OpenAI Navier–Stokes: Proof Dispute

OpenAI announced on September 8, 2026, that an unreleased internal model had proved finite-time blowup for the three-dimensional Navier–Stokes equations.
It published a 166-page paper and a repository formalizing the proof in Lean at the same time.
About 12 hours earlier, New York University mathematician Tristan Buckmaster had posted three papers coauthored with Anthropic employee Levent Alpöge (Matei Coiculescu also coauthored the porous-medium paper), along with a statement describing their dealings with OpenAI.
Over the following week, mathematicians argued about that sequence of events and how the proof should be treated. On September 10, OpenAI reported that Buckmaster’s Codex prompts in the two months before its September 8 announcement could not have influenced its result.

What OpenAI says it proved

The Navier–Stokes equations are partial differential equations describing how the velocity and pressure of a viscous fluid change over time. They are used in numerical calculations for weather forecasting and aircraft design.
The Clay Mathematics Institute assigned a $1 million prize in 2000 to the Millennium Prize Problem called “Navier–Stokes Existence and Smoothness.”
It asks whether a solution starting from smooth initial conditions in three dimensions can undergo “blowup”: its velocity becoming unbounded in finite time.
Charles Fefferman of Princeton University, who wrote the problem statement, specifies in the official description that a proof of any one of four alternatives would solve it.

Alternative	Domain	External force	What to prove
(A)	Full space	None	A smooth solution exists for all time
(B)	Periodic	None	A smooth solution exists for all time
(C)	Full space	Smooth forcing	A smooth solution breaks down in finite time
(D)	Periodic	Smooth forcing	A smooth solution breaks down in finite time

In Theorem 1.1, OpenAI constructs, for any positive viscosity, a solution starting from fluid at rest under a smooth external force supported within bounded regions of space and time.
The solution’s kinetic energy stays bounded, but its velocity becomes unbounded at time 1.
The paper presents this as a solution to (C) and derives the periodic case (D) in Corollary 10.6.
It does not address Clay’s unforced alternatives (A) and (B).

A separate 57-page Euler equations paper, published at the same time, claims to construct an unforced solution whose velocity gradient and vorticity diverge when viscosity is zero. The Euler equations are outside the Millennium Prize Problem.

Buckmaster, Córdoba, and Martínez-Zoroa had pursued the approach earlier

According to Buckmaster’s statement, Diego Córdoba in Madrid and Luis Martínez-Zoroa at CUNEF University spent years developing the approach of producing blowup with smooth forcing.
They constructed blowup for the three-dimensional Euler equations under rough forcing, among other results. Buckmaster says he used LLMs to extend that program to smooth forcing and three-dimensional Euler.
The three papers he posted on September 7 claim finite-time blowup under smooth forcing for the incompressible porous medium equation, the two-dimensional Boussinesq equations, and the three-dimensional incompressible Euler equations, respectively. He says all three results were formalized in Lean.

Buckmaster describes his collaboration with Alpöge as personal research unrelated to either person’s employer.
They used Anthropic’s Claude and OpenAI’s Codex. Within Codex, he mainly used GPT-5.6 Sol; he used Astra only to check writing and discussions. He paid the costs from his research funds.
Progress was slow for nearly a year, but they obtained the Boussinesq and Euler blowup results on August 15 and completed Lean verification on August 22.
He says the first LLM-generated proof was the worst he had ever read and that much of the subsequent work went into making it readable by humans.
He also writes that he believes they have a blowup result for “low-dissipation Navier–Stokes,” a variant with a weaker effect from viscosity, but will not publish it until Lean verification is complete.

In a September 7 blog post, Terence Tao describes Córdoba and his collaborators’ method as repeatedly adding small, high-frequency corrections to a low-frequency solution so the solution grows while the force remains small.
Tao considers it likely that Alpöge and Buckmaster’s refinements could be extended to Navier–Stokes. He says there are many intricate technical points, but the extension could be completed in the near future. He also notes that he has not fully understood all the differences in the details himself.

The two accounts of the timeline

OpenAI says it began work after hearing a rumor on September 1. Buckmaster says OpenAI’s first prompt came after information about his team’s work had reached the company.

Date	Event	Source
August 15	Buckmaster and Alpöge obtained the Boussinesq and Euler blowup results	Buckmaster’s statement
August 22	Lean verification of those results passed	Buckmaster’s statement
August 28	Training of OpenAI’s new model began	OpenAI’s announcement
Late August	A rumor circulated that Anthropic had solved a major problem	Buckmaster’s statement (OpenAI describes a rumor that two Millennium problems had been solved)
September 1	OpenAI heard the rumor and began work on a Millennium Prize Problem	OpenAI’s announcement
September 3	Buckmaster heard that OpenAI had learned of their progress and emailed an OpenAI mathematician	Buckmaster’s statement
September 5	Roughly 10,000 OpenAI AI agents reached a Navier–Stokes proof after 88 hours	OpenAI’s announcement
September 6	Buckmaster spoke twice with OpenAI’s Sébastien Bubeck and colleagues, who disclosed the proof and made two proposals	Buckmaster’s statement
Late on September 7	Buckmaster posted three papers and his statement	Buckmaster’s statement
September 8	OpenAI announced the result; Bubeck held a press conference that day	OpenAI’s announcement; Scientific American
September 11	Clay said the problem appeared to have been settled	Clay

Buckmaster writes that hearing the word “forced” on the September 6 call immediately raised a red flag for him.
He says almost no one apart from his team and Córdoba’s collaborators was pursuing (C) and (D) through smooth forcing, and that the approach would not simply emerge within days from giving a model the problem statement.
When Buckmaster asked when OpenAI had sent its first prompt, he says there was no immediate answer. He writes that OpenAI eventually acknowledged it had been sent in the previous few days, after information about his team’s work had reached the company.
Buckmaster stresses that he is not accusing anyone; he says he is reporting what he was told, when it happened, and what was proposed.

OpenAI says it did begin work after hearing the September 1 rumor, but did not see the other team’s results before publication and obtained its proof independently.
An OpenAI statement quoted by TechCrunch says it had not seen their work by any means before publication. It adds that, although unlikely, it could not rule out the possibility that de-identified data from product use had contributed to improving the model.
In an interview with Quanta Magazine, OpenAI also acknowledged that rumors about Alpöge and Buckmaster solving a Millennium Prize Problem helped spur its work.

Drafts entered into Codex and the question of training data

Buckmaster and his collaborators had put all of their drafts into OpenAI Codex sessions during nearly a year of work.
On the September 6 call, they asked whether OpenAI’s model had consulted those sessions or been trained on them. According to Buckmaster, they were told that models do not access user data, but received no answer about training.

OpenAI’s initial September 8 statement said it could not rule out a contribution from de-identified product data. In a September 10 update, it reported that Buckmaster’s Codex prompts in the two months before its September 8 announcement could not have influenced the result in any way, including through training.
At a press conference on September 8, Bubeck said the team had not used Alpöge and Buckmaster’s prompts or proofs to direct the model. OpenAI also told the New York Times on September 10 that any influence from Codex prompts in the two months before the September 8 announcement was “categorically impossible.”

In a September 9 post, mathematician Andreas Thom said there was too little transparency about whether unpublished mathematics had entered training data or been accessible during the solution process.
In a September 8 post, Buckmaster asked whether it was ethical to use customer data to outpace a customer. OpenAI concluded on September 10 that Buckmaster’s Codex prompts in the two months before its September 8 announcement had no influence, but its public update does not present the records or method behind its investigation.

The claim of very little human input

On the September 6 call, OpenAI showed Buckmaster one prompt and, in his account, described the process as merely giving its internal research model the problem statement.
Alpöge also says Bubeck had told him there was “very little human input.”
During the call, however, other OpenAI team members sent Bubeck corrections and additional context through an internal chat.
Buckmaster says that revealed a wider team effort and that the prompt shown was only one of several attempts. The team had initially tried the unforced problem and had asked the model to solve easier problems, including Euler, first.
His statement also says Codex wrote the prompt that was shown and that the effort used enormous computing resources.

Bubeck responded that several people were indeed involved, but no one on the team had research-level expertise in fluid dynamics or could handle the mathematical part of the proof. On that basis, he maintains that the description of very little human input was accurate.

According to OpenAI’s announcement, it ran approximately 10,000 agents using an unreleased model whose training began on August 28. They reached the proof in 88 hours. OpenAI’s GPT-6 Astra completed the Lean formalization in 17 hours.
Bubeck estimated the computation cost at several million dollars in the Quanta interview. OpenAI says it will not claim the $1 million prize.

The proposal to exclude Alpöge and the “career” remark

Buckmaster’s statement describes two proposals from OpenAI on the September 6 call.
Under the first, Buckmaster’s team would post its Euler result, then OpenAI would post the Navier–Stokes result the following day.
Under the second, after the Euler result was posted, Buckmaster would write a Navier–Stokes paper alone and state that an internal OpenAI model had solved it.
According to Buckmaster, Bubeck said twice that he wanted Alpöge excluded as an author and remarked that everything would be simple if Alpöge did not work at Anthropic.
Buckmaster says he rejected both proposals and said he would disclose the sequence of events if OpenAI published as proposed. The response, according to his statement, was: “Why would you ruin your career?”
When Buckmaster asked why that would ruin his career as an academic, he says Bubeck replied: “If you don’t want me to be nice, then I don’t have to be nice.”

On X, Bubeck has said he never asked for Alpöge to be removed as an author from the pair’s own work. He says they discussed Buckmaster rewriting OpenAI’s proof as lead author. Including an Anthropic employee in an internal OpenAI project, he says, would have been difficult.
Bubeck apologized for his choice of words about Buckmaster’s career, saying he withdrew the remark on the call.
Sam Altman also rejected the narrative of OpenAI trying to outpace the other team. He posted that OpenAI did not rush to publish even while the others did not contact it, and that the other side threatened it with an unfounded accusation of plagiarism.

Peter Sarnak of the Institute for Advanced Study told the New York Times that he had never heard of bargaining like this and praised Buckmaster for acting as a mathematician.
Buckmaster’s statement and Bubeck’s explanation still differ on what was said during the calls.

How the forced proof differs from the unforced problem

Clay’s problem statement accepts a proof of (C) or (D) as a solution, so OpenAI’s claimed result may formally satisfy its requirements.
With forcing, one can first choose a flow that blows up and define the external force as the residual left by the equation.
OpenAI’s paper describes that approach in Section 2 and explains that the difficult part is keeping the residual smooth all the way through.
Without forcing, that method is unavailable: one must construct a fluid flow that blows up on its own.

Fefferman told Quanta he was thrilled that the problem had been solved, while observing that, in the setting without a boundary, it is the fluid itself that does the strange thing.
On September 8, Princeton mathematician Stan Palasek suggested to Tao that, for unforced Navier–Stokes, energy loss from viscosity should outweigh the amplification produced by this construction’s instabilities at separated frequencies. Tao welcomed the observation.

Lean guarantees only that a formalized proposition follows from the chosen axioms. Humans still have to check whether that proposition faithfully captures every condition in Fefferman’s problem statement.

Tao’s criticism

In his September 7 post, Tao writes that the value of this kind of work lies in mathematical insights people can understand. Without those insights, he says, the Navier–Stokes regularity problem (the question of whether solutions stay smooth) does not have the fundamental mathematical significance often attributed to it.

In posts after September 8, Tao also wrote about how the research was conducted.
He worries that if companies repeatedly use massive computing resources to produce results first after hearing rumors, researchers will stop sharing promising directions and the principles of open, reproducible science will suffer.
He criticized issuing a press release while withholding failed attempts and the route to a solution, and called for criteria that value mathematical insight alongside the answer.

Buckmaster writes in his statement that he had wanted to discuss what it means for one mathematician working with an LLM to achieve this much in a month. He says the community needs time to consider how to train students, attribute credit, review papers, and decide where researchers spend their lives.
He calls it a Deep Blue–Kasparov moment and says Martínez-Zoroa deserves a Fields Medal.
Córdoba told Scientific American that he was somewhat shocked and that, if the result was real, it would be a major surprise.

Missing citations

OpenAI’s initial paper cited neither Córdoba and Martínez-Zoroa’s papers nor Alpöge and Buckmaster’s work.
After Gonzalo Cao-Labora of EPFL pointed out the sparse references, an updated version on September 8 added Córdoba and his collaborators’ papers.
The introduction of the September 8 revision describes Córdoba and Martínez-Zoroa’s construction for forced Euler, and their extension to low-dissipation Navier–Stokes with Zheng, as establishing a strategy for controlling the regularity of forcing through amplification across scales. It says that the perturbations being amplified play a different role in OpenAI’s construction.
Alpöge and Buckmaster’s three papers were still absent from the references in that September 8 revision.

Clay Mathematics Institute’s response

On September 11, the Clay Mathematics Institute said the Navier–Stokes problem had “apparently been settled.”
Its statement says the process of checking what was achieved under the rules and deciding who deserves credit is “deliberately unhurried.” It does not say Clay has verified the proof or awarded the prize.
The prize rules generally require publication in a well-regarded, refereed mathematics venue (or one approved by its board), at least two years of scrutiny after publication, and broad acceptance in the mathematical community before a result can be considered for the prize.
OpenAI’s Navier–Stokes paper is currently available as a PDF and preprint, a manuscript released before peer review. The two-year scrutiny period for the prize begins only after this paper is published in a qualifying outlet under Clay’s rules.

As an ITmedia article notes, the Poincaré conjecture is the only Millennium Prize Problem previously recognized as solved.
Apart from Clay’s unforced Navier–Stokes alternatives (A) and (B), Buckmaster describes his unpublished low-dissipation Navier–Stokes result as suggesting that unforced Euler may be possible.
OpenAI says it has made major progress on another Millennium Prize Problem since Navier–Stokes and is considering how to share it.