OpenAI solves 90-year-old math problem and reveals the leak that matters

JEAN LERAY proved in 1934 that the Navier–Stokes equations always have solutions, provided one is willing to accept what mathematicians call the weak kind. On September 8th OpenAI produced something similar for the question of who solved them. The company announced that an internal model, run as a swarm of roughly 10,000 agents for 88 hours, had shown that a smooth fluid at rest can be driven to infinite speed in finite time. That resolves one of the seven Millennium Prize problems the Clay Mathematics Institute posed in 2000, a bounty of $1m that OpenAI says it will not collect.

The proof arrived with a rival account stapled to it. Tristan Buckmaster, a mathematician at NYU's Courant Institute, and Levent Alpöge, a mathematician employed by Anthropic, had spent the better part of a year pushing a program devised by Diego Córdoba and Luis Martínez-Zoroa, two Madrid-based analysts, toward the same destination. On August 15th, with heavy help from Anthropic's Claude and OpenAI's Codex, they obtained blowup for the Euler equations, the frictionless cousin of Navier–Stokes, and verified it in Lean a week later. OpenAI's own account says its effort began on September 1st, after "hearing a rumor," and finished on the 5th. The whole story sits between those two dates.

Buckmaster's public statement asks whether OpenAI's model was trained on his Codex sessions, into which the pair had fed every draft for a year. OpenAI says no specific user data was touched, and that it cannot rule out de-identified data having improved its models. The question deserves an answer, but it settles less than it seems. OpenAI did not need the drafts. It needed a rumor, a research direction, and roughly 1,000 times the compute it had spent on its previous mathematical results. The lab that sells the tool can outrun anyone using it, and that holds whether or not it has read a word of their work.

Satya Nadella, Microsoft's boss, made a version of this argument in July, calling it a "reverse information paradox": customers pay for AI twice, once in money and again in the exhaust of prompts and corrections the model learns from. Alex Karp, Palantir's boss, calls the second payment a firm's "alpha." Their version is about data. The Navier–Stokes version is harder on the customer, because Buckmaster's own email to OpenAI on September 3rd shows the terms of trade. He wrote that he paid for the tools from his own research funds, "footing a large bill to OpenAI." The reply, sent the same day, said any details he could share "would be useful to avoid competing here." Read generously, that is an offer not to duplicate his effort. Read as Buckmaster did, the vendor had put the word on the table before the customer had.

Weak solutions

True, the two proofs differ, and OpenAI has been careful to say so. Its Euler result covers the unforced case, Buckmaster's the forced; its Navier–Stokes argument runs by a route the company says it offered to show the pair prompt by prompt; and it has recognized their priority on Euler in writing. Buckmaster himself ran his program partly on OpenAI's models, which is some evidence that a swarm given the smooth-forcing question could find the smooth-forcing answer without a leak. The mathematics, moreover, is real: a Lean-verified singularity is more than the field produced in ninety years.

True, too, priority races predate the transformer. Elisha Gray and Alexander Graham Bell reached the patent office on the same day in February 1876, and historians still argue about the order. Newton and Leibniz spent decades on the calculus. Science has always paid the fast, and a lab that heard a rumor on a Tuesday and had a proof by Saturday is, on this account, merely faster.

Yet speed on this scale changes what counts as a leak. OpenAI's week-long sweep of the open problems used about 300bn output tokens, which TechCrunch prices at $22.5m at list rates; Navier–Stokes alone took 130bn of them, and the company itself puts the bill in the "millions." At the lowest estimate on the table, the compute cost more than the $1m prize OpenAI declined; at the highest, it cost twenty of them. When one side can spend that on hearsay, the hearsay is the leak. Buckmaster's rumor traveled, by his account, from a Courant colleague to an analyst in Britain to "somewhere further upstream," and no data-use policy, enterprise tier or training opt-out closes a channel that runs through a faculty lounge. The thing customers have spent a year negotiating over, training on prompts, turns out to be the smaller exposure.

Terence Tao, a mathematician at UCLA, called the Alpöge–Buckmaster result "a milestone advance in human knowledge" on his blog on September 7th, having endorsed a warning the week before from Hugo Duminil-Copin, a Fields medallist, against the "strip-mining" of open problems by automated solvers: answers collected without insight, dead ends left unpublished. For mathematics the loss is the ecosystem of technique that grows around a problem while it stays open. For anyone using a frontier model on unpublished work, the loss is narrower and more personal: the option value of being first, which was the only asset Buckmaster had. He called the month a Deep Blue–Kasparov moment. Kasparov, at least, was allowed to finish the match.

What OpenAI proposed on the September 6th calls shows how a lab prices credit. Either the pair posted Euler and OpenAI posted Navier–Stokes the next day, with a note that the humans were "closest" to the problem, or Buckmaster alone wrote up the full result, acknowledging OpenAI's model, with Alpöge's name removed because he works for Anthropic. Sébastien Bubeck, the OpenAI researcher running the project, twice asked for the removal, according to Buckmaster, and when the professor said he would go public, asked why he would "ruin your career." Bubeck denies the characterization and says he will say more. On Buckmaster's account, an academic's byline was on the table as a bargaining chip, which tells one what the theorem is to the lab: a benchmark result with a press cycle. Declining the Clay money, whose rules require publication and a two-year wait for general acceptance, is the same tell. OpenAI wanted the week, not the wait.

Leray's weak solutions turned out to be the right idea, and the field spent nine decades trying to strengthen them. OpenAI's claim to have solved the problem is the strong kind. Its claim to have done so on its own is the weak kind, and it will need more than a swarm to make it hold. Buckmaster wrote that he would much rather be talking about mathematics. He is not, and that too was decided by the side with the compute.