# Affine maximal graph rigidity reference

Family 353: Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Geometry researchers can track the dimension-three-to-nine rigidity threshold and dimension-ten smooth counterexample.

## Applicability and commercial boundary

Local uniform convexity, completeness and the selected affine or Euclidean metric are strong exact conditions. No finite shape reconstruction method follows.

## Initial business decision

Research or conditional engineering only. A practical implementation, recurring buyer problem and willingness to pay have not been established. Mathematical usefulness alone is insufficient evidence for a standalone lucrative product.

## Next verification action

Extract explicit paraboloid and counterexample models for research comparison.

## Evidence scope

The catalog statement was individually reviewed. The manuscript proof and formal-scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[A Smooth Nonquadratic Entire Affine Maximal Graph in Dimension Ten](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Smooth-Nonquadratic-Affine-Maximal-Graph-in-Dimension-Ten-October-5-2026/affine-maximal-dimension-ten.pdf).