The affine Bernstein problem
From OpenAI Aftermath
The affine Bernstein problem asks whether a complete locally uniformly convex affine maximal hypersurface, a critical point of affine surface area, must be an elliptic paraboloid; the answer depends on the dimension and on the notion of completeness, Euclidean or affine. Two manuscripts in OpenAI's public mathematics collection (github.com/openai/math), dated 24 September and 5 October 2026, claim that every smooth locally uniformly convex affine maximal graph of dimension three through nine that is complete for the induced Euclidean metric is an elliptic paraboloid, and that a smooth nonquadratic entire affine maximal graph exists in dimension ten. The collection includes a Lean formalization of the rigidity theorem. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introductions of the manuscripts:
- Chern asked whether an entire locally uniformly convex affine maximal surface must be an elliptic paraboloid. Calabi proved the surface case under both Euclidean and affine completeness, Trudinger and Wang proved the Euclidean-complete surface theorem without affine completeness, and Li and Jia gave a proof under affine completeness alone. Trudinger and Wang also proved that affine completeness implies Euclidean completeness for locally uniformly convex hypersurfaces.
- In higher dimensions, the interior estimates and rescaling argument of Trudinger and Wang give rigidity provided a modulus of strict convexity stays uniform under normalization, and they exhibited a singular entire affine maximal graph in dimension ten that is not smooth. Wang and Zhou gave a proof of the surface theorem through the partial Legendre transform and higher-dimensional rigidity under integral bounds on the Hessian and the inverse Hessian determinant, and Sun, Xing and Xu constructed nonquadratic Euclidean-complete examples for affine maximal type equations with other determinant exponents, excluding the classical one.
The original results
- Claim. Theorem 1.1 of The affine Bernstein theorem in dimensions three through nine states that for 3 ≤ n ≤ 9, if a smooth function on a nonempty open convex domain in Rn has positive-definite Hessian, satisfies the affine maximal equation and has a graph complete for the induced Euclidean metric, then the domain is all of Rn and the function is a positive-definite quadratic polynomial plus an affine term; a corollary extends the conclusion to connected open affine-complete locally uniformly convex immersed hypersurfaces that are affine maximal. Theorem 1.1 of A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten gives a nonquadratic smooth function on R10 with positive-definite Hessian at every point that solves the affine maximal equation; the manuscript states that the Hessian positivity is pointwise and that no growth assumption or completeness of the affine metric is asserted.
- Method, as described: the rigidity manuscript studies full-dimensional convex limits of affine images of the epigraph, obtains a uniform centering estimate by maximizing the number of nonnegative model directions, and excludes two or more such directions by an integral inequality in logarithmic coordinates; the dimension restriction enters through a quadratic coercivity estimate that is strictly positive through dimension nine. The dimension-ten manuscript keeps the relation suggested by the singular example of Trudinger and Wang with a smooth radial profile, and proves global existence of the resulting radial system with an invariant box for logarithmic derivatives.
- Sources. The manuscripts The affine Bernstein theorem in dimensions three through nine and A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten at commit
1581d4c. - Formalization, as described by the repository. The scope note states that the formalization proves the Euclidean-complete affine Bernstein theorem for graph dimensions three through nine: a smooth function on a nonempty open convex domain with positive-definite Hessian that satisfies the affine maximal equation and has a graph complete in the induced Euclidean metric is defined on all of Rn and is a positive-definite quadratic polynomial plus an affine term. The extension to immersed hypersurfaces without an initial graph assumption is outside this statement, and the dimension-ten manuscript is not listed. The Comparator statement is
AffineBernstein.lean.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 20 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | N. S. Trudinger and X.-J. Wang, The higher dimensional affine Bernstein and Plateau problems, arXiv:2610.09385v1 | Parallel work (independent) | Proves interior regularity for the affine Plateau problem in all dimensions, presents a smooth counterexample to the affine Bernstein problem in dimensions ten and higher, and proves an affine Bernstein theorem in all dimensions under a growth condition. A note added on 7 October 2026 states that the dimension-ten counterexample in the OpenAI manuscript is already included in Section 3 of the paper, which the note says was submitted on 2 February 2026 and presented at conferences in March and May 2026. | arXiv preprint, version 1 of 7 October 2026. |
| 2026-10-08 | MathVet (GitHub; review of openai/math, tables at repository commit c76f1e4, 14 October 2026) |
Review (fidelity of the formal statements) | Rates this family "partial": the Lean statement gives the classification of Euclidean-complete graphs in dimensions three through nine, while the dimension-ten example and the extension to hypersurfaces complete for the affine metric have no statement. It notes that Euclidean completeness is defined through a hand-built path metric on the graph. It records a sandboxed Comparator pass for AffineBernstein. |
MathVet states that it does not claim any theorem false and reports what was machine-checked as stated. |
| 2026-10-10 | cert-machine, openai/math lane K, run 38017370880 | Verification (two kernels) | Records that Lean's default kernel and the nanoda kernel accept the AffineBernstein solution (1,264 s). The project reports two-kernel certification of 398 of 416 challenges of the collection. |
Repository. |
| 2026-10-10 to 2026-10-14 | con-leche development journal (task #348) | Verification (second kernel) | In its run over all Comparator challenges of the collection at commit fd4aeeb, the journal's table of exceptions (time-outs, memory, errors) does not list AffineBernstein, so it was accepted by con-leche and by Lean's official kernel. |
Development journal. |
| 2026-10-13 | M. Colbrook and contributors, AIM — Open Applied Problems (GitHub), problem 391 | Catalogue | The collection records its problem 391, the entire-graph case of the affine Bernstein conjecture in dimensions three through nine without growth, bounded-Hessian or affine-completeness assumptions, as solved by the rigidity manuscript, noting that an entire graph is complete for the induced Euclidean metric. | The status review is described as "an AI statement-level review of a published manuscript, not a fresh independent audit of its complete proof or a Lean kernel replay". The collection is not affiliated with the American Institute of Mathematics. |
| 2026-10-16 | J. Jardine, Lattice24, referee reviews of the rigidity manuscript and the dimension-ten manuscript in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Both reviews report no gap. The review of the dimension-ten manuscript states that every step was checked by hand, with a numerical check of the trajectory, and has no minor notes. The review of the rigidity manuscript has one minor note: an interior estimate of Trudinger and Wang is quoted in a repackaged form whose match with the original hypotheses is not shown, and the final rescaling step depends on it; it also remarks that the dimension restriction comes from a single case of the coercivity estimate, as the manuscript states, and that the cited theorems of Trudinger and Wang were not checked against their sources. | Part of an audit of the collection's manuscripts without Lean that the audit describes as complete; its list includes the rigidity manuscript, which the scope note names as the paper accompanying the formalization. The reviews are published under CC BY 4.0. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscripts.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
