OpenAI says Astra, an internal version of its next major model family, has helped solve ten long-standing problems in mathematics and theoretical computer science, including one that had remained open for nearly 30 years.
The reported advances span geometry, group theory, computational complexity, quantum computing, cryptography, and graph theory:
- 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 prescribed minimum distances, with similar results for high-dimensional spherical codes.
- Non-sofic groups: A construction demonstrating the existence of non-sofic groups, addressing a central open question in group theory.
- Connes’s rigidity conjecture: A disproof of the conjecture that certain groups are uniquely determined by their von Neumann algebras.
- Arithmetic circuit complexity: New lower bounds for computing the permanent using arithmetic circuits and formulas, including a lower bound of roughly n⁴/log n for arithmetic formulas.
- Quantum parallel repetition: An exponential parallel repetition theorem for general two-player quantum games, extending a key principle from classical complexity theory.
- Closest vector problem: Polynomial-factor hardness-of-approximation results for a foundational lattice problem connected to post-quantum cryptography.
- Ehrhart’s volume conjecture: A determination of the maximum volume, in every dimension, of a convex body whose centroid is its only interior lattice point.
- Multicolor Ramsey numbers: A superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.
- Extremal number conjectures: Results resolving the compactness and degeneracy conjectures in extremal graph theory, known as Erdős problems 146 and 180.
The announcement suggests that advanced AI systems may increasingly contribute to research-level mathematical reasoning rather than only assisting with calculations or literature review.
What remains unclear is how independently Astra produced the results, how much human guidance was required, and when OpenAI will publish detailed proofs and external evaluations.
