OpenAI Reported to Claim Progress on a Second Millennium Prize Problem, with Rumors Pointing to the Hodge Conjecture

A New York Times follow-up has filled in previously undisclosed details of the conflict between OpenAI and New York University mathematician Tristan Buckmaster, most notably an updated statement from OpenAI about how user data was used.

In that account, OpenAI now says the Codex prompts Buckmaster submitted over the past two months could not have influenced its internal model in any way, and that no user input after July 3 could have entered the system. That is far more explicit than its earlier position that it could not fully rule out that anonymous user data had helped improve the model, and it effectively draws a timeline around whether user data reached the research results published in August. The report also notes that this remains OpenAI's own account, with no public technical material that outsiders can independently check.

Before the data dispute settled, OpenAI raised a new claim: after the Navier-Stokes equations, it says it has made substantial progress on another Millennium Prize problem and is preparing to publish results. The latest rumor points to the Hodge conjecture as the next target, but that comes from unverified threads that OpenAI has not confirmed, and had earlier been treated as a rumor of a rumor.

The dispute began when Buckmaster said that after he rejected a collaboration proposal and indicated he might make their communications public, OpenAI research lead Sébastien Bubeck asked him, in effect, why he wanted to destroy his own career. Bubeck later acknowledged the first remark was his, denied it was a threat, described it as extremely bad wording and said he retracted it and apologized on the spot.

The New York Times added a fuller picture of the terms discussed. At that point OpenAI had already completed the Navier-Stokes proof while Buckmaster and his collaborator Levent Alpöge had only advanced to the simpler Euler equations. OpenAI's options included having the two mathematicians publish the Euler results first with OpenAI disclosing the Navier-Stokes proof the next day; having Buckmaster serve as lead author to reorganize the model-generated paper; providing him with ample compute to continue his own research; and publicly recommending that the $1 million Millennium Prize recognition go to Buckmaster and his collaborator. The report also notes OpenAI was not offering to pay the prize directly and that there is no evidence the two sides discussed a non-disclosure arrangement, yet Buckmaster still felt he was being bought off.

The sticking point was Alpöge. He works at Anthropic, but the research was his personal side project rather than an Anthropic initiative, and Bubeck confirmed in the interview that OpenAI did not want Alpöge on the final paper, asking how an Anthropic employee could take part in an internal OpenAI project. Buckmaster argued the work used both Anthropic's Claude and OpenAI's Codex, so the reasonable path was for the two rivals to set aside competition and handle the result together.

The deeper sting may be about credit rather than data. Buckmaster stresses that human mathematicians laid the road: several years ago Spanish mathematicians Diego Córdoba and Luis Martínez-Zoroa proposed a strategy that uses a series of precise external-force kicks to push a fluid toward extreme states until a singularity with near-infinite velocity emerges. He set up the problems, Alpöge fed them to a model, and the two revised the work based on model output. Under the old research rhythm, that path might have been carried forward by different mathematicians for another decade; OpenAI mobilized tens of thousands of agents and compressed it into days. Shortly before midnight on Monday, Buckmaster rushed out three lightly polished papers and a 1,870-word account of events, and hours later OpenAI formally announced the Navier-Stokes proof.

The scale of that problem-solving system has also emerged. According to OpenAI, the internal model began training on August 28 and already significantly surpasses GPT-6 Astra in mathematics, even though it had not finished training while the Navier-Stokes project was underway. It was wired into a system of tens of thousands of agents attempting several unsolved Millennium problems at once, and the full run produced 4.9 million agent messages and roughly 300 billion output tokens. Navier-Stokes alone used about 10,000 parallel agents, 2.7 million messages and about 130 billion output tokens, going from agent launch to a found solution in 88 hours, after which GPT-6 Astra spent another 17 hours completing Lean formalization and verification.

Taken together, the past two weeks look less like an isolated breakthrough than a rumor-fueled math race between OpenAI and Anthropic, with the hardest open problems in mathematics as the test set. Only the Poincaré conjecture among the seven Millennium Prize problems has been formally solved; if the Navier-Stokes proof clears review, five will remain. For mathematicians the other side of the coin is just as real: they pose the problems, accumulate the routes and hand unpublished drafts to AI tools, and may then find the company behind those tools racing them to the finish. Buckmaster told the New York Times that going up against a trillion-dollar company is very frightening. What to watch next is whether the Navier-Stokes paper survives peer review, when OpenAI publishes the results it says it is preparing, and whether the Hodge conjecture rumor holds up.