OpenAI has published a claimed 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 company says an internal AI system, described as significantly more capable than GPT-6 Astra, produced both an analytical proof and a formal verification in the Lean proof assistant — and it chose the same day that two academic mathematicians released their own closely related papers to announce the result. Within hours, the story became as much about credit as about computation.
What the model actually proved For more context on this story, see our ongoing AI industry coverage.
The Navier-Stokes equations, which grew out of nineteenth-century work by Claude-Louis Navier and George Gabriel Stokes, describe how fluids move using Newton's second law of motion. They underpin aircraft design, weather forecasting and the study of blood flow. Since Jean Leray proved in 1934 that solutions exist in a generalized sense, mathematicians have not known whether smooth three-dimensional incompressible flow can break down — whether fluid speeds can grow without bound within a finite time, creating what mathematicians call a singularity.
OpenAI's post states that its system proved an initially smooth fluid at rest can develop exactly such a singularity in finite time when a smooth external force is applied, with the fluid's energy remaining finite throughout. According to the company, this establishes statements "C" and "D" of the official Clay Mathematics Institute formulation of the problem. The proof centers on a vortex — a spinning swirl that spirals inward and elongates, in OpenAI's description, "like spaghetti," shrinking and speeding up in a way that keeps total energy bounded even as velocity grows without bound.
A system more capable than Astra
OpenAI says the work was done by an internal model that has been in training since August 28 and is "significantly more capable than GPT-6 Astra," the frontier model the company rolled out to select customers earlier this summer. The model's training, the company says, is ongoing and its performance continues to improve.
Inside the 10,000-agent run
The effort began on September 1, when OpenAI says it heard rumors that two Millennium Prize problems had been resolved. The company launched an evaluation campaign across all open Millennium Prize problems using a system of coordinating agents with access to a cached copy of the internet and code execution tools. Agents were subdivided into communicating groups; the group that produced the Navier-Stokes resolution involved roughly 10,000 concurrent agents, according to OpenAI.
Different groups received different variants of each problem — versions "A" and "B," which would yield a proof of smoothness, and versions "C" and "D," which would yield a disproof, were assigned to separate groups. Along the way, the agents resolved a related question OpenAI had classified as easier: the regularity problem for the Euler equations in the unforced case, where no external force acts on the fluid. Nearly 100 agents worked for approximately 50 hours on that result.
After the Euler resolution, OpenAI shifted agents from the other Millennium problems onto Navier-Stokes. The company says the agents arrived at their resolution on Saturday, September 5 — about 88 hours after launch — and that Lean formalization and verification took an additional 17 hours using GPT-6 Astra. Across all attempted problems, OpenAI says the agents exchanged 4.9 million messages and consumed roughly 300 billion output tokens; the Navier-Stokes effort alone accounted for 2.7 million messages and approximately 130 billion output tokens.
The mathematicians who published first
The announcement landed on the same day that Tristan Buckmaster, a mathematics professor at NYU's Courant Institute, and Levent Alpoge made public three of their own results: finite-time blowup with smooth forcing for the incompressible porous media equation, the Boussinesq equations, and the three-dimensional incompressible Euler equations. Buckmaster writes that the pair also believe they have blowup for hypo-dissipative Navier-Stokes, though the Lean verification of that result has not yet finished.
In a public statement, Buckmaster is careful about credit. The research program, he writes, was neither started by him and Alpoge nor proposed by a large language model — the foundational idea belongs to Diego Cordoba and Luis Martinez-Zoroa, who spent years constructing forced blowups. Buckmaster argues that Martinez-Zoroa's contribution is strong enough to deserve a Fields Medal. He and Alpoge then used large language models — Anthropic's Claude, OpenAI's Codex running GPT-5.6 Sol, and more recently Astra for writeups and auditing — to push that program to smooth forcing and to Euler. Their breakthrough came on August 15, with the first LLM-generated proof verified in Lean on August 22, Buckmaster says.
Buckmaster also apologized for the presentation quality of the papers, writing that "the Euler writeup, in particular, can only be described as AI slop."
The credit controversy
OpenAI's post addresses the overlap directly. The company says its effort began September 1 after a rumor it "later realized was related to" Alpoge — described as an Anthropic employee — and Buckmaster. After completing its own project and Lean verification on September 6, OpenAI says it contacted the pair to offer a concurrent release and to recognize their priority in a joint announcement, at which point it learned they had resolved the forced Euler problem.
The company denies seeing any of their work before the public release, stating that "no specific user data was accessed in order to solve this problem." But OpenAI concedes a subtler channel: "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models." It notes that the proofs differ significantly and that the Euler results are distinct — forced versus unforced.
Coverage has been pointed. WIRED reported that "some academics are crying foul," and Axios described OpenAI's "historic math solution overshadowed by credit controversy." The core tension: OpenAI's own timeline shows it launched its effort only after hearing that mathematicians were on the verge of announcing resolutions, then published a stronger-compute result seven days later.
No prize claimed — for now
OpenAI states plainly that it does not intend to claim the Millennium Prize for the result. Any official recognition would rest with the Clay Mathematics Institute and the scrutiny of the mathematical community, which will now dissect both the OpenAI proof and the Buckmaster-Alpoge papers. Notably, the mathematicians' own results stop short of the Millennium formulation — their published blowups cover related equations, with their hypo-dissipative Navier-Stokes work still awaiting formal verification.
Either way, the episode marks a new chapter for AI-accelerated mathematics. Terence Tao, among the most followed mathematicians working with AI tools, publicly discussed the competing results on Mastodon within hours. Two independent teams — one academic, one corporate — used large language models as working collaborators on century-old problems, published within a day of each other, and promptly quarreled over who arrived first. The math will be checked in Lean. The credit will be argued everywhere else.
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