On the Navier–Stokes Millennium Prize Problem

OpenAI says an internal AI system produced an analytical proof, with a Lean formalization, of a fluid flow that starts smooth and at rest, is driven by a smooth external force, and develops a finite-time singularity: its speed grows without bound while it keeps finite energy. The result concerns the Navier–Stokes equations for viscous fluid motion and is described in the company's announcement and paper.

OpenAI describes the construction as an inward-spiraling, increasingly elongated vortex whose velocity grows without bound, with acceleration, pressure, momentum-transfer and viscosity terms cancelling so that the external force stays smooth. Because the flow is driven by an imposed force, the claim is about a breakdown of the mathematical model under those stated conditions, and OpenAI says it is not claiming the Millennium Prize. These are the company's own claims about its own proof; it has published the Lean formalization at github.com/openai/NavierStokesAndEuler.