OpenAI said on September 8, 2026 that an internal AI system had produced a solution to the Navier–Stokes existence and smoothness problem, one of the seven Clay Mathematics Institute Millennium Prize Problems. The company published both a writeup of the proof and a formalization in the Lean proof assistant.
According to OpenAI, the result shows that the dynamics of the three-dimensional incompressible Navier–Stokes equations can develop a singularity in finite time: an initially smooth fluid at rest, driven by a smooth external force and with finite energy throughout, can reach unbounded velocity. OpenAI says this establishes statements “C” and “D” in the official Millennium Prize formulation.
The company described the constructed solution as a vortex that spirals inward and stretches, with carefully balanced acceleration, pressure, momentum transfer, and viscosity terms so the external force stays smooth even as fluid speed grows without bound.
OpenAI said the work used an internal model it describes as significantly more capable than GPT‑6 Astra. After hearing rumors on September 1 that Millennium problems might have been resolved elsewhere, researchers launched a multi-agent evaluation across open Millennium problems. The group that produced the Navier–Stokes resolution involved on the order of 10,000 concurrent agents. Agents reached a resolution on September 5, about 88 hours after the effort began; Lean formalization and verification took about another 17 hours using GPT‑6 Astra, OpenAI said.
In related concurrent work, OpenAI recognized priority for Levent Alpöge (Anthropic) and Tristan Buckmaster (NYU) on a forced Euler result and said the teams’ proofs and precise statements differ.
OpenAI was explicit that its goal is to report progress in AI capability and that it does not intend to claim the Millennium Prize. Major outlets including the BBC and The Guardian covered the announcement, noting that independent mathematical acceptance and Clay Institute prize processes are separate from a company research release. DigiEditorial is reporting OpenAI’s verified public claims and corroborating coverage; community review of the paper and Lean artifacts will determine how the result is ultimately received.