OpenAI announced on September 8, 2026 that an internal model, described as stronger than GPT-6 Astra, coordinated roughly 10,000 agents over about 88 hours to produce a 166-page proof of the Navier-Stokes existence and smoothness problem, accompanied by Lean theorem prover verification code. The Wall Street Journal described it as only the second Millennium Prize Problem solved since the seven were established in 2000. The Clay Mathematics Institute has not accepted the proof, and it has not been independently verified by the broader mathematics community. Clay Mathematics Institute president Martin Bridson called it 'an exciting day' representing 'major advances in the human understanding of mathematics.' Researcher Sébastien Bubeck estimated the computational cost at several million dollars.
The announcement is contested on multiple fronts. Mathematicians Buckmaster and Alpöge had achieved a Lean-verified proof for the 3D Euler equations by August 22, with LLM-generated proofs playing a role; Buckmaster said the first LLM-generated proof he received was 'the most horrendous I have ever read.' OpenAI concedes priority for the 3D Euler result to Buckmaster and Alpöge but claims priority for the Navier-Stokes result. OpenAI's Bubeck denied using Buckmaster and Alpöge's prompt, models, or proof, calling Buckmaster's account 'false and inflammatory,' but acknowledged the Navier-Stokes solution used a similar method to Buckmaster and Alpöge's approach. Buckmaster alleges OpenAI learned of his and Alpöge's unpublished work and adopted their method, and suggests OpenAI researchers or their AI agents may have improperly accessed that work via OpenAI's own platform. Buckmaster also alleges OpenAI asked him to publish the Euler result as sole author, omitting Alpöge, who is an Anthropic employee, and that he was warned 'Why would you ruin your career?' when he threatened to go public. OpenAI told Buckmaster during a call that an internal model had produced a proof requiring 'very little human input,' a characterization Buckmaster says 'unraveled over the call.' Quanta Magazine noted that the details of interactions among Buckmaster, Alpöge, and OpenAI remain murky, with different parties presenting different versions.
OpenAI told the New York Times it had made 'substantial progress' on another Millennium Prize Problem and was 'working out how to share the result.' Commentator Andrew Curran said the second problem is rumored to be the Hodge Conjecture and the model may be called Aeon; OpenAI has confirmed neither. A separate report states a new OpenAI model began training on August 28 with mathematical capabilities described as significantly exceeding GPT-6 Astra.