NYU mathematics professor Tristan Buckmaster announced three proofs on Tuesday, including a preliminary finding on the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize problems carrying a $1 million bounty from the Clay Mathematics Institute. The Navier-Stokes equations are widely used in fluid mechanics but poorly understood in theoretical terms; a full solution would represent a significant advance in the collective understanding of mathematical physics. The work was done with Levent Alpöge, a mathematician employed by Anthropic but not acting on its behalf; the duo used a mix of models, relying primarily on OpenAI‘s Codex and also using Claude. The findings are significant in themselves, but they are also accompanied by an unusual controversy surrounding OpenAI‘s attempts to solve the same problem.

Buckmaster says that while he and Alpöge were finalizing their results, information about their progress was passed to OpenAI. When they contacted the company, they were told OpenAI had already achieved a full proof of the central problem. Follow-up questions about when OpenAI began its research and how much human input was involved produced evasive answers, according to Buckmaster. It emerged that an entire team had been working on the problem with an ‘insane amount of compute,’ and eventually it was agreed that the first prompt had been sent only after information about the pair’s work reached OpenAI. If true, that would suggest OpenAI became convinced the duo’s approach was correct and then used its computing advantage to reach a formal proof first.

Although the problem is widely pursued among mathematicians, the specific tactic taken by Buckmaster and his collaborator is far less common. Buckmaster writes that the path through a smooth force, options c and d in Fefferman’s statement of the problem, was opened by Luis and Diego, and he and Alpöge had quietly chosen to attack it. ‘Almost nobody else I know of was working on it,’ he wrote, and it is not a direction one arrives at in a few days by giving a model the problem statement. As a result, Buckmaster found it suspicious that OpenAI ended up taking the same approach at the same time. Alpöge’s affiliation with a rival lab seems to have been a sore point for OpenAI, and Buckmaster alleges that Sebastian Bubeck, who leads OpenAI‘s mathematical research, asked him to remove Alpöge’s credit as part of a proposed compromise. When Buckmaster pushed to make the dispute public, Bubeck allegedly replied, ‘Why would you ruin your career?’ and then, ‘If you don’t want me to be nice, then I don’t have to be nice.’

Bubeck disputed the account on the record, calling the claims ‘false and inflammatory.’ He said he came into the discussion following academic norms and was disappointed it had come to this, adding that academic standards are of the highest importance to him and promising a more complete statement soon. Buckmaster also raised a specific concern about his extensive use of Codex. OpenAI reserves the right to train models on Codex interactions, although users can opt out. If the OpenAI team used a model trained on Buckmaster’s own Codex interactions, it is plausible that it could have regurgitated his work when faced with a similar problem. OpenAI did not respond to a request for comment on that possibility.

Regardless, the issue is likely to reignite the ongoing debate about AI’s role in mathematical research and OpenAI‘s specific incentives. Buckmaster says he has not seen OpenAI‘s proof, does not know what their model did or how, and does not know whether the duo’s data was used. ‘I am not accusing anyone of anything,’ he wrote. ‘I am stating what I was told, when, and what was proposed to me. I am stating it because the alternative is to let a sequence of announcements say something I know to be false.’