OpenAI said an internal version of its next-generation flagship model, Astra, solved 10 open problems in mathematics and theoretical computer science that had been stalled for more than a decade. The post marks the first public disclosure of the Astra codename. OpenAI said the total token cost to solve all 10 problems was about $2,000 based on Sol API pricing.
In its official blog post, OpenAI argued that progress in mathematics is usually measured in years, with a proof often taking two to three years to move from submission to a final peer-review decision. Against that backdrop, the company said Astra handled several long-standing public problems within a single compute batch.
How OpenAI says the results were checked
OpenAI described a two-layer verification process. First, human researchers worked with the same model to organize the arguments into a 249-page manuscript. Second, the model reformulated each argument as a Lean proof, producing machine-checkable verification that can be inspected line by line and does not allow skipped steps.
The company said the full Lean proof repository has been open-sourced on GitHub for anyone to download and verify. Each problem also includes the model’s own description of its reasoning process.
The 10 problems Astra reportedly solved
According to OpenAI, all 10 results address public problems that had remained open for more than 10 years, spanning pure mathematics and theoretical computer science:
- High-dimensional sphere packing: upper bounds approaching the Cohn-Elkies threshold
- Binary codes and spherical codes: exponential improvements in upper bounds
- Non-sofic groups: construction of a non-sofic group, addressing a core open question in group theory
- Connes rigidity conjecture: disproved, showing some groups are no longer uniquely determined by von Neumann algebras
- Arithmetic circuit complexity: a new lower bound for the permanent
- Quantum parallel repetition: an exponential parallel repetition theorem for two-player quantum games
- Closest Vector Problem, or CVP: a hardness result for polynomial-factor approximation, with implications for post-quantum cryptography
- Ehrhart volume conjecture: determination of the maximum volume in every dimension
- Multicolor Ramsey numbers: a superexponential lower bound that resolves Erdős Problem 183
- Extremal graph theory: results that resolve Erdős Problems 146 and 180 in one release
Earlier OpenAI research and academic rollout
OpenAI said that in May it had already announced a model result disproving the Erdős unit distance conjecture, which it said led to at least five follow-up papers on arXiv. More recently, the company launched ChatGPT for Academic Researchers, offering its strongest model free of charge to 100,000 scientists.
Praise from mathematicians, but also resistance
Fields Medal winner Timothy Gowers called the development a “milestone for AI-assisted mathematics.” Thomas Bloom, who maintains the Erdős problem list, said it was big news in terms of constructive results.
Still, the release arrives in the middle of a broader dispute over how AI-generated mathematical work should be handled. In June, an international group of mathematicians published the Leiden AI and Mathematics Declaration, backed by the International Mathematical Union. The declaration gathered more than 1,000 signatures within 24 hours, including Kevin Buzzard and Peter Scholze.
The declaration opposes three practices: using papers as training data without consent, bypassing peer review to publish directly to media outlets, and failing to state whose prior work a result builds on. OpenAI’s choice to publish the Astra results through its blog instead of submitting them to a journal fits one of the patterns criticized in that declaration.
OpenAI responded by saying, “The questions raised by AI participating in mathematical research are not ones a single technology company can answer on its own,” while also saying it is responsible for correctness even though the arguments themselves were generated by the model.
As the pace of problem solving starts to outrun the pace of peer review, the pressure is shifting toward the academic system itself: verification, authorship, and whether existing trust mechanisms can keep up.