On the Navier–Stokes Millennium Prize Problem

OpenAI announced that an internal AI-driven multiagent system has generated a solution to the Navier–Stokes existence and smoothness problem, one of the seven Millennium Prize Problems designated by the Clay Mathematics Institute. The proof, produced by a group of approximately 10,000 concurrent agents powered by an internal model significantly more capable than GPT-6 Astra, shows that the dynamics of Navier–Stokes equations for three-dimensional fluid motion can develop a singularity in finite time. According to OpenAI, the system produced both an analytical proof and a formalization in Lean, resolving the problem by establishing statements C and D in the official Millennium Prize formulation. The solution describes a vortex that spirals inward and elongates while maintaining finite energy. The effort began on September 1, following rumors of mathematical breakthroughs, and the agents arrived at their resolution in about 88 hours, with Lean formalization taking an additional 17 hours via GPT-6 Astra. Across the Navier–Stokes effort, the agents sent 2.7 million messages and used approximately 130 billion output tokens. OpenAI also noted that their agents successfully resolved the unforced regularity problem for the Euler equations prior to tackling Navier–Stokes. After completing their work and verification on September 6, OpenAI reached out to Levent Alpöge and Tristan Buckmaster regarding rumors of their work, discovering that they had produced a resolution of the forced Euler problem using an internal Anthropic model, and OpenAI acknowledged their priority on that work.