OpenAI Said It Has Solved One of the Seven Millennium Prize Problems

• OpenAI’s internal system produced a proof of finite-time singularity formation in the three-dimensional Navier — Stokes equations.
• About 10,000 AI agents worked on the solution in parallel, and its formalization and verification in Lean took another 17 hours.
• OpenAI said it is not claiming the Millennium Prize for this result.

OpenAI announced that it has solved the existence and smoothness problem for solutions to the Navier — Stokes equations — one of the seven Millennium Prize Problems of the Clay Mathematics Institute.

The company claims its internal system produced an analytical proof that an initially smooth, stationary flow of an incompressible fluid can develop a singularity in finite time. The result was also formalized in the Lean language.

What did OpenAI’s system prove?

The Navier — Stokes equations describe the motion of liquids and gases as a continuous medium. They are used, among other things, to design aircraft, forecast the weather, and model blood flow.

One of the main open questions is whether smooth solutions to the three-dimensional equations for an incompressible fluid can lose smoothness over a finite time interval.

According to OpenAI, its system proved exactly such a scenario. In the proposed solution, a vortex forms, winds inward, and stretches, causing the central region to compress and accelerate. At the same time, the system’s energy remains finite.

OpenAI notes that the technical challenge was to show the emergence of a singularity through the fluid’s own dynamics, rather than as a result of applying an infinite external force.

Thousands of agents worked on the solution

OpenAI began assessing its capabilities on the Millennium Prize Problems on September 1, after rumors emerged about a possible solution to two of them.

For the work, it used a system of coordinated AI agents with access to a cached version of the internet and tools for running code. The agents worked in separate groups and could communicate within them.

The group working on Navier — Stokes consisted of about 10,000 concurrent agents. Different groups used different formulations of the problem, including variants aimed at proving smoothness or, conversely, the existence of a singularity.

Before tackling Navier — Stokes, the agents solved a simpler problem. OpenAI then used the resulting solution as a hint for the Navier — Stokes work. The most useful ideas from different groups were further consolidated with the help of Codex.

The agents obtained a solution on September 5 — roughly 88 hours after the first agents were launched. Formalization and verification in Lean using GPT-6 Astra took another 17 hours.

Overall, while working on all tasks, the agents sent 4.9 million messages and used about 300 billion output tokens. Specifically during the Navier — Stokes work, they sent 2.7 million messages and used about 130 billion output tokens.

OpenAI said the result was independent

The company also described parallel work on the Euler problems by Anthropic researchers and New York University researchers Levent Alpoge and Tristan Buckmaster.

OpenAI said that before publishing its work, it had not seen any materials from the researchers and did not use their results while solving Navier — Stokes. The company also noted that it did not use specific user data in its work on the problem.

At the same time, OpenAI acknowledged the researchers’ priority on the forced Euler equation and congratulated them on the result. The company emphasized that its work is a snapshot of current AI progress, not the end of the development of such systems.

OpenAI also stated outright that it is not claiming the Millennium Prize for solving the Navier — Stokes problem. The result is presented as evidence of major progress in AI capabilities and as material for further study by mathematicians.

As a reminder, earlier OpenAI Chief Scientist Jakub Pachocki called on AI companies to be prepared to slow down the development of the most powerful models.

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