On the morning of September 8, mathematicians at OpenAI announced that roughly 10,000 autonomous agents, directed by researchers and running on an internal model the public cannot access, had located a singularity in the three-dimensional Navier-Stokes equations. Within twelve hours the mathematics community was arguing less about the fluid dynamics than about who deserved the credit.
Key takeaways
- OpenAI says about 10,000 agents worked for 88 hours and exchanged nearly 5 million messages before producing a blowup proof, which a separate model then formalized in Lean over another 17 hours.
- The company will not claim the $1 million Clay prize, because its construction relies on an external forcing term that the official Millennium formulation does not permit.
- NYU's Tristan Buckmaster, who announced related results twelve hours earlier with Anthropic's Levent Alpoge, has suggested OpenAI benefited from seeing their work; OpenAI's Sebastien Bubeck denies it.
What the agents actually proved
The Navier-Stokes equations, written down in the mid-19th century, apply Newton's second law to fluids. The open question the Clay Mathematics Institute attached a $1 million prize to in 2000 is whether their solutions always stay well-behaved, or whether some vanishingly small parcel of fluid can begin moving infinitely fast β a singularity, or in the field's own shorthand, a blowup.
OpenAI's answer is that blowup happens. According to the company's technical post, it first pointed a group of roughly 100 agents at the related Euler equations, which describe frictionless fluids, and got a regularity disproof after about 50 hours. It then scaled up to some 10,000 agents for Navier-Stokes. Those agents ran 88 hours and traded close to 5 million messages; a second model spent 17 more hours converting the argument into Lean, the proof assistant that lets a machine certify each logical step. Bubeck has put the compute bill at several million dollars.
Formal verification is what separates this from the usual round of model-solves-math claims. Lean does not care whether an argument sounds persuasive. What it cannot check is whether the statement fed into it is the statement mathematicians care about β and that is precisely where the asterisk sits. OpenAI's construction introduces a forcing term, an outside push on the fluid. The prize version asks what the fluid does on its own. The company's decision not to file for the prize is the clearest signal available about how it rates its own claim against the formal criteria.
Why the credit fight erupted
Twelve hours before OpenAI published, Buckmaster of New York University released a statement describing work with Alpoge, now at Anthropic, that resolved several closely related problems using a mix of models. The pair had a Lean-verified Euler proof by August 22 and had planned a more careful write-up, but moved their timeline up after word of their progress reached OpenAI. Buckmaster was blunt about the cost of rushing, describing one of the three papers they released as slop and apologizing for it.
OpenAI concedes priority on the three-dimensional Euler result to Buckmaster and Alpoge while claiming Navier-Stokes for itself. It also acknowledges that rumors of their progress are what prompted it to attack the problem at all. Buckmaster's statement goes further, implying that OpenAI researchers or their agents may have seen and benefited from work the pair carried out using OpenAI's own models. Bubeck has rejected that account, saying the internal model reached the Euler result independently and by different means. As of publication the participants are telling materially different versions of the same week.
The humans behind the method
Both teams stand on work by Diego Cordoba of Madrid's Institute for Mathematical Sciences and Luis Martinez-Zoroa of CUNEF University. In his 2021 dissertation, Martinez-Zoroa developed analytic techniques that avoid computers entirely, an unfashionable choice in a field that had leaned on computer-assisted error control since Thomas Hou and Guo Luo's 2013 cylinder result. By 2023 the pair had shown that a version of Euler with an ugly forcing function admits singularities, using an infinite cascade of layered non-singular solutions. Making that cascade produce a smooth forcing function was the remaining hurdle, and it is the step both AI groups appear to have cleared.
Charles Fefferman of Princeton, who wrote Clay's official description of the problem, called Cordoba and Martinez-Zoroa the heroes of the story, as Quanta Magazine reported. Buckmaster said publicly that he believes Martinez-Zoroa deserves a Fields Medal. Cordoba's own line on the AI boom is drier: he jokes that he does not use AI because he has Luis.
What this means for AI-assisted mathematics
Strip out the priority dispute and a structural claim remains: a coordinated swarm of AI agents running for days, checked by a formal proof system, is now a viable instrument for research-level mathematics. That is a different proposition from a large language model answering a competition question, and it echoes the multi-agent scaling patterns described in OpenAI's earlier 1,200-agent experiments. The open questions are now sociological as much as technical β how priority is assigned when a lab can point compute at a rumor, and how credit flows back to the people whose methods made the search tractable.
FAQ
Did OpenAI win the Millennium Prize?
No. OpenAI has said it will not claim the prize. Its proof adds an external forcing term to the equations, while the Clay Mathematics Institute's official formulation concerns the unforced case, so the result does not satisfy the prize criteria as written.
Has the proof been verified?
The argument was formally checked in Lean, which gives strong confidence that the stated theorem follows from its premises. Human reviewers still have to confirm that the formalized statement is mathematically equivalent to the blowup claim, and that review is ongoing.
Who is Luis Martinez-Zoroa?
He is a mathematician at CUNEF University in Madrid who, with his doctoral adviser Diego Cordoba, developed the infinite-cascade technique both AI teams built on. Buckmaster has argued publicly that the work merits a Fields Medal.
