OpenAI's Unreleased Model Astra Solves 10 Open Math Problems

OpenAI announced that it used Astra, an internal version of its next-generation model, to produce new results on 10 open math problems. The total token cost required for the solution search was about $2,000 at Sol API rates, and after human researchers compiled the results into paper form, the model formalized each argument as a Lean certificate.

The problems covered are as follows.

• Improving the upper bound on high-dimensional sphere packing density
• New bounds on the maximum size of binary and spherical codes
• A proof of the existence of non-sofic groups
• A counterexample to the Connes rigidity conjecture
• Improving the lower bound for computing the permanent in arithmetic circuit complexity
• An exponential parallel repetition theorem for general two-player quantum games
• Approximation hardness results for the closest vector problem
• A solution to the Ehrhart volume conjecture in all dimensions
• A super-exponential lower bound on multicolor Ramsey numbers
• Results on conjectures in extremal graph theory and Erdős Problems 146 and 180

OpenAI explained that the fact that some results were verified in Lean does not mean all claimed content has been completely proven. However, it added that the results have so far generally held up well under review, and that a Millennium Prize Problem has not yet been solved.

It was also confirmed that Sol and Fable proved the existence of non-sofic groups within a conversational interface, without separate tools. Researchers believe that with greater investment in test-time compute going forward, AI could achieve similar results in other areas of mathematics.