OpenAI's Astra Solved Ten Open Problems in Mathematics — What It Means
OpenAI says an internal version of its next model, Astra, produced ten results in mathematics and theoretical computer science, with Lean certificates on GitHub for roughly $2,000 in compute.
What happened: ten open problems, ten results
On August 1, 2026, OpenAI said an internal version of Astra — its next major model, still unreleased — produced ten new results in mathematics and theoretical computer science, each advancing a problem open for at least a decade. The company published a 249-page manuscript and machine-checkable Lean 4 certificates for all ten on GitHub.
The headline claim is the first explicit construction of a non-sofic group: a question open since Mikhail Gromov introduced soficity in 1999, and unresolved for 27 years.
Why it matters
Astra also disproved Connes's rigidity conjecture, proved Ehrhart's volume conjecture and settled three problems in Paul Erdős's catalogue, including number 183. It delivered the first improvement since 1978 to the upper bound on high-dimensional sphere-packing density.
OpenAI's head of mathematics research, Sebastien Bubeck, called the results "beautiful" on X; each ships with a Lean certificate and a reasoning log. OpenAI says all ten solutions cost about $2,000 in compute at Sol API rates.
What it means
The news lands amid a dispute with mathematicians: June's Leiden Declaration, endorsed by the International Mathematical Union, warned that results are announced through blog posts instead of peer-reviewed journals. OpenAI's answer is verifiability — anyone with the Lean compiler can check the proofs. Thomas Bloom, who runs the erdosproblems site, called the ten results "big news". OpenAI has not said when Astra will be released.