California, USA, September 9, 2026 — OpenAI says an internal artificial intelligence system has produced a solution to the 90-year-old Navier–Stokes existence and smoothness problem, using a coordinated network of about 10,000 AI agents that reached the result in roughly 88 hours.
The company published its research on September 8, describing the achievement as a solution to one of the seven Millennium Prize Problems established by the Clay Mathematics Institute. OpenAI said its system produced an analytical proof showing that certain three-dimensional fluid dynamics governed by the Navier–Stokes equations can develop a singularity in finite time. The company also released a formalised version of the proof in Lean, a programming language used for formally checking mathematical arguments.
The Navier–Stokes equations are fundamental to the mathematical description of fluid motion and are used in areas including aircraft design, weather forecasting and the study of blood flow. The unresolved problem has centred on whether smooth solutions in three-dimensional fluid dynamics can remain smooth indefinitely or instead develop a singularity, where quantities such as fluid velocity become unbounded within a finite period.
OpenAI said it began training the internal model on August 28 and turned it toward the Millennium Prize Problems after hearing rumours on September 1 that two of the problems had been resolved. The company then deployed groups of AI agents to investigate several open problems, eventually concentrating its computing resources on Navier–Stokes.
The group working on the Navier–Stokes problem involved about 10,000 concurrent agents. According to OpenAI, the agents communicated through millions of messages, explored different mathematical approaches and used Codex to consolidate useful findings between groups. They reached their result on September 5, about 88 hours after the first agents were launched. Formalisation and verification of the proof in Lean took another 17 hours using GPT-6 Astra.
