.RuledApproximation.fullMain Challenge statement: Finite cylinder approximation for the selected ruled families sorry line 215
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.Geometry.RuledCovering.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Geometry/RuledCovering/Main.lean:13 (same statement text; different proof or body). Qualify references as OAI.RuledApproximation.fullMain@path.
```
theorem fullMain : FullMain := by
…
```
Mentions: OAI.RuledApproximation.FullMain