Credit dispute between OpenAI and an NYU mathematician after a historic Navier-Stokes proof

New York University mathematics professor Tristan Buckmaster announced on Tuesday three proofs containing a preliminary finding on one of the great unsolved problems in theoretical mathematics. The work, carried out with Anthropic mathematician Levent Alpöge and supported by Codex (OpenAI) and Claude models, is significant in its own right, but it arrived accompanied by an unusual controversy over how OpenAI became involved in the same problem.

According to Buckmaster's own statement, a parallel effort by OpenAI was built on his work before it was made public. Shortly after Buckmaster released his statement, OpenAI published a complete proof of the Navier-Stokes existence and smoothness problem, one of the Clay Mathematics Institute's seven Millennium Problems, each carrying a one-million-dollar reward for whoever solves it. The Navier-Stokes equations are widely used in fluid mechanics but very poorly understood at the theoretical level, so a formal proof would be a major advance.

OpenAI says its proof was obtained by an as-yet unreleased next-generation model that over the past week tackled several unsolved problems. According to the company, that effort consumed 300 billion output tokens, equivalent to $22.5 million of compute at current Astra rates.

Buckmaster recounts that, while he and Alpöge were finalizing their own results, they learned that information about their advance had reached OpenAI. When they contacted the company, they were told it already had a complete proof of the central problem. But when they asked when OpenAI had begun its research and how much human intervention was involved, the answers turned evasive. According to Buckmaster, it eventually emerged that an entire team had worked on the problem with "an insane amount" of compute, and it was finally agreed that the first prompt had been sent "in the last few days," after information about Buckmaster and Alpöge's work reached OpenAI. If so, that would suggest OpenAI's team became convinced their approach was the right one and used its advantage in compute resources to reach a formal proof first.

OpenAI's own statement confirms much of this timeline: the most recent effort began on September 1, prompted by rumors that two Millennium Problems had been solved, and it acknowledges the conversations held with Buckmaster and Alpöge.

Buckmaster argues that it is suspicious for OpenAI to have arrived at the same line of attack at the same time, since it is an unusual route among mathematicians working on this problem: the one running through options c and d of Fefferman's formulation, with "soft force," opened up by other collaborators and which he and Alpöge had quietly chosen. According to Buckmaster, "almost no one else" he knew of was working in that direction, and it is not a route one reaches in a few days simply by handing a model the statement of the problem.

Although Alpöge works at Anthropic, he did not carry out this research on the company's behalf, and in fact the pair of researchers relied above all on OpenAI's Codex. Even so, Alpöge's affiliation with a rival lab appears to have been a sensitive point for OpenAI. Buckmaster says OpenAI mathematician Sébastien Bubeck asked him to drop Alpöge's credit as part of a proposed agreement, and that when Buckmaster insisted on making the dispute public, Bubeck replied: "Why would you ruin your career?" When he persisted, Bubeck reportedly added: "If you don't want me to be nice, I don't have to be."

Buckmaster also raised a further concern: because he used Codex extensively to develop his project, information from that work could have unknowingly fed OpenAI's own efforts to solve the problem. OpenAI reserves the right to train its models on Codex interactions, though users can opt out. Had OpenAI's team used a model trained on Buckmaster's Codex interactions, it is plausible that it would reproduce part of his work when confronted with a similar problem.

In its statement, OpenAI plays down that possibility: it says neither its researchers nor its agents saw Buckmaster and Alpöge's work by any means until it was published, and that no specific user data was accessed to solve the problem. The company acknowledges that, while unlikely, it cannot rule out that de-identified data derived from the use of its products contributed to improving its models. It also stresses that its proofs differ significantly and that even the exact results proved are different in the Euler case (forced versus unforced).

The episode revives the debate over the role of artificial intelligence in mathematical research and over OpenAI's specific incentives in that race. Buckmaster, for his part, appears to believe that the best response is to publicly disclose as much information as possible about what happened.