OpenAI and Navier-Stokes: mathematical feat or method theft?

If I have seen further, it is by standing on the shoulders of giants. This Tuesday, OpenAI made a resounding announcement: an internal model reputed to be even more robust than GPT-6 Astra would have demonstrated a 90-year-old mathematical property on the equations that govern the movement of fluids. Enough, in theory, to win a prize of 1 million dollars and ensure significant buzz with unprecedented publicity. However, it took less than 48 hours for the party to turn into scandal: a mathematician from New York University accuses the company of having siphoned off his method before it was even published. Enough to give Newton's quote above a completely different meaning?

Navier-Stokes, a $1 million math problem that has been dormant for 90 years

Let's go back to 1934. The French mathematician Jean Leray demonstrates that the Navier-Stokes equations, those which describe the movement of air around an airplane wing, a cloud that forms or the blood that circulates in your veins, have many solutions. Apparently good news.

Except that it does not prove that they remain “smooth”: nothing excludes that a fluid, while scrupulously obeying these equations, one day begins to accelerate without limit in a finite time. A point where speed becomes infinite? A physical absurdity since it is forbidden in theory that no one has managed to resolve for 90 years.

In 2000, the Clay Mathematics Institute even included this question among its seven problems of the millennium, each with a prize of one million dollars for those who could solve it. But only one has given way to date: the Poincaré conjecture, in 2003, thanks to the Russian mathematician Grigori Perelman. The other six are holding on. And no one, until this week, imagined that artificial intelligence would tackle it in a few days.

Yet this is what OpenAI claims. His proof describes a vortex formed as a whirlpool that wraps around itself and stretches, a bit like a spaghetti that twists infinitely, while accelerating without its energy becoming infinite. On paper, the long-awaited crack in the equation.

OpenAI exploit would rely on 10,000 agents and $22.5 million in computing

However, we should not imagine that AI would have solved the problem in the blink of an eye. Work began on September 1 after OpenAI heard rumors that two millennium problems had just been solved elsewhere. Sam Altman makes no secret of it: “We were curious to know if ours could also” achieve it, he slips. Around 10,000 agents work in parallel for no less than 88 hours. Result: a proof, formalized and verified in Lean (an assistant which guarantees the logical rigor of reasoning) by the Astra model, in 17 more hours.

According to OpenAI's official announcement, all of the week's attempts mobilized nearly 300 billion exit tokens. Not really a straw since it would amount to a bill of 22.5 million dollars calculated at the Astra rate, if the bill was paid in hard currency. Twenty-two times Clay's reward! An operation that is not really profitable if we omit the commercial benefits of such an announcement.

Buckmaster accuses OpenAI of having siphoned off its method

But there is a dark side to the picture. Let's go back to Sunday September 6 in the evening, two days before the announcement. Tristan Buckmaster, a mathematician at NYU, and Levent Alpöge, who works at Anthropic (but led this project independently) publish their own progress on this problem. Not Navier-Stokes directly but a cousin problem, the Euler equation in three dimensions: a partial resolution, considered more affordable, obtained thanks to a method called “smooth force”, originally developed by the mathematicians Diego Córdoba and Luis Martínez-Zoroa.

Buckmaster steps up to the plate the day after the announcement and tensions rise. According to Buckmaster's written testimony, published September 8, Buckmaster claims that OpenAI learned of his unpublished approach three days earlier and launched a team to replicate it, with computing power that neither he nor anyone else could seriously compete with. “Information about our progress has been transmitted to OpenAI,” he writes. Almost no one else was working on this track.

The exchange gets tougher with Sébastien Bubeck, OpenAI researcher on the front line on the file. Threatened with having their messages made public, he reportedly replied: “Why would you ruin your career? » And, at Buckmaster’s insistence: “If you don’t want me to be nice, I don’t have to be nice. » OpenAI even proposed to remove Alpöge's name from the publication, on the grounds that he works... for the main rival.

OpenAI denies… but doubt hangs over scientific authorship

Tuesday, at a press conference, OpenAI denies point by point. “We did not use their prompt or their evidence to question our models,” assures Bubeck. If the company recognizes (or rather, does not deny) that deidentified data from the use of its products by the two researchers could have fed its models. however, she asserts that this remains unlikely. Moreover, according to her, the two proofs differ in substance between versions of the problem with or without external forcing. Finally, clarification that the chronology also raises questions: it does indeed take time to collect, feed and train a model on data and the incident as reported by the various protagonists makes this hypothesis quite unlikely in our opinion.

In any case, to date, no peer review has yet taken place. It is therefore urgent to wait before drawing any conclusions whatsoever. And we will have to be patient: tracing the intellectual lineage of a proof produced by 10,000 agents in 88 hours has nothing to do with auditing an article signed by a single mathematician. The problem will arise with each new breakthrough signed by a machine rather than by a researcher in the flesh…