OpenAI's Internal 'Astra' Model Advances Ten Open Problems in Mathematics, Including an Erdős Problem and a Disproof of Connes's Rigidity Conjecture

OpenAI says its internal experimental model, Astra, contributed to new results on ten long-standing open problems in mathematics and theoretical computer science. The results range from tighter bounds on high-dimensional sphere packing to a disproof of Connes’s rigidity conjecture and a resolution of a decades-old Erdős problem, with some proofs formalized in Lean.

Details

• High-dimensional sphere packing: New upper bounds on sphere-packing density, pushing down toward the Cohn–Elkies threshold
• Coding theory: Substantially improved upper bounds on the maximum size of binary codes, with analogous results for spherical codes
• Non-sofic groups: A constructive proof settling a central open question in group theory — whether non-sofic groups exist
• Disproof of Connes’s rigidity conjecture: Disproves the long-standing conjecture that a group is uniquely determined by its von Neumann algebra
• Arithmetic circuit complexity: New lower bounds for computing the permanent using arithmetic circuits and formulas
• Quantum parallel repetition: An exponential parallel repetition theorem extending a classical complexity-theory principle to quantum games
• Closest vector problem: A proof of polynomial-factor hardness of approximation for this foundational problem in lattice cryptography
• Ehrhart’s volume conjecture: Determines the maximum volume of a convex body under specific lattice-point constraints
• Multicolor Ramsey numbers: A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183
• Extremal number conjectures: Results on compactness and degeneracy conjectures, resolving Erdős problems 146 and 180
• Methodology: Each result reportedly required around $2,000 in compute, with human mathematicians preparing manuscripts and formalizing key arguments in Lean

How to try it

• This is a research announcement about an internal experimental model, not a publicly available product feature
• Detailed proofs and papers for each result are expected to be published progressively by OpenAI and its collaborators
• Full details are available on OpenAI’s blog at https://openai.com/index/ten-advances-in-mathematics