OpenAI dropped something unusual on the maths world this weekend: it says an internal version of Astra — its next big model, still unreleased — has cracked ten open problems in mathematics and theoretical computer science. None of them was low-hanging fruit. Every problem on the list had sat unsolved for a decade or more.
And the company didn't just claim victory in a blog post. It published a 249-page manuscript plus Lean 4 proof certificates for all ten results on GitHub, which means the proofs can be checked by a compiler rather than taken on faith.
The 27-year question
The headline result answers a question that's been open since 1999, when Mikhail Gromov introduced the idea of sofic groups. Mathematicians have wondered ever since whether non-sofic groups exist at all, and nobody could prove it either way. Astra produced the first explicit construction of one, and with it, the answer.
The rest of the list is scattered across fields. A disproof of Connes's rigidity conjecture in von Neumann algebras. A proof of Ehrhart's volume conjecture. Three problems from Erdős's famous catalogue, including number 183 on multicoloured Ramsey numbers. The first improvement to the general high-dimensional sphere-packing bound since 1978. A parallel repetition theorem for two-player quantum games. New lower bounds on the circuit complexity of the permanent. Any one of these would have been a career highlight for a working mathematician.
The $2,000 bill
Here's the detail that raised eyebrows: OpenAI says the compute for all ten solutions cost about $2,000 at Sol API rates. Sebastien Bubeck, who runs the company's maths research, confirmed the results on X, called them "beautiful," and pointed out that each comes with a Lean certificate and a chain-of-thought walkthrough of how the model got there.
Those certificates are doing a lot of work. The standard complaint about AI-generated proofs is that checking them can take experts months. A machine-checkable certificate changes the equation — anyone with a Lean compiler can validate the result without trusting OpenAI's word, its model, or its press office.
Not everyone's applauding
The timing is spiky. In June, mathematicians published the Leiden Declaration, backed by the International Mathematical Union, accusing AI companies of using published research without consent, sidestepping peer review, and undermining how proof and attribution are supposed to work. One of its specific complaints: results announced by press release instead of journal. This announcement is, well, exactly that.
OpenAI knows the tension firsthand. In May it said the same model family had disproved the Erdős unit distance conjecture, an 80-year-old problem in discrete geometry. Fields Medalist Timothy Gowers said then that he'd recommend that proof to Annals of Mathematics without hesitation. And Thomas Bloom, who runs the erdosproblems site, called the new results "big news" — in his view, bigger than the unit distance counterexample.
Where this goes
There's still no release date for Astra. OpenAI calls it only its "next major model," and some observers — investor Mark Kretschmann among them — suspect it's the GPT-6 series. Meanwhile the company is handing 100,000 academic researchers free access to its frontier models through 2027, pulling the research world closer to its platform even while parts of that world push back.
So the proofs can be verified by machine. Whether mathematics accepts results that arrive by blog post is a question no compiler can settle — and after this weekend, it's a lot harder to postpone.
Image: Vitaly Gariev, via Pexels
