Let’s start from something obvious. I am not a mathematician. I am just an entrepreneur who spends his days with AI models, writing with them, building a company around them, teaching other lawyers how to use them without losing their heads. So when OpenAI announced on Tuesday that its models had solved the Navier–Stokes problem, my first reaction was the one I have every few weeks now: here we go again.
First, the problem, for those who like me never opened Landau–Lifshitz.
Navier–Stokes is a set of equations, written in the nineteenth century, that describes how fluids move: water in a pipe, air around a wing, blood in an artery. What nobody knew was whether the equations themselves could break. Take a smooth fluid at rest in three dimensions, let it evolve: does it stay regular forever, or can it produce a singularity, a point where the velocity grows without limit in a finite time? In 2000 the Clay Mathematics Institute put that question on its list of seven Millennium Prize Problems, one million dollars each. Until last week only one of the seven had fallen.
The facts, as OpenAI reports them. On September 1st the company heard rumours that two Millennium problems had been resolved and pointed an internal model, more capable than its public flagship, at all of them. About ten thousand agents ran in parallel and produced 130 billion tokens. Eighty-eight hours later they had a proof that a smooth fluid, pushed by a smooth force, can blow up while its energy stays finite. Seventeen more hours went into formalising it in Lean, where a computer checks every step. TechCrunch put the compute bill for the week at around 22 million dollars.
The idea at the centre of the proof is not new, and this is the part I find most human. Luis Martínez-Zoroa developed it in his doctoral work in Madrid, and by 2023 he and his advisor Diego Córdoba had used it on the Euler equations: a cascade of well-behaved solutions, each faster and smaller than the one before, stacked until the sum explodes. Years of work, two people, a blackboard. Charles Fefferman of Princeton, who wrote the official statement of the problem for Clay, told Quanta that the heroes of the story are Córdoba and Martínez-Zoroa. OpenAI says it will not claim the prize. It could not anyway: Clay’s rules require publication, two years of waiting and general acceptance by the community, and the Institute has promised an evaluation that will be deliberately unhurried. An announcement, in science, is where the conversation starts.
I find it hard not to be moved. A question that resisted the best minds of ninety years, closed in a long weekend by machines talking to each other. I have written before about slow minds and the long game, about how the things that matter compound over decades. Eighty-eight hours is a strange number to put next to that.
Now the part that makes this a Dystopia post.
Six weeks earlier, on July 29th, OpenAI had launched a programme offering free access to its models to one hundred thousand academic researchers, 250 million dollars through 2027. Researchers, the launch text said, would remain in control of their work, and their data would not be used for training by default.
In those same weeks, Tristan Buckmaster, a mathematician at NYU, and Levent Alpöge, now at Anthropic, were climbing the same cascade with AI models as collaborators, among them OpenAI’s own Codex. By August 15th, by their account, they had blow-up with a smooth force for the Euler equations, one step from the summit. Terence Tao, writing on his blog on September 7th, noted that the method had a high likelihood of extending to Navier–Stokes, and that the authors had been forced to release their preprints before they were polished, because of external events.
The external events are what is now in dispute, and I want to be careful here, partly because of my lawyerish attitude and partly because I only know what the parties have said in public. Buckmaster published a statement on his NYU page describing a call with OpenAI, two proposals for publication that he declined, and a tense discussion about authorship. He writes that he asked whether the internal model had been trained on, or had access to, their Codex sessions. OpenAI recognises their priority on the Euler result, says that neither its researchers nor its agents saw any of their work until it was public, and adds that, while unlikely, it cannot rule out that de-identified data derived from their use of its products helped improve its models. The researcher involved has called the allegations false.
I have no way of knowing who is right, and neither do you. It may all be a coincidence of the kind that mathematics produces from time to time, when an idea is in the air and two groups reach for it in the same week.
Still, the shape of the story is worth sitting with, whatever the facts turn out to be. A company promises researchers control over their work. A researcher works inside that company’s tools. The company hears he is close, and reaches the summit first, with compute he could never afford. Even if every action was clean, the structure invites the doubt. When the same company sells you the laboratory and competes with you for the discovery, your unfinished work sits on the competitive surface, whether anyone looks at it or not. I think about this every time I open a chat window to draft something I care about, and I suspect I am not alone. Someone said that the indiscriminate strip-mining of open problems may destroy the ecosystem from which new techniques grow. You get the object and lose the site.
Since the first versions of GPT, IP and copyright lawyers have been struggling with questions like these. How will we react when machines start producing new knowledge? What will the rules be for who owns a session, an idea, a half-finished proof? And, more important, can we find a way for the people who ask the questions to still be in the room when the answer arrives?
This blog has always been more a place for questions than for answers. I have made my peace with that. Some questions deserve to stay open a little longer.
