Has GPT Really Solved a 90-Year-Old Million-Dollar Math Puzzle?

If I have seen further, it is by standing on the shoulders of giants. This Tuesday, OpenAI made a shocking announcement: an internal model reputed to be even more powerful than GPT-6 Astra has purportedly demonstrated a 90-year-old mathematical property regarding the equations governing fluid motion. This could theoretically win a $1 million prize and generate significant buzz, leading to unprecedented publicity. However, it took less than 48 hours for the celebration to turn into a scandal: a mathematician from New York University accused the company of siphoning his method before it was even published. Does this give Newton's quote above a whole new meaning?
Key points of this article:

• OpenAI claimed to have demonstrated a 90-year-old mathematical property regarding the Navier-Stokes equations, potentially capable of winning a $1 million prize.
• A mathematician from New York University accused OpenAI of using his method before its publication, sparking a controversy over scientific authorship.

Navier-Stokes, a Million-Dollar Math Problem That Has Been Dormant for 90 Years {#h-navier-stokes-a-million-dollar-math-problem-that-has-been-dormant-for-90-years}

Let’s go back to 1934. The French mathematician Jean Leray demonstrated that the Navier-Stokes equations, which describe the movement of air around an airplane wing, a forming cloud, or blood circulating in your veins, indeed have solutions. A good piece of news on the surface.

Except that he did not prove that they remain <>: nothing excludes the possibility that a fluid, while strictly obeying these equations, could one day start to accelerate without limit in a finite time. A point where the speed would become infinite? A physical absurdity since it is theoretically forbidden, and no one has managed to resolve this for 90 years.

In 2000, the Clay Mathematics Institute even categorized this question among its seven Millennium Problems and offered a prize of one million dollars for anyone who could solve it. But only one has been solved to date: the Poincaré conjecture, in 2003, by the Russian mathematician Grigori Perelman. The other six remain unsolved. And until this week, no one imagined that an artificial intelligence would tackle it in just a few days.

Yet this is what OpenAI claims. Its proof describes a vortex formed like a whirlpool that wraps around itself and stretches, somewhat like spaghetti being twisted infinitely, while accelerating without its energy becoming infinite. On paper, the sought-after crack in the equation. The vortex described by OpenAI's proof: it wraps inward while stretching, without its energy becoming infinite. Source: OpenAI.

OpenAI's feat would rely on 10,000 agents and $22.5 million in computation {#h-openais-feat-would-rely-on-10000-agents-and-225-million-in-computation}

It should not be imagined that AI would solve the problem with a snap of the fingers. The project starts on September 1 after OpenAI heard rumors that two millennium problems had been solved elsewhere. Sam Altman does not hide it: << We were curious to see if ours could also >> succeed, he hints. About 10,000 agents are working in parallel for no less than 88 hours. The result: a proof, formalized and verified in Lean (an assistant that ensures the logical rigor of reasoning) by the Astra model, in an additional 17 hours.

According to OpenAI's official announcement, the entire week's attempts mobilized nearly 300 billion output tokens. Not exactly a small amount, as this would amount to a bill of $22.5 million in Astra computing fees, if the bill were to be paid in cash. Twenty-two times Clay's reward! An operation not really profitable if we ignore the commercial fallout of such an announcement.

Buckmaster Accuses OpenAI of Siphoning His Method

But there is a shadow over the picture. Let's return 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 was leading this project independently), publish their own advancement on this problem. Not Navier-Stokes directly but a related problem, the three-dimensional Euler equation: a partial resolution, deemed more approachable, obtained through a method called << smooth force >>, originally developed by mathematicians Diego Córdoba and Luis Martínez-Zoroa.

Buckmaster steps up the next day after the announcement and tensions rise. According to Buckmaster's written testimony, published on September 8, he claims that OpenAI had gotten wind of his unpublished approach three days earlier and launched a team to reproduce it, with a computing power that neither he nor anyone else could seriously compete with. << The information about our advancement was transmitted to OpenAI >>, he writes. Almost no one else was working on this track.

The exchange intensifies with Sébastien Bubeck, an OpenAI researcher on the front line of the case. Threatened with having their messages made public, he allegedly responded: << Why would you ruin your career? >> And, in response to Buckmaster's insistence: << If you don't want me to be nice, I don't have to be nice. >> OpenAI even allegedly offered to remove Alpöge's name from the publication, on the grounds that he works... at the main rival.

OpenAI Denies... But Doubts Linger Over Scientific Authorship

On Tuesday, at a press conference, OpenAI denies point by point. << We did not use their prompt or their proofs to query our models >>, assures Bubeck. While the company acknowledges (or rather, does not deny) that de-identified data from the use of its products by the two researchers may have fed its models, it claims that this remains unlikely. Furthermore, according to them, the two proofs differ in substance between versions of the problem with or without external forcing. Finally, it should be noted that the timeline also raises questions: it indeed takes 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.

Regardless, to date, no peer review has yet taken place. It is therefore urgent to wait before drawing any conclusions. And patience will be required: tracing the intellectual lineage of a proof produced by 10,000 agents in 88 hours is nothing like auditing an article signed by a single mathematician. The issue will arise again with each new breakthrough signed by a machine rather than by a flesh-and-blood researcher...

Whether the accusations are true or false, this should serve as a wake-up call for everyone: unless working locally, conversations with LLMs are, by default, not private and are always likely to serve as training data.

The history of science is full of disputes over authorship, from Newton and Leibniz on calculus to Watson and Franklin on DNA. What changes with Navier-Stokes is the speed: a method published on a Sunday, siphoned or not, caught up in 48 hours by a computing power that almost no one else possesses. Anthropic, where Alpöge works, likely understands better than anyone what it feels like to see a direct competitor scoop the prize on its own research turf. Irony of fate. The controversy arises in the midst of the race for AGI, where OpenAI has never hidden its desire to move faster than everyone else, even if it means stepping on a few shoulders along the way.

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