OpenAI said on August 1 that ten breakthroughs in mathematics and theoretical computer science were achieved by an internal version of its next-generation flagship model, Astra. The company said the total token cost of finding solutions to the problems was about $2,000, according to 36Kr. The results cover topics including upper bounds on high-dimensional sphere-packing density, improved bounds for binary and spherical codes, a proof of the existence of non-pure simple groups, a refutation of the Connes rigidity conjecture, new lower bounds in arithmetic circuit complexity, the quantum parallel repetition theorem, a proof of polynomial-factor approximation hardness for the nearest vector problem, the Ehrhart volume conjecture, super-exponential lower bounds for multicolor Ramsey numbers, and the Erdos extremal number conjecture.