Tristan Buckmaster, a mathematics professor at New York University, announced on Tuesday three proofs connected to the Navier–Stokes existence and smoothness problem — one of the seven Millennium Prize problems designated by the Clay Mathematics Institute, each carrying a $1 million award. Buckmaster worked with Levent Alpöge of Anthropic, and their research made use of AI models including OpenAI’s Codex and Anthropic’s Claude.
In the statement accompanying the proofs, Buckmaster said there was an additional issue: he learned that information about their progress had been transmitted to OpenAI. When he and his collaborator contacted OpenAI, they were told that OpenAI had already produced a full proof of the central problem. Follow-up questions about when OpenAI had begun its work on the problem and how much human involvement was involved reportedly received less clear answers.
Buckmaster wrote that “an entire team had been working on the problem” and that an “insane amount of compute” had been used. He said that it was eventually agreed that the first prompt had been sent only in the past few days, after OpenAI had been informed of Buckmaster and Alpöge’s work. If accurate, that timeline would suggest OpenAI’s team reached the same approach and then used substantial computational resources to attempt to produce a formal proof quickly.
OpenAI response and denial
Sebastian Bubeck, who leads OpenAI’s mathematical research, described those claims as “false and inflammatory.” Bubeck said he entered the discussion following academic norms and expressed disappointment that the situation had escalated; he promised a fuller statement to come.
The Navier–Stokes existence and smoothness problem concerns the mathematical foundations of the equations used in fluid mechanics. Although the equations are widely applied in science and engineering, rigorous theoretical understanding of existence and smoothness of solutions remains incomplete. A valid proof would be a major advance for mathematical physics.
Why Buckmaster found the timing suspicious
Buckmaster emphasized that the particular route he and Alpöge adopted — introducing a smooth force term, corresponding to options c and d in Charles Fefferman’s framing of the problem — was not a common approach among researchers. He argued this made it unlikely that a model would arrive at the same direction in just a few days after being given the problem statement.
He wrote: “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. Almost nobody else I know of was working on it. It is not the direction one arrives at in a few days by giving a model the problem statement.”
Authorship and interpersonal disputes
Although Alpöge is employed by Anthropic, Buckmaster said Alpöge was not conducting this work on Anthropic’s behalf. The collaborators nonetheless relied primarily on OpenAI’s Codex for the work. Buckmaster alleges that Alpöge’s affiliation with a rival lab was an issue for OpenAI and that Bubeck asked him to remove Alpöge’s credit as part of a proposed compromise.
Buckmaster says that when he pushed to make the dispute public, Bubeck replied: “Why would you ruin your career?” and later, when Buckmaster pushed back, “If you don’t want me to be nice, then I don’t have to be nice.”
Data provenance and model-training concerns
Buckmaster also expressed worry that his heavy use of Codex could have exposed aspects of his work that were then used in OpenAI’s own attempts. OpenAI reserves the right to use Codex interactions to train models, although users can opt out. If the OpenAI team used a model that had been trained in part on Buckmaster’s Codex interactions, then the model might reproduce elements of his approach when given a similar problem.
OpenAI did not respond to a request for comment about that possibility.
Broader implications
The episode is likely to reawaken debate over the role of AI in frontier mathematical research, and the incentives large AI labs face when company resources can be deployed to pursue high-profile discoveries. Buckmaster stressed that he is not accusing particular people, but wanted to report what he had been told and how events unfolded, because he fears a sequence of announcements could create a misleading public record.
“I have not seen OpenAI’s proof,” he 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.”
