On September 8, OpenAI published a proof of the Navier-Stokes existence and smoothness problem, releasing both the paper and a Lean formalization. It is one of the seven Millennium Prize Problems set out by the Clay Mathematics Institute in 2000, each carrying a $1 million prize. OpenAI said in its announcement that it does not intend to claim the prize.
The proof was produced by a group of agents running an as-yet-unreleased model, described by the company as noticeably more capable than GPT-6 Astra. The conclusion: a fluid can start from a smooth initial state and develop a singularity in finite time.
88 Hours, Ten Thousand Agents
Rarer than the conclusion itself is the fact that the process numbers were made public at all. The agents started on September 1 and produced a result by September 5 — roughly 88 hours — with about 10,000 agents running concurrently at peak. This single problem consumed 2.7 million messages and roughly 130 billion output tokens; across every problem attempted in this round, the total came to 4.9 million messages and about 300 billion output tokens. Converted at public API prices, outside estimates put the cost at around $15 million.
After obtaining the proof in analytic form, GPT-6 Astra spent another 17 hours encoding it in Lean and running the checker. The formalization file is now public, and this layer is on firmer ground than the paper itself: long proofs in analysis have historically been hard to review, so compiling successfully in Lean at least removes the question of logical self-consistency from human peer review.
That remaining question cannot be removed the same way. The Millennium Problem has an official problem statement, and whether a proof actually solves it depends on which set of conditions the construction satisfies. Until a mathematician has read the full paper, the word "solved" stays on hold. The public accounts released so far do not clearly state whether the construction includes a forcing term — and whether or not it does has been the single most decisive fork in this problem for the past two decades.
The Official Announcement, and Another Statement
The same day brought a second half to the story. Tristan Buckmaster of NYU's Courant Institute and Levent Alpöge of Anthropic published their own finite-time blow-up results for three equations, including the forced Euler equation. Buckmaster also released a separate statement describing his communications with OpenAI.
OpenAI's account is that it initially believed the other team had also solved Navier-Stokes and proposed a joint release, only later realizing the other side had worked on the forced Euler equation, and it acknowledged the other team's priority on that problem. On the data question, the announcement states that neither its researchers nor its agents saw the other team's work through any channel before it was made public — then adds that it cannot rule out that de-identified data derived from the other team's usage behavior may have played a role, while stressing that the two proofs differ substantially.
In TechCrunch's report on September 8, Buckmaster's account was blunter. He said his progress had been passed along to OpenAI, that the company then took the same route he had, and that "almost no one else was working on this." He also quoted Sébastien Bubeck as telling him on a call, "why would you want to ruin your career," and "if you don't want me to be nice, I can not be nice." Bubeck had previously denied the account on X, calling the circulating allegations false and inflammatory. The two sides' descriptions of the same phone call currently do not line up.
How This Differs From Previous Cases
The milestones over the past year on this front are easy to trace: first a complete machine proof of Fermat's Last Theorem compiled on a 96-core machine, then a batch of results around the Erdős conjectures, after which Lean formalization became a standard part of any release. What they had in common was that the disputes were about the mathematics, backed by a reproducible chain of evidence.
What's new this time is that the process itself has become the point of dispute. Who started first, whose draft passed through whose system, what was said on the phone — these kinds of disputes have happened in purely human mathematics circles too, but in the past, preprint timestamps and conference records could at least sort out the sequence of events. When one side's toolchain is the other side's product, the timestamp stops being neutral. Simon Willison put it more bluntly on September 8: if he used ChatGPT to complete part of a Millennium Problem, what are the odds that his work feeds back into training, letting a later model help someone else finish it first?
OpenAI's announcement doesn't answer that question, and right now no lab's data terms answer it either.
Sources: OpenAI's Millennium Problem announcement, TechCrunch, CocoLoop, Simon Willison's personal blog, Tristan Buckmaster's public statement; figures on agent scale, messages, and tokens follow the official announcement and its reporting, and the cost conversion is an external estimate based on public API pricing.