10 Results in Mathematics and Theoretical Computer Science

OpenAI has released new results on 10 problems in mathematics and theoretical computer science that had seen no major progress for at least 10 years. The research spans high-dimensional geometry, coding theory, group theory, operator algebras, quantum complexity, lattice cryptography, extremal combinatorics, and more.

An internal version of the next-generation model Astra was used in the solving process. OpenAI explained that the total token cost required to find the solution for each problem was about $2,000 based on the Sol API, after which humans organized the manuscripts and the model formalized each argument as a Lean certificate.

The major results released are as follows.

• Improved the upper bound on high-dimensional sphere packing density up to the Cohn–Elkies threshold
• Exponential improvement on the maximum size of binary codes and high-dimensional spherical codes
• Presented a construction showing the existence of non-sofic groups
• Disproved the Connes rigidity conjecture, which states that a certain group is uniquely determined by its von Neumann algebra
• Derived new lower bounds for arithmetic circuits/formulas for computing the permanent
• Presented an exponential parallel repetition theorem for general two-player quantum games
• Proved polynomial-level approximation hardness of the closest vector problem
• Determined, in all dimensions, the maximum volume of a convex body with a unique interior lattice point
• Derived a super-exponential lower bound on multicolor triangle Ramsey numbers
• Presented results on the compactness and degeneracy conjectures in extremal graph theory

OpenAI stated that attributing AI-generated proofs to human authors alone could distort both the system's contribution and the nature of human intellectual work. The company explained that it supported manuscript writing and Lean formalization and takes responsibility for the accuracy of the results, but the mathematical arguments themselves were generated by the model.