An AI Model Solved Ten Open Math Problems for About $2,000 in Compute, and Published the Proofs

OpenAI disclosed that an internal version of Astra, described as a successor to its current frontier line, produced formal, machine-checkable proofs for ten open problems spanning group theory and combinatorics, including a construction bearing on the existence of non-sofic groups and a new upper bound tightening toward the Cohn-Elkies sphere-packing threshold. The proofs were published as verified Lean code on GitHub rather than asserted in prose, which is the detail doing the real work here — a wrong proof in a formal verification language fails to compile, so the claim is checkable by anyone, not just trusted on the lab's word. The headline number — roughly $2,000 in compute for results that would represent a strong year's output for a working mathematician — is the part likely to get repeated without its caveats. Formal verification systems like Lean are good at confirming a proof is internally valid; they say nothing about whether the problem was well-chosen, already close to solved by existing techniques, or genuinely as hard as its reputation suggested. Mathematicians will spend the next few months establishing which of the ten results are actually significant advances versus solvable-but-unglamorous corners of open problem lists. What's not in dispute is the method: pointing a frontier model at a curated set of open problems with a formal proof assistant as the grading function is now a repeatable research pipeline, not a one-off demo.