OpenAI Says It Made Progress on Another Millennium Prize Problem as Authorship Dispute Escalates

The Times reported that after Buckmaster refused a cooperation proposal and said he might publicize their communications, Sébastien Bubeck, an OpenAI research leader, asked him: 'Why are you trying to destroy your career?' Bubeck acknowledged using that phrase but denied it was a threat.

Bubeck later shared chat screenshots with Buckmaster's collaborator Levent Alpöge in which he proposed coordinating publication, recognizing both researchers' priority, and sharing proofs and human prompts. The messages showed OpenAI expressed willingness to cooperate before a phone call, but could not reconstruct the call that triggered the conflict, according to the Times.

The Times added details of OpenAI's offers. At the time, OpenAI had completed the Navier-Stokes proof while Buckmaster and Alpöge had advanced only to the simpler Euler equations. OpenAI offered options: Buckmaster and Alpöge could publish Euler results first and OpenAI would announce Navier-Stokes the next day; Buckmaster could be lead author and reorganize the paper generated by OpenAI's model; OpenAI would provide enough computing power for him to continue his own research; and OpenAI would recommend that the $1 million Millennium Prize honor go to Buckmaster and his collaborators. OpenAI was not offering to pay the $1 million directly, and there was no evidence of a gag agreement.

Buckmaster still felt OpenAI was trying to buy him off. The sticking point was Alpöge, who works at Anthropic but conducted the research as a personal side project not initiated by Anthropic. Bubeck confirmed to the Times that OpenAI did not want Alpöge in the final paper, saying: 'How can an Anthropic employee be part of an OpenAI internal project?' Buckmaster argued that the work used both Anthropic's Claude and OpenAI's Codex and that the two companies should set aside competition and handle the result together. He said OpenAI could not understand why he would not 'throw Levent under the bus.'

Bubeck said OpenAI had reached similar arrangements with outside mathematicians after other math proofs and was surprised by Buckmaster's refusal. Regarding the career warning, Bubeck said Buckmaster had told him he would immediately make a series of allegations to the media if OpenAI published the proof, and that he meant to ask why Buckmaster would risk his career with unverified allegations. Bubeck called it 'extremely bad wording' and said he retracted it and apologized on the spot. Buckmaster did not perceive goodwill.

On Sunday, Buckmaster turned his NYU office into a temporary war room, according to the Times. A blackboard was covered with a timeline, colleagues sat nearby, and he called Peter Sarnak, a mathematician at the Institute for Advanced Study. After hearing the terms, Sarnak said, 'I have never heard of such bargaining,' and praised Buckmaster for acting 'the way a mathematician acts.' Shortly before midnight on Monday, Buckmaster published three unpolished papers and a 1,870-word account. He left the office at 2:30 a.m. Hours later, OpenAI formally announced the Navier-Stokes proof.

The data dispute remains unresolved. OpenAI updated its account to say that Buckmaster's Codex prompts over the past two months could not have affected the internal model in any way, and that no user input after July 3 could enter the system. That is more definitive than its earlier statement that it could not completely rule out that anonymous user data had helped improve the model. The claim is still one-sided, with no public technical material for independent verification, according to the report.

Buckmaster said the fight has obscured the human mathematicians who paved the way. Years ago, Spanish mathematicians Diego Córdoba and Luis Martínez-Zoroa proposed a strategy for attacking the relevant fluid equations: a series of precise external-force 'kicks' that push the fluid toward extreme states until a singularity with velocity approaching infinity emerges. Buckmaster set up the problem, and Alpöge fed it into Anthropic's model. The two then used the model's output to iterate and solve more complex equations. Buckmaster stressed that the main intellectual credit should go to Luis and Diego. According to the report, the work that might have taken mathematicians another decade was compressed by OpenAI into days using more than 10,000 agents.

OpenAI has not identified the next target, but the latest rumors point to the Hodge conjecture. Before the Navier-Stokes result was announced, Elliot Glazer, math lead at Epoch AI, was sorting through confusing sources when he mentioned that a math professor had heard from a 'credible channel' that the problem solved was not Navier-Stokes but the Hodge conjecture. At the time it was treated as a rumor of a rumor. After OpenAI confirmed progress on another Millennium Prize problem, the old clue resurfaced. The Hodge conjecture asks whether certain topological 'holes' in a class of complex geometric spaces can always be matched by algebraic structures defined by polynomial equations. Proposed by British mathematician William Hodge in the 20th century, it connects topology, complex geometry and algebraic geometry and is one of the seven Millennium Prize Problems.

The model behind the effort still has no official name. OpenAI says only that it began training on Aug. 28 and that its math ability already significantly exceeds GPT-6 Astra, and that it had not finished training during the Navier-Stokes project. Outside speculation has called it Aeon, a name OpenAI has not confirmed. OpenAI says the model was connected to a system of more than 10,000 agents that tried multiple unsolved Millennium Prize problems at once. It disclosed 4.9 million agent messages and about 300 billion output tokens. The Navier-Stokes problem alone used about 10,000 parallel agents, produced 2.7 million messages and consumed about 130 billion output tokens. It took 88 hours from agent launch to finding a solution, followed by 17 hours for GPT-6 Astra to complete Lean formalization and verification.

Placed alongside recent news, the episode looks less like an isolated mathematical breakthrough than an escalating AI problem-solving race between OpenAI and Anthropic, with both aiming at Millennium Prize problems. Only the Poincaré conjecture has been formally solved among the seven; if OpenAI's Navier-Stokes proof passes review, five would remain. Buckmaster told the Times he does not care about the $1 million prize, that he loves mathematics and wanted to be part of this history. He has become part of it, but not in the way he imagined. 'It is terrifying to go up against a trillion-dollar company,' he said. 'We are all afraid of OpenAI.'