Panel review: A smooth nonquadratic entire affine maximal graph in dimension ten

Independent model assessments of the frozen manuscript. Model conclusions are not human proof grades. Each model received only the supplied manuscript; external-reference inspection was not enabled. Baseline-v2 and source-first-v2 share the typed reference schema and differ in experimental review strategy.

Returned findings

gpt-6.1-sol / baseline-v2

All six stated claims have complete proofs, and the decisive reduction, invariant-box calculation and continuation argument check out. The construction establishes the advertised pointwise Hessian positivity and classical dimension-ten equation without asserting uniform Hessian bounds or affine-metric completeness. The historical and bibliographic comparisons were not independently inspected and are not needed for the construction.

gpt-6.1-sol / source-first-v2

All six formal claims have correct and complete proofs under the stated pointwise Hessian-positivity hypothesis. The reduction, smooth origin construction, invariant box, and finite-radius continuation close the main existence argument without relying on specialized imported results. Historical comparisons and novelty claims have not been checked against the cited papers.

claude-opus-5-5 / baseline-v2

I re-derived every computation and found no mathematical errors: the Hessian and determinant formulas, the h-identity relying on k=m+2, the radial reduction, the fixed-point setup, the autonomous system, the vanishing of the vector field at the corner, the nonnegativity of the cross derivatives, and the continuation bounds. The argument is self-contained and uses only standard tools (contraction mapping, Gronwall, ODE continuation); the literature is cited only as background. Remaining issues are expository.

claude-opus-5-5 / source-first-v2

Every computation and proof step I checked holds: the Hessian and determinant reduction, the trace identity, the radial ODE, the fixed-point construction, the derivation of the autonomous system, the stationary corner, the cooperative cross-derivatives, the invariance argument and the continuation. The main theorem follows within the stated formulation: smooth, with pointwise positive-definite Hessian, and with no completeness or uniform-convexity assumption. The remaining issues are expository brevity in routine steps and literature claims that cannot be verified from the manuscript.

claude-fable-5-1 / baseline-v2

The paper is short, self-contained, and every computational step (Hessian formula, determinant, w=Ph, the trace identity, radial reduction, the fixed-point setup, the autonomous system, the equilibrium a_*, the cross-derivative signs, and the compactness argument) was independently rederived and found correct. The argument relies only on standard ODE facts (contraction mapping, Gronwall, Picard-Lindelof) and no external theorems are used as proof inputs. Given the significance of the claimed result (a smooth entire counterexample to the affine Bernstein assertion in dimension ten), it deserves independent scrutiny, but I found no mathematical error or gap.

claude-fable-5-1 / source-first-v2

The paper is short, self-contained and, on independent recomputation of every identity (Hessian block structure, determinant and w formulas, the h-identity, the autonomous system, the equilibrium a_*, the cross-derivative signs, and the compactness bounds), the proofs are correct and complete. The only external citations serve as background (identification of the classical equation and of the singular example); no imported theorem is a proof input. The main theorem follows from the verified chain of lemmas.

gemini-4-argon / baseline-v2

This is an exceptionally clear, self-contained, and rigorous paper that constructs a smooth nonquadratic entire affine maximal graph in dimension ten with everywhere positive-definite Hessian. Every calculation and logical step—from the block-diagonal Hessian reduction and the desingularized local fixed-point argument at the origin to the cooperative invariant box for the logarithmic derivatives—has been verified and is completely sound.

gemini-4-argon / source-first-v2

This is a remarkably clear, self-contained, and rigorous paper that resolves the smooth entire-graph affine Bernstein question in dimension ten (under pointwise Hessian positivity) by smoothing the singular Trudinger-Wang example. Every step of the reduction, local fixed-point argument, and global continuation via a cooperative invariant box has been verified and holds without any gaps.

Remaining assessment

Human substantiation, localization, evidence, uncertainty and coverage grades remain pending in the separate anonymous comparison packet. Finite output caps and failed responses are resource outcomes, not mathematical verdicts. External publications and claimed formalizations have not been independently audited.