OpenAI announced on September 8 that an unreleased internal model produced a solution to the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems, after roughly 10,000 AI agents worked on it in parallel for 88 hours. Two mathematicians immediately disputed the account, alleging the company built on their unpublished work, and the proof has not been independently verified.
The announcement landed as a technical paper and a formal proof written in Lean, a theorem-proving language that lets computers check every step of an argument. The claim itself is narrow but heavy: that a smooth fluid at rest can develop a singularity in finite time, a so-called blowup. Jean Leray showed in 1934 that solutions exist in a generalized sense, and mathematicians have argued ever since about whether smooth solutions can fail. The Clay Mathematics Institute listed the problem in 2000 with a $1 million prize, and it is one of only six still open.
How the run worked
According to OpenAI, the model entered training on August 28 and quickly showed it was far stronger at mathematics than previous systems. On September 1, after rumors spread that Anthropic had solved two Millennium Prize Problems, the company aimed the model at all six remaining open problems, including the Riemann hypothesis and P versus NP.
On Navier-Stokes, the system first answered a simplified version of the question in about 50 hours using 1,000 agents. OpenAI then scaled up to roughly 10,000 agents working in groups. Some searched for a proof, others hunted for a counterexample, and the groups could exchange findings. Agents exchanged millions of messages over the run. By September 5, about 88 hours in, the company described the problem as resolved. Executives told reporters the compute bill reached millions of dollars.
| Milestone | Date | Detail |
|---|---|---|
| Model training begins | Aug 28 | Internal model, stronger at math than prior systems |
| Run on all Millennium problems | Sept 1 | After rumors of Anthropic solving two problems |
| Simplified version solved | ~50 hours in | 1,000 agents |
| Full claim published | Sept 8 | Paper plus Lean formalization, ~10,000 agents, 88 hours |
The credit dispute
The trouble started fast. Tristan Buckmaster, a mathematician at New York University, and Levent Alpoge, who works at Anthropic, had spent about a year studying a related version of the Navier-Stokes equations with AI assistance. Buckmaster alleges OpenAI may have used data from their work. OpenAI denies seeing the researchers’ unpublished material. The dispute has escalated to the point where legal preservation orders have been reported, and several mathematicians have called for a full independent audit of how the model was trained.
Sebastien Bubeck, a researcher at OpenAI, framed the model’s power in sweeping terms: “Basically, you throw at it almost any open problem, and it’s a coin flip whether the model can solve it.” The company says the unreleased model solved about 50 percent of problems on its internal math benchmark, against roughly 10 percent for GPT-6 Astra, the system the company confirmed as its next major model family on August 1.
“Basically, you throw at it almost any open problem, and it’s a coin flip whether the model can solve it.” – Sebastien Bubeck, OpenAI researcher
Why this problem is hard for machines
Navier-Stokes describes how fluids like air and water move, and the open question is whether solutions can always stay smooth or can instead develop infinite values in finite time. There is no known example that settles the question, which is what separates it from most computational wins. Chess had Stockfish to train against. Protein folding had the Protein Data Bank. Here, a model cannot check its work against a known answer, because no such answer exists in any textbook.
That is why the Lean formalization is the strongest part of the package. A machine-checked proof removes an entire class of errors, since every inference has to satisfy the proof assistant. But Lean verifies the argument as written. It cannot verify that the setup matches the actual Navier-Stokes problem, that definitions were not quietly weakened, or that a hidden assumption crept in during translation. Specialists in formalization will need weeks to check those choices, and the Clay Mathematics Institute has said nothing so far about whether it would even entertain a machine-generated claim.
What a proof actually proves
OpenAI says it will not claim the prize. That is a safe move politically, but it also sidesteps the normal peer process, where a claimed solution to a Millennium problem would be posted publicly, picked apart in seminars, and scrutinized for months before anyone calls it solved. Grigori Perelman’s proof of the Poincare conjecture sat on the arXiv for years while other teams verified it, and that was a human-written argument with a known community of experts. An AI-generated proof of comparable weight has no established review path at all.
Expert reaction so far has ranged from skeptical to unimpressed. Mathematicians quoted in coverage say the document has not been made public in full, so the claim cannot be examined at all, let alone verified. A short roundup of expert commentary put it plainly: the claim requires rigorous independent scrutiny to determine whether the proof is valid and free of hidden assumptions or fatal errors.
A pattern, not an accident
This is the second such fight in three months. In June, an internal OpenAI reasoning model disproved a 1946 conjecture of Paul Erdos, the first time AI settled a major open math problem, and that result went through with comparatively little friction because the problem had a known, checkable answer. In July, OpenAI hired Fields Medalist Jacob Tsimerman, who said AI will soon outperform human mathematicians and could accelerate the field a hundredfold. The labs are hiring the arbiters of these disputes even as they generate the disputes.
The competitive dynamics are also unusual. OpenAI moved after hearing rumors about Anthropic, meaning two of the three frontier labs are now racing on pure mathematics as a benchmark. Mathematicians who collaborate with these companies face a choice that did not exist two years ago: share work in progress with a lab that can deploy a thousand times more compute on it, or keep it private and risk being scooped by a machine. Buckmaster’s allegation, whatever it turns out to be worth, is now part of that calculation for every researcher considering a lab partnership.
For the AI industry, the episode is a preview of coming verification problems. When a lab claims a result in a field where checks take months, the public record is a press release until someone independent reproduces it. In cryptography that cycle is days. In number theory it can be years. Fluid dynamics sits somewhere in between, and the mathematicians who know the difference are, for now, not convinced. The next real signal will be whether any independent group, academic or commercial, attempts to replay the Lean proof and check the translation from the paper to the formal system. Until that happens, this is a claim about a claim.