`ComparatorChallenges/AffineBernstein.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

12 declarations, 1 contain sorry, 0 declared as axioms

abbrev OAI.AffineBernstein.Space line 12
Also declared: OAI/Analysis/AffineBernstein/Basic.lean:10 (same declaration text).
def OAI.AffineBernstein.coordinateVector line 14
Also declared: OAI/Analysis/AffineBernstein/Basic.lean:12 (same declaration text).
def OAI.AffineBernstein.hessian Coordinate Hessian line 19
Source documentation
Coordinate Hessian. On the open domain it depends only on the restriction of u. The order of the two differentiations is immaterial under the smoothness hypothesis.
L4DEL explanation
Coordinate Hessian. On the open domain it depends only on the restriction of u. The order of the two differentiations is immaterial under the smoothness hypothesis.
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/AffineBernstein/Basic.lean:17 (same declaration text).
```
def hessian {n : ℕ} (u : Space n → ℝ) (x : Space n) : Matrix (Fin n) (Fin n) ℝ :=
…
```
Mentions: OAI.AffineBernstein.Space
def OAI.AffineBernstein.cofactorHessian The coefficient U^{ij} = det(D²u) (D²u)^{-1}_{ij} line 24
Source documentation
The coefficient U^{ij} = det(D²u) (D²u)^{-1}_{ij}.
L4DEL explanation
The coefficient U^{ij} = det(D²u) (D²u)^{-1}_{ij}.
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/AffineBernstein/Basic.lean:22 (same declaration text).
```
def cofactorHessian {n : ℕ} (u : Space n → ℝ) (x : Space n) :
    Matrix (Fin n) (Fin n) ℝ :=
…
```
Mentions: OAI.AffineBernstein.Space
def OAI.AffineBernstein.affineWeight The classical determinant weight; in particular, this is not a generalized exponent line 29
Source documentation
The classical determinant weight; in particular, this is not a generalized exponent.
L4DEL explanation
The classical determinant weight; in particular, this is not a generalized exponent.
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/AffineBernstein/Basic.lean:27 (same declaration text).
```
def affineWeight {n : ℕ} (u : Space n → ℝ) (x : Space n) : ℝ :=
…
```
Mentions: OAI.AffineBernstein.Space
def OAI.AffineBernstein.AffineMaximalOn line 32
Also declared: OAI/Analysis/AffineBernstein/Basic.lean:30 (same declaration text).
def OAI.AffineBernstein.graphPathLength Length on [0,1] for g_x(v,v) = ‖v‖² + (Du_x v)², the metric induced by the Euclidean graph embedding x ↦ (x,u(x)) line 39
Source documentation
Length on [0,1] for g_x(v,v) = ‖v‖² + (Du_x v)², the metric induced by the Euclidean graph embedding x ↦ (x,u(x)). The product norm on Lean's ordinary product type is NOT used (it would be the maximum norm).
L4DEL explanation
Length on [0,1] for g_x(v,v) = ‖v‖² + (Du_x v)², the metric induced by the Euclidean graph embedding x ↦ (x,u(x)). The product norm on Lean's ordinary product type is NOT used (it would be the maximum norm).
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/AffineBernstein/Basic.lean:37 (same declaration text).
```
def graphPathLength {n : ℕ} (u : Space n → ℝ) (γ : ℝ → Space n) : ℝ≥0∞ :=
…
```
Mentions: OAI.AffineBernstein.Space
def OAI.AffineBernstein.graphEDist The intrinsic extended distance: infimum of lengths of C¹ paths in the domain line 48
Source documentation
The intrinsic extended distance: infimum of lengths of C¹ paths in the domain. Using a single C¹ path gives the same intrinsic distance as piecewise C¹ paths, by smooth endpoint reparameterization and concatenation.
L4DEL explanation
The intrinsic extended distance: infimum of lengths of C¹ paths in the domain. Using a single C¹ path gives the same intrinsic distance as piecewise C¹ paths, by smooth endpoint reparameterization and concatenation.
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/AffineBernstein/Basic.lean:46 (same declaration text).
```
def graphEDist {n : ℕ} (Ω : Set (Space n)) (u : Space n → ℝ)
    (x y : Space n) : ℝ≥0∞ :=
…
```
Mentions: OAI.AffineBernstein.Space
def OAI.AffineBernstein.EuclideanGraphComplete Sequential completeness for the induced Euclidean path metric line 58
Source documentation
Sequential completeness for the induced Euclidean path metric. This states that every intrinsic Cauchy sequence of points in Ω has an intrinsic limit in Ω. It imposes no growth condition and no completeness condition on the affine metric.
L4DEL explanation
Sequential completeness for the induced Euclidean path metric. This states that every intrinsic Cauchy sequence of points in Ω has an intrinsic limit in Ω. It imposes no growth condition and no completeness condition on the affine metric.
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/AffineBernstein/Basic.lean:56 (same declaration text).
```
def EuclideanGraphComplete {n : ℕ} (Ω : Set (Space n)) (u : Space n → ℝ) : Prop :=
…
```
Mentions: OAI.AffineBernstein.Space
def OAI.AffineBernstein.graph The graph, regarded as an affine subset of R^{n+1} line 66
Source documentation
The graph, regarded as an affine subset of R^{n+1}.
L4DEL explanation
The graph, regarded as an affine subset of R^{n+1}.
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/AffineBernstein/Basic.lean:64 (same declaration text).
```
def graph {n : ℕ} (Ω : Set (Space n)) (u : Space n → ℝ) : Set (Space n × ℝ) :=
…
```
Mentions: OAI.AffineBernstein.Space
def OAI.AffineBernstein.standardParaboloid line 69
Also declared: OAI/Analysis/AffineBernstein/Basic.lean:67 (same declaration text).
theorem OAI.AffineBernstein.affine_bernstein Challenge statement: Affine Bernstein theorem in dimensions 3 to 9 sorry line 73
Source documentation
Literal main theorem, with its stated affine-image consequence.
L4DEL explanation
a smooth locally uniformly convex affine maximal graph is an elliptic paraboloid whenever its induced Euclidean metric is complete
— OpenAI affine-Bernstein manuscript (abstract)
an affine maximal graph of a smooth, locally uniformly convex function on two dimensional Euclidean space, R2, must be a paraboloid
— Neil S. Trudinger and Xu-Jia Wang (2000), abstract
A complete, smooth, uniformly convex solution of the affine maximal surface equation on a convex domain must be a paraboloid.
1. The hypotheses fix n between 3 and 9, an open convex nonempty domain, smooth u with positive definite Hessian, the affine maximal equation holding, and completeness for the graph's Euclidean path metric.
2. The conclusion has three parts: the domain is the whole space, u is a quadratic polynomial with symmetric positive definite matrix, and an affine equivalence carries the standard paraboloid onto the graph of u.
3. The challenge file leaves the proof as sorry.
Prerequisites: OAI.AffineBernstein.AffineMaximalOn, OAI.AffineBernstein.EuclideanGraphComplete
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI affine-Bernstein manuscript (abstract), Pinned OpenAI formalization note 353, Neil S. Trudinger and Xu-Jia Wang (2000), abstract
Distinct definitions of this name: OAI/Analysis/AffineBernstein/Main.lean:18 (same statement text; different proof or body). Qualify references as OAI.AffineBernstein.affine_bernstein@path.

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)
2 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.

affine maximal: satisfying the fourth-order equation that the cofactor-Hessian contracted with the Hessian of det(D2u)^(-(n+1)/(n+2)) vanishes
```
theorem affine_bernstein
    (n : ℕ) (hn₃ : 3 ≤ n) (hn₉ : n ≤ 9)
    (Ω : Set (Space n)) (hΩne : Ω.Nonempty) (hΩopen : IsOpen Ω)
    (hΩconvex : Convex ℝ Ω) (u : Space n → ℝ)
    (hu : ContDiffOn ℝ ∞ u Ω)
    (hpos : ∀ x ∈ Ω, (hessian u x).PosDef)
    (hmax : AffineMaximalOn Ω u)
    (hcomplete : EuclideanGraphComplete Ω u) :
    Ω = univ ∧
      (∃ (A : Matrix (Fin n) (Fin n) ℝ) (b : Space n) (c : ℝ),
        A.IsSymm ∧ A.PosDef ∧
        ∀ x, u x = (1 / 2 : ℝ) * (∑ i : Fin n, ∑ j : Fin n, x i * A i j * x j)
…
```
Mentions: OAI.AffineBernstein.Space, OAI.AffineBernstein.hessian, OAI.AffineBernstein.AffineMaximalOn, OAI.AffineBernstein.EuclideanGraphComplete