Internal OpenAI Model Reports a Solution to the Navier–Stokes Millennium Prize Problem
OpenAI says an internal model has produced a solution to the Navier–Stokes existence and smoothness problem, one of the seven Millennium Prize Problems. The work addresses whether smooth three-dimensional fluid motion can develop a singularity in finite time, a question that has remained unresolved for roughly 90 years. OpenAI has released both a written proof and a formalization in the Lean proof assistant.
The Navier–Stokes equations describe fluid motion by applying Newton’s second law to a fluid treated as a continuous medium. They are used in areas including aircraft design, weather forecasting, and the study of blood flow. The problem concerns an incompressible fluid with constant density that begins in a smooth state. Specifically, it asks whether the equations can produce speeds that grow without bound in a finite amount of time, despite viscosity, which normally smooths motion.
OpenAI’s reported result establishes that an initially smooth fluid at rest can develop such a singularity. The system’s proof uses a smooth force, while the fluid’s energy remains finite throughout the evolution. According to the explanation, the solution takes the form of a vortex that spirals inward and becomes increasingly elongated. Its central region contracts while its speed increases, creating the breakdown through the fluid’s own dynamics rather than through an externally imposed infinite force.
The proof depends on a precise balance among the equations’ terms for acceleration, pressure gradients, momentum transfer, and viscosity. These terms become large but cancel in a way that allows the external force to remain smooth while the fluid velocity grows without bound. OpenAI says this establishes statements C and D in the official Millennium Prize formulation.
To develop the result, OpenAI used a coordinating system of agents powered by its internal model. The agents could read a cached version of the internet, run code, and communicate in groups. The group working on Navier–Stokes involved roughly 10,000 concurrent agents. They explored different versions of the problem and approaches before useful insights were consolidated. The agents reached their resolution about 88 hours after the effort began, and Lean formalization and verification required an additional 17 hours.
OpenAI also reports that the model resolved an unforced Euler-equation regularity question during the same effort. The company says its work was independent of concurrent research by Levent Alpöge and Tristan Buckmaster, whose result concerned forced Euler equations.
OpenAI describes the Navier–Stokes work as evidence of progress in AI-assisted mathematics, but says it does not intend to claim the Millennium Prize for the result. The release presents an analytical proof, a Lean formalization, and an account of how an internal multi-agent system produced them.
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