OpenAI’s Proof Is Under Evaluation. Its Conduct Is Already on Record.

On September 6, NYU mathematician Tristan Buckmaster got on a call with OpenAI researchers and learned the lab was about to announce a solution to the Navier-Stokes problem using the same approach he and Levent Alpöge had spent most of a year developing. Then, according to Buckmaster’s four-page statement published the following day, OpenAI researcher Sebastien Bubeck offered him a choice: participate in a joint announcement — but only if Alpöge’s name was removed from the paper. The stated reason, Buckmaster wrote, was Alpöge’s employer. Levent Alpöge works for Anthropic.

Buckmaster refused. According to his statement, Bubeck then said: “Why would you ruin your career?” And later: “If you don’t want me to be nice, then I don’t have to be nice.”

OpenAI disputes the characterization. Bubeck, in a statement to Scientific American, said the company “did not use their prompts or proofs to prompt our models or direct our agents.” That denial is about technical use of their work. It is not a denial that the conversation happened, that multiple publication options were discussed, or that Alpöge’s Anthropic affiliation was offered as a reason to remove his name. OpenAI has not addressed those specifics in any public statement.

This is the part of the Navier-Stokes story that doesn’t depend on whether the proof holds.

The Race That Wasn’t Supposed to Be One

When OpenAI published its 166-page manuscript on September 8 — produced by a 10,000-agent system running for 88 hours, at a compute cost Fortune reported as somewhere between $2 million and $22.5 million — the company framed it as a landmark in AI capability. What that framing buried was also in the Fortune report: OpenAI, by its own admission, only entered the competition after rumors circulated about Anthropic’s impending announcement. The 10,000 agents ran because a rival was close.

Buckmaster and Alpöge had been building toward this for most of a year. They used both Claude and GPT-5.6 Sol as collaborative instruments for human-directed mathematical research — AI tools from both Anthropic and OpenAI. Alpöge held a position as a mathematician at Anthropic. Buckmaster was at NYU’s Courant Institute and had been the one interfacing directly with OpenAI’s platform, including pushing drafts into Codex. By mid-August the pace of their work had accelerated sharply. When OpenAI’s researchers first reached out on September 3, Buckmaster asked twice on the September 6 call whether his Codex sessions had contributed to training the model that had now apparently found the same result. He says he did not get an answer on the training question.

OpenAI’s response to that specific concern has been narrow: the company said only that “de-identified platform usage could have helped train its models.” That is not a denial. It is a hedge that covers the exact mechanism Buckmaster flagged.

None of this is hypothetical. The Leiden Declaration on Artificial Intelligence and Mathematics, published June 2 by sixteen mathematicians — now carrying 3,937 signatures, including Terence Tao, Peter Scholze, and IMU President-Elect Robbert Dijkgraaf, with endorsement from the International Mathematical Union — had named this dynamic five months ago. Its fourth identified threat was publication integrity: “Results announced via press releases bypass peer review, oversimplifying significance and misrepresenting AI contributions.” OpenAI announced September 8 via blog post and press briefing, with no peer review and a manuscript the mathematical community has not yet evaluated. The mechanism the declaration described ran exactly as predicted.

What Tao’s September 7 Post Actually Measured

The day before OpenAI’s announcement, Terence Tao published a blog post about the work of Alpöge, Buckmaster, Córdoba, and Martínez-Zoroa — the mathematicians whose approach to finite-time blowup results in forced fluid equations Tao had been tracking. He noted that their work was “heavily AI-assisted” and that their proofs had been formalized in Lean, making the mathematical argument machine-checkable. Then he wrote the sentence that defines what the Navier-Stokes moment actually represents to mathematicians: “the actual solving of these problems is only a proxy goal for the primary goal of developing mathematical understanding and insight.”

What Tao values about the Alpöge/Buckmaster approach is that it was isolatable — the techniques could be extracted, understood, applied elsewhere, taught to the next mathematician who needs them. That is the accumulated knowledge a mathematical research culture produces. Tao’s stated position on AI in mathematics is explicit about what the competitive race destroys: “The indiscriminate strip-mining of open problems for solutions may destroy the ecosystem from which the next generation of mathematical techniques, problems, and practitioners would have developed.” He compared it to archaeological looting — the artifact obtained, the context destroyed. He also disclosed that OpenAI had recorded an hour-long interview with him in March 2026 at an IPAM workshop and used clips in advertising; he said he “should have pushed back harder” on that use of his footage. The extraction dynamic he describes in mathematics is not metaphorical. It has already happened to him directly.

On September 11, twenty-five Fields Medal winners signed an open letter warning that AI labs are threatening mathematical research culture. The letter said AI companies announce solutions “in a rush, leaving no time for a proper writeup, the isolation of new methods and ideas, and citing relevant previous work.” That sentence describes a structure, not a conspiracy. It describes what happens when a competitive announcement cycle hits a research community built around deliberate verification. OpenAI withdrew its sponsorship of a CalTech math event after criticism from CalTech researchers the same week — a minor retreat, but one that confirms the pressure the announcement created in the institutional relationships OpenAI had been cultivating.

What the Clay Math Institute’s Silence Means

The Clay Mathematics Institute issued its statement September 11. It said “the Navier-Stokes problem has apparently been settled” and continued: “The rules governing the prizes describe the process for evaluating what has been achieved and for assigning credit. The process is deliberately unhurried, but we will provide updates.” The CMI names no proof authors in the statement. Names no methods. Names no company. It hopes to “see waves of new human understanding unleashed as the innovations behind this work are analysed.”

“Deliberately unhurried” is the Millennium Prize standard in miniature. A proof wins the prize when the result is “generally accepted” by the mathematics community — a threshold that requires published, peer-reviewed work and sustained scrutiny. The fields community’s response this week — twenty-five Fields Medalists, 3,937 signatures on the Leiden Declaration, the CMI refusing to name anyone in its statement — is not the “generally accepted” standard. It is something closer to the opposite. Whatever OpenAI’s 166-page manuscript eventually turns out to contain, it is not yet past the evaluation the math community uses to distinguish a real proof from an impressive computation.

It is worth naming what OpenAI’s proof might genuinely be: an independent discovery of the same mathematical approach, reached by a different path, without exposure to Buckmaster’s work. Convergent discovery in mathematics is real. Two groups working similar ideas can arrive at similar results. OpenAI has denied using Buckmaster and Alpöge’s work directly, and that denial may be true. The CMI’s evaluation will eventually speak to whether the proof is original, whether it is correct, and how credit gets assigned under the formal prize rules. That process will take time.

What happened on September 6 does not need that process. OpenAI has since shifted to recognizing “the priority of Levent Alpöge and Tristan Buckmaster’s work” in press briefings — a phrase Bubeck offered at a public event after calling Buckmaster’s initial allegations “false and inflammatory.” The shift in language from “false and inflammatory” to “we recognize their priority” does not resolve what Buckmaster documented. He says he was presented with a publication option contingent on removing his co-author’s name because of that co-author’s employer. He says he was asked why he would ruin his career. He published his statement. That record exists independent of what OpenAI’s proof eventually turns out to be worth.

The Navier-Stokes evaluation will take months. Buckmaster’s four-page statement took one day. Tao’s blog post was published before OpenAI made its announcement. The Leiden Declaration was drafted in September 2025, published in June, three months before any of this. The mathematicians had already described exactly what was coming, in specific terms, with an institutional endorsement from the body that governs their profession. Then it came. The conversation on September 6 already happened. The proof is what the CMI is evaluating. The call is already on record.

AI-generated editorial illustration · TemperatureZero · September 12, 2026