OpenAI says it resolved Navier-Stokes. Buckmaster says Bubeck wanted his co-author removed.

Tristan Buckmaster, a mathematics professor at NYU, published a four-page signed statement on 8 September 2026 which says that on two calls with OpenAI on 6 September, Sebastien Bubeck twice asserted that he wanted Buckmaster's collaborator Levent Alpöge removed from authorship. OpenAI published a page the same day saying an internal model resolved the Navier-Stokes Millennium Prize problem, and that page mentions no proposal about authorship and does not name Bubeck. Alpöge, described on OpenAI's page as an Anthropic employee, wrote on his own account that he had heard, in a loud hallway conversation, "the part where a millennium prize was offered if i'd just be removed from the paper" (the lowercase is his); his post does not say who said it.

What follows is a report of what those three documents say. We read all three on 8 September 2026. We did not open OpenAI's proof, its Lean formalization, or the three preprints and formalizations Buckmaster and Alpöge released the same day. Every claim below is attributed to the document it comes from, and where only one document supports something, the item says so.

The program, the tools, and a Lean check

Buckmaster's statement gives credit away before it defends anything. The route to the Clay problem through a smooth force belongs to two other mathematicians, and he names them on the first page. The same statement dates the August results and lists the tools that produced them.

The statement says the program "was not started by us nor was it proposed by a Large Language Model." It credits the basic idea to Diego Córdoba and Luis Martínez-Zoroa, "who for several years have been exploring the construction of forced blow ups," and describes what he and Alpöge did as taking a program that had achieved blowup with rough forcing and pushing it, with a great deal of help from language models, to smooth forcing and to the incompressible Euler equations. "The ideas making this line of attack possible are due to Córdoba and Martínez-Zoroa." That attribution does double duty later, because the same route is the one he says he was told an OpenAI model had taken.

Then comes a sentence the statement says he had until now made only to colleagues in private: "in view of this body of work, I believe Luis Martínez-Zoroa deserves a Fields Medal."

Which tools were used, and for what, is not a side detail in this story. It is the ground of the question Buckmaster later puts to OpenAI about training, because the drafts lived inside one company's product for the whole project. His statement lists the models by name.

We used several LLMs throughout: Anthropic's Claude, OpenAI's Codex, especially with GPT-5.6 Sol and, more recently, Astra. The latter was only used for writeups and auditing our arguments.

The statement calls the collaboration "a purely personal collaboration, free of any institutional agreements or official involvement by either of our employers." It describes about a year of slow progress, working through the literature and upgrading preliminary results as far as finite time blowup for the incompressible porous media equation with smooth forcing. Then, roughly a month before publication: "on August 15th, we obtained the blow up results, with smooth forcing, for both Boussinesq and Euler." On what arrived first: "I can say the first LLM generated proof Levent sent me was the most horrendous I have ever read; we verified it on Lean on August 22nd." Since that point, the statement says, they had been working around the clock to understand the proof and turn it into something readable.

OpenAI trains an internal model, then hears a rumour

OpenAI's page dates its own work first, and its account of the week starts before Buckmaster's does. The order of events is the least contested part of this story: the two documents describe the same run of days from opposite ends, and on dates they do not contradict each other. The page gives two dates for how its effort began.

It says that since 28 August it has been training a new internal model, and then:

On Tuesday, September 1, we heard rumors that two Millennium Prize problems had been resolved. Inspired by these rumors and by the step change in performance of our internal model, we launched an effort to evaluate it on all open Millennium Prize problems and a few other high-impact problems.

Its "Concurrent work" section dates the same start and says who the rumour turned out to concern. The effort, it says, "began on September 1st after hearing a rumor which we later realized was related to Levent Alpoge, an Anthropic employee, and Tristan Buckmaster, a math professor at NYU."

Buckmaster's email, and the reply that came the same day

Buckmaster's account begins two days after OpenAI's. His statement is the only document among the three that describes the first contact between the two sides. The reply he quotes came back the same day.

On Thursday 3 September, with a rumour circulating that Anthropic had resolved a major open problem, and with Alpöge having received tips that information about their progress had been passed to OpenAI, he wrote to a prominent mathematician at OpenAI whom he does not name. He reproduces the email in full, including the lines that this "is not an institutional effort. It is a strictly personal collaboration between the two of us, and there is no formal agreement behind it," and that "I pay for the tools my group uses out of my own research funds, including footing a large bill to OpenAI." The email also says why they had not posted yet: "We intentionally decided against rushing out a Lean certificate alongside an unpolished preprint." He quotes the reply, sent the same day:

If you are willing to give any details it would be useful to avoid competing here and in general we are always thrilled when mathematician make progress with our models. Additionally if there is anything in terms of compute from OpenAI's end we would be happy to provide it.

A same-day meeting request, and the agents reach their resolution

Two things happen over these two days, one in each account. Buckmaster is being asked to meet sooner than he wants to, and OpenAI's agents are finishing. Its page is specific about its machinery in a way that can be quoted rather than inferred, so the figures for the run are set out here as it gives them.

Buckmaster says he asked to speak the following week. On Friday 4 September he was asked whether he could meet that day, and again said the following week.

OpenAI's page covers the same weekend from its side. It says the agents "arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched." The page states its claim in one sentence:

This resolves the Navier-Stokes Millennium Prize problem by establishing statement "C" (and also "D") in the official Millennium Prize formulation

Under the heading "How we found the proof" it gives the run in figures. Those figures describe scale and effort; none of them is evidence that the proof is correct, and none is a reading of it by anyone outside the company. The numbers below are the page's own, transcribed from a browser read on 8 September 2026.

The page also separates its own Euler result from theirs. Its "Concurrent work" section says that "our proofs differ significantly and even the precise results proved are different in the Euler case (forced vs unforced)." The Euler result Buckmaster and Alpöge have is the forced one; the page describes OpenAI's as an Euler regularity disproof.

The project completes, and two calls that afternoon

Sunday is the day the two accounts touch. OpenAI's page says its own work finished and that it then reached out; Buckmaster's statement says he was asked at short notice to get on a call, and sets out what he was told on it. From the point the calls begin, only one document is speaking.

OpenAI's page dates the end of its work to this day. After completing the project and the Lean verification on 6 September, the page says, it reached out "to offer a concurrent release of our result and to recognize their priority in a joint announcement."

Buckmaster's statement describes the same approach from the receiving end. At 12:45 on Sunday 6 September he was asked whether he could meet "at any point today." Sebastien Bubeck joined. The three of them spoke twice that afternoon, and Alpöge was not on the calls.

Everything from here to the end of this section rests on one document. OpenAI's page does not describe the two calls, and we did not read an account from anyone else who was on them. Buckmaster's statement is the only source here and should be read as one.

He says he was told an internal OpenAI model had produced a proof of finite time blowup for the forced Navier-Stokes equations. When Alpöge asked by text for the precise statement, the answer he records is "Existence of forced blowup in R3 and T3," with "the forcing function is smooth option c and d in Fefferman." He was told the proof is about 100 pages, and writes: "I have not seen it."

The statement explains why the word forced mattered to him:

The route to the Clay problem through a smooth force, options c and d in Fefferman's statement of the problem, is the route Luis and Diego opened and the one Levent and I had quietly chosen to attack. Almost nobody else I know of was working on it. It is not the direction one arrives at in a few days by giving a model the problem statement. When I heard "forced," it was a bright red flag.

He says he was shown a prompt and told the internal research model had simply been given the problem statement, and that Alpöge had been told by Bubeck that "very little human input" had been used. His statement then says: "This turned out not to be true." What he says came out over the course of the call:

Over the course of the call, as members of their team sent Sebastien corrections and details over their internal chat, it emerged that an entire team had been working on the problem, that this was one of a number of things that was tried, that work had started on the unforced problem, that the team first set the model on easier problems, including Euler, that even the prompt that had been shown to me had been written by prompting Codex, and that an insane amount of compute had been used.

The two accounts put the same events in the same order. Buckmaster's statement says he asked when OpenAI's first prompt had been sent, that the question "was not answered directly by OpenAI for some time," and that "Eventually it was agreed that it had been sent in the past few days, after information about our work had reached OpenAI." OpenAI's page says the same thing from the other end: the effort began on 1 September, after the rumour.

Three blowup papers, and the three accounts published the same day

Buckmaster and Alpöge put out their mathematics before either of them said anything about OpenAI, and what they put out is not the Navier-Stokes result OpenAI announced. They released three finite-time blowup results with smooth forcing and held a fourth back, and the same day carried OpenAI's page, Buckmaster's four-page statement and Alpöge's post. Which papers exist and which do not is what makes the rest of the week readable.

Buckmaster's statement opens by saying that he and Alpöge have "made public three results: finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3d incompressible Euler." It adds that they believe they also have blowup for hypo-dissipative Navier-Stokes, that they are not releasing that paper because "the Lean verification has not yet finished," and that he mentions it "because it is suggestive of a path to unforced Euler."

He is unhappy with the papers as released and says so before explaining why. The Boussinesq and Euler writeups, the statement says, are "much closer to what models produce under human direction than to a paper written by a person." Of the Euler paper in particular:

The Euler writeup, in particular, can only be described as AI slop. I am sorry for this. The reasons are below, and they involve our being pressured by outside factors.

He adds that he says this "not as an excuse but as an explanation."

Where OpenAI's page and Buckmaster's statement diverge

There are three places where the accounts do not line up, and the rest of the record is consistent. In two of them, OpenAI's page is silent about something Buckmaster's statement describes at length. In the third, both documents speak, and they do not say the same thing.

The data question: Codex sessions and what the model was trained on

This is the one point where all three documents speak. OpenAI's page draws a line between accessing user data and training on it. Buckmaster says he asked about both and got an answer to only one of them, and Alpöge's post is a direct reply to OpenAI's sentence about training.

OpenAI's page addresses it in two consecutive sentences, in this order:

We (the researchers and the agents) did not see any of their work through any means until they released it publicly — in particular, no specific user data was accessed in order to solve this problem. While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models.

Buckmaster's statement gives his own account of asking:

I asked whether the model had been trained on, or had access to, our sessions in Codex, into which we had been putting all our drafts for the whole of this project. I was told the model did not look up user data. I asked again, about training, and I did not get an answer.

Alpöge quotes the second of OpenAI's two sentences back, "we cannot rule out that de-identified data derived from their usage of our products helped improve our models," and writes: "i mean props to them for straight coming clean." The lowercase is his. His post then adds, in parentheses, "(so far the proof looks more along the lines of another euler blowup proof we had, off of whose ansatz naming we were making really stupid puns like 'smooth criminale', unlike the much better 'ideal fluids explode', Tristan)". The post does not say what he had seen, how much of it, or when.

The authorship dispute: two proposals, and Bubeck on Alpöge's name

This is the part of Buckmaster's statement that OpenAI's page does not touch at any point. The statement sets out two proposals, names the person it says made a request about authorship, and records that Buckmaster declined. It is also the passage the headline of this item rests on, so it is quoted rather than summarised.

Two proposals were offered to me. The first was that we post our Euler result, and that OpenAI post its Navier-Stokes result the next day. The second was that, after posting Euler, I alone write a paper presenting the Navier-Stokes result, acknowledging that an internal OpenAI model had resolved it. Sebastien twice asserted that he wanted Levent removed from authorship, and said it would all be simple if only it were not the case that, and it was so annoying that, Levent works at Anthropic.

The statement adds that "It was also said that if OpenAI posted after us, they would say that we deserved the Clay Prize, and that we were the 'closest humans to the problem'," and that he declined both offers.

Alpöge's post describes hearing something similar from a different position. He writes that he "woulda been pumped to collaborate on this," that there are "a lot of people at oai i like," and that he does not care about authorship on that step. Then: "but on hearing the loud convo in the hallway, especially the part where a millennium prize was offered if i'd just be removed from the paper, it was kinda clear the die had been cast and things were locked." His post does not say who said it. He describes his own side of the work as "mostly me and claude having a good time yoloing random stuff in the corner rather than anything institutional," and calls the outcome "pretty wacky, unstrategic, and unnecessary."

OpenAI's page describes the same outreach without any of this. It says the offer was "a concurrent release of our result and to recognize their priority in a joint announcement," that "In these discussions we offered them visibility into all of the prompts we used and later to see the proof," and that "We recognize the priority of their work on forced Euler and congratulate them on their remarkable mathematical achievement." The page says nothing about authorship and does not name Sebastien Bubeck. That absence was checked by reading the whole page rather than searching it.

Two quoted sentences, and the speaker the statement does not name

Two short replies in Buckmaster's statement carry more weight than anything else in it, and they are the least attributed lines in the document. The statement gives both as "The reply" and names no speaker in that passage. That limit is worth stating before the passage is read, not after.

I said that if OpenAI released its result in the way proposed I would go public with what happened. The reply was, "Why would you ruin your career?" I replied that I am an academic, and asked why he thought going public would ruin my career. The reply was, "If you don't want me to be nice, then I don't have to be nice."

By the statement's own account two people from OpenAI were on the calls: the mathematician Buckmaster emailed first, whom he does not name, and Bubeck. OpenAI's page does not address either sentence.

The statement closes the sequence there. It says that some time later Alpöge received a text proposing that he and Bubeck speak one on one, saying "I don't know if Tristan is being fully rational right now," that Alpöge declined and said conversations should be with Buckmaster, that Bubeck sent a follow-up email that night requesting to speak on Monday 7 September, and that Buckmaster did not respond.

Who claims the Millennium Prize, and what each document says about it

The prize appears in all three documents, in three different registers. OpenAI's page renounces it in a single sentence. Buckmaster records what he says he was told OpenAI would say about who deserved it, and Alpöge describes hearing a prize mentioned in a hallway without saying who mentioned it.

OpenAI's page says: "We do not intend to claim the Millennium Prize for this result." That sits alongside its claim that the result "resolves the Navier-Stokes Millennium Prize problem by establishing statement 'C' (and also 'D') in the official Millennium Prize formulation."

Buckmaster's statement records the prize as something raised during the calls: if OpenAI posted after them, "they would say that we deserved the Clay Prize, and that we were the 'closest humans to the problem'." He declined the proposals that came with it.

Alpöge's account is the only source among these documents for a prize being tied to his name coming off a paper, and it is second-hand within his own post. He places it in a loud hallway conversation he overheard, and does not say who was speaking or who they were speaking to. Nothing in OpenAI's page or in Buckmaster's statement describes such an offer.

What Buckmaster says he is not claiming

The statement puts an explicit limit on itself in a paragraph near the end. For a reader deciding how much weight the rest of it carries, this is the paragraph that sets the terms, so it is reproduced in full rather than paraphrased.

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. I am stating it because the alternative is to let a sequence of announcements say something I know to be false.

The paragraph after it reads: "If indeed an OpenAI model did close the gap to Navier-Stokes, that is a remarkable thing and it should be said loudly, by them, with the history intact."

That is where the record stands on 8 September 2026. None of the three documents contains a reading of OpenAI's proof by anyone outside OpenAI. Buckmaster says he has not seen it. We have not seen it either.

The model that resolved it is not the GPT-6 Astra anyone can buy

Two different models appear in OpenAI's account, and running them together is the easiest mistake to make with this announcement. One is the internal model the page credits with the resolution, which it does not price, does not offer and describes only in its own words. The other is GPT-6 Astra, the released model, which the page says did the Lean verification.

OpenAI's page says: "To solve the Navier-Stokes problem, we used an internal model that is significantly more capable than GPT-6 Astra." It then says that the Lean formalization and verification of the result ran "via GPT-6 Astra."

Astra is the released model our GPT-6 Astra review covers, at published prices and against published outside evaluations. The internal model described in this announcement is not that model. Buckmaster's statement puts Astra on his side of the story as well: it is listed among the tools he and Alpöge used, and the statement says it "was only used for writeups and auditing our arguments."

America's AI Action Plan, and the line Buckmaster ends on

One more document belongs beside this one, and it is not about mathematics. The administration's own AI Action Plan says what the competition between American labs is understood to be for, in published words that can be checked. Buckmaster's statement ends somewhere else entirely, and the two are set here without commentary.

America's AI Action Plan, dated July 2025, opens on page 1:

The United States is in a race to achieve global dominance in artificial intelligence (AI). Whoever has the largest AI ecosystem will set global AI standards and reap broad economic and military benefits. Just like we won the space race, it is imperative that the United States and its allies win this race.

Buckmaster's four-page statement says he had planned to announce the work by saying that the results were not the important thing, and that what mattered was "the significance that a mathematician and an LLM model can now do all this work in a month." He calls that a "Deep Blue-Kasparov moment" and writes that the community "needs to have serious and unhurried discussion about where to go from here." Near the end of the statement, this line:

I would much rather be talking about mathematics, Luis and Diego's ideas, and what this all means for the rest of us.

What we read on 8 September, and what we did not

This item rests on four documents, all first-party, and on nothing else. The list below says which ones and how each was read. A reader who wants to check any sentence above can go to the same place we did.

We read three accounts of this dispute on 8 September 2026: Buckmaster's four-page statement as published on his own site, the passages of OpenAI's announcement that bear on the dispute, and Alpöge's post. OpenAI's site refuses automated retrieval, so its page was read in a browser and the passages above were transcribed from that read. America's AI Action Plan is a fourth document and not an account of this dispute. It was read the same day as a PDF published at whitehouse.gov, and the passage quoted is the opening of its introduction, on page 1.

We did not read OpenAI's proof paper or its Lean formalization, the three preprints and formalizations Buckmaster and Alpöge released, any statement published by Sebastien Bubeck himself, or any statement from Anthropic. A sentence attributed to Bubeck has been quoted in secondary coverage; because we have not read it at a source he published, it is not quoted here. We also did not read the Córdoba and Martínez-Zoroa papers the program builds on.

If any of the parties publishes further, this item will be updated and the change dated.