OpenAI reports that an internal AI system has produced a proof that smooth solutions to the three-dimensional incompressible Navier–Stokes equations can develop a singularity in finite time when driven by a smooth external force. The result is accompanied by a formalization in the Lean proof assistant. The work addresses one of the seven Millennium Prize Problems, specifically statement C (and also D) of the official formulation: that there exist smooth, finite-energy initial conditions and a smooth forcing term for which the velocity field becomes unbounded in finite time despite the smoothing effect of viscosity.

The proof was generated by a multiagent system built on an internal frontier model, described as significantly more capable than GPT‑6 Astra. The effort began on September 1 after the team heard a rumor that two other researchers, Levent Alpöge of Anthropic and Tristan Buckmaster of NYU, had a related resolution. The OpenAI agents were organized into groups, prompted with different variants of the problem, and later cross-pollinated using Codex to consolidate insights from intermediate results. A further trained version of the model was deployed as it became available. The group that solved Navier–Stokes arrived at the resolution on September 5, roughly 88 hours after the first agents were launched; Lean formalization and verification took an additional 17 hours.

According to the post, across all attempted problems the agents exchanged 4.9 million messages and used about 300 billion output tokens. For the Navier–Stokes problem specifically, the agents exchanged 2.7 million messages and used approximately 130 billion output tokens. Before reaching Navier–Stokes, the agents also resolved the analogous regularity question for the unforced Euler equations, where viscosity is absent; that result came earlier in the effort and helped guide the later work.

The post also discloses that the team initially believed Alpöge and Buckmaster had solved Navier–Stokes and reached out to propose a concurrent release. In fact, their work resolved the forced Euler problem. OpenAI says its researchers and agents did not see any of the other group’s work until it was publicly released, and that no specific user data was accessed to solve the problem. The post acknowledges the possibility that de-identified data derived from product usage may have helped improve the models, but notes that the proofs differ significantly and that even the exact results differ in the Euler case (forced versus unforced).

OpenAI states that the goal in releasing the result is to report substantial progress in AI mathematics, not to claim the Millennium Prize. The post frames the result as evidence of accelerating AI capability and emphasizes the need for steerable, accountable AI systems and deliberate choices about the pace of progress. It closes by describing the milestone as a snapshot in time rather than a culmination, and reiterates the mission to ensure AGI benefits all of humanity.