OpenAI Claims AI Solved a Millennium Prize Math Problem, Sparking a Credit Dispute

OpenAI said Tuesday that a swarm of roughly 10,000 autonomous AI agents, running on an unreleased internal model, found a way for the Navier-Stokes equations to break down under precise mathematical conditions — a result that resolves one of the seven Millennium Prize Problems posed by the Clay Mathematics Institute in 2000, each carrying a $1 million award. The company published a 165-page proof along with a formal verification written in Lean, a programming language mathematicians use to check that every logical step in an argument holds.

The Navier-Stokes equations, first written down in the mid-19th century, use Newton's second law to describe how fluids move, from ocean currents to air over a wing. The open question was whether their solutions always stay smooth, or whether some infinitesimally small part of a fluid could begin moving infinitely fast, creating what mathematicians call a singularity. OpenAI's agents produced a configuration of a tightening, ever-faster vortex whose energy nonetheless stays bounded — a proof, if it survives scrutiny, that a singularity is possible even under the smooth, unforced conditions the prize problem specifies.

OpenAI described the process in a press release, saying that nearly 100 agents first worked for about 50 hours to produce a related disproof involving the simpler Euler equations, before a larger group of roughly 10,000 agents spent 88 hours reaching the Navier-Stokes result. A further 17 hours went into formalizing the proof in Lean. Over the full effort, the agents exchanged nearly 5 million messages with one another. OpenAI mathematician Sébastien Bubeck estimated the computing cost at several million dollars.

A Foundation Built by Human Mathematicians

The breakthrough did not emerge from nowhere. Both OpenAI's effort and a rival one trace their core technique to Diego Córdoba and Luis Martínez-Zoroa of Madrid's CUNEF University, who spent years developing an analytic, computer-free approach to fluid singularities — an approach that stood apart from the simulation-heavy methods most of the field had used since a landmark 2013 result. Martínez-Zoroa's 2021 doctoral dissertation and a 2023 paper with Córdoba showed that the Euler equations could blow up under a "messy" forcing function; the open problem was making that function smooth enough to satisfy the Millennium Prize's criteria, first for Euler and then for the more difficult Navier-Stokes case.

"Ten years ago, nobody believed there was a singularity for Navier-Stokes," Córdoba told Quanta Magazine. The result caps a summer in which large language models from OpenAI and Anthropic have been credited with a string of proofs across mathematics, often accompanied by Lean verification.

A Credit Dispute Erupts

Just before midnight on September 7 — roughly twelve hours before OpenAI's announcement — New York University mathematician Tristan Buckmaster published his own results, describing a collaboration with Levent Alpöge, a mathematician at Anthropic, that used a mix of AI models to attack the same family of problems. Buckmaster said the pair had a Lean-verified proof for the Euler case by August 22, and an unverified proof of blowup for an easier version of Navier-Stokes. Racing to publish after word of their progress reached OpenAI, Buckmaster acknowledged one of the three papers he and Alpöge released "can only be described as AI slop. I am sorry for this."

OpenAI says its own push began September 1, after researchers heard rumors that Buckmaster and Alpöge were closing in on a Millennium Prize result. Buckmaster has suggested OpenAI's agents may have benefited from work he and Alpöge did using OpenAI's own models during their research process; OpenAI has said it did not access their work before publication but has acknowledged it cannot entirely rule out that de-identified user data influenced its systems. OpenAI now cedes priority on the Euler result to Buckmaster and Alpöge while claiming the Navier-Stokes result for itself — a division neither side has fully accepted, and which outlets including Semafor and Axios reported was still unresolved as of Wednesday.

Let me make plain what I have said to colleagues in private: in view of this body of work, I believe Luis Martínez-Zoroa deserves a Fields Medal.

Tristan Buckmaster, NYU mathematician

Buckmaster separately described the pace of AI-assisted progress this year as "a Deep Blue-Kasparov moment" for mathematics. Martínez-Zoroa, whose earlier work underpins both results, struck a more measured note: "I'm very happy for Tristan. It would have been nice to do this ourselves, but I'm very happy for him." Under Clay Mathematics Institute rules, any Millennium Prize claim must be published in a recognized journal and withstand roughly two years of scrutiny before money changes hands, so neither OpenAI's nor Buckmaster's results are close to formal resolution. For now, mathematicians are left to referee both the proofs and the question of who deserves credit for them.