OpenAI Agents Resolve Navier-Stokes Millennium Prize Problem in 88 Hours

OpenAI announced that autonomous agents running on an unreleased internal model produced a solution to the Navier–Stokes existence and smoothness problem, one of the seven Clay Mathematics Institute Millennium Prize Problems. The proof shows that the dynamics of the Navier–Stokes equations for a three-dimensional incompressible fluid can develop a singularity in finite time, even when the motion starts smoothly and energy remains finite. OpenAI is sharing both a writeup of the proof and a formalization in Lean. The company said it used an internal model significantly more capable than GPT-6 Astra and that a major goal was to inform the world about the pace of AI progress. The effort began on Tuesday, September 1, after OpenAI heard rumors that two Millennium Prize problems had been resolved and after a step change in the performance of its internal model. Agents were launched to evaluate the model on all open Millennium Prize problems and a few other high-impact problems. The agents arrived at their Navier–Stokes resolution on Saturday, September 5, about 88 hours after the first agents were launched. Lean formalization and verification took an additional 17 hours via GPT-6 Astra. Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens. In resolving Navier–Stokes they sent 2.7 million messages and used approximately 130 billion output tokens. The solution is a vortex, a spinning swirl of fluid, that spirals inward. It establishes statements C and D in the official Millennium Prize formulation, with a smooth force applied to an initially smooth fluid at rest. The discovery is overshadowed by accusations from Tristan Buckmaster, an NYU mathematics professor who had been collaborating for almost a year with Levent Alpöge, a mathematician at Anthropic. The pair made extensive use of Claude and Codex, mainly GPT-5.6 Sol, and had a breakthrough on August 15. Buckmaster said OpenAI’s first prompt was sent in the past few days after information about their work reached OpenAI, that he was told the model did not look up user data, and that he did not get an answer about whether the model had been trained on their Codex sessions. OpenAI offered to wait for Buckmaster to publish or to have him author a paper about their result, but was clear that Alpöge would not be invited as a co-author because of OpenAI’s competitive relationship with Anthropic.