Navier-Stokes Announcement
Archived — this story has rotated out of today’s deck. It is kept here in full.
The gist
OpenAI says an internal AI system solved the Navier-Stokes Millennium problem, with a Lean-verified proof.
Outside researchers dispute the claim and the $1 million prize is not yet awarded.
Background
The Navier-Stokes existence and smoothness problem asks whether solutions to the equations of fluid motion in three dimensions always stay smooth, or can blow up in finite time; it is one of seven Millennium Prize Problems posed by the Clay Mathematics Institute in 2000, each carrying a $1 million prize. Before OpenAI's announcement only one of the seven had been solved. Anticipation had been building as AI models made a string of mathematical discoveries in recent months, and CMI says the process for evaluating what has been achieved and assigning credit is deliberately unhurried.
How it unfolded
- 2000The Clay Mathematics Institute unveiled the Millennium Prize Problems in Paris, attaching a $1 million prize to each; Navier-Stokes existence and smoothness was among the seven.
- Aug 28, 2026OpenAI began training a new AI model with advanced mathematical capabilities, according to Sebastien Bubeck in a press briefing.
- Sep 1, 2026Word spread online that researchers associated with Anthropic might have solved two Millennium Prize Problems; OpenAI says it began working on the Millennium problems because of those rumors.
- Sep 6, 2026OpenAI says its full project and Lean verification were completed.
- Sep 7, 2026NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge posted documents claiming key advances in an area relevant to the Navier-Stokes problem, saying they used several AI models including Claude and Codex; Buckmaster alleged OpenAI had learned of their progress and adopted the same method.
- Sep 8, 2026OpenAI announced that an internal AI system, not released to the public, solved the problem using roughly 10,000 concurrent AI agents, and shared a writeup and a Lean formalization; OpenAI said running the same problem for a customer would cost around $15 million.
- Sep 10, 2026CMI said it shares in the excitement of the global mathematical community as it contemplates the announcement that the Navier-Stokes problem has apparently been settled.
Who’s saying what
- OpenAI
- An internal AI system solved the problem, producing a proof that under certain conditions a smoothly moving three-dimensional fluid can reach a point where its calculated speed grows without limit in finite time, certified in Lean.
- Caution
- Buckmaster alleged OpenAI found out about his and Alpöge's progress and adopted the same method; OpenAI denied it, saying no people or AI systems searched user data, though it cannot rule out that de-identified data from the researchers' product usage helped improve its models.
- Official
- CMI says the problem has apparently been settled and that the process for evaluating what has been achieved and assigning credit is deliberately unhurried.
- Expert
- Charles Fefferman of Princeton, who wrote the Clay Institute's official description of the problem, said he was thrilled it was solved and that the heroes of the story are Córdoba and Martínez-Zoroa.
Still unverified
CMI describes the announcement as the problem having 'apparently been settled'. OpenAI says it 'cannot rule out' that de-identified data derived from the outside researchers' usage of its products helped improve its models. Buckmaster alleged OpenAI found out about the pair's progress and adopted the same method; OpenAI has denied the allegations. The episode began with a rumor: on September 1, word spread online that researchers associated with Anthropic might have solved two Millennium Prize Problems. WIRED reports documents including one in which Buckmaster could publish a paper announcing the problem had been solved by an internal OpenAI model, but without Alpöge's name included.