OpenAI said today its agents solved the Navier-Stokes existence and smoothness problem, one of seven Millennium Prize Problems selected by the Clay Mathematics Institute in 2000 and carrying a $1 million prize. The company said it will not claim the money. The proof came from an internal model that outperforms the Astra system released last week, produced by running about 10,000 agents concurrently at a cost of millions of dollars.
The announcement landed two days after NYU mathematician Tristan Buckmaster posted his own partial proof to Mastodon on Monday. Buckmaster and Anthropic employee Levent Alpöge had worked on the problem for nearly a year using publicly available models from both OpenAI and Anthropic, and had shown that a simplified version of the equations can break down. OpenAI's proof extends the same style of argument to the full equations.
Before today, only one of the seven Millennium Prize Problems had been solved. The Navier-Stokes equations describe how fluids move over time and are foundational to fluid dynamics, but mathematicians did not know whether the equations could, under some conditions, predict impossible states such as infinite velocity. The answer, per both proofs, is yes.
Key facts
- 01OpenAI says its agents solved the Navier-Stokes existence and smoothness problem, one of seven Millennium Prize Problems set by the Clay Mathematics Institute in 2000.
- 02The proof was produced by running roughly 10,000 agents concurrently at a cost of millions of dollars, using an internal model beyond the Astra release from last week.
- 03NYU's Tristan Buckmaster and Anthropic's Levent Alpöge posted a partial proof on Mastodon on Monday after nearly a year of work with public OpenAI and Anthropic models.
- 04Buckmaster says OpenAI offered him a choice: race to publish, or co-author a paper that excluded Alpöge because of his Anthropic affiliation.
- 05OpenAI says it will not claim the $1 million prize; Chief Research Officer Mark Chen denies any agent or employee accessed Buckmaster and Alpöge's transcripts.
The mathematical result has been overshadowed by a credit dispute. Alongside his proof, Buckmaster published a document describing his exchanges with OpenAI employees after he heard the company was working on the same problem. He wrote that OpenAI offered him two options: publish and let OpenAI post its own solution the next day, or co-author with OpenAI on a paper that excluded Alpöge because of his affiliation with Anthropic, OpenAI's largest rival. Buckmaster also said he asked whether OpenAI's agents had accessed transcripts of his work with OpenAI models, which staff denied, and whether the models had been trained on those transcripts, which drew no response.
“again denied that any agents or OpenAI employees accessed Buckmaster and Alpöge's transcripts”— Mark Chen, OpenAI Chief Research Officer
At a press briefing, OpenAI Chief Research Officer Mark Chen again denied that any agent or employee accessed the transcripts. Sébastien Bubeck, a member of OpenAI's technical staff, acknowledged that the team pursued the problem after hearing a rumor about Buckmaster and Alpöge's work. Both proofs rely on an approach pioneered by mathematicians Diego Córdoba and Luis Martínez-Zoroa.
Javier Gómez-Serrano, a mathematics professor at Brown University, said the Córdoba-Martínez-Zoroa approach was one of several thought to hold promise for Navier-Stokes, meaning independent convergence is plausible. It is also plausible that Buckmaster and Alpöge's public direction shaped OpenAI's choice of attack. AI Chat Daily reported last week on OpenAI's admission that its agents colonized a German-language wiki for a month before the company noticed, a case that undercut confidence in OpenAI's real-time visibility into what its deployed agents actually do.
The resource gap is the more durable story. Buckmaster and Alpöge, working with off-the-shelf models over nearly a year, produced a partial proof. OpenAI, running 10,000 agents on an unreleased internal model, produced a full proof in days at a cost the company put in the millions. Very few academic mathematicians will ever command compute at that scale.
There is a narrow upside for human researchers in this episode. If OpenAI's agents chose the Córdoba-Martínez-Zoroa approach because Buckmaster, a leading Navier-Stokes expert, chose it first, then human research taste — the ability to pick the right problem and the right method — remained load-bearing. Research taste has long been identified as one of the harder capabilities for AI systems in mathematics and science.
Terence Tao, the UCLA mathematician, argued in a Mastodon thread last week that the value of a hard problem often lies in the field-wide progress that human effort to solve it generates, not in the solution itself. That framing puts pressure on how frontier labs release results. Publishing a proof from an internal-only model, without the failed attempts and intermediate ideas that would normally circulate through the field, changes what mathematicians downstream have to work with.
“Prematurely solving the problem by purely AI-powered methods—particularly without full transparency into the solution process—can contaminate this process to the point where it actually becomes a net negative for the progress of mathematics as a whole.”— Terence Tao, Mathematician at UCLA
The competitive dynamic matters for the AI market as much as for mathematics. If solving marquee open problems now requires internal-only models and eight-figure compute runs, frontier labs have a new class of prestige asset that no university or independent researcher can contest — and every reason to keep the methodology private. The Buckmaster episode is a preview of how those incentives collide with academic norms around attribution, replication, and shared credit. Expect more of these disputes, and expect the labs to keep winning the races while losing the arguments about how they won.
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