OpenAI says AI agents cracked part of the Navier-Stokes problem, but a credit fight erupts first

OpenAI said on Sept. 8 that an unreleased internal AI system, working through a coordinated swarm of roughly 10,000 software agents, produced a formally verified proof describing a finite-time singularity in the three-dimensional Navier-Stokes equations — a partial resolution of one of the seven Millennium Prize Problems the Clay Mathematics Institute has offered $1 million apiece to solve since 2000. The equations, which describe how fluids like water and air move, are one of the oldest open questions in mathematical physics, and the specific question OpenAI says its system answered — whether a perfectly smooth flow can spontaneously blow up in finite time — has resisted proof for roughly 90 years.

Within hours the announcement was overtaken by a dispute over credit. Tristan Buckmaster, a mathematician at New York University, said that he and Levent Alpöge, a mathematician who works at the rival AI lab Anthropic, had been quietly close to a related result on the Euler equations — a simpler cousin of Navier-Stokes — when OpenAI's effort surfaced days later using what Buckmaster describes as the identical mathematical route. Buckmaster has publicly accused an OpenAI researcher of pressuring him to drop Alpöge as a co-author because of where Alpöge works, an allegation OpenAI disputes.

The numbers

According to OpenAI and reporting on the effort, the company ran up to 10,000 AI agents in parallel for about 88 hours, exchanging nearly five million messages and consuming roughly 300 billion output tokens, at a compute cost estimated near $22.5 million. The system used was not a commercially available product but what OpenAI describes internally as a research model more capable than its current public flagship. Separately, Buckmaster and Alpöge have posted three preprints — covering the incompressible porous medium equation, the two-dimensional Boussinesq system and the three-dimensional Euler equations — each accompanied by a machine-checked formalization in the Lean proof assistant, a tool mathematicians increasingly use to verify that a written argument is logically airtight line by line. One of those preprints, on the Boussinesq system, is posted directly on Buckmaster's NYU faculty page. Notably, none of the parties involved is claiming the underlying $1 million Clay Prize; OpenAI has said explicitly that it is not seeking the award, and the Euler and forced-Navier-Stokes results are, by the researchers' own account, steps toward — not the same as — the original, unforced Clay problem.

How the field got here

The Navier-Stokes existence-and-smoothness problem was one of seven problems the Clay Mathematics Institute named in 2000; only one, the Poincaré conjecture, has been resolved since, by Grigori Perelman in 2003. The core question for Navier-Stokes is whether solutions that start out smooth can always be continued smoothly forever, or whether they can develop a singularity — a point where velocity or vorticity becomes infinite — in finite time. Progress had been slow for decades until a 2022 paper by the mathematicians Diego Córdoba and Luis Martínez-Zoroa introduced an analytic technique for constructing blowup solutions to a "forced" version of the equations, in which an external smooth force is added to make the mathematics more tractable. That paper, posted to the preprint server arXiv, is the foundation both the Buckmaster-Alpöge and OpenAI efforts say they built on this month, extending the same forcing technique from a toy model to the Euler equations and, in OpenAI's case, on to the full three-dimensional Navier-Stokes system. UCLA mathematician Terence Tao, one of the field's most prominent figures, wrote on his research blog on Sept. 7 — a day before OpenAI's announcement — that Buckmaster and Alpöge's unpublished results looked to him like a genuine breakthrough, adding that reaching the full Navier-Stokes case "looks very feasible ... in the near future."

Who is affected

The immediate audience is a small, specialized community of mathematicians working on partial differential equations, who now face the unusual task of auditing thousands of pages of AI-generated argument and Lean code rather than a traditional handwritten paper. Buckmaster and Alpöge have said openly that they used Anthropic's Claude models to help identify and reproduce elements of the Córdoba-Martínez-Zoroa proof, and used Claude alongside OpenAI's Codex to draft much of the written argument — text Buckmaster has separately called, in an earlier draft, "the worst writeup we had ever seen in the history of mathematics" before the authors rewrote it. More broadly, the episode lands in the middle of an increasingly public contest between AI labs to demonstrate that their models can generate original scientific results, not just summarize or verify them — a contest in which OpenAI, Anthropic and Google DeepMind have each publicized math and science claims in 2026. It also touches graduate students and early-career researchers in the field, whose slower, unassisted work on the same open problems could be overtaken by well-resourced corporate labs running thousands of parallel agents.

Reaction

Buckmaster's account, laid out in a public statement, says an OpenAI mathematician contacted him after learning of his and Alpöge's progress and pushed him to remove Alpöge from credit on a paper because Alpöge is employed by Anthropic.

"Why would you ruin your career?" Buckmaster says the OpenAI researcher told him when he declined to drop Alpöge as a co-author.

OpenAI has denied that it obtained or used the pair's unpublished work, saying in a statement reported by outlets including TechCrunch that its researchers and agents "did not see any of their work through any means until they released it publicly," while acknowledging the company "cannot rule out" that de-identified usage data from its own products indirectly informed its models. Princeton mathematician Charles Fefferman, who has studied the Navier-Stokes problem for decades, told Quanta Magazine he was pleased to see progress but was careful about where the credit belongs, saying, according to Quanta's report, that "the heroes of the story" are Córdoba and Martínez-Zoroa, whose earlier analytic method made the later results possible. Coverage of the dispute, including in Fortune, has framed the fight as a preview of how credit and priority disputes may increasingly play out when AI companies compete to be first on hard scientific problems.

What happens next

Under the Clay Mathematics Institute's own published rules, no Millennium Prize can be awarded until a proposed solution has been published in a refereed mathematics journal and has then held up to community scrutiny for two full years — a deliberately slow process meant to let errors surface. None of this month's results has cleared that bar, and none of the researchers involved is asking the institute to start the clock; both sides describe their work as progress toward, rather than a completed solution of, the original unforced problem. In the coming weeks, mathematicians outside both groups are expected to work through the Lean formalizations and the underlying papers to check whether the arguments hold, a process that in past AI-assisted claims has sometimes uncovered gaps missed by automated verification of the wrong statement. The credit dispute between Buckmaster, Alpöge and OpenAI shows no sign of resolving quickly either; Buckmaster has said he plans to publish a fuller account of his interactions with the company, and Anthropic has yet to issue its own detailed statement on Alpöge's role. Whatever the final verdict on Navier-Stokes itself, the episode has already become a case study in how the AI industry's rivalries are starting to spill into the mathematics community's own credit and authorship norms.