Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald–Blaschke metric. (OpenAI’s Lean note)
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Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Let 3\le n\le9. Suppose u\in C^\infty(\Omega) has positive-definite Hessian and satisfies (1.1). 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.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
We construct a smooth nonquadratic entire graph in dimension ten that solves the classical affine maximal equation and has positive-definite Hessian everywhere. This gives a smooth counterexample to the entire-graph affine Bernstein assertion in dimension ten. Hessian positivity is pointwise; no global uniform lower bound or completeness of the Berwald–Blaschke metric is asserted.
We prove the Euclidean-complete affine Bernstein conjecture in dimensions three through nine: a smooth locally uniformly convex affine maximal graph is an elliptic paraboloid whenever its induced Euclidean metric is complete. The same conclusion holds, without an initial graph assumption, for connected smooth open (noncompact and without boundary) affine-complete locally uniformly convex immersed hypersurfaces that are classically affine maximal in these dimensions.
One Lean statement in this family
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Let 3\le n\le9. Suppose u\in C^\infty(\Omega) has positive-definite Hessian and satisfies (1.1). 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.
The words they share: 1/2, vector, induced, matrix, complete, paraboloid, standard, particular, graph, ij, euclidean, hessian, image, metric, maximal, affine, bernstein, elliptic, definite, positive.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
AffineBernstein.lean, the release lists it as a supporting statement.
theoremaffine_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) := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
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
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: 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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNLINKED”).
The gap they name:Not stated: the dimension-ten counterexample (clause 2, undeclared) and the affine-complete hypersurface extension (clause 3, declared).
“NARROWER — UNLINKED”, in their words: The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
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.
The paper’s extension to immersed hypersurfaces without an initial graph assumption is outside this selected statement.
AI-drafted comparisons, with no recorded human review
Affine Bernstein classification for complete graphs in dimensions three through nineAffineBernstein
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for 3 ≤ n ≤ 9, Euclidean-complete affine maximal graphs are elliptic paraboloids. The extension to immersed hypersurfaces is outside the statement.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Affine Bernstein classification for complete graphs in dimensions three through nineAffineBernstein
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
These reports examine claims in the papers using HOL Light, a different proof assistant, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
The algebra of the invariant box: stationary corner (9/2, 7/2, 33/7), face signs, the D-equation from the C-equation
HOL Light card 353
Reported by
Ben Knill, openai-math-hol-cards
Method
Proof of a selected step in HOL Light
Their verdict
“PASS” The report records a proof of a supporting fact.
What it covers
Checks algebra inside a result that is paper-only in the release (Lean covers dimensions 3–9 only); the ODE and PDE reduction are not covered
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
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Explain a result, compare statements or propose a correction
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
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Neil S. Trudinger, Xu-Jia Wang, The higher dimensional affine Bernstein and Plateau problems, 7 October 2026. Source
Trudinger and Wang state that the dimension-ten affine Bernstein counterexample in OpenAI’s release is already in Section 3 of their paper, submitted on 2 February 2026.
Questions awaiting review
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Read the Lean statement against the paper
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Identify the theorem the catalogue describes
Family 353The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method
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.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.