OpenAI's Astra model solves 10 math problems

OpenAI has announced that an early, internal iteration of Astra, its upcoming flagship model, has successfully resolved or made significant progress on ten long-standing open problems in mathematics and theoretical computer science. This announcement follows OpenAI's recent launch of ChatGPT for Academic Researchers, which provides 100,000 scientists and mathematicians with free access to its top models. The computational resources required to find these solutions were surprisingly modest, totaling a volume of tokens that would cost roughly $2,000 at Sol API rates. Human researchers worked alongside the same model to compile the mathematical arguments into formal manuscripts. Following this, the AI system translated each proof into a Lean certificate to verify its logical correctness, and OpenAI has published the model's step-by-step reasoning walkthroughs alongside the final papers.

The ten breakthroughs span several highly specialized fields of mathematics. In arithmetic circuit complexity, Astra established new lower bounds for computing the permanent, including an arithmetic-formula lower bound of order n 4 /log n. In extremal combinatorics and graph theory, the model resolved Erdős problems 146, 180, and 183, the latter by proving a superexponential lower bound for multicolor triangle Ramsey numbers. The model also disproved Connes's rigidity conjecture regarding von Neumann algebras and constructed a non-sofic group. Other achievements include establishing new upper bounds on sphere-packing density down to the Cohn-Elkies threshold, creating exponentially improved bounds for binary and spherical codes, proving an exponential parallel repetition theorem for quantum games, demonstrating polynomial-factor hardness for the closest vector problem, and solving Ehrhart's volume conjecture across all dimensions.

For research mathematicians and computer scientists, this development signals a profound shift in how theoretical research is conducted. Rather than relying solely on human intuition, practitioners can now leverage advanced AI models as active collaborators capable of generating rigorous proofs. The integration of Lean formalization ensures that these AI-generated arguments are mathematically sound, addressing historical concerns about machine-learning hallucinations. By reducing the cost of complex mathematical discovery, this paradigm opens up new pathways for rapid advancement in cryptography, quantum computing, and algorithmic complexity.