We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal Open AI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.

The question of whether smooth three-dimensional fluid motion can break down has remained unresolved for roughly 90 years. To solve the Navier–Stokes problem, we used an internal model that is significantly more capable than GPT‑6 Astra.

The Navier–Stokes equations use Newton’s second law to describe fluid motion, treating fluid as a continuous medium. In 2000, the Clay Mathematics Institute named the problem one of seven Millennium Prize Problems. Our system produced an analytical proof and a Lean formalization that an initially smooth fluid at rest can develop a singularity in finite time, with energy remaining finite.

The solution is a vortex that spirals inward and elongates, with speed growing unbounded while energy stays finite. This resolves the problem by establishing statement “C” (and also “D”) in the official formulation.