OpenAI said on August 1 that an internal, unreleased version of Astra — the model family it has previously described as coming after the GPT-5.6 line — generated solutions to ten open problems in mathematics and theoretical computer science, several of which had stood for decades. The company published every result as a machine-checkable Lean 4 certificate on GitHub under an Apache 2.0 license, alongside a 249-page technical manuscript and a separate account of how the model arrived at each argument.

What the model solved

The ten results span high-dimensional sphere packing, binary and spherical coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice-based cryptography and extremal combinatorics. The headline result is an explicit construction of a non-sofic group, resolving a question left open since Mikhail Gromov introduced the concept of soficity in 1999. Other results include an improved general bound on high-dimensional sphere-packing density — the first such improvement since 1978 — and disproofs or partial resolutions of several problems from Paul Erdős's catalogue of combinatorics questions, including Erdős problem 183 on multicolored Ramsey numbers. OpenAI said the full set of results was produced using roughly $2,000 of API compute.

Why the Lean verification matters

What distinguishes the announcement from earlier claims of AI-assisted mathematical progress is that each proof compiles in Lean 4, a formal proof assistant whose kernel returns a strict pass/fail verdict rather than a plausibility judgment — OpenAI reported a "sorry" count of zero across all ten certificates, meaning no step was left unproven. That removes the need to trust the model's own explanation of its reasoning, since outside mathematicians can independently verify the certificates compile. Coverage of the release noted that mathematicians reviewing the results, including at least one Fields Medalist, described some of the proofs as strong enough to submit to a top journal.

Why it matters

OpenAI has not set a release date for Astra or said whether it will ship as GPT-6 or as a variant within the existing GPT-5 line, but has described the family as designed to let multiple agents work on a single hard problem for extended stretches. Publishing genuine, independently verifiable mathematical advances — rather than benchmark scores — gives outside researchers a harder data point to evaluate frontier progress by, and raises the bar other labs will be measured against as they make their own claims about AI-assisted research.