OpenAI says an internal AI system has produced a proof resolving the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems in mathematics.
The problem asks whether three-dimensional fluid motion governed by the Navier-Stokes equations can develop a singularity, where speeds become unbounded in a finite period. The equations are used to model phenomena including aircraft aerodynamics, weather and blood flow.
According to OpenAI, the system found the proof after about 88 hours of work by roughly 10,000 concurrent AI agents. The agents explored different approaches, exchanged intermediate findings and used Codex to consolidate promising ideas. They generated 2.7 million messages and approximately 130 billion output tokens during the Navier-Stokes effort.
OpenAI said the proof shows that an initially smooth fluid can develop a finite-time singularity while its energy remains finite. The company also released a formalized version in Lean, with verification taking an additional 17 hours through GPT-6 Astra.
The result still requires assessment by the wider mathematical community. OpenAI said it does not intend to claim the $1 million Millennium Prize for the work, positioning the release as evidence of progress in AI-assisted scientific research.
Source: OpenAI
