OpenAI says an internal artificial intelligence (AI) system has produced a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. Those are seven especially hard unsolved questions listed in 2000 by the Clay Mathematics Institute. The Navier–Stokes equations describe how fluids move by treating the fluid as a continuous medium. The open question is whether smooth three-dimensional motion can break down.
The claimed result is that a fluid that starts at rest, with a smooth push from outside, can develop a singularity in a finite time while its energy stays finite. A singularity here means the calculated speeds grow without bound. Viscosity, the internal friction that usually smooths motion, is still present. Because a real fluid cannot move infinitely fast, that breakdown would mean the continuum equations have stopped being a good model. OpenAI says this matches statements in the official prize text. The constructed motion is a vortex.
How the argument was produced
The work used an internal AI model described as stronger than GPT-6 Astra, together with many coordinating agents. The agents first settled a related blow-up question for the Euler equations, which are Navier–Stokes without viscosity. Effort then shifted to the full problem. One group of about ten thousand agents reached a resolution about eighty-eight hours after the first agents started. A further seventeen hours of checking in Lean, a language used to write proofs so a computer can verify each step, was done with GPT-6 Astra. That Navier–Stokes run involved about 2.7 million messages and about 130 billion output tokens, the chunks of text a model writes.
OpenAI later learned that Levent Alpöge and Tristan Buckmaster had related results on forced Euler flow, meaning Euler equations with an outside force. The company says it did not see their unpublished work, that the precise Euler statements differ, and that it offered a joint announcement. It does not plan to claim the million-dollar prize and presents the release as a report on model progress. Independent mathematical review is still required.
News coverage treated the claim as potentially historic if the Lean proof holds, while also reporting a public credit dispute with Alpöge and Buckmaster.
