OpenAI says its Astra model solved 10 open maths problems
- Vichitra Mohan
- Aug 3
- 2 min read
In a major milestone for artificial intelligence and fundamental science, OpenAI announced on 1 August that an internal version of Astra—its next major model family—has produced solutions to ten long-standing open problems in mathematics and theoretical computer science. Remarkably, the total inference cost to generate these solutions came out to roughly $2,000 at Sol API rates.
These problems, which had resisted progress by human researchers for at least a decade, span a wide range of disciplines, including high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics.
To ensure strict rigor, human researchers assisted in preparing the generated arguments into manuscripts using the Astra model itself. Astra then went a step further, formalising each proof in Lean, a specialized programming language designed for machine-verified theorems. OpenAI has publicly released all certificates and reasoning records for community review.
What Astra Solved
Astra’s breakthroughs address several foundational questions in mathematics and theoretical computer science:
Group Theory & Operator Algebras: Astra constructed a proof for the existence of non-sofic groups—a central question open since 1999—and produced a disproof of Connes's rigidity conjecture on von Neumann algebras.
Geometry & Cryptography: The model provided new upper bounds on sphere-packing density down to the Cohn-Elkies threshold, resolved Ehrhart's volume conjecture in every dimension, and established polynomial-factor hardness of approximation for the closest vector problem (which holds significant implications for post-quantum cryptography).
Complexity Theory: Astra delivered new lower bounds on arithmetic circuit complexity for computing the permanent, as well as an exponential parallel repetition theorem for two-player quantum games.
Combinatorics: The model produced a superexponential lower bound for multicolour triangle Ramsey numbers (resolving Erdős problem 183) and addressed compactness and degeneracy conjectures in extremal graph theory (resolving Erdős problems 146 and 180).
Reactions from the mathematical community have been overwhelmingly attentive. Thomas Bloom, a mathematician at the University of Manchester, described the breakthrough as "big news" and a "significant step" forward for AI-assisted research.
Astra Remains Unreleased
Despite these impressive results, Astra has not yet been released to the public.
In late July, OpenAI CEO Sam Altman demonstrated the model to US senators and regulators in Washington. Reports indicate Astra could be among the first major systems submitted under the Trump administration's planned pre-release evaluation framework for AI models. Furthermore, according to The Information, OpenAI has not yet decided whether Astra will eventually launch as GPT-6 or as a advanced variant within the GPT-5 lineup, and no official release date has been set.
While Astra marks a monumental leap, it still has its limits. Noam Brown, co-author of Astra's reasoning technology, noted that the model failed to solve any of the prestigious Millennium Prize Problems—the seven monumental questions identified by the Clay Mathematics Institute in 2000.
OpenAI also emphasized the importance of ethical attribution, stating plainly that "claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system's contribution and the nature of genuine human intellectual work."
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