OpenAI says internal AI model solved Navier-Stokes Millennium Problem

September 9, 2026 at 09:46 AM2 min

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OpenAI said its internal AI model solved the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems. The company published both the proof description and its formalization in the Lean proof assistant. The claim drew an immediate priority dispute from mathematician Tristan Buckmaster.

Key Facts

OpenAI said its internal model solved the Navier-Stokes existence and smoothness problem, one of seven Millennium Prize Problems, on September 5.
The proof was formalized and verified in Lean by GPT-6 Astra in 17 hours, with 2.7 million messages and 130 billion output tokens used.
Mathematician Tristan Buckmaster of New York University accused OpenAI of relying on his and Levent Alpöge's unpublished work before it was made public.
Buckmaster presented preliminary results on September 8, obtained with Alpöge using AI models Codex and Claude.
OpenAI said it contacted Buckmaster and Alpöge to propose simultaneous publication and recognize their priority in a joint statement.

The Claimed Proof

OpenAI said its internal AI model solved the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems. The company published both a description of the proof and its formalization in the Lean proof assistant. The solution was reached on Saturday, September 5, after approximately 88 hours of work by about 10,000 AI agents. GPT-6 Astra then required 17 hours to formalize and verify the proof in Lean. The agents sent 2.7 million messages and used 130 billion output tokens, roughly the volume of a million books.

Priority Dispute

Mathematician Tristan Buckmaster of New York University accused OpenAI of relying on his and Levent Alpöge's work before it was made public. Buckmaster presented preliminary results on September 8, obtained with Alpöge using AI models Codex and Claude. OpenAI said it heard rumors on September 1 that two Millennium Problems had been solved, later learning the solutions belonged to Alpöge and Buckmaster. The company said it contacted them to propose simultaneous publication and recognize their priority in a joint statement. Buckmaster said he would rather not deal with this side of the story.