OpenAI · 353-01 · 2026-10-05 · Differential geometry
A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten
preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026
Family 353. The sharp dimension threshold for affine Bernstein rigidity.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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.
Preprint source and Lean scope
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A Lean scope note is stored for family 353, but it does not name this preprint. It names 353-02. This page still has the full preprint source. No compilation is asserted. O.1 stays NOT_CLOSED.