OpenAI says an unreleased AI model has solved the Navier-Stokes existence and smoothness problem, a mathematical challenge that has resisted researchers for roughly 90 years and carries a $1 million prize from the Clay Mathematics Institute.
The company deployed around 10,000 AI agents on the problem, producing a proof in 88 hours. The agents exchanged approximately 2.7 million messages and generated about 130 billion output tokens before arriving at the proof on September 5. Verification using the Lean proof language took another 17 hours. Estimates for the wider computing effort range from $10 million to $22.5 million, although OpenAI says it will not claim the prize.
The Navier-Stokes equations describe how liquids and gases move and are used in aircraft design, weather forecasting and blood-flow simulations. The longstanding mathematical question is whether a fluid that begins smoothly can develop a singularity, where its behaviour becomes unbounded in finite time. OpenAI says its proof demonstrates such a breakdown, describing an inward-spiralling, stretching vortex that accelerates while retaining finite energy.
The announcement was complicated by findings published hours earlier by Tristan Buckmaster, a mathematics professor at New York University. Buckmaster and collaborator Levent Alpöge had spent months working on a closely related problem using OpenAI’s Codex and Anthropic’s Claude. On August 15, they proved that the Euler equations, the frictionless counterpart of Navier-Stokes, can blow up.
Their approach used a forcing technique associated with Diego Córdoba and Luis Martínez-Zoroa. Buckmaster said very few researchers were pursuing this route and accused OpenAI of racing down a research path derived from his unpublished work.
Buckmaster described conversations with OpenAI researcher Sébastien Bubeck, saying he repeatedly asked whether his Codex sessions had influenced training but received no answer. He also alleged he was offered sole authorship on a paper excluding Alpöge and was warned against going public.
OpenAI has denied that its researchers or agents accessed the pair’s work before publication, while saying it cannot rule out de-identified user data improving its models. Sam Altman has stated that the approaches appear different, but the Clay Institute has not verified OpenAI’s proof. Some mathematicians have also questioned its formulation, leaving both the breakthrough and the research dispute unresolved.
