Lean statement

Affine Bernstein classification for complete graphs in dimensions three through nine

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.AffineBernstein.affine_bernstein

Lean statement file: AffineBernstein.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 353 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

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)          + (∑ i : Fin n, b i * x i) + c) ∧      (∃ e : (Space n × ℝ) ≃ᵃ[ℝ] (Space n × ℝ),        e '' standardParaboloid n = graph Ω u) := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.AffineBernstein.Space line 12

abbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)

definition OAI.AffineBernstein.coordinateVector line 14

def coordinateVector (n : ℕ) (i : Fin n) : Space n :=  EuclideanSpace.single i 1

definition OAI.AffineBernstein.hessian line 19

def hessian {n : ℕ} (u : Space n → ℝ) (x : Space n) : Matrix (Fin n) (Fin n) ℝ :=  fun i j => fderiv ℝ (fun y => fderiv ℝ u y (coordinateVector n j)) x    (coordinateVector n i)

definition OAI.AffineBernstein.cofactorHessian line 24

def cofactorHessian {n : ℕ} (u : Space n → ℝ) (x : Space n) :    Matrix (Fin n) (Fin n) ℝ :=  (hessian u x).det • (hessian u x)⁻¹

definition OAI.AffineBernstein.affineWeight line 29

def affineWeight {n : ℕ} (u : Space n → ℝ) (x : Space n) : ℝ :=  Real.rpow (hessian u x).det (-(((n : ℝ) + 1) / ((n : ℝ) + 2)))

definition OAI.AffineBernstein.AffineMaximalOn line 32

def AffineMaximalOn {n : ℕ} (Ω : Set (Space n)) (u : Space n → ℝ) : Prop :=  ∀ x ∈ Ω, ∑ i : Fin n, ∑ j : Fin n,    cofactorHessian u x i j * hessian (affineWeight u) x i j = 0

definition OAI.AffineBernstein.graphPathLength line 39

def graphPathLength {n : ℕ} (u : Space n → ℝ) (γ : ℝ → Space n) : ℝ≥0∞ :=  ∫⁻ t in Icc (0 : ℝ) 1,    ENNReal.ofReal (Real.sqrt      (‖derivWithin γ (Icc (0 : ℝ) 1) t‖ ^ 2 +       (fderiv ℝ u (γ t) (derivWithin γ (Icc (0 : ℝ) 1) t)) ^ 2))

definition OAI.AffineBernstein.graphEDist line 48

def graphEDist {n : ℕ} (Ω : Set (Space n)) (u : Space n → ℝ)    (x y : Space n) : ℝ≥0∞ :=  ⨅ (γ : ℝ → Space n)    (_ : ContDiffOn ℝ 1 γ (Icc (0 : ℝ) 1))    (_ : MapsTo γ (Icc (0 : ℝ) 1) Ω)    (_ : γ 0 = x) (_ : γ 1 = y), graphPathLength u γ

definition OAI.AffineBernstein.EuclideanGraphComplete line 58

def EuclideanGraphComplete {n : ℕ} (Ω : Set (Space n)) (u : Space n → ℝ) : Prop :=  ∀ x : ℕ → Space n, (∀ i, x i ∈ Ω) →    (∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ i ≥ N, ∀ j ≥ N,      graphEDist Ω u (x i) (x j) < ENNReal.ofReal ε) →    ∃ a ∈ Ω, ∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ i ≥ N,      graphEDist Ω u (x i) a < ENNReal.ofReal ε

definition OAI.AffineBernstein.graph line 66

def graph {n : ℕ} (Ω : Set (Space n)) (u : Space n → ℝ) : Set (Space n × ℝ) :=  {p | p.1 ∈ Ω ∧ p.2 = u p.1}

definition OAI.AffineBernstein.standardParaboloid line 69

def standardParaboloid (n : ℕ) : Set (Space n × ℝ) :=  {p | p.2 = ∑ i : Fin n, (p.1 i) ^ 2}

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: 1/2, vector, induced, matrix, complete, paraboloid, standard, particular and eight more. Found in the paper’s title, which its main theorem states: bernstein. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: elliptic, definite and positive.

Statements in The affine Bernstein theorem in dimensions three through nine come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of The affine Bernstein theorem in dimensions three through nine

Theorem

Let 3\le n\le9. Suppose u\in C^\infty(\Omega) has positive-definite Hessian and satisfies (eq:affine-maximal). If its graph is complete for the induced Euclidean metric

g_{ij}=\delta_{ij}+u_i u_j,

then \Omega=\R^n and

u(x)=\tfrac12 x^{\mathsf T}Ax+b\cdot x+c

for a positive-definite symmetric matrix A, a vector b, and a constant c. In particular, the graph is an invertible affine image of the standard elliptic paraboloid.

Section 1, Introduction Read it in the paper TeX source, line 82

Show the TeX

Let \(3\le n\le9\). Suppose \(u\in C^\infty(\Omega)\) has
positive-definite Hessian and satisfies \eqref{eq:affine-maximal}.
If its graph is complete for the induced Euclidean metric
\[
 g_{ij}=\delta_{ij}+u_i u_j,
\]
then \(\Omega=\R^n\) and
\[
 u(x)=\tfrac12 x^{\mathsf T}Ax+b\cdot x+c
\]
for a positive-definite symmetric matrix \(A\), a vector \(b\), and a
constant \(c\). In particular, the graph is an invertible affine image
of the standard elliptic paraboloid.

Other possible paper matches, closest wording first

Theorem 6.4 of The affine Bernstein theorem in dimensions three through nine. Words of the Lean statement, its names split into words, that also appear in the theorem: domain, integer, gives, dist, convex, hessian and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: function, definite and positive.
Corollary 6.5 of The affine Bernstein theorem in dimensions three through nine. Words of the Lean statement, its names split into words, that also appear in the theorem: complete, paraboloid, convex, standard, image, metric and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: elliptic.
Proposition 2.6 of The affine Bernstein theorem in dimensions three through nine. Words of the Lean statement, its names split into words, that also appear in the theorem: coordinates, particular, point, depending and maximal.
Theorem 1.1 of A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten. Words of the Lean statement, its names split into words, that also appear in the theorem: det, ij and point. Found in the paper’s title, which its main theorem states: graph, smooth, maximal and affine. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: function, definite and positive.
Proposition 4.1 of The affine Bernstein theorem in dimensions three through nine. Words of the Lean statement, its names split into words, that also appear in the theorem: completeness, euclidean, metric and depending.
Proposition 2.1 of A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten. Words of the Lean statement, its names split into words, that also appear in the theorem: sqrt, det, hessian, smooth and affine. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: functions, dimension, definite and positive.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 353’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: affine_bernstein (3 ≤ n ≤ 9): a smooth function with positive-definite Hessian on a nonempty open convex Ω solving the affine-maximal equation with graph complete for the induced Euclidean path metric forces Ω = ℝⁿ, u a positive-definite quadratic plus affine term, and the graph an affine image of the standard paraboloid. The sharpness claim (smooth entire nonquadratic example in dimension ten) and the extension to hypersurfaces complete for the affine Berwald-Blaschke metric have no statement (docs: immersed extension outside the statement).
Definitions the group asks readers to compare with the paper: AffineMaximalOn := Σ U^{ij} ∂_ij ((det D²u)^(-(n+1)/(n+2))) = 0 with U the cofactor matrix (the classical affine maximal equation); EuclideanGraphComplete is sequential completeness for the hand-built path metric graphEDist = inf of lengths ∫√(|γ'|²+(Du·γ')²) of C¹ paths in Ω
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: Not stated: the dimension-ten counterexample (clause 2, undeclared) and the affine-complete hypersurface extension (clause 3, declared).

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves the Euclidean-complete affine Bernstein theorem for graph dimensions $3\le n\le9$. If a smooth function on a nonempty open convex domain has positive-definite Hessian, satisfies the affine maximal equation, and its graph is complete in the induced Euclidean metric, then the domain is all of $\mathbb R^n$ and the function is a positive-definite quadratic polynomial plus an affine term. Its graph is therefore an elliptic paraboloid.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

94 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 43 to its own declarations, 47 to Mathlib’s documentation. 12 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves affine_bernstein exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 11 other declarations of the file; with the theorem itself they span 51 lines. 6 of them are named in the statement itself.

Names in the namespace OAI.AffineBernstein are listed without it.

abbrev Spaceline 12in the statementlibrary
def coordinateVectorline 14library
def hessianline 19in the statementlibrary
def cofactorHessianline 24library
def affineWeightline 29library
def AffineMaximalOnline 32in the statementlibrary
def graphPathLengthline 39library
def graphEDistline 48library
def EuclideanGraphCompleteline 58in the statementlibrary
def graphline 66in the statementlibrary
def standardParaboloidline 69in the statementlibrary
theorem affine_bernsteinline 73selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen Set MeasureTheoryopen scoped BigOperators ContDiff ENNRealnamespace AffineBernsteinabbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)def coordinateVector (n : ℕ) (i : Fin n) : Space n :=  EuclideanSpace.single i 1/-- 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. -/def hessian {n : ℕ} (u : Space n → ℝ) (x : Space n) : Matrix (Fin n) (Fin n) ℝ :=  fun i j => fderiv ℝ (fun y => fderiv ℝ u y (coordinateVector n j)) x    (coordinateVector n i)/-- The coefficient U^{ij} = det(D²u) (D²u)^{-1}_{ij}. -/def cofactorHessian {n : ℕ} (u : Space n → ℝ) (x : Space n) :    Matrix (Fin n) (Fin n) ℝ :=  (hessian u x).det • (hessian u x)⁻¹/-- The classical determinant weight; in particular, this is not a generalized exponent. -/def affineWeight {n : ℕ} (u : Space n → ℝ) (x : Space n) : ℝ :=  Real.rpow (hessian u x).det (-(((n : ℝ) + 1) / ((n : ℝ) + 2)))def AffineMaximalOn {n : ℕ} (Ω : Set (Space n)) (u : Space n → ℝ) : Prop :=  ∀ x ∈ Ω, ∑ i : Fin n, ∑ j : Fin n,    cofactorHessian u x i j * hessian (affineWeight u) x i j = 0/-- Length on [0,1] for g_x(v,v) = ‖v‖² + (Du_x v)², the metric inducedby the Euclidean graph embedding x ↦ (x,u(x)). The product norm on Lean's ordinaryproduct type is NOT used (it would be the maximum norm). -/def graphPathLength {n : ℕ} (u : Space n → ℝ) (γ : ℝ → Space n) : ℝ≥0∞ :=  ∫⁻ t in Icc (0 : ℝ) 1,    ENNReal.ofReal (Real.sqrt      (‖derivWithin γ (Icc (0 : ℝ) 1) t‖ ^ 2 +       (fderiv ℝ u (γ t) (derivWithin γ (Icc (0 : ℝ) 1) t)) ^ 2))/-- 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. -/def graphEDist {n : ℕ} (Ω : Set (Space n)) (u : Space n → ℝ)    (x y : Space n) : ℝ≥0∞ :=  ⨅ (γ : ℝ → Space n)    (_ : ContDiffOn ℝ 1 γ (Icc (0 : ℝ) 1))    (_ : MapsTo γ (Icc (0 : ℝ) 1) Ω)    (_ : γ 0 = x) (_ : γ 1 = y), graphPathLength u γ/-- Sequential completeness for the induced Euclidean path metric. This statesthat 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. -/def EuclideanGraphComplete {n : ℕ} (Ω : Set (Space n)) (u : Space n → ℝ) : Prop :=  ∀ x : ℕ → Space n, (∀ i, x i ∈ Ω) →    (∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ i ≥ N, ∀ j ≥ N,      graphEDist Ω u (x i) (x j) < ENNReal.ofReal ε) →    ∃ a ∈ Ω, ∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ i ≥ N,      graphEDist Ω u (x i) a < ENNReal.ofReal ε/-- The graph, regarded as an affine subset of R^{n+1}. -/def graph {n : ℕ} (Ω : Set (Space n)) (u : Space n → ℝ) : Set (Space n × ℝ) :=  {p | p.1 ∈ Ω ∧ p.2 = u p.1}def standardParaboloid (n : ℕ) : Set (Space n × ℝ) :=  {p | p.2 = ∑ i : Fin n, (p.1 i) ^ 2}/-- Literal main theorem, with its stated affine-image consequence. -/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)          + (∑ i : Fin n, b i * x i) + c) ∧      (∃ e : (Space n × ℝ) ≃ᵃ[ℝ] (Space n × ℝ),        e '' standardParaboloid n = graph Ω u) := by  sorryend AffineBernsteinendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

The affine Bernstein theorem in dimensions three through nine