OpenAI says an unreleased internal model has produced an analytical proof and a Lean formalisation for the Navier–Stokes existence and smoothness problem, showing that an initially smooth fluid can develop a singularity in finite time. The company says this question has remained unresolved for roughly 90 years.[1]
The Navier–Stokes equations describe the motion of fluids such as water and air.[2] OpenAI’s proposed result concerns whether a smooth three-dimensional flow can break down, with velocity growing without bound in finite time despite viscosity.[1] It is one of the seven Millennium Prize Problems, each associated with a US$1 million prize.[1]
The scale and speed of the attempt are striking. OpenAI reports that roughly 10,000 concurrent agents reached the result about 88 hours after the project began, exchanging 2.7 million messages and producing around 130 billion output tokens for the Navier–Stokes work. GPT-6 Astra then spent another 17 hours on Lean formalisation and verification.[1]
That does not make the result settled mathematics yet. The Clay Mathematics Institute still lists the problem as active, and its rules require a proposed solution to appear in a qualifying outlet, survive at least two years and gain general acceptance in the global mathematics community before it will be considered.[2][3] Nature reported encouraging initial reactions from mathematicians, but the published proof still needs sustained independent scrutiny.[4]
If the proof survives that process, the achievement will be difficult to overstate. The important point is not merely that an AI system produced a plausible answer quickly, but that it generated an analytical argument and a machine-checkable formalisation that specialists can inspect. OpenAI has published both the [paper and Lean proof](https://openai.com/index/navier-stokes-solution/) for that examination.[1]
Supporting sources:
[1] https://openai.com/index/navier-stokes-solution
[2] https://www.claymath.org/millennium/navier-stokes-equation
[3] https://www.claymath.org/millennium-problems/rules
[4] https://www.nature.com/articles/d41586-026-02842-5
Source: OpenAI