OpenAI’s Unreleased Model Astra Solves Ten Major Open Mathematics Problems

Math is hard. Math used to be strangely hard for LLMs. People used to gloat about that. Remember? Math is getting easier. AI is getting more capable. Life comes at you fast. Remember this meme? Why yes. Yes it is. We don’t know the extent to which Astra is a big jump over Fable and Sol in this realm. We do know that Astra can do math. As in real math. OpenAI : We provide new results for the following problems. The results were achieved by an internal version of Astra, our next major model. The total number of tokens needed to find solutions to these problems would cost roughly $2,000 at Sol API rates. These arguments were then prepared into manuscripts by humans with the same model. Afterward, the model formalized each argument in a Lean certificate⁠(opens in a new window) . We are also releasing for each solution a model’s narration of its thinking process. High-dimensional sphere packing. New upper bounds on sphere-packing density down to the Cohn–Elkies threshold. Binary and spherical codes: Exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous results for high-dimensional spherical codes. Non-sofic groups. A construction establishing the existence of non-sofic groups, addressing a central open question in group theory. Connes’s rigidity conjecture. Disproof of a longstanding conjecture that certain groups are uniquely determined by their von Neumann algebras. Arithmetic circuit complexity. New lower bounds fo…