OpenAI said Tuesday that an internal AI system solved the Navier–Stokes existence and smoothness problem, one of the Clay Mathematics Institute’s seven Millennium Prize Problems, according to a company research post and coverage from The New York Times and New Scientist. The writeup and a Lean formalization argue that an initially smooth three-dimensional incompressible fluid at rest can develop a finite-time singularity under a smooth external force while keeping finite energy — Clay statements C and D in the official prize formulation.
The company said the proof came from an unnamed internal model “significantly more capable” than GPT-6 Astra. OpenAI described a multi-agent run on the order of 10,000 concurrent agents that reached a resolution on Saturday, Sept. 5, about 88 hours after launch, with Lean formalization and verification via Astra taking roughly 17 more hours. New Scientist reported from a press conference that OpenAI put the customer cost of a comparable run at about $15 million. The company said it does not intend to claim the $1 million Millennium Prize, and Clay president Martin Bridson told New Scientist the institute’s evaluation process is deliberately unhurried.
OpenAI also said its agents earlier produced a blowup result for the unforced Euler equations, a viscosity-free cousin of Navier–Stokes, with nearly 100 agents working about 50 hours. The company framed the release as evidence of accelerating model progress rather than a finished chapter for mathematics.
The announcement landed hours after NYU mathematician Tristan Buckmaster and Anthropic employee Levent Alpöge posted stepping-stone blowup results for forced Euler and related equations, with Lean checks and heavy LLM assistance. Buckmaster’s statement, covered by Scientific American and New Scientist, raises questions about timeline and whether Codex usage informed OpenAI’s push, while saying he is not accusing anyone of using their proof. OpenAI said it began its Millennium effort after hearing related rumors, recognizes the pair’s priority on forced Euler, denies seeing their work before public release, and says its Euler result (unforced) differs; it also said humans did not access Codex customer work and that it cannot rule out de-identified product-usage data helping its models.
Clay has not awarded the prize. Independent mathematicians still have to digest the Lean artifacts, and some already note that the forcing route leans on a term many researchers treat as secondary in informal statements of the problem.