OpenAI says it has solved the existence and smoothness problem for the Navier–Stokes equations—one of the seven famous Millennium Prize Problems from the Clay Mathematics Institute. According to the company, its internal system built an analytic proof showing that a smooth, stationary flow of incompressible fluid can actually develop a singularity in finite time. OpenAI also states the result has been formalized in the Lean proof assistant.

What OpenAI Is Claiming

The company describes the result as an analytic proof that a singularity can form in finite time for an initially smooth, motionless flow of incompressible fluid. The formalization in Lean is presented as an extra layer of verification for the logical steps in the proof.

The Problem in Context

The question of existence and smoothness for solutions to the Navier–Stokes equations is one of the seven Millennium Prize Problems and has long been a central challenge in mathematical fluid dynamics. OpenAI's announcement specifically highlights the achievement of demonstrating the possibility of singularity formation.

What We Know So Far

OpenAI notes that the proof was developed internally and subsequently formalized in Lean. No further details or supporting materials have been released at this time.