OpenAI says its Astra model solved 10 open math problems

OpenAI announced on August 1 that an internal version of Astra, its next major model family, produced solutions to ten long-standing open problems in mathematics and theoretical computer science at a total inference cost of roughly $2,000 at Sol API rates.openai

The problems had resisted progress for at least a decade and span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics. Human researchers prepared the arguments into manuscripts using the same model, which then formalized each proof in Lean, a language for machine-verified theorems. OpenAI has published all certificates and reasoning records publicly.forklog+1

What Astra Solved

The ten results include new upper bounds on sphere-packing density down to the Cohn-Elkies threshold, a construction proving the existence of non-sofic groups — a central question in group theory open since 1999 — and a disproof of Connes's rigidity conjecture on von Neumann algebras. Astra also delivered new lower bounds on arithmetic circuit complexity for computing the permanent, an exponential parallel repetition theorem for two-player quantum games, and polynomial-factor hardness of approximation for the closest vector problem, which has implications for post-quantum cryptography.forklog+1

The model resolved Ehrhart's volume conjecture in every dimension, produced a superexponential lower bound for multicolor triangle Ramsey numbers resolving Erdős problem 183, and addressed compactness and degeneracy conjectures in extremal graph theory resolving Erdős problems 146 and 180.openai

Thomas Bloom, a mathematician at the University of Manchester, described the results as "big news" and a "significant step."forklog

Astra Remains Unreleased

Astra has not been released publicly. Sam Altman demonstrated the model to US senators and regulators in Washington in late July, and it could be the first system submitted under the Trump administration's planned pre-release evaluation framework for AI models. According to The Information News Corp ↘ 1.67%, OpenAI has not decided whether Astra will ship as GPT-6 or as a variant within the GPT-5 lineup, and no release date has been set.softonic+2

Noam Brown, co-author of Astra's reasoning technology, noted the model failed to solve the Millennium Prize Problems, seven questions identified by the Clay Mathematics Institute in 2000 as the most important in mathematics. OpenAI stated plainly that "claiming human authorship for a proof generated entirely by an AI system would misrepresent both the system's contribution and the nature of genuine human intellectual work."forklog+1