Navier-Stokes by AI: Why a Mathematician Is Leveling Serious Charges at OpenAI

On September 8, OpenAI announced that it had solved one of the hardest problems in mathematics: the existence and smoothness question for the Navier-Stokes equations, one of the seven Millennium Prize Problems of the Clay Mathematics Institute, each carrying a one-million-dollar award. Within hours, the announcement was overshadowed by a public accusation. New York mathematician Tristan Buckmaster says the company pressured him and tried to make a co-author unwelcome because that co-author works at Anthropic. At its core, the dispute is about authorship, priority in time, and the question of what happens to the drafts researchers upload into a Codex session.

Key takeaways

• OpenAI reports a roughly 100-page proof of the Navier-Stokes problem, produced by an unreleased internal model that coordinated a swarm system of up to 10,000 sub-agents.
• The company says it will not claim the one-million-dollar prize and concedes that it only took the work up in earnest after rumors of a competitor’s success.
• NYU mathematician Tristan Buckmaster and Anthropic employee Levent Alpöge had previously worked on the same question with several AI models and stored drafts in OpenAI’s Codex.
• Buckmaster describes two options he was offered and statements by OpenAI researcher Sébastien Bubeck that he reads as a threat.
• OpenAI denies any access to the outside work but cannot rule out that de-identified usage data fed into its models. The proof has not yet been independently checked.

What OpenAI announced

The Navier-Stokes equations describe how liquids and gases move. They sit inside every weather forecast and every wing calculation, yet mathematically it remains unresolved whether their solutions stay smooth for all time or can become singular in finite time, that is, blow up toward infinity. That is exactly the question OpenAI says it addressed: an internal, still unreleased model steered a multi-agent system with up to 10,000 parallel sub-agents and showed that there are conditions under which the equations blow up. The route via a smooth external force, called smooth force in the original, is considered unusual and is rarely pursued.

OpenAI puts the compute effort at at least a thousand times that of previous mathematical AI projects, which by some estimates means costs of around two million dollars, against a few thousand dollars for comparable earlier work. The company explicitly says it will not claim the Clay Institute prize. One admission in the announcement itself stands out: the project was only begun in earnest in the final days of August, after rumors of an imminent success by a team involving Anthropic had made the rounds. The move fits OpenAI’s stated ambition to automate research itself, as seen recently in the presentation of an automated research intern.

The mathematician’s allegations

Buckmaster, a professor at New York University, has laid out his account in a public document. According to it, he and Levent Alpöge, a mathematician at Anthropic, had worked since mid-August with several language models on a nearly identical solution, among them Claude and OpenAI’s GPT-5.6 Sol, and had continuously stored their drafts in Codex sessions. In early September, as rumors about the progress spread, he says he reached out to OpenAI on his own initiative.

In the conversations, he says he was offered two ways to preserve his priority: either a partial publication, after which OpenAI would publish its own solution the next day; or a sole-author paper that credits the use of an OpenAI model but leaves out Alpöge’s name. He rejected both offers. The fact that Alpöge works at Anthropic was, he says, described to him as so annoying. When he refused to drop the co-author, OpenAI researcher Sébastien Bubeck allegedly asked, “Why would you ruin your career?” and later said, “If you don’t want me to be nice, then I don’t have to be nice.” At the same time, Buckmaster stresses that he is expressly not accusing OpenAI of directly accessing his data; he only wants to put the facts on record to counter false narratives about the origin and order of the work. When he asked whether the model could access the Codex sessions or was trained on them, he says he received no clear answer on the training question.

OpenAI’s version and the open data question

OpenAI rejects the charge that it saw outside work: “We (the researchers and the agents) did not see any of their work through any means until they released it publicly,” a statement reads. In the same text, however, the company concedes that it cannot rule out that de-identified data from product usage contributed to improving the models. It says it acknowledges Buckmaster’s priority in time.

Bubeck himself called the circulating allegations “false and inflammatory” on the platform X and denied demanding the removal of Alpöge’s name. He says he did use the phrasing about ruining a career, regrets the wording, and retracted it immediately; he announced a detailed response. OpenAI chief Sam Altman also weighed in with his version. An independent mathematical check of the roughly hundred-page proof is still pending, unlike a machine-verified result of the kind recently discussed around a proof of Fermat’s Last Theorem.

Perspective: what is at stake

The case bundles three conflicts that reach beyond this one proof. First, the confidentiality of commercial AI tools: anyone who uploads unpublished research into the cloud session of a provider that also does its own research and analyzes usage data enters a dependency that academic work has so far had no equivalent for. Second, the competition between labs: the fact that OpenAI, by its own account, takes on a decades-old problem within days once a competitor is close to a solution shifts mathematics from patient foundational work to a resource race.

Third, the question of what such a proof is worth. Fields Medalist Terence Tao criticizes that AI systems deliver answers without insight: they disclose neither their lines of reasoning nor the discarded dead ends from which the next generation of mathematical methods would otherwise develop. Tao warns of an indiscriminate strip-mining of open problems that destroys the ecosystem in which new techniques arise. If the proof holds up, it remains a milestone for machine reasoning. Yet the dispute already shows that the rules for authorship, data protection, and fairness in the interplay of academic and industrial AI research have not yet been written, and that they are urgently needed.