The Rumour Was the Prompt: OpenAI's Navier-Stokes Proof, 10,000 Agents, and the Team It Scooped

The Navier-Stokes existence and smoothness problem has carried a $1 million Clay Institute bounty since 2000. The question is whether smooth solutions to the equations for 3D incompressible fluid can blow up in finite time. On September 8 OpenAI announced that an internal model had found one: under a smooth external force, a tightening vortex develops a singularity while the energy stays finite. The proof is formalised in Lean. It resolves two of the four routes the Clay formulation allows.

I have no reason to doubt the mathematics, and neither does anyone I can find. The story is how it was made.

Eighty-Eight Hours

OpenAI’s post is candid about the mechanism, and the numbers are its own.

• The trigger was a rumour. “Our effort began on September 1st after hearing a rumor which we later realized was related to Levent Alpöge, an Anthropic employee, and Tristan Buckmaster, a math professor at NYU.” Sébastien Bubeck told Science: “Just like everyone, we saw rumors on Twitter that Anthropic might have solved two Millennium Prize problems. So we thought to ourselves: ‘We have such a strong model. Why don’t we try to solve also a Millennium Prize problem?’”
• The model is unreleased. “Significantly more capable than GPT-6 Astra”, with training that started August 28. Astra did only the 17-hour Lean verification.
• The scale. About 100 agents took roughly 50 hours to disprove unforced Euler regularity first. Then “on the order of 10,000 concurrent agents” ran on Navier-Stokes, arriving at the result “about 88 hours after the first agents were launched”. Across all Millennium problems attempted: 4.9 million messages, about 300 billion output tokens. For Navier-Stokes alone: 2.7 million messages, about 130 billion output tokens.
• The bill. Mark Chen put compute in “the ballpark of millions of dollars”. Karthik Duraisamy at Michigan told Science about $6 million at retail and about $1 million internal. My own arithmetic on GPT-6 Astra’s list price of $50 per million output tokens gives $6.5 million for the Navier-Stokes run, and it is a proxy, because the real model has no price.
• The prize. “We do not intend to claim the Millennium Prize for this result.”

Jakub Pachocki, at the press event: “This is not the result of some long-going effort at OpenAI to solve the Millennium Prize problem.” That is the sentence to hold onto. Seven days from rumour to Lean-verified proof, from a standing start, because someone on Twitter said Anthropic might have done it.

The Year Before the Week

Tristan Buckmaster published a statement on his NYU page. It is four pages and it is the most careful document in this whole affair.

He and Alpöge had worked “for most of the past year” on fluid blowup, building on Diego Córdoba and Luis Martínez-Zoroa’s program. Their breakthrough came on August 15: finite-time blowup with smooth forcing for three systems including 3D Euler, the viscosity-free cousin of Navier-Stokes. Lean verification finished August 22. They believed they had Navier-Stokes itself in a weaker form and had not released it because “the Lean verification has not yet finished.” Their tools: “Anthropic’s Claude, OpenAI’s Codex, especially with GPT-5.6 Sol and, more recently, Astra.” Buckmaster paid his own OpenAI bill.

On September 3 he emailed an OpenAI mathematician to say the work was personal, not institutional. The reply was warm and offered compute. On September 6, Bubeck asked to meet “at any point today”. On the calls, by Buckmaster’s account, he learned OpenAI had a 100-page forced-blowup proof he had never seen, and was offered two options: a joint release, or a solo paper under his name crediting OpenAI’s model, with Alpöge left off because “it was so annoying that Levent works at Anthropic.”

I said that if OpenAI released its result in the way proposed I would go public with what happened. The reply was, “Why would you ruin your career?”

— Tristan Buckmaster, statement, September 2026

Axios confirmed that quote independently. OpenAI has not disputed it. Buckmaster is explicit about what he is not saying: “I have not seen OpenAI’s proof. I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything.”

Whose Data

OpenAI’s post carries a September 10 update headed “Concurrent work”. It says no specific user data was accessed, then: “While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.” A later paragraph says an investigation confirmed Buckmaster’s Codex prompts over the preceding two months “could not have influenced the system in any way, including through training.”

Both sentences can be true. Two months of prompts are ruled out. A year of de-identified usage feeding a model whose training started August 28 is “unlikely”. Sam Altman on X, once Buckmaster’s work was public: “Now that we can see their work, the approaches appear to be different.” I take him at his word on the approaches. The question Simon Willison asked survives that answer: “If I use ChatGPT to help me partially solve a Millennium Prize problem, what are the chances that my work will influence training such that a later model helps someone else solve it first?”

What This Doesn’t Settle

• The proof stands. Lean-checked, forced case, two Clay routes. Terence Tao called the human forced-Euler result “a remarkable achievement” and said full Navier-Stokes “looks very feasible to complete these goals in the near future”. OpenAI completed it. Credit for the approach, per Charles Fefferman, belongs to Córdoba and Martínez-Zoroa, and Buckmaster says the same.
• The pressure account is one side. Buckmaster’s statement is detailed and dated, and Axios confirmed the key quote, but OpenAI has not given its version of the September 6 calls. Tao noted on Mathstodon that the story is “missing context from both sides”.
• The cost is a proxy. $6.5 million prices an unreleased model at Astra’s rate. The internal number is closer to $1 million, and OpenAI can run it again tomorrow.
• “Unlikely” is not “no”. OpenAI ruled out two months of one person’s prompts. It has not ruled out the year, and it has said so itself.

The Takeaway

• Seven days, 10,000 agents, 130 billion tokens. From rumour to Lean proof, with no prior program. Pachocki said so himself.
• The humans used the same vendor. A year of Codex and Claude, paid for personally, and the vendor announced first.
• The one line nobody has denied is “Why would you ruin your career?” Whatever the September 6 calls were, that is the sentence that will be quoted about them.
• The rumour was the prompt. Being known to be close is now the expensive thing to leak. Plan your disclosures accordingly.