OpenAI's Navier-Stokes Announcement, and the Mathematician Who Says It Followed His Unpublished Work
OpenAI published a paper and a public Lean repository on September 8 claiming that an internal model had resolved a piece of the Navier-Stokes existence and smoothness problem, one of the seven Millennium Prize Problems the Clay Mathematics Institute set out in 2000. Within the same 48 hours, NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge published their own AI-assisted, Lean-verified proofs of finite-time blowup for related fluid equations, then accused OpenAI of learning about their unpublished work and moving to claim credit before they could release it themselves. The mathematics is genuinely significant. The dispute over how OpenAI arrived at its result, using tools built by the same company that Buckmaster and Alpöge had used to draft their own work in private, is now getting as much attention as the proof itself.
What OpenAI Published
OpenAI's writeup describes a three-dimensional incompressible fluid that starts at rest, has a smooth external force applied to it, and develops an unbounded velocity within finite time while its total kinetic energy stays finite throughout. The mechanism is a vortex that spirals inward and stretches along its axis, shrinking and accelerating until the local velocity exceeds any bound even as the total energy in the system remains finite. OpenAI says this satisfies statements "C" and "D" of the Clay Institute's official 2000 problem formulation, the versions that ask whether a breakdown can occur under a driving force rather than the unforced case most people think of when they picture "solving Navier-Stokes." Coverage from Neowin puts the writeup at roughly 165 pages. OpenAI says the work was produced by a model "significantly more capable than GPT-6 Astra" and posted both the proof and its Lean formalization publicly alongside the announcement.
The Clay Institute's own problem page still lists Navier-Stokes as unsolved, and the institute's rules require publication, at least two years of general acceptance in the mathematical community, and a recommendation from its advisory board before it will consider a prize. OpenAI has said explicitly that it does not intend to claim the $1 million award.
How OpenAI Says It Got There
According to the company's account, a new internal model began training around August 28 and showed unusual gains on math benchmarks. On September 1, after hearing rumors that two Millennium Prize problems had been resolved, OpenAI pointed roughly 10,000 concurrent agents at every open Millennium problem plus a handful of other high-profile questions, running separate groups on different variants of each statement. A smaller group of agents produced an unforced Euler blowup result as a warmup in about 50 hours, and OpenAI then redirected most of its compute toward Navier-Stokes, periodically consolidating insights across agent groups through Codex.
The winning group reached its result on September 5, roughly 88 hours after the first agents launched, and Lean verification took another 17 hours. Across the full effort, OpenAI says its agents sent 4.9 million messages and generated about 300 billion output tokens, with the Navier-Stokes branch alone accounting for 2.7 million messages and 130 billion tokens. Company representatives have described the compute cost in the millions of dollars, with press estimates ranging from roughly $15 million to $22.5 million depending on which token rate is applied.
The Buckmaster and Alpöge Results
Buckmaster and Alpöge released three papers of their own on September 8, each Lean-verified: finite-time blowup with smooth forcing for the incompressible porous media equation, the Boussinesq system, and three-dimensional incompressible Euler. The work extends a multi-year research program by Diego Córdoba and Luis Martínez-Zoroa, which had previously achieved forced blowup only with rougher, less physically realistic forcing terms. Buckmaster and Alpöge say they pushed the same approach all the way to smooth forcing, and that they used Claude, Codex, and other models extensively over roughly a year of work, largely funded out of Buckmaster's own research budget. Fields Medalist Terence Tao called the result "a remarkable achievement" and said he sees no obvious obstacle to eventually extending the same method to the unforced Navier-Stokes case, while noting that real technical difficulty remains between forced Euler and the full Millennium Prize question. Buckmaster says the pair also believe they have a blowup result for hypo-dissipative Navier-Stokes but are withholding it because Lean verification is not finished.
The Disputed Contacts With OpenAI
The controversy stems from Buckmaster's own written statement, published alongside his papers. He says that on September 3, he learned rumors of his and Alpöge's unpublished work had reached OpenAI, and he proactively emailed an OpenAI-affiliated mathematician to clarify that the project was a personal collaboration with no institutional backing from Anthropic or NYU. Three days later, on Sunday September 6, he had two calls involving OpenAI's math lead Sébastien Bubeck. According to Buckmaster, he was told during those calls that an internal OpenAI model had already produced a roughly 100-page proof of forced Navier-Stokes blowup, following essentially the route he and Alpöge had been pursuing quietly for a year, and that the model had been given only the bare problem statement with minimal human involvement.
Buckmaster writes that this account did not hold up over the course of the same call. He says it emerged that an entire team had worked the problem, that the model had first been warmed up on Euler, that the prompt he had been shown was itself generated by Codex, and that the first prompt to agents had gone out only after word of his and Alpöge's work reached OpenAI, not before. He asked whether his and Alpöge's private Codex sessions, which they had used to draft the work, had been accessed. He says he was told the model does not look up user data during use, but that he received no answer on whether training had drawn on it. He was then offered two options: a coordinated release in which OpenAI's Navier-Stokes result would follow his Euler paper by a day, or a version in which he alone would write up the Navier-Stokes result crediting an internal OpenAI model, with Bubeck reportedly pushing twice to exclude Alpöge from authorship because he works at Anthropic. Buckmaster declined both and said he would go public. He says Bubeck responded, "Why would you ruin your career?" Buckmaster is careful to note in his statement that he has not seen OpenAI's proof, does not know what the model actually did, and is not accusing anyone of a specific wrongdoing, only laying out what he was told and when.
OpenAI's Response
OpenAI's own writeup addresses the overlap directly, saying its effort began September 1 after hearing rumors it later realized concerned Buckmaster and Alpöge, and that after finishing its own proof and Lean verification on September 6 it reached out to offer a joint announcement, only learning at that point that the pair's result covered forced Euler rather than the full Navier-Stokes problem. The company states that neither its researchers nor its agents saw any of Buckmaster and Alpöge's work through any means before it was released publicly, and that no specific user data was accessed to solve the problem, while adding a caveat that it cannot rule out that de-identified data derived from product usage helped improve its models generally. Bubeck, in a press briefing reported by Nature and Scientific American, said the Euler result was reached by a completely different method than Buckmaster and Alpöge's, that the Navier-Stokes proof followed a similar route to theirs, and that it was developed after the weekend calls, not before. He separately called Buckmaster's allegations against him "false and inflammatory" and said he had followed academic norms, promising a fuller response.
The dispute has since drawn in other researchers from both companies trading pointed posts, including Alpöge sharing an email he sent Buckmaster in September 2025, a full year before this week's events, under a subject line describing their collaboration as ordinary personal work.
What Would Actually Have to Happen for This to Count as Solved
Even taking OpenAI's proof at face value, resolving statements C and D under a smooth forcing term is not the same as resolving the unforced global-regularity question that most people mean when they invoke Navier-Stokes casually, and it is a step short of what the Clay Institute's rules require for prize consideration regardless. The forced version is the same category of result Buckmaster, Alpöge, Córdoba, and Martínez-Zoroa have been building toward, which is precisely why the overlap in timing and method drew scrutiny rather than simple applause. Independent scrutiny of OpenAI's specific 165-page proof has barely begun, and the mathematicians best positioned to evaluate it, the same small community assessing Buckmaster and Alpöge's Euler work, have not yet weighed in publicly on OpenAI's version specifically.
Practical Implications
The story that will outlast this week's headlines is not really about whether a smooth force can make a fluid break down in finite time. It is about what happens when researchers draft unpublished, career-defining work inside a lab's own coding assistant, and that same lab later claims to have independently reached an adjacent result on the identical technical route within days of hearing a rumor about their progress. OpenAI's answer, that no specific user data was accessed while adding a hedge about de-identified training data, will not satisfy critics who point out that a hedge is not a guarantee. Mathematicians doing unpublished work with any lab's tools now have a concrete recent example of what can go wrong, whatever the eventual verdict on this particular case turns out to be.
Author bio: David Borish is the author of The Tony Hawk Paradox: When Video Games Predict Reality and publishes long-form analysis on frontier AI at davidborish.com
