In early August 2026, OpenAI published something unusual: a 249-page mathematical manuscript containing proofs for ten open problems — each one unsolved for at least a decade, some for much longer. The work was produced by an internal build of Astra, OpenAI's next major model. Total compute cost: approximately $2,000.
The number that makes this story strange isn't the $2,000. It's the ten.
What Was Actually Solved
The problems span high-dimensional geometry, group theory, quantum complexity, lattice cryptography, and extremal combinatorics. OpenAI hasn't released the full list with individual attribution yet, but the most significant result — and the one that has mathematicians paying closest attention — is a proof that non-sofic groups exist.
Soficity is a property related to how well a group can be approximated by finite symmetric groups. The question of whether every group is sofic was posed by mathematician Mikhail Gromov in 1999 and became one of the more persistent open questions in geometric group theory. Astra's proof, if it holds up to peer review, would resolve it negatively: there exist groups that are genuinely non-sofic, meaning the approximation fails in a fundamental way.
Machine-Checkable Proofs
OpenAI published the manuscript alongside machine-checkable Lean 4 certificates on GitHub. Lean 4 is a formal proof verification system — feeding a proof into Lean either confirms it is logically valid or identifies exactly where it breaks down, with no room for human error in the checking process. This is significant: it means any mathematician with a computer can verify the proofs themselves in minutes, without having to trust OpenAI's claims or wait years for traditional peer review.
The ability to generate trustless, instantly-verifiable mathematical proofs is arguably more important than any individual result. It changes the economics of mathematical verification entirely.
Why $2,000 Matters
The cost figure is deliberately highlighted by OpenAI, and for good reason. Mathematical research at the frontier typically requires years of expert human effort. A Fields Medal winner might spend a decade on a single hard problem. The cost per solved problem here — roughly $200 — isn't just cheap. It reframes the question of what "doing mathematics" looks like at scale.
If Astra can reliably solve decade-old open problems at this cost, the implication isn't just that AI is good at math. It's that the backlog of unsolved mathematical problems — there are thousands of them, at varying difficulty levels — could begin to clear at a pace that would have seemed absurd two years ago.
What Comes Next
OpenAI hasn't announced a public release date for Astra. The math manuscript appears to be a deliberate capability preview, similar to how DeepMind used AlphaFold protein structure results before the full system became available. The Lean 4 certificates are live and human mathematicians are already working through them.
The non-sofic groups proof in particular is drawing attention. Several group theorists contacted by Forbes described the result as credible pending review, while noting that the Lean verification provides a level of assurance that doesn't exist for most claimed proofs. Whether every result survives close scrutiny remains to be seen — but the bar for dismissal is now much higher when the proof checker already said yes.
Sources: The Decoder — Astra math announcement · Forbes — $2,000 and ten problems