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Math-2650 · mathematics · Geometry

Affine Bernstein assertion for entire graphs in dimension ten

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Must every smooth function $u:R^{10}\to R$ with everywhere positive-definite Hessian, satisfying $\Sigma _ij$ $U^ij$ $\partial _ij(det$ $D^{2}u)^{-11/12}=0$ and $U=det(D^{2}u)(D^{2}u)^{-1}$, be a quadratic polynomial plus an affine term?

Short English statement

Must every smooth function $u:R^{10}\to R$ with everywhere positive-definite Hessian, satisfying $\Sigma _ij$ $U^ij$ $\partial _ij(det$ $D^{2}u)^{-11/12}=0$ and $U=det(D^{2}u)(D^{2}u)^{-1}$, be a quadratic polynomial plus an affine term?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2650-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2650

Smooth affine-maximal graph in dimension ten: The entire graph is smooth and has pointwise positive Hessian, so the dimension-ten example lies in the asserted class. Uniform convexity with a global numerical lower bound and completeness of the affine metric are not part of this question.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

Affine Bernstein assertion for entire graphs in dimension ten

OpenAI's claim

Smooth affine-maximal graph in dimension ten: In dimension ten, the claimed affine-maximal example is an entire smooth convex graph whose Hessian is positive at every point and whose defining function is not quadratic. The assertion does not require a uniform Hessian lower bound or affine-metric completeness.

Mathematical statement of the claimed result

Mathematical claim 1.
The manuscript claims a nonquadratic $u\in C^\infty(\mathbb R^{10})$ with positive-definite Hessian everywhere and \[\sum_{i,j=1}^{10}U^{ij}\partial_{ij}(\det D^2u)^{-11/12}=0,\qquad U=\det(D^2u)(D^2u)^{-1}.\] Thus it is an entire strictly convex smooth affine-maximal graph not given by a quadratic polynomial.

Outline of the argument

Smooth affine-maximal graph in dimension ten: An ansatz of the form $u(x,t)=\sqrt{F(x)^2+t^2}$ reduces the fourth-order affine equation to a coupled system for two functions in one fewer variable. The reduced determinant and inverse-Hessian identities are solved while retaining smooth positivity. Substitution gives an everywhere convex entire graph, and its explicit nonquadratic behavior contradicts the universal dimension-ten assertion.

Formal verification

Family 353 describes Lean coverage for the Euclidean-complete graph rigidity in dimensions three through nine; the nongraph and dimension-ten assertions are separate.

Smooth affine-maximal graph in dimension ten: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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