NYU mathematics professor Tristan Buckmaster announced three proofs on Tuesday, including preliminary findings on one of the most significant unsolved problems in theoretical mathematics. The work, carried out with Anthropic mathematician Levent Alpöge and leveraging both Codex and Claude AI models, is notable in its own right—but it also comes wrapped in an unusual controversy involving OpenAI’s attempts to tackle the same problem. “There is another part of this story,” Buckmaster wrote in his statement announcing the proofs, “and one that, honestly, I very much wish I did not have to be concerned with.”

As Buckmaster and Alpöge were putting the finishing touches on their results, they discovered that “information about our progress had been passed to OpenAI.” When they reached out to OpenAI about this, they were told the company had already achieved a full proof of the central problem. But when they pressed for details—such as when OpenAI had started its research and how much human input was involved—the answers grew increasingly vague. “It emerged that an entire team had been working on the problem,” Buckmaster says, “and that an insane amount of compute had been used…. Eventually, it was agreed that [the first prompt] had been sent in the past few days, after information about our work had reached OpenAI.”

If that account holds, it would suggest the OpenAI team had become convinced that Buckmaster and Alpöge’s approach was the right one and decided to leverage its substantial computing resources to reach a formal proof first. Sebastian Bubeck, who leads OpenAI’s mathematical research, calls those claims “false and inflammatory.”

“To clarify, I came into the discussion following academic norms, and I’m disappointed that it has come to this,” he wrote in a post following Buckmaster’s statement. “Anyone who knows me knows that academic standards are of the highest importance to me.” Bubeck said he would release a more complete statement in the near future.

The dispute centers on the “Navier-Stokes existence and smoothness” problem, one of the seven “Millennium Prize problems”—a collection of major unsolved math challenges, each offering a $1 million prize from the Clay Mathematics Institute to the first person or group to provide a solution. The Navier-Stokes equations are widely used in fluid mechanics but remain poorly understood from a theoretical standpoint. A solution would mark a major leap forward in the collective understanding of mathematical physics.

While the problem is pursued by many mathematicians, the specific strategy taken by Buckmaster and his collaborator is far less common. That made it suspicious to Buckmaster that OpenAI ended up following the same path at the same time. “The route to the Clay problem through a smooth force, options c and d in Fefferman’s statement of the problem, is the route Luis and Diego opened and the one Levent and I had quietly chosen to attack,” Buckmaster wrote. “Almost nobody else I know of was working on it,” he continued. “It is not the direction one arrives at in a few days by giving a model the problem statement.”

Although Alpöge is employed by Anthropic, he was not conducting this research on behalf of the company. As a result, the pair used a mix of models, relying mainly on OpenAI’s Codex for their work. Even so, Alpöge’s affiliation with a rival lab appears to have been a point of friction for OpenAI. Buckmaster alleges that Bubeck asked him to remove Alpöge’s credit as part of a proposed compromise. When Buckmaster pushed to make the dispute public, he says Bubeck replied: “Why would you ruin your career.” Buckmaster adds that when he pushed back, Bubeck followed up with: “If you don’t want me to be nice, then I don’t have to be nice.”

Buckmaster also voiced concerns that, given his extensive use of Codex in assembling the project, information from his work could have informed OpenAI’s own efforts on the problem. OpenAI retains the right to train models on Codex interactions, though users can opt out. If the OpenAI team used a model trained on Buckmaster’s Codex interactions, it’s conceivable that the model could have reproduced his work when faced with a similar problem. OpenAI did not respond to a request for comment on this possibility.

Regardless, the issue is likely to reignite the ongoing debate about AI’s role in mathematical research and OpenAI’s particular incentives. For his part, Buckmaster seems to believe the best course is to get as much information about the research into the public eye as possible. “I have not seen OpenAI’s proof,” Buckmaster wrote. “I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything. 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.”