On September 8, 2026, OpenAI said an internal model produced a proof of finite-time blowup for the forced version of the Navier-Stokes equations, a fluid-dynamics problem that's been open since the Clay Mathematics Institute put a $1 million bounty on it in 2000. OpenAI says the effort ran about 88 hours, used roughly 10,000 AI agents working in parallel, and cost an estimated $22.5 million in compute. NYU mathematician Tristan Buckmaster says he and Anthropic researcher Levent Alpöge had been working on closely related math in the weeks before OpenAI's announcement, using AI tools from both companies. He says OpenAI used his and Alpöge's unpublished work to get there, and pressured him over credit before going public.

1. OpenAI Says It Won Fair, and It's Not Even Taking the Prize Money (Sébastien Bubeck, Sam Altman, Mark Chen, OpenAI)

OpenAI denies touching the rival team's work and says its own effort stands on its own.

It never saw the rival team's work, OpenAI says. Researcher Sébastien Bubeck put it directly: "We did not use their prompts or proofs to prompt our models," and separately, "We did not see any of their work through any means until they released it publicly." OpenAI's own statement adds a caveat: "no specific user data was accessed in order to solve this problem," but "while unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models." Bubeck called Buckmaster's broader account "a series of false and inflammatory allegations."

A widely circulated quote wasn't accurate, Bubeck now says. He says he actually told Buckmaster "why would you risk your career," not the harsher line that circulated. He's since apologized: "I deeply apologize for this extremely poor choice of words, it is the opposite of what I was trying to convey... I retracted them on the spot." He describes it as a frustrated, in-the-moment remark made amid what he calls a barrage of accusations and a threat to go to the press.

OpenAI CEO Sam Altman admits the timing wasn't a coincidence. He said Bubeck "acted with integrity and generosity throughout." He also confirmed OpenAI moved on the problem because "there were rumors on the internet last week that Anthropic's models had solved a millennium problem and we were curious if ours could do it too."

OpenAI isn't claiming the $1 million prize. The company's result covers the forced version of Navier-Stokes, not the unforced version the Clay Institute's rules actually require, so OpenAI is framing this purely as a demonstration of what its model can do. Bubeck has since said OpenAI "recognize[s] the priority of Levent Alpöge and Tristan Buckmaster's work," even while maintaining the two proofs "differ significantly."

2. Buckmaster Says OpenAI Fought Dirty on a Career-Making Problem (Tristan Buckmaster, NYU; Levent Alpöge, Anthropic)

Buckmaster says OpenAI used his private work and then threatened him when he pushed back.

He never got a straight answer, Buckmaster says. He'd asked OpenAI point-blank whether its model trained on his private sessions. His account: "I was told the model did not look up user data. I asked again, about training, and I did not get an answer."

OpenAI offered him a deal that erased his collaborator, Buckmaster says. He says Bubeck gave him two options before the announcement. One: publish his and Alpöge's partial results, with OpenAI's fuller proof following the next day. Two: write a solo paper that credited OpenAI but dropped Alpöge's name, since Alpöge works at Anthropic, a competitor. Buckmaster turned down both.

Bubeck then threatened his career, Buckmaster says. When Buckmaster said he'd go public instead, he alleges Bubeck told him: "Why would you ruin your career?" and "If you don't want me to be nice, then I don't have to be nice."

Buckmaster doesn't buy that OpenAI got there independently. He argues that his and OpenAI's proofs took the same uncommon mathematical route. He says he and Alpöge had already discussed that approach with Bubeck directly, before OpenAI's public release. On whether OpenAI's model actually trained on his private sessions, though, Buckmaster says he doesn't know. He's "not accusing anyone of anything" on that specific point.

Buckmaster and Alpöge backed up their allegations with published proofs. The pair built on a "forcing" approach from mathematicians Diego Córdoba and Luis Martínez-Zoroa. Around September 7-8, they posted three preprints with Lean-verified proofs, establishing finite-time blowup for the incompressible porous medium equation, the 2D Boussinesq system, and the 3D incompressible Euler equations — a close cousin of Navier-Stokes, though not the same problem.

3. Outside Mathematicians Say Nobody's Actually Checked This Yet (Terence Tao, UCLA; Luis Silvestre, University of Chicago)

Mathematicians outside the dispute are asking whether the proof even counts as solved and whether AI is good for the field.

By the Institute's own rules, OpenAI hasn't solved the problem. The prize covers the unforced Navier-Stokes equations; OpenAI's result covers the forced version, which doesn't meet the award criteria. Clay Institute rules also require peer-reviewed publication and roughly two years of community validation before the Institute even considers a prize, and that process hasn't started.

The Institute's own president has already weighed in. Clay Mathematics Institute president Martin Bridson told AFP: "The process of evaluation is deliberately unhurried, and we shall ensure that it is absolutely rigorous." The Institute still lists Navier-Stokes as unsolved.

Three days in, nobody outside OpenAI has confirmed the proof holds up. OpenAI ran its proof through the Lean 4 theorem prover, so the code compiles mechanically against Lean's math library. But no outside mathematician has independently reproduced that check or confirmed the proof's approach actually matches what the Clay Institute's problem statement requires.

AI can give answers without real understanding, UCLA mathematician Terence Tao warns. He said flooding open problems with AI effort "may destroy the ecosystem from which the next generation of mathematical techniques ... would have developed." He argues it discourages researchers from finding their own approaches, and it buries the failed attempts that normally teach the field something. He compared it to "using excavators to loot an archaeological site."

Silvestre, a University of Chicago mathematician, put it simply. He said: "Yesterday and today are crazy days."

Where This Lands

OpenAI says its model reached the result on its own and is walking away from the prize money to prove the point. Buckmaster says OpenAI drew on his and Alpöge's unpublished work and pressured him over credit, though he stops short of accusing OpenAI of training on his private sessions specifically. Neither camp fully agrees on what happened on that call. Bubeck disputes his own quoted words and has apologized for the version that circulated. Meanwhile, by the Clay Institute's own definition, nobody has actually solved the problem the $1 million prize was for, and three days on, no outside mathematician has independently confirmed OpenAI's proof holds up.

Sources