--- title: "OpenAI says an internal model solved the Navier-Stokes problem" url: https://blog.vomini.com/blog/openai-navier-stokes-millennium-problem date: 2026-09-09T13:15:00+00:00 updated: 2026-10-27T22:42:11.977654+00:00 author: "Roberto R." section: "News" language: en site: "Vomini" license: free to quote with attribution --- # OpenAI says an internal model solved the Navier-Stokes problem > OpenAI published a proof, with a machine-checkable version in Lean, that the Navier-Stokes equations for fluids can break down in finite time, one of the seven Millennium Prize Problems. It came from about 10,000… On 8 September, OpenAI published what it says is a solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems named by the Clay Mathematics Institute in 2000. The Navier-Stokes equations describe how fluids move, and engineers use them for aircraft design, weather forecasts and blood flow. The open question was whether a fluid that starts out moving smoothly can, under these equations, reach infinite speed in a finite amount of time. OpenAI's answer is yes. Its proof describes a spinning vortex that spirals inward and stretches out, speeding up while its energy stays finite, until the equations break down. OpenAI says this settles the problem by proving the statements labeled C and D in Clay's official formulation, which are the disproof side. ## How the proof was produced According to OpenAI, the work came from an internal model it has been training since 28 August, which it describes as significantly more capable than GPT-6 Astra. That model ran as a system of cooperating agents with code execution and a cached copy of the internet. The group that found the Navier-Stokes result had on the order of 10,000 agents working at once. The agents reached the result on Saturday, 5 September, about 88 hours after the first ones were launched. GPT-6 Astra then spent another 17 hours writing and checking a version of the proof in Lean, a programming language for proofs in which a computer verifies every step. For this problem alone, OpenAI counts 2.7 million messages between agents and about 130 billion output tokens. ## Another team got part of the way first OpenAI says its effort started on 1 September, after hearing a rumor that turned out to involve Levent Alpöge, an Anthropic employee, and Tristan Buckmaster, a professor at NYU. Using an internal Anthropic model, they had solved a related problem, the Euler equations with an external force. OpenAI recognizes their priority on that result. Its own system separately solved the Euler case without an external force, and OpenAI says the two proofs differ and that none of their work was seen before they released it. ## What is still open OpenAI says it does not intend to claim the Millennium Prize. Its post does not say whether outside mathematicians have reviewed the proof yet, and it does not mention any response from the Clay Mathematics Institute. The Lean version helps here, because anyone can run the checker on it, though experts still need to confirm that the formal statement matches the problem as Clay wrote it. ## What to watch The next few weeks will show whether mathematicians who work on fluid equations accept the proof. Watch for their comments on the Lean files, and for any response from the Clay Mathematics Institute. ## What we think If the proof holds, it is the strongest evidence yet that AI systems can produce new mathematics at the highest level. What surprised us more is that OpenAI used an unreleased model it says is well beyond [GPT-6 Astra](/blog/gpt-6-astra-launch), which is only a few days old. That tells everyone how much distance there is between what labs have inside and what the rest of us can use. ## Sources - [OpenAI](https://openai.com/index/navier-stokes-solution/), 8 Sep 2026 --- Originally published at https://blog.vomini.com/blog/openai-navier-stokes-millennium-problem