Illustrating OpenAI's AI Proof of the Mahler Conjecture

Summary

A mathematical expository blog examines OpenAI’s reported proof of the symmetric Mahler conjecture, focusing on its symplectic-geometric construction rather than presenting the result as independently verified. For an n-dimensional symmetric convex body K and its polar K° , the proof constructs a symplectic embedding of a ball of width 4 into K × K°, which implies the volume bound Vol(K × K°) ≥ 4^n/n! and settles the conjectured lower bound when combined with a known upper bound on the Gromov width. The article explains the connection to polar duality, symplectic capacities, the Gromov width, and the Viterbo conjecture. Its central technical strategy has two parts: a symplectic monotonicity result for holomorphic submanifolds intersected with balls, and a construction of holomorphic submanifolds from complexified strip-defining linear functions. A conformal map from a lens-shaped domain to the unit disk is used to straighten the pulled-back symplectic form; in the large-power limit, the resulting domains approach K × (πk/4)K°. The author illustrates the construction with interactive visualizations, including sliced four-dimensional shapes and the gradient-flow images of symplectic balls. The post also discusses how the method may characterize equality cases associated with Hanner polytopes. The author emphasizes that the AI paper is 45 pages, gives only a short discussion of the symmetric case, and lacks a Lean proof; the exposition treats the proof as correct for purposes of explanation but explicitly says its line-by-line details have not been checked. The author also describes using Claude Code and Claude for visualizations, layout, reading assistance, and proofreading, while stating that the article’s text and hand-drawn figures were made by the author.