The OpenAI blog post describes a collection of ten results generated by an internal version of the company’s forthcoming Astra model. Each result addresses a long‑standing open problem in mathematics or theoretical computer science. The post notes that the work builds on an earlier AI‑generated disproof of the Erdős unit‑distance conjecture shared in May, which has already inspired further research.
The advances cover a range of fields. In high‑dimensional sphere packing, the model produced new upper bounds on sphere‑packing density that approach the Cohn–Elkies threshold. For binary and spherical codes, it achieved exponentially improved bounds on the maximum size of binary codes at any prescribed minimum distance, with analogous improvements for high‑dimensional spherical codes.
In group theory, the model constructed an example establishing the existence of non‑sofic groups, addressing a central open question. Regarding operator algebras, it disproved Connes’s rigidity conjecture, which posited that certain groups are uniquely determined by their von Neumann algebras.
For arithmetic circuit complexity, the model derived new lower bounds for computing the permanent using arithmetic circuits and formulas, including an arithmetic‑formula lower bound of order n⁴⁄log n. In quantum complexity, it proved an exponential parallel repetition theorem for general two‑player quantum games, extending a classical principle.
The closest vector problem, a foundational lattice question relevant to post‑quantum cryptography, received a polynomial‑factor hardness of approximation result. For Ehrhart’s volume conjecture, the model determined, in every dimension, the maximum possible volume of a convex body whose centroid is its only interior lattice point.
In extremal combinatorics, the model obtained a superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183. It also produced results on the compactness and degeneracy conjectures in extremal graph theory, resolving Erdős problems 146 and 180.
The source explains that the total number of tokens required to find these solutions would cost roughly $2,000 at Sol API rates. After the model generated the arguments, humans prepared the manuscripts using the same model, and then the model formalized each proof in a Lean certificate. For each solution, OpenAI is also releasing a narration of the model’s thinking process.
The post emphasizes that attribution should reflect the model’s role, and that the mathematical arguments themselves were generated by the AI system, while humans helped prepare the manuscripts and verify the proofs in Lean. OpenAI hopes the community will engage with the results, place them in context, and build on the ideas through further research.