10,000 Agents, 88 Hours, and a $1M Prize OpenAI Won't Claim

TL;DR

On September 8, OpenAI published a claimed resolution of one direction of the Navier-Stokes Millennium Prize Problem: an unreleased internal model, described as more capable than GPT-6 Astra, produced a proof that 3D Navier-Stokes flow can blow up in finite time. Roughly 10,000 concurrent agents, 88 hours of wall clock, about 130 billion output tokens, then 17 more hours for Lean to certify it. The certificates are public under Apache-2.0, so anyone with a Lean toolchain can re-check the formal argument today. Hours earlier the same day, NYU's Tristan Buckmaster and Anthropic's Levent Alpöge posted three Lean-verified blowup results of their own, plus a four-page statement describing what happened when OpenAI found out what they were working on. Nobody is getting a million dollars, and the interesting part is not the math.

What OpenAI's Lean file actually says

The Clay problem is not "prove Navier-Stokes is fine." Charles Fefferman's official problem description offers four statements and says a proof of any one of them wins, "to give reasonable leeway to solvers while retaining the heart of the problem." Statements (A) and (B) assert smooth solutions exist forever with the external force set to zero. Statements (C) and (D) assert breakdown, and they explicitly permit a smooth external forcing term.

OpenAI's repository formalizes exactly those two. For any positive viscosity, there exist smooth initial data and smooth forcing on R3 with no global smooth solution of uniformly bounded kinetic energy, and a periodic version of the same on R3/Z3. There is a bonus Euler theorem in the repo too: smooth, compactly supported, divergence-free initial velocity whose unforced Euler solution goes singular in finite time. Euler is not a Clay problem, but as Fefferman notes in the same document, it is "also open and very important."

The physical picture OpenAI describes is a vortex that spirals inward and stretches like spaghetti, spinning faster as it shrinks, while total energy stays finite the whole way down. Speed goes to infinity at a point in finite time. Fluids do not do this; the equations, apparently, do.

The other proof, which landed first

Alpöge and Buckmaster published finite-time blowup with smooth forcing for the incompressible porous medium equation, the 2D Boussinesq system, and 3D incompressible Euler, with Lean formalizations in a public repo and the papers on Buckmaster's NYU page. Terence Tao called it "a remarkable achievement."

Both efforts sit on the same foundation, and Buckmaster is emphatic about whose it is: Diego Córdoba and Luis Martínez-Zoroa, who spent years building forced-blowup constructions. "I believe Luis Martínez-Zoroa deserves a Fields Medal," Buckmaster writes. "The program this fits into was not started by us nor was it proposed by a Large Language Model."

The mechanism, briefly

Tao's summary of the Córdoba and Martínez-Zoroa strategy: you iteratively add small, localized high-frequency corrections to an existing solution, and the low-frequency part exponentially amplifies the high-frequency part until it takes over the dynamics, at which point the next iteration begins. Think of a relay where each runner sprints a shorter leg faster than the last and hands off just before collapsing. Every leg is finite, every leg is quicker, and the whole race finishes in finite time with the baton moving infinitely fast.

The compute bill

Across every problem it attempted in this run, OpenAI reports about 4.9 million inter-agent messages and roughly 300 billion output tokens. TechCrunch puts a price on that: $22.5 million of compute. The Navier-Stokes problem alone accounted for about 2.7 million messages and 130 billion tokens.

Buckmaster and Alpöge ran the same class of attack on a personal budget. "I pay for the tools my group uses out of my own research funds, including footing a large bill to OpenAI," Buckmaster wrote in an email he reproduces in full. They used Claude, Codex with GPT-5.6 Sol, and Astra for writeups and auditing. They got smooth-forcing blowup for Boussinesq and Euler on August 15 and had Lean confirm it on August 22, two weeks before OpenAI's run started.

The part that should worry you

Buckmaster's statement is worth reading in full, because he is careful in a way summaries are not. On September 3, hearing that word of their progress had reached OpenAI, he emailed a mathematician there to set the record straight. He was offered compute. On Sunday September 6 at 12:45 he was asked to meet "at any point today." Sébastien Bubeck joined; they spoke twice that afternoon; Alpöge was on neither call.

He was told an internal model had produced a roughly 100-page proof of forced Navier-Stokes blowup, on R3 and T3, with a smooth forcing function, "option c and d in Fefferman." That specificity is why he reacted the way he did. The smooth-forcing route is the one Córdoba and Martínez-Zoroa opened and the one he and Alpöge had quietly chosen. "Almost nobody else I know of was working on it," he writes. "It is not the direction one arrives at in a few days by giving a model the problem statement."

He was shown a prompt and told very little human input had been involved. Over the course of the call, as colleagues fed Bubeck corrections in an internal chat, that account came apart: an entire team had been on it, easier problems including Euler had been run first, an insane amount of compute had been used, and the prompt he was shown had itself been written by prompting Codex. The demonstration of minimal human involvement was, it turned out, machine-generated.

Then the question that matters to everyone reading this. He asked whether the model had been trained on, or had access to, their Codex sessions, where they had put every draft of the project for a year. He was told the model does not look up user data. He asked again about training. "I did not get an answer."

"I would like to be clear about what I am not claiming. 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. I am stating what I was told, when, and what was proposed to me."

He also reports two offers: publish Euler and let OpenAI publish Navier-Stokes the next day, or write the Navier-Stokes paper himself crediting an internal OpenAI model. Bubeck, he says, twice asserted that Alpöge should be removed from authorship because Alpöge works at Anthropic. When Buckmaster said he would go public, the reply was "Why would you ruin your career?" and then "If you don't want me to be nice, then I don't have to be nice."

OpenAI's answer

OpenAI says its effort started September 1 after hearing a rumour, that "we did not see any of their work through any means until they released it publicly," and, per chief research officer Mark Chen, that no people or AI systems searched user data to solve the problem. It confirms Alpöge would not have been invited as a co-author, citing its competitive relationship with his employer. And it includes one sentence that is going to outlive the rest of this story: "While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models."

Why nobody gets $1 million

The Clay Institute's rules require publication in a qualifying outlet, then at least two years, then general acceptance in the global mathematics community. Nothing here is published in a journal yet, so the earliest any of this can even be considered is 2028. OpenAI says it does not intend to claim the prize.

Lean verification is not the same as community acceptance either. A machine-checked certificate tells you the argument is valid given the definitions in the file. It does not tell you the definitions faithfully encode Fefferman's problem, and that translation is exactly where a formalization can quietly cheat. Independent readers still have to check the statement, not just the proof, which is precisely the work now underway.

Key Takeaways

• OpenAI claims Lean-verified finite-time blowup for forced 3D Navier-Stokes on R3 and R3/Z3, matching statements (C) and (D) in Fefferman's official Clay problem description. The certificates are public and Apache-2.0.
• The run took roughly 10,000 concurrent agents and 88 hours, about 130 billion output tokens on this problem, plus 17 hours of GPT-6 Astra formalizing it. Across all problems attempted: 4.9 million messages, ~300 billion tokens, $22.5 million of compute per TechCrunch.
• Alpöge and Buckmaster got smooth-forcing blowup for Boussinesq and 3D Euler on August 15, Lean-verified August 22, on personal research funds. Both efforts build on Córdoba and Martínez-Zoroa's construction.
• Buckmaster documents being told the proof came from a bare problem statement, then learning during the call that a full team, prior easier problems, and a Codex-written prompt were involved.
• He asked whether the model was trained on their Codex sessions and got no answer. OpenAI denies looking up user data but concedes it "cannot rule out" that de-identified data from their product usage improved its models.
• If you are doing unpublished work inside a frontier lab's coding agent, your recourse when the lab ships an adjacent result is currently a strongly worded PDF. That is the actual news for builders.
• No prize is possible before 2028 under Clay's rules, and OpenAI says it will not claim it.

Sources: OpenAI: On the Navier-Stokes Millennium Prize Problem, openai/NavierStokesAndEuler (Lean certificates), Tristan Buckmaster's statement (PDF), tristanbuckmaster/fluid_lean, Terence Tao: Finite time blowup with smooth forcing term, Clay Mathematics Institute: Fefferman's problem description (PDF), Clay Millennium Prize rules, TechCrunch, Quanta Magazine, Scientific American, Simon Willison