OpenAI publishes a paper claiming to prove singularity formation in the Navier-Stokes equations

OpenAI publishes a paper claiming to prove singularity formation in the Navier-Stokes equations

On September 8, OpenAI released the full 166-page paper and a formalization in Lean, a language that automatically checks the steps of a formally written argument. In the proposed construction, a smooth flow starts from rest, retains finite kinetic energy, and develops unbounded velocity in finite time.

The Navier-Stokes equations describe fluid motion in a continuum; they are used to calculate blood flow, weather, and airflow around a wing. In 1934, Jean Leray constructed generalized solutions with finite energy, but the question of smoothness in three-dimensional flow remained open. In 2000, the Clay Mathematics Institute included it among the seven Millennium Prize Problems.

In the preprint by Levent Alpöge and Tristan Buckmaster, a singularity is constructed for the Euler equation, which has no viscosity. In OpenAI's construction, viscosity remains positive: the equation accounts for internal friction in the fluid.

According to the stated theorem, for any positive viscosity, the authors specify a smooth external force confined to a bounded region of space and time, together with a smooth flow in three-dimensional space with finite energy. The fluid is initially at rest. A vortex forms around an axis: the fluid spirals inward toward the axis and flows outward along it. The radius of the fast-moving core shrinks faster than its height, so the velocity increases while the region's volume decreases quickly enough to keep the energy finite.

At the boundary of the core, the vortex alone would leave a growing residual in the equation, causing the external force to lose smoothness. The authors add short-wavelength velocity waves around the core. Shear in the background vortex amplifies these waves, and their average transport of momentum compensates for the residual. Subsequent corrections and spatial confinement of the construction preserve the smoothness of the external force even as the maximum velocity grows without bound.

OpenAI reports that a group of approximately 10 thousand agents working simultaneously searched for the construction, taking about 88 hours. It says that formalization and verification in Lean with GPT-6 Astra took another 17 hours. The public repository contains Lean formalizations of the results and instructions for checking them with Comparator, a separate program for this purpose. Mathematicians can examine the full argument in the paper, and the program can rebuild and verify its formal representation.