The proof was produced by an internal OpenAI system and demonstrates how the equations governing fluid motion can develop a singularity in finite time. OpenAI said the work was intended to demonstrate progress in its AI systems and help scientists advance research and technology.
The company said the model used to solve Navier–Stokes is “significantly more capable than GPT-6 Astra”. OpenAI said it wanted to provide an indication of the pace of AI progress and what could be expected from future models.
According to OpenAI, the system produced both an analytical proof and a Lean formalisation showing that an initially smooth fluid at rest can develop a singularity in finite time. In the version addressed by the system, a smooth external force is applied to the fluid, while its total energy remains finite throughout the process, from the initial state to the formation of the singularity.
OpenAI described the resulting structure as a vortex, or spinning swirl of fluid, that spirals inward while becoming increasingly elongated. As the central region shrinks, the fluid accelerates in a way that allows its energy to remain finite, as required by the equations.
The central challenge is to demonstrate that the equations can break down because of the fluid’s own dynamics rather than through the introduction of an infinite force externally.
After learning that Millennium Prize Problems were being resolved, OpenAI launched an effort to evaluate its internal model on all of the open Millennium Prize Problems, along with several other high-impact mathematical problems.
The company used a multi-agent system in which groups of agents worked on different versions of the problems. The agents had access to tools including cached internet information and code execution, with around 10,000 agents working concurrently on the Navier–Stokes problem.
Before turning its full attention to Navier–Stokes, OpenAI gave the system several related problems. Nearly 100 agents worked for about 50 hours to produce a disproof of the unforced Euler regularity problem.
The result convinced OpenAI that Navier–Stokes was the most promising problem to pursue, prompting it to shift additional resources towards it.
OpenAI’s claimed proof marks the latest stage of its effort to test increasingly capable AI systems against major unresolved mathematical problems, with the Navier–Stokes existence and smoothness problem at the centre of the project.
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