OpenAI's Astra model cracks ten longstanding math problems

OpenAI said on 1 August 2026 that an internal model — described across the industry as a version of its Astra system — solved ten longstanding problems in mathematics and theoretical computer science. The company published a full paper with the proofs, their Lean 4 formalizations, and the compute cost: under $2,000 per problem.

Key takeaways

Ten problems across pure mathematics and theoretical computer science, each with no progress for at least a decade
Cost under $2,000 per problem, priced at GPT-5.6 Sol token rates
Proofs formalized in Lean 4 and released in the openai/ten-proofs GitHub repository
A separate PDF reconstructs the reasoning that led to the results
For comparison, Anthropic's earlier cryptography research with Claude cost around $100,000

What the model solved

The paper “Ten Advances in Mathematics and Theoretical Computer Science" spans two worlds. On the pure-math side, the model disproved Connes's rigidity conjecture by constructing infinitely many non-isomorphic groups sharing the same von Neumann algebra, and proved the sharp form of Ehrhart's volume conjecture. It also pinned down the exact exponential rate of the Cohn-Elkies linear program for high-dimensional sphere packing — a problem whose asymptotic behavior had remained unknown.

On the theoretical-CS side, the results concern computational complexity. The model established lower bounds for arithmetic circuits computing the permanentpermanent: A matrix function like the determinant but without alternating signs — famously hard to compute., proved exponential parallel repetition for every finite entangled game, and obtained a new hardness result for the closest vector problem (CVP) via a reduction from 3SAT3SAT: The canonical NP-complete problem: whether a Boolean formula with three-literal clauses is satisfiable.. A superexponential lower bound for multicolor Ramsey numbers and the refutation of two conjectures in extremal graph theory round out the list.

Verifiable, not just claimed

The key detail is how the results were shared. The proofs were formalized in Lean 4 — a language where correctness is checked by a machine, not a referee. That separates the announcement from mere claims of having “solved" the problems. OpenAI included the full paper plus a model-generated document explaining how it reached each proof. Transparency has limits, though — as Simon Willison notes, the company did not release the prompts it used.

That's a decent level of transparency, but I want to see the prompts they used!

Simon Willison, AI blogger and co-creator of the Datasette framework.

Cost as a signal

The most telling number is not the problem list but the price. Under $2,000 per problem — at GPT-5.6 Sol token rates — puts this kind of work within reach of a single lab, not just a large corporation. By comparison, Anthropic's earlier study, in which Claude found weaknesses in the HAWK and AES ciphers, reportedly cost around $100,000. A two-order-of-magnitude gap shows how fast the cost of machine research is falling.

< $2,000cost to solve one problem (GPT-5.6 Sol token prices)OpenAI

It fits Terence Tao's vision of “big mathematics" — large-scale, decentralized human-machine collaboration in which AI handles the technical grunt work while humans keep the creative part.

Why it matters

The announcement shifts the reference point for AI in science. Until now language models were treated mainly as assistants — here they produced new, verifiable, research-grade results, and cheaply. The Lean 4 formalization removes the most common objection to such claims, namely the lack of checkability. If the cost has truly dropped to thousands of dollars per problem, mathematics and theoretical CS may become the first fields where models generate original knowledge faster than the community can review it. That changes the question from “does AI help" to “who verifies".

What's next

The math community must independently review the ten proofs — Lean 4 formalization speeds this up but does not replace judging their significance
OpenAI released neither the prompts nor full model details, leaving reproducibility an open question
A per-problem cost under $2,000 points to more such work from other labs in the coming months