`ComparatorChallenges/SmoothYau.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

20 declarations, 2 contain sorry, 0 declared as axioms

abbrev OAI.YauCounterexamples.Euclidean line 12
Also declared: OAI/Geometry/SmoothYau/Model.lean:11 (same declaration text).
abbrev OAI.YauCounterexamples.Sphere line 13
Also declared: OAI/Geometry/SmoothYau/Model.lean:12 (same declaration text).
abbrev OAI.YauCounterexamples.FourManifold line 14
Also declared: OAI/Geometry/SmoothYau/Model.lean:13 (same declaration text).
abbrev OAI.YauCounterexamples.FourModel line 15
Also declared: OAI/Geometry/SmoothYau/Model.lean:14 (same declaration text).
instance OAI.YauCounterexamples.torusChartedSpace line 17
Also declared: OAI/Geometry/SmoothYau/Model.lean:16 (same declaration text).
instance OAI.YauCounterexamples.torusIsManifold line 20
Also declared: OAI/Geometry/SmoothYau/Model.lean:19 (same declaration text).
instance OAI.YauCounterexamples.fourChartedSpace line 24
Also declared: OAI/Geometry/SmoothYau/Model.lean:23 (same declaration text).
instance OAI.YauCounterexamples.fourIsManifold line 27
Also declared: OAI/Geometry/SmoothYau/Model.lean:26 (same declaration text).
abbrev OAI.YauCounterexamples.SmoothMetric Ordinary smooth positive-definite Riemannian metrics, with the existing tangent topology line 39
Source documentation
Ordinary smooth positive-definite Riemannian metrics, with the existing tangent topology.
L4DEL explanation
Ordinary smooth positive-definite Riemannian metrics, with the existing tangent topology.
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/SmoothYau/Model.lean:38 (same declaration text).
```
abbrev SmoothMetric (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E]
    (M : Type*) [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ, E) ∞ M] :=
…
```
abbrev OAI.YauCounterexamples.CoordIndex line 43
Also declared: OAI/Geometry/SmoothYau/Model.lean:42 (same declaration text).
def OAI.YauCounterexamples.coordinateVector A coordinate vector, pushed from a chart back into the tangent bundle line 47
Source documentation
A coordinate vector, pushed from a chart back into the tangent bundle.
L4DEL explanation
A coordinate vector, pushed from a chart back into the tangent bundle.
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/SmoothYau/Model.lean:46 (same declaration text).
```
def coordinateVector (p : M) (y : E) (i : CoordIndex E) :
    TangentSpace 𝓘(ℝ, E) ((chartAt E p).symm y) :=
…
```
Mentions: OAI.YauCounterexamples.CoordIndex
def OAI.YauCounterexamples.metricCoefficients Metric coefficients in a chart line 52
Source documentation
Metric coefficients in a chart. Only values in the chart target are used.
L4DEL explanation
Metric coefficients in a chart. Only values in the chart target are used.
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/SmoothYau/Model.lean:51 (same declaration text).
```
def metricCoefficients (g : SmoothMetric E M) (p : M) (y : E) :
    Matrix (CoordIndex E) (CoordIndex E) ℝ :=
…
```
Mentions: OAI.YauCounterexamples.SmoothMetric, OAI.YauCounterexamples.CoordIndex
def OAI.YauCounterexamples.laplaceBeltrami The local coordinate formula det(g)^(-1/2) d_i(det(g)^(1/2) g^{ij} d_j u) line 60
Source documentation
The local coordinate formula det(g)^(-1/2) d_i(det(g)^(1/2) g^{ij} d_j u). The chart at the evaluation point is used, so all derivatives are taken in its open target.
L4DEL explanation
The local coordinate formula det(g)^(-1/2) d_i(det(g)^(1/2) g^{ij} d_j u). The chart at the evaluation point is used, so all derivatives are taken in its open target.
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/SmoothYau/Model.lean:59 (same declaration text).
```
def laplaceBeltrami (g : SmoothMetric E M) (u : M → ℝ) (x : M) : ℝ :=
…
```
Mentions: OAI.YauCounterexamples.SmoothMetric
structure OAI.YauCounterexamples.CoordinateTest One compact-chart seminorm test for the usual smooth topology of metrics line 71
Source documentation
One compact-chart seminorm test for the usual smooth topology of metrics.
L4DEL explanation
One compact-chart seminorm test for the usual smooth topology of metrics.
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/SmoothYau/Model.lean:70 (same declaration text).
```
structure CoordinateTest (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E]
    [FiniteDimensional ℝ E] (M : Type*) [TopologicalSpace M] [ChartedSpace E M] where
…
```
def OAI.YauCounterexamples.IsSmoothNeighborhood The finite compact-chart derivative tests defining a C∞ neighborhood line 83
Source documentation
The finite compact-chart derivative tests defining a C∞ neighborhood. A single positive tolerance suffices for the finitely many tests.
L4DEL explanation
The finite compact-chart derivative tests defining a C∞ neighborhood. A single positive tolerance suffices for the finitely many tests.
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/SmoothYau/Model.lean:82 (same declaration text).
```
def IsSmoothNeighborhood (g₀ : SmoothMetric E M) (N : Set (SmoothMetric E M)) : Prop :=
…
```
Mentions: OAI.YauCounterexamples.SmoothMetric
def OAI.YauCounterexamples.nodalMeasure Hausdorff measure for the path distance of the actual Riemannian metric, not the metric inherited from any ambient embedding line 93
Source documentation
Hausdorff measure for the path distance of the actual Riemannian metric, not the metric inherited from any ambient embedding.
L4DEL explanation
Hausdorff measure for the path distance of the actual Riemannian metric, not the metric inherited from any ambient embedding.
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/SmoothYau/Model.lean:92 (same declaration text).
```
def nodalMeasure [T3Space M] (g : SmoothMetric E M) (d : ℝ) (u : M → ℝ) : ℝ≥0∞ :=
…
```
Mentions: OAI.YauCounterexamples.SmoothMetric
def OAI.YauCounterexamples.HasUnboundedNodalRatio The eigensequence belongs to one fixed metric and has finite nodal measure line 104
Source documentation
The eigensequence belongs to one fixed metric and has finite nodal measure.
L4DEL explanation
The eigensequence belongs to one fixed metric and has finite nodal measure.
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/SmoothYau/Model.lean:103 (same declaration text).
```
def HasUnboundedNodalRatio [T3Space M] (g : SmoothMetric E M) (d : ℕ) : Prop :=
…
```
Mentions: OAI.YauCounterexamples.SmoothMetric
def OAI.YauCounterexamples.IsRound The round metric is the pullback of the Euclidean scalar product under inclusion of the unit sphere line 117
Source documentation
The round metric is the pullback of the Euclidean scalar product under inclusion of the unit sphere. This predicate specifies exactly that tensor.
L4DEL explanation
The round metric is the pullback of the Euclidean scalar product under inclusion of the unit sphere. This predicate specifies exactly that tensor.
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/SmoothYau/Model.lean:116 (same declaration text).
```
def IsRound (g : SmoothMetric (Euclidean 3) (Sphere 3)) : Prop :=
…
```
Mentions: OAI.YauCounterexamples.SmoothMetric, OAI.YauCounterexamples.Euclidean, OAI.YauCounterexamples.Sphere
theorem OAI.YauCounterexamples.sphere_three Challenge statement: Nodal counterexamples in dimensions three and four sorry line 132
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.SmoothYau.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Geometry/SmoothYau/SphereMetric/ThreeSphere.lean:12 (same statement text; different proof or body). Qualify references as OAI.YauCounterexamples.sphere_three@path.
```
theorem sphere_three (gRound : SmoothMetric (Euclidean 3) (Sphere 3))
    (hRound : IsRound gRound) :
    ∀ N : Set (SmoothMetric (Euclidean 3) (Sphere 3)),
      IsSmoothNeighborhood gRound N →
      ∃ g ∈ N, HasUnboundedNodalRatio g 3 := by
…
```
Mentions: OAI.YauCounterexamples.SmoothMetric, OAI.YauCounterexamples.Euclidean, OAI.YauCounterexamples.Sphere, OAI.YauCounterexamples.IsRound, OAI.YauCounterexamples.IsSmoothNeighborhood, OAI.YauCounterexamples.HasUnboundedNodalRatio
theorem OAI.YauCounterexamples.sphere_two_torus_two Challenge statement: Nodal counterexamples in dimensions three and four sorry line 139
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.SmoothYau.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Geometry/SmoothYau/ProductMetric/SphereProduct.lean:75 (same statement text; different proof or body). Qualify references as OAI.YauCounterexamples.sphere_two_torus_two@path.
```
theorem sphere_two_torus_two :
    ∃ g : SmoothMetric FourModel FourManifold,
      HasUnboundedNodalRatio g 4 := by
…
```
Mentions: OAI.YauCounterexamples.SmoothMetric, OAI.YauCounterexamples.FourModel, OAI.YauCounterexamples.FourManifold, OAI.YauCounterexamples.HasUnboundedNodalRatio