On August 1, 2026, OpenAI announced that an internal version of its forthcoming Astra model family had produced solutions to ten open problems in mathematics and theoretical computer science. These problems, some of which had remained unsolved for decades, include the first explicit construction of a "non-sofic group," a question initially posed in 1999.
OpenAI stated that the computational cost for finding all ten solutions was approximately $2,000, based on its Sol API rates. Crucially, the company did not merely present the answers; it published formal proofs written in Lean, a proof-checking language, on GitHub. These machine-checkable proofs, accompanied by a 249-page manuscript, enable independent verification of the results without reliance on OpenAI's claims.
The problems tackled by Astra span various fields, including group theory, high-dimensional geometry, coding theory, quantum complexity, lattice cryptography, and extremal combinatorics. One notable achievement is Astra's proof of a tighter ceiling on sphere packing density in high dimensions, representing the first improvement to this specific bound since 1978. Another involves the explicit construction of a non-sofic group, which had been an open question in group theory since Mikhail Gromov introduced the concept of "soficity" in 1999.
The announcement follows earlier reports of OpenAI's AI models making progress in mathematics. In May 2026, OpenAI reported that an internal AI model had disproved a part of the 80-year-old unit distance problem, originally proposed by mathematician Paul Erdős. That result was also accompanied by an accompanying paper from external experts who reviewed the AI's findings.
Mathematicians, including Fields Medal winner James Maynard of the University of Oxford, have been engaging with the increasing capabilities of AI in their field. Maynard, who received the Fields Medal in 2022 for his contributions to analytic number theory, has expressed that he has been "soul searching" regarding the future of mathematics as the discipline adapts to AI.
The verifiable nature of Astra's outputs addresses a key challenge in the deployment of AI, particularly in fields where precision and correctness are paramount. This approach allows for immediate, trustless verification, potentially altering traditional peer review processes by collapsing the timeline for proof validation.
While the results have been met with positive reactions from some mathematicians, such as Fields Medalist Timothy Gowers, others caution that the work is still undergoing review. Some observers have raised questions regarding OpenAI's selection of problems and the fact that the model itself is not publicly accessible, meaning the results cannot be independently reproduced, only checked. OpenAI has stated that its staff assisted in preparing the papers and formalizing the arguments, while asserting that the mathematical content originated from Astra.
OpenAI has been actively working to integrate AI into scientific research, having launched a program in July 2026 to provide 100,000 researchers with free access to its frontier models, including GPT-5.6 Sol Pro. The company's GPT-5.2 model, released in December 2025, was also described as its strongest model for mathematical and scientific work, demonstrating improvements in reasoning and abstraction.