The Navier-Stokes equations are a system of partial differential equations that describe the velocity and pressure of certain fluids in space, occasionally in the presence of external forces like gravity. The Navier-Stokes existence and smoothness problem is a million-dollar Clay Mathematics Institute Millennium Prize Problem that asks whether "physically reasonable solutions exist.” It questions whether continuous solutions to the equations, given smooth initial data, continue to exist for all time, or whether they can instead "blow up" and develop a singularity in finite time. The Millennium Prize Problems were announced in 2000 at the Collège de France in Paris, as seven mathematical problems that had perturbed mathematicians into the turn of the new millennium. Creating an inarguable proof of any of these problems would earn the solver one million dollars.
On September 8, OpenAI announced that an internal artificial intelligence system produced a proof addressing a version of this problem. Per their release, a model “more capable than GPT-6 Astra” coordinated 10,000 AI agents over an 88-hour run to produce a proof establishing finite-time blowup for a forced version of Navier-Stokes, corresponding to two of the four alternatives laid out in Clay Institute's official problem statement. The proof came with a machine-checked formalization in the logic-based Lean4 proof assistant, verified in 17 hours with GPT-6 Astra. OpenAI said it will not seek the prize since the particular version it addressed is narrower than the full problem.
Dr. Tristan Buckmaster, a mathematician in the Courant Institute at New York University, and Dr. Levent Alpöge, a mathematician who also works at Anthropic, had spent roughly a year extending a forcing technique developed by Dr. Diego Córdoba and Dr. Luis Martínez-Zoroa. On August 15, using large language models from both Anthropic and OpenAI, they proved that the Euler equations, the frictionless special case of Navier-Stokes, can blow up in finite time. Fields Medalist Dr. Terence Tao called the result "a remarkable achievement,” and that he saw no obstacle in principle to extending the method to the full Navier-Stokes problem, which is what OpenAI’s system sought to do next.
Buckmaster's account of what happened alleges that OpenAI learned of his and Alpöge's unpublished Euler result about a week before its own announcement and pursued the same underlying method. He said that when he raised the possibility of going public himself, OpenAI proposed a joint paper structured in a way that would have left Alpöge off it because of his employment by Anthropic.
OpenAI says its effort began September 1 after their researchers heard an unattributed rumor that the Navier-Stokes Millennium Prize problem had been solved. Once their Navier-Stokes proof and Lean verification were finished on September 6, OpenAI says it reached out to the Buckmaster and Alpöge to propose a “concurrent” release that would credit their priority on the Euler result, and it was only in those conversations that they learned Buckmaster and Alpöge had used an internal Anthropic model to resolve the forced Euler case. Dr. Sébastien Bubeck, OpenAI's math lead, had denied on X that their own proof relied on the pair’s unpublished work.
The Clay Mathematics Institute issued a statement on September 11 acknowledging the apparent resolution of the forced solutions, while noting that formal prize review remains a separate process. A joint declaration has been signed by 25 living Fields Medalists criticizing a broader pattern of AI labs to claim credit for famous open problems.
The mathematics and solutions are not in dispute. Independent mathematicians have not finished scrutinizing either proof, but the Lean formalizations mean that the logical structure of both has been checked. This does force the mathematics, scientific, and larger academic research communities to question their best practices regarding integrating AI in their work in order to navigate this new age of discovery and proof.
