SAN FRANCISCO — OpenAI said Tuesday, Sept. 8, 2026, that an internal AI system produced a solution to the Navier–Stokes existence and smoothness problem — one of the seven Clay Mathematics Institute Millennium Prize Problems — showing that three-dimensional incompressible fluid dynamics can develop a singularity in finite time, according to the company’s announcement.

OpenAI published both an analytical writeup and a formalization in the Lean proof assistant. The claimed result: an initially smooth fluid at rest, under a smooth external force and with finite energy throughout, can develop unbounded speeds in finite time — establishing statements “C” and “D” in the official Millennium Prize formulation, the company said. OpenAI described the singularity as a vortex that spirals inward and elongates “like spaghetti,” with motion terms growing large yet canceling so the external force stays smooth.

The company said agents powered by an internal model “significantly more capable than GPT‑6 Astra” reached the resolution on Saturday, Sept. 5, about 88 hours after launch, with Lean formalization and verification taking about 17 hours via GPT‑6 Astra. Across Navier–Stokes work alone, agents sent roughly 2.7 million messages and used about 130 billion output tokens; overall attempts across problems totaled about 4.9 million messages and 300 billion output tokens, OpenAI wrote.

OpenAI said it does not intend to claim the Millennium Prize for the result, framing the release as evidence of AI research progress. It also recognized concurrent work by NYU mathematician Tristan Buckmaster and Anthropic employee Levent Alpöge on the forced Euler problem after initially believing their rumor pointed to Navier–Stokes; OpenAI said its proofs differ and that it did not see their work before public release.

Whether mathematicians accept an AI-produced Lean formalization as settling a century-old continuum question will play out in peer review — not a press post. For now, the claim is public, the prize is not claimed, and the pace of capability OpenAI is advertising is the other story.

Sources