On a video call convened by the French Mathematical Society, Hugo Duminil-Copin, who won a Fields Medal in 2022, told colleagues he had wept the night before, hard enough to wake his daughter. Sixteen of the problems OpenAI claims to have settled in a mass release of machine-generated mathematics sit in the field he has spent his career on.
The company posted more than 700 manuscripts to a public repository this month, describing results it attributes to an unreleased internal model. OpenAI says the collection answers more than 300 problems that human mathematicians had not solved, spread across algebra, logic, geometry and probability. Three of the manuscripts were withdrawn after errors surfaced within a day.
The response from the field is split, and the split runs through a single sentence. Mathematicians who have started reading describe the work as better than they expected from the company’s earlier write-ups, and describe the verification job as one no institution is staffed to do.
Cedric Villani, another Fields Medal winner, told France Inter that the release amounted to the most radical change in mathematics in at least four centuries. In the same interviews he warned that its effect on academic careers could be destructive.
OpenAI acknowledges in notes published with the repository that the results sit at different stages of verification. Fewer than half the manuscripts arrive with a machine-checkable proof written in Lean, a proof assistant that lets a computer confirm a chain of logic without a human reading the argument. Formalization is not the end of the job. Someone still has to establish that the machine-checked theorem states what the prose manuscript claims, and researchers who have begun that work say the two do not always line up.
“I either have to read possibly-not-correct slop, or wait for others to do the same, or wait for someone to formalise them before I can say for sure that the results are even correct,” Kevin Buzzard, a mathematics professor at Imperial College London, told The Verge. Buzzard said that in his own area, algebraic number theory, only about six results immediately stood out, and that few, if any, of those carried a formal proof he could lean on.
Brendan Hassett, a mathematics professor at Brown, said a quick read of the write-up on the problem he knew best made little sense, adding that if a person had submitted it he would not have spent more time on it.
Where researchers have been able to reach the substance, the top end holds up. Jared Duker Lichtman, a mathematician at Stanford, put the number of results he would rank as first-rate in the tens, including progress toward the Riemann hypothesis, a special case of the Hodge conjecture and a solution to the four-dimensional Kakeya conjecture. Those are problems that consume careers. In a world without AI, researchers said, several of the manuscripts would have cleared the bar for a top journal, and a handful would have put their author in the conversation for a Fields Medal.
The bottleneck is everything downstream. The manuscripts run short by the conventions of the field, with arguments that would ordinarily fill hundreds of pages compressed into dozens, and their bibliographies are thin enough that mathematicians have asked publicly where the credit went. Attribution is the working currency of the discipline, and a result whose precedents are unclear is hard to place in a literature.
The retractions hint at the error rate that remains when nobody reads the argument. OpenAI withdrew three manuscripts after other mathematicians flagged them, and has said more formalizations are on the way. The repository functions, in effect, as a working paper series for a model the company has not released.
Scale compounds it. A field that reviews work through volunteer referees at a pace set by journals has received, in one drop, more claimed results than it can process in a year. Several researchers said that reading only the forty-page table of contents and the abstracts took most of an hour.
Careers are already moving. Scott Armstrong, a mathematician who describes himself as an optimist about the technology, said his priority was to finish work in progress before OpenAI scooped him, and he worried about what the next few years hold for younger researchers entering a field where a model can produce a decade of results overnight. Others said the larger risk falls on the culture rather than on any individual: a literature that grows faster than it can be checked degrades into a body of claims nobody can rely on.
OpenAI said in a statement that it will update the repository as it obtains more formalizations, and it published guidance on how to navigate the collection. The company has not said who is expected to pay for the rest of the checking. The three retractions, which came within a day, show what happens when that work is left undone.
That is the open argument. The mathematics is the part researchers say they can eventually assess, given years and money. The pricing of verification is the part nobody has settled, and the models are not waiting for an answer.
