OpenAI announced Tuesday that one of its internal models had demonstrated finite-time "blow-up" for the forced Navier-Stokes equations, one of seven Millennium Prize Problems worth $1 million each. If verified, it would be only the second Millennium problem ever solved and the first credited to an AI company.
Days before OpenAI's announcement, NYU mathematician Tristan Buckmaster and Levent Alpöge, a mathematician at Anthropic, had published their own results on finite-time blow-up, which Fields Medalist Terence Tao called a "remarkable achievement."
Buckmaster says the account OpenAI gave him during a September 6 phone call differed significantly from the company's public announcement. According to Buckmaster's account, he was initially told an internal OpenAI model had produced a roughly 100-page proof with "very little human input." During the call, he says this description changed as additional details emerged.
Buckmaster wrote that he learned "an entire team had been working on the problem, that this was one of a number of things that was tried, that work had started on the unforced problem, that the team first set the model on easier problems, including Euler," and that a significant amount of compute had been used.
Buckmaster also raised questions about whether OpenAI's model might have accessed his and Alpöge's draft proofs, which he says were kept in OpenAI Codex sessions. He stated he received no answer to this question, and emphasized he was not claiming access occurred, only that the question went unanswered.
Buckmaster attributed the core mathematical ideas to Diego Córdoba and Luis Martínez-Zoroa, saying he and Alpöge used large language models to extend that work. He stated he believes Martínez-Zoroa should be awarded a Fields Medal.
OpenAI denied Buckmaster's allegations. Sébastien Bubeck, who participated in the call, called the claims "false and inflammatory" on social media and said he had proceeded "following academic norms." At a briefing Tuesday, Bubeck said the model reached the Euler problem differently than the mathematicians but took a similar route for the complete Navier-Stokes proof, which was built over the weekend.
Buckmaster also alleged that Bubeck twice pressed to remove Alpöge as an author due to his Anthropic affiliation.
