OpenAI said on Sunday that one of its general-purpose reasoning models had autonomously produced a proof of a long-standing problem in classical convex geometry that had eluded mathematicians for roughly 80 years. The company published the proof together with a verification by a small panel of academic mathematicians, who independently confirmed that the argument was both novel and correct.
The problem, which originated in a 1944 paper and has since become a touchstone in the geometry of convex bodies, asks whether a particular extremal inequality on volumes of projections holds for all dimensions above three. Partial results going back decades had established the inequality for dimensions four through seven, but a general argument had remained out of reach.
OpenAI declined to identify which of its reasoning models produced the result but said the system had been deployed with broad tool use, including a long-context theorem-proving harness and access to existing formal proof libraries. The company said the model arrived at its argument without explicit human prompting beyond a single instruction to attempt the problem, and without retrieving any prior published material on it during the session.
Independent academic reaction was cautiously enthusiastic. Three mathematicians named on the verification panel — from Princeton, the Collège de France and the University of Tokyo — described the proof as "structurally interesting" and "not a recombination of known techniques", and said they had spent more than two months examining it before agreeing to publish their endorsement. A more thorough peer-reviewed write-up is expected in a forthcoming issue of the Annals of Mathematics.
The result is the most concrete evidence yet that the current generation of large reasoning models can contribute to research-level mathematics, an area in which prior model claims have often shaded into overstatement. Both Anthropic and Google DeepMind have published results in formal-verification benchmarks earlier this year; Sunday’s announcement is the first time a frontier lab has claimed an open-problem proof that was not previously reduced to a benchmark format.
OpenAI did not say whether the model that produced the proof will be released to the public. The company has been cautious about exposing its most capable reasoning systems outside enterprise customers, citing both compute cost and safety review. Sam Altman, in a post on X, said the result was "a hint of what’s coming next year" and credited the model with "good taste in choosing which arguments to chase".
The geometry result lands two weeks after Anthropic disclosed that its operating profit had turned positive on revenue that more than doubled, in a quarter that has seen the AI sector’s narrative shift decisively from infrastructure spend to applied capability.