Ten Proofs in Lean: OpenAI's Astra Cracks a Decade of Stalled Mathematics
OpenAI's August 1 release documents ten stalled math and complexity problems solved by an internal Astra model, formalized in Lean — with explicit attribution ethics and a ~$2,000 inference cost estimate.
On August 1, OpenAI did something unusual for a frontier lab: it published not a benchmark chart, but ten new mathematical and complexity-theory results — each with a Lean certificate and a narrated reasoning walkthrough — and named the internal system that produced them: Astra.
The release lands amid a crowded year for AI-assisted mathematics. Google DeepMind reported autonomous Erdős progress in May; Anthropic and Harmonic have published formal proofs; even skeptical mathematicians have flagged surprising results. OpenAI's post is distinct in scope: ten problems across sphere packing, group theory, quantum complexity, lattice cryptography, and extremal combinatorics — all with no progress on the main result for at least a decade.
What Astra produced
According to OpenAI's announcement, an internal Astra version achieved new results including: upper bounds on high-dimensional sphere packing; exponentially improved binary and spherical code bounds; a construction establishing non-sofic groups; a disproof of Connes's rigidity conjecture; arithmetic circuit lower bounds for the permanent; an exponential parallel repetition theorem for quantum games; polynomial-factor CVP hardness; resolution of Ehrhart's volume conjecture; a superexponential multicolor Ramsey lower bound (Erdős problem 183); and extremal graph results resolving Erdős problems 146 and 180.
OpenAI said humans helped prepare manuscripts and formalize proofs in Lean while taking responsibility for correctness; the mathematical arguments themselves were generated by the system.
Cost, attribution, and community response
OpenAI estimated the total tokens needed to find all ten solutions would cost roughly $2,000 at Sol API rates — a concrete datapoint for the "knowledge per dollar" framing Sam Altman has been advancing in Washington.
The post addressed attribution ethics directly, citing respect for signers of the Leiden declaration on AI and Mathematics and arguing against misrepresenting AI-generated proofs as purely human work.
What Astra is — and is not
OpenAI describes Astra as its "next major model" — an internal development evaluation, not a shipping product. The company has not published release timelines, public API pricing, or head-to-head benchmark tables.
What is verified: Lean certificates, published walkthroughs, and a lab taking responsibility for ten results spanning fields where stagnation had become the norm.
The open question is verification culture at scale: Lean formalization is becoming table stakes for credible math claims from labs, but peer review and community context still belong to mathematicians — not marketing departments.
### Sources
- OpenAI — Ten advances in mathematics and theoretical computer science (August 1, 2026)
- The Decoder — OpenAI announces its next major model Astra (August 1, 2026)
- OfficeChai — OpenAI Says It Has Solved 10 Open Math Problems Using Astra (August 1, 2026)