THE TEN RESULTS
Ten places where OpenAI says Astra moved a research frontier.
The collection includes stronger bounds, new constructions, exact extremal results, and counterexamples. “Breakthrough” is the editorial shorthand; the mathematics is more specific.
High-dimensional geometry
Sphere packing
OpenAI reports new upper bounds on sphere-packing density that reach the Cohn–Elkies threshold in the asymptotic setting.
Coding theory
Codes
OpenAI reports exponentially stronger upper bounds for binary codes at every prescribed minimum distance, with analogous spherical-code bounds.
Group theory
Non-sofic groups
OpenAI reports a construction of a non-sofic group, resolving whether every group admits finite permutation approximations.
Operator algebras
Connes rigidity
OpenAI reports a counterexample to the idea that certain rigid groups are uniquely determined by their group von Neumann algebras.
Theoretical computer science
The permanent
OpenAI reports new lower bounds for computing the permanent, including an arithmetic-formula lower bound of order n⁴/log n.
Quantum complexity
Quantum repetition
OpenAI reports an exponential parallel-repetition theorem for general finite two-player quantum games.
Lattice theory
Closest vector
OpenAI reports polynomial-factor hardness of approximation for the closest vector problem, with related consequences for decoding and lattice problems.
Geometry of numbers
Ehrhart volume
OpenAI reports the exact maximum volume, in every dimension, under the centroid-and-lattice-point constraint.
Extremal combinatorics
Ramsey numbers
OpenAI reports a superexponential lower bound for multicolor triangle Ramsey numbers, resolving Erdős problem 183.
Extremal graph theory
Extremal graphs
OpenAI reports counterexamples to the Erdős–Simonovits compactness conjecture and an Erdős degeneracy conjecture, resolving problems 146 and 180.