OpenAI's tweet said it was "sharing a solution to the Navier-Stokes Millennium Prize Problem." OpenAI's fuller writeup is considerably more careful about what was actually established. Neither compares to what's now available directly: Tristan Buckmaster's own full public statement, dated, quoted verbatim, and considerably more specific than any secondhand account of it — including an earlier version of this post.
What OpenAI's writeup actually establishes
OpenAI's result resolves the Navier-Stokes Millennium Prize problem "by establishing statement 'C' (and also 'D') in the official Millennium Prize formulation" — one of the four sub-statements the Clay Mathematics Institute itself defined as a valid resolution path in 2000, not an easier problem outside the prize's scope. Concretely: an initially smooth, resting fluid, under a smooth external force, develops a singularity in finite time while its total energy stays finite throughout, established for both the whole space and the periodic case. OpenAI's agents also separately resolved the unforced Euler regularity problem as a stepping stone. OpenAI states directly that it does not intend to claim the $1 million prize, and that this is "a snapshot in time" rather than a culmination — genuinely careful framing that the tweet alone didn't carry. The process: the effort began September 1 after OpenAI "heard rumors" that two Millennium Prize problems had been resolved; agent groups were prompted with proof-oriented ("A"/"B") and disproof-oriented ("C"/"D") variants in parallel; nearly 100 agents spent about 50 hours resolving unforced Euler first; the Navier-Stokes group ran on the order of 10,000 concurrent agents for about 88 hours, exchanging 2.7 million messages and roughly 130 billion tokens, with a further 17 hours of Lean formalization run through GPT-6 Astra.
Buckmaster's own statement, dated and quoted
Buckmaster's account starts with attribution, not grievance: the underlying program, he writes, "was not started by us nor was it proposed by a Large Language Model." Credit for the basic idea belongs to Diego Córdoba and Luis Martínez-Zoroa, who spent years developing forced blow-up constructions; Buckmaster states plainly that he believes "Luis Martínez-Zoroa deserves a Fields Medal" for that body of work. What he and Alpöge did — a purely personal collaboration, he stresses, with no institutional agreement on either side, using Anthropic's Claude and OpenAI's Codex (running GPT-5.6 Sol, and more recently Astra for writeups and auditing only) — was push that program from rough forcing to smooth forcing and to the Euler equations. Progress was slow for most of a year, until August 15, when they obtained smooth-forcing blow-up for both Boussinesq and Euler; they Lean-verified the result on August 22. Buckmaster is candid that the writeups are rougher than he'd like — he calls the Euler write-up "AI slop" outright — and says that's a consequence of being rushed by the events that follow, not an excuse.
On September 3, with a rumor circulating that "Anthropic had resolved a major open problem" and Alpöge separately hearing that information about their progress had reached OpenAI, Buckmaster contacted OpenAI first — emailing a mathematician there to explain, on the record, that he and Alpöge had unpublished work they were confident in and would post "shortly," and that the collaboration was personal, not institutional. That mathematician replied the same day, friendly, offering compute if useful. Buckmaster asked to talk the following week; asked again on September 4, he again deferred. At 12:45pm on Sunday, September 6, he was asked to meet "at any point today." Sébastien Bubeck joined; the three spoke twice that afternoon.
What he says he was told, and what he says turned out not to be true
Buckmaster was told an internal OpenAI model had produced a roughly 100-page proof of forced Navier-Stokes blow-up, with the precise statement given as "options c and d in Fefferman" — the exact route through a smooth force that Córdoba and Martínez-Zoroa had opened and that he and Alpöge had "quietly chosen to attack," which he says almost nobody else was working on. "It is not the direction one arrives at in a few days by giving a model the problem statement," he writes. "When I heard 'forced,' it was a bright red flag." He and Alpöge were initially told "very little human input" had been used — which he says turned out not to be true: over the course of the call, as OpenAI team members fed Bubeck corrections in real time, it emerged that an entire team had tried multiple approaches, that work had also started on the unforced problem, that the model had first been set on easier problems including Euler, that even the prompt shown to Buckmaster had itself been written by prompting Codex, and that an "insane amount of compute" had been used. Asked when OpenAI's first prompt had actually been sent, Buckmaster says he wasn't given a direct answer for some time; eventually OpenAI agreed it was sent in the days just before, after information about Buckmaster and Alpöge's own work had reached them.
Buckmaster then asked the direct question this dispute turns on: had the model been trained on, or had access to, the Codex sessions where he and Alpöge had been drafting their entire project? He says he was told the model "did not look up user data" — addressing real-time access, not training. He asked again specifically about training. He did not get an answer.
Two proposals, and an exchange that reads like a threat
Buckmaster describes being offered two options: post the Euler result while OpenAI posted its Navier-Stokes result the next day, or — after posting Euler — write up the Navier-Stokes result himself, alone, crediting an internal OpenAI model, with no involvement from Alpöge at all. He says Bubeck twice asserted he wanted Alpöge removed from authorship, and said it was "annoying" that Alpöge works at Anthropic; Buckmaster was also told that if OpenAI posted after them, OpenAI would say they deserved the Clay Prize and were "the closest humans to the problem." Buckmaster declined both offers and said he'd go public if OpenAI proceeded as proposed. The reply, he says, was: "Why would you ruin your career?" When he asked why, as an academic, going public would ruin his career, the reply was: "If you don't want me to be nice, then I don't have to be nice." Separately, he says Alpöge received a text proposing the two of them speak one-on-one, without Buckmaster: "I don't know if Tristan is being fully rational right now." Alpöge declined. Bubeck emailed that night asking to speak again on Monday, September 7. Buckmaster didn't respond.
He's careful about the limits of what he's 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." Bubeck has separately called Buckmaster's characterization "false and inflammatory" and said he acted "following academic norms," without addressing these specific claims point by point. OpenAI's own writeup describes reaching out only after finishing its Navier-Stokes work on September 6 "to offer a concurrent release... and recognize their priority" — which, set against dated emails and a timestamped Sunday-afternoon call, is a materially different story than the one Buckmaster tells about who contacted whom first, and doesn't engage with the proposals or quotes at all.
The question OpenAI's own timeline raises but doesn't answer
Buckmaster followed up separately on Mastodon (@tristanbuckmaster@mastodon.social), pointing directly at a line in OpenAI's own writeup: "Since August 28 we have been training a new internal model." He notes plainly that this is OpenAI "openly admitting they used training data from a period after we found our result" — Buckmaster and Alpöge's own core result landed August 15, Lean-verified August 22, six days before OpenAI's stated training start, during which time Buckmaster says he was paying OpenAI directly for Codex usage on the project. His question: "Is it ethical to use customer's data to try to scoop their customer?" That's not proof of anything — Buckmaster says so himself, repeatedly — but it's a specific, dated overlap that OpenAI's own disclosure created, on top of a direct question about training data that Buckmaster says went unanswered on the call.
Why this isn't OpenAI's first math-credibility problem this year
This blog covered the last one in detail: last October, an OpenAI VP tweeted that GPT-5 had "found solutions to 10 (!) previously unsolved Erdős problems," which turned out to be false, and the tweet was deleted after mathematician Thomas Bloom called it "a dramatic misrepresentation." OpenAI's response to that embarrassment, this blog noted at the time, was visibly building a habit of getting the field's specific skeptics on record before a headline claim landed. This episode inverts that pattern in the starkest way available: instead of an endorsement, the field's most detailed account of contact with OpenAI's math team is a dated, quoted statement alleging pressure to erase a coauthor, a misrepresentation about human involvement, and an unanswered question about training data — from a mathematician who spent much of his own statement crediting the actual originators of the underlying idea rather than claiming more than his share. The Leiden Declaration on AI and Mathematics — cited by OpenAI itself in July — warns about unreliable results, missing attribution, and exaggerated claims. This episode now has a dated, sourced example of all three from the same week.
What to expect next
- Watch for OpenAI's promised fuller response, and specifically whether it answers the training-data question directly rather than repeating "false and inflammatory."
- Watch whether OpenAI discloses what training data actually went into the model trained from August 28, and whether Codex session content from paying customers is excluded, included, or simply undisclosed.
- Watch for independent mathematical review of OpenAI's actual proof, still unseen by Buckmaster or, as far as is publicly known, anyone outside OpenAI.
- Watch whether Anthropic comments. Alpöge is a named party in a dispute over his own authorship with a direct rival lab; Anthropic's silence or response is itself a data point.