OpenAI has released results produced by an internal version of Astra, its next major artificial intelligence model. The work addresses long-standing open problems that had seen little progress for at least ten years. The problems come from high-dimensional geometry, efficient encoding of information, the difficulty of certain calculations, mathematical groups that describe symmetries, related algebraic structures, quantum information tasks, regular grids of points used in cryptography, and the maximum number of connections in networks that avoid specific patterns.
The computation needed to find the solutions would cost roughly two thousand dollars at current rates. Human researchers prepared the arguments into manuscripts with help from the same model. The model then produced formal certificates in the Lean proof-checking system so the reasoning can be verified by machine. Narrations of the model’s thinking have also been released.
Key findings
Key findings include better upper limits on how densely equal spheres can pack into high-dimensional space. Stronger limits were found for the largest sets of binary sequences that stay a minimum distance apart. An explicit non-sofic group was constructed; this is a symmetry structure that cannot be closely approximated by rearrangements of finite sets. A conjecture of Connes, claiming certain groups are uniquely fixed by an associated algebraic object called a von Neumann algebra, was disproved.
New lower bounds were set on the number of arithmetic operations needed to compute the permanent, a matrix quantity similar to the determinant but without alternating signs. An exponential parallel-repetition result was proved for general two-player quantum games. Hardness results appeared for finding the nearest point in a lattice, with implications for cryptography. The maximum volume of a convex body whose only interior lattice point is its center was settled in every dimension. A strong lower bound was obtained for multicolor numbers that guarantee a single-color triangle, and two further conjectures about forbidden patterns in networks were resolved.
OpenAI states that the mathematical arguments were generated by the system and takes responsibility for the formal versions. The company invites mathematicians to examine and extend the ideas..