Did OpenAI Copy Buckmaster's Navier-Stokes Proof?
A viral reading says the OpenAI Navier-Stokes plagiarism row was a misunderstanding. The dated record backs half of it, and the proof itself is still unconfirmed.
Elena Marsh covers science policy for Stanford Tech Review, with a focus on federal research funding, biosecurity oversight, and how Washington governs the labs it pays for.
Six weeks after OpenAI said it had found a finite-time singularity in the three-dimensional Navier-Stokes equations, the popular verdict has quietly flipped. The story that spread in September was that OpenAI had lifted an unpublished idea from the mathematician it was racing. The story spreading now is that the whole thing was a misunderstanding. The dated record backs about half of that, and the half it does not back is the half everyone is repeating.
The primary document in the dispute is still the post Sébastien Bubeck, who leads OpenAI's math effort, published on the afternoon of September 8. It ran to four numbered points, drew 3.8 million views, and is the only place either side set out a full account of what was said in the call that started everything.
I would like to clarify a few things: 1) The screenshot is my reaching out to Levent to coordinate our releases. I hope it's clear from the message that we came in with the best possible intentions. 2) I never ever asked for Levent to be removed from authorship of his own work… I deeply apologize for this extremely poor choice of words, it is the opposite of what I was trying to convey. (I should say that I retracted them on the spot by the way.)
— Sebastien Bubeck (@SebastienBubeck) September 8, 2026
Read closely, that post concedes one thing and denies two. It concedes the phrasing — Bubeck apologized for telling Tristan Buckmaster he did not understand why anyone would risk their career, called it an extremely poor choice of words, and said he retracted it during the call itself. It denies asking for Levent Alpöge to be dropped from his own work, and it denies that anything was taken. What it adds, and what almost none of the secondhand summaries carry, is Bubeck's account of why Alpöge's name came up at all: he says he was surprised to learn on the call that the pair had solved Euler rather than Navier-Stokes, that one option discussed was Buckmaster leading a rewrite of OpenAI's Navier-Stokes proof, and that it was in that narrow context he said it would be simpler if Alpöge were not an Anthropic employee.
What the record actually says
The underlying events are tightly dated, which makes them checkable. OpenAI started its run on September 1 after what Bubeck describes as viral rumors that Anthropic had resolved two Millennium Prize Problems. Buckmaster, a mathematician at New York University, went public late on September 7 with blow-up results for Euler, IPM and Boussinesq obtained with Alpöge, a researcher affiliated with Anthropic, on August 15 and verified in Lean about a week later. OpenAI announced its own result at noon the following day: 10,000 agents, 2.7 million messages, 130 billion output tokens, 88 hours of wall clock, a 166-page paper and a Lean formalization.
Then came the part that produced the plagiarism story.
| Date (2026) | OpenAI's position on whether Buckmaster's data touched the model |
|---|---|
| Sep 8, morning | "We cannot rule out that de-identified data derived from their usage… helped improve our models." |
| Sep 9, evening | A spokesperson calls it "categorically impossible" for Buckmaster's Codex prompts to have influenced the system. |
| Sep 10, late evening | Statement updated: the prompts "could not have influenced the system in any way." |
| Sep 13 | The company dates the cutoff, saying no user input after July 3 could have reached the system. |
That is four public positions in five days. OpenAI changed its account of the training-data question three times in the 120 hours after the announcement, and the version that went viral 36 days later is the one it abandoned first. The denominator is the four dated statements above, counted from the September 8 noon announcement; the method is nothing more than reading them in order.
The shape of that chart is the story. Everything contested happened inside a five-day window; everything that resolved it arrived weeks later, to an audience that had already stopped reading.
Where the popular version breaks
Three claims now circulating as settled deserve separating.
"OpenAI admitted plagiarism." It did not. The September 8 line was a hedge about de-identified usage data improving models in general, which is a true description of how the products work and a very long way from conceding that a competitor's unpublished idea was lifted. Within roughly 30 hours it was retracted and replaced with a flat denial. A hedge withdrawn the next day is a communications failure, not a confession.
"OpenAI could not understand its own model's proof." Nothing in the public record supports this. What is on the record is Bubeck arguing from day two that the two proofs were constructed differently, which is not the behavior of a company that cannot read its own output. He did apologize, but for the career remark specifically, while disputing Buckmaster's wider account as a series of false and inflammatory allegations.
"It was proved the two approaches are completely different." This is OpenAI's position, and it is the strongest of the three, but it is a position rather than a finding. Both efforts worked inside the programme pioneered by Diego Córdoba and Luis Martínez-Zoroa. Where they separate is in what was actually proved.
| Buckmaster and Alpöge | OpenAI | |
|---|---|---|
| Euler result | Blow-up with external forcing | Blow-up without external forcing |
| Navier-Stokes | Unverified proof of an easier variant | Finite-time blow-up with a smooth, compactly supported force |
| Verification route | Lean, posted before the announcement | Lean, released with a 166-page paper |
| Priority as of Sep | Conceded to them by OpenAI for 3D Euler | Claimed by OpenAI for Navier-Stokes |
That forcing distinction is not a technicality, and it is the single thing most of the commentary drops. The Millennium Prize question asks whether a three-dimensional fluid, left alone with smooth initial data and no outside push, can run away to infinite velocity in finite time — a singularity, or "blow-up". Add an external force to the equations and you hand the construction a lever: you are allowed to engineer the push that drives the solution to explode, which makes the result easier to reach and weaker as an answer to the prize question. OpenAI's Navier-Stokes claim is of the forced kind, with a smooth, compactly supported force. Its Euler claim is the unforced one, which is why Bubeck leaned on it when arguing the two efforts had diverged, and why Buckmaster and Alpöge's forced Euler result sits on the other side of the same line. Two groups can share a programme, share a vocabulary, and still be proving statements that are not interchangeable.
Alpöge's own reading is less tidy than the exoneration now circulating. He has said the OpenAI proof "looks more along the lines of another Euler blowup proof we had" — which is a claim of resemblance, not of theft, and it is not what "completely different" means.
The replies asked the one question that mattered
The thread under Bubeck's post was hostile, and most of it was about the career line rather than the mathematics. "Which part are you disappointed with? The one where you threatened to end his career?" ran one of the most-liked replies; another noted that threatening someone's career is famously known to be part of top-tier academic standards. That is the internet doing what it does with an apology attached to a denial.
One reply did better, and it is the one the next five days turned on. "Look in the training data and confirm or deny that Buckmaster's data is or isn't in there," wrote @QRDL. "Until you do that, it's very very hard to trust anything OAI says or does." That is precisely the question OpenAI then answered four different ways between September 8 and September 13 — and the fact that it took four attempts, rather than one audit result on day one, is why the suspicion outlived the denial. A company that can account for 130 billion output tokens can account for its training cutoff on the first try.
The complication nobody is posting about
Eight days before the exoneration narrative took hold, three researchers put a different problem on the table. In Navier-Stokes lost in translation, Alexander Bastounis, Fabian Circelli and Anders C. Hansen argue that OpenAI's Lean-verified formalization does not correspond to the natural-language proof it was supposed to formalize — and, more broadly, that semantically faithful autoformalization is harder than the halting problem. The Clay Mathematics Institute, which has to make the call, said on September 11 only that the problem had "apparently been settled" and moved its status from unsolved to active, with verification "deliberately unhurried."
So the honest summary is narrower than either side's headline. The plagiarism charge has not held up. The proof has not yet been confirmed. Those are two different questions, and the version going around answers the first while reading as though it answered both.
Which is the thing worth taking away. The clearing of OpenAI's name was always going to travel on a delay, because an allegation is an event and a retraction is an absence; whoever writes the first statement in an AI priority dispute writes the version that survives. The mathematics, meanwhile, is still being checked — and that part has no deadline at all. For the wider argument about what attribution even means when a model does the work, see our earlier piece on Terence Tao, OpenAI, and who owns mathematics.
Further reading: the day-one account of the announcement and the credit fight in Quanta, what the controversy says about the future of mathematics in MIT Technology Review, and the consolidated timeline of the priority controversy.
Cover image: NASA wake vortex study, public domain, via Wikimedia Commons.