`ComparatorChallenges/BiholderTransport.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

14 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.WeakMTWTransport.Model line 12
Also declared: OAI/Analysis/BiholderTransport/Model.lean:12 (same declaration text).
def OAI.WeakMTWTransport.cost Squared-distance cost with factor one half line 19
Source documentation
Squared-distance cost with factor one half.
L4DEL explanation
Squared-distance cost with factor one half.
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/Analysis/BiholderTransport/Model.lean:19 (same declaration text).
```
def cost (x y : M) : ℝ := dist x y ^ 2 / 2
```
def OAI.WeakMTWTransport.euclideanVolumeFactor Calibration of Hausdorff measure to Euclidean volume line 22
Source documentation
Calibration of Hausdorff measure to Euclidean volume.
L4DEL explanation
Calibration of Hausdorff measure to Euclidean volume.
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/Analysis/BiholderTransport/Model.lean:22 (same declaration text).
```
def euclideanVolumeFactor (n : ℕ) : ℝ≥0∞ :=
…
```
def OAI.WeakMTWTransport.metricVolume Riemannian volume calibrated to Lebesgue measure in Euclidean space line 30
Source documentation
Riemannian volume calibrated to Lebesgue measure in Euclidean space.
L4DEL explanation
Riemannian volume calibrated to Lebesgue measure in Euclidean space.
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/Analysis/BiholderTransport/Model.lean:30 (same declaration text).
```
def metricVolume [BorelSpace M] (n : ℕ) : Measure M :=
…
```
def OAI.WeakMTWTransport.AdmissibleDensity Almost-everywhere measurable probability densities between fixed bounds line 34
Source documentation
Almost-everywhere measurable probability densities between fixed bounds.
L4DEL explanation
Almost-everywhere measurable probability densities between fixed bounds.
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/Analysis/BiholderTransport/Model.lean:34 (same declaration text).
```
def AdmissibleDensity (vol : Measure M) (lam cap : ℝ) (rho : M → ℝ) : Prop :=
…
```
def OAI.WeakMTWTransport.densityMeasure Multiplication by a density, ignoring exceptional negative values on null sets line 39
Source documentation
Multiplication by a density, ignoring exceptional negative values on null sets.
L4DEL explanation
Multiplication by a density, ignoring exceptional negative values on null sets.
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/Analysis/BiholderTransport/Model.lean:39 (same declaration text).
```
def densityMeasure (vol : Measure M) (rho : M → ℝ) : Measure M :=
…
```
def OAI.WeakMTWTransport.IsOptimalMap Optimality for the Monge minimization problem line 43
Source documentation
Optimality for the Monge minimization problem.
L4DEL explanation
Optimality for the Monge minimization problem.
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/Analysis/BiholderTransport/Model.lean:43 (same declaration text).
```
def IsOptimalMap (vol : Measure M) (rho0 rho1 : M → ℝ) (T : M → M) : Prop :=
…
```
def OAI.WeakMTWTransport.BiHolderEstimate The two global estimates have the very same exponent and constant line 51
Source documentation
The two global estimates have the very same exponent and constant.
L4DEL explanation
The two global estimates have the very same exponent and constant.
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/Analysis/BiholderTransport/Model.lean:51 (same declaration text).
```
def BiHolderEstimate (alpha C : ℝ) (T : M ≃ₜ M) : Prop :=
…
```
def OAI.WeakMTWTransport.IsGeodesicWithInitialData Complete affinely parametrized geodesic with prescribed initial tangent line 65
Source documentation
Complete affinely parametrized geodesic with prescribed initial tangent. Local metric minimization is an intrinsic definition of Riemannian geodesics.
L4DEL explanation
Complete affinely parametrized geodesic with prescribed initial tangent. Local metric minimization is an intrinsic definition of Riemannian geodesics.
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/Analysis/BiholderTransport/Model.lean:63 (same declaration text).
```
def IsGeodesicWithInitialData (x : M) (v : TangentSpace 𝓘(ℝ, Model n) x)
    (gamma : ℝ → M) : Prop :=
…
```
Mentions: OAI.WeakMTWTransport.Model
def OAI.WeakMTWTransport.riemannianExp Evaluation at time one of the geodesic determined by initial data line 75
Source documentation
Evaluation at time one of the geodesic determined by initial data.
L4DEL explanation
Evaluation at time one of the geodesic determined by initial data.
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/Analysis/BiholderTransport/Model.lean:73 (same declaration text).
```
def riemannianExp (x : M) (v : TangentSpace 𝓘(ℝ, Model n) x) : M := by
…
```
Mentions: OAI.WeakMTWTransport.Model
def OAI.WeakMTWTransport.injectivityDomain The open injectivity domain, defined by extendible minimizing vectors line 80
Source documentation
The open injectivity domain, defined by extendible minimizing vectors.
L4DEL explanation
The open injectivity domain, defined by extendible minimizing vectors.
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/Analysis/BiholderTransport/Model.lean:78 (same declaration text).
```
def injectivityDomain (x : M) : Set (TangentSpace 𝓘(ℝ, Model n) x) :=
…
```
Mentions: OAI.WeakMTWTransport.Model
def OAI.WeakMTWTransport.mtw The Ma–Trudinger–Wang fourth derivative with factor -3/2 line 84
Source documentation
The Ma–Trudinger–Wang fourth derivative with factor -3/2.
L4DEL explanation
The Ma–Trudinger–Wang fourth derivative with factor -3/2.
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/Analysis/BiholderTransport/Model.lean:82 (same declaration text).
```
def mtw (x : M) (p xi eta : TangentSpace 𝓘(ℝ, Model n) x) : ℝ :=
…
```
Mentions: OAI.WeakMTWTransport.Model
def OAI.WeakMTWTransport.WeakMTW Weak MTW only at interior velocities and only for orthogonal test vectors line 91
Source documentation
Weak MTW only at interior velocities and only for orthogonal test vectors.
L4DEL explanation
Weak MTW only at interior velocities and only for orthogonal test vectors.
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/Analysis/BiholderTransport/Model.lean:89 (same declaration text).
```
def WeakMTW : Prop :=
…
```
theorem OAI.WeakMTWTransport.uniform_biHolder_transport Challenge statement: Uniform bi-Hölder optimal transport sorry line 100
Source documentation
Uniform bi-Hölder optimal transport for all densities between fixed bounds on a compact connected smooth Riemannian manifold satisfying weak MTW.
L4DEL explanation
Uniform bi-Hölder optimal transport for all densities between fixed bounds on a compact connected smooth Riemannian manifold satisfying weak MTW.
1. This explanation is a condensed rendering of this declaration's own doc comment at the pinned source commit.
2. The linked formalization note covers 2 distinct comparator results, so its combined Scope prose is not copied onto this declaration.
3. The repository's comparator configuration selects OAI.Analysis.BiholderTransport.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Analysis/BiholderTransport/Main.lean:27 (same statement text; different proof or body). Qualify references as OAI.WeakMTWTransport.uniform_biHolder_transport@path.
```
theorem uniform_biHolder_transport
    {n : ℕ} (hn : 2 ≤ n) {M : Type*}
    [MetricSpace M] [CompactSpace M] [ConnectedSpace M]
    [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ, Model n) ∞ M]
    [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ, Model n) x)]
    [IsContMDiffRiemannianBundle 𝓘(ℝ, Model n) ∞ (Model n)
      (fun x : M => TangentSpace 𝓘(ℝ, Model n) x)]
    [IsRiemannianManifold 𝓘(ℝ, Model n) M]
    [MeasurableSpace M] [BorelSpace M]
    (hmtw : WeakMTW (n := n) (M := M))
…
```
Mentions: OAI.WeakMTWTransport.Model, OAI.WeakMTWTransport.WeakMTW