OpenAI Astra Proves 10 Open Math Problems in Lean 4

Every few months an AI system is described as having done mathematics, and the claim usually rests on a benchmark score. This one is different in a way that deserves careful attention: OpenAI has published solutions to ten long-standing open problems in mathematics and theoretical computer science produced by an internal model called Astra, and every one of them ships with a Lean 4 certificate that anyone can verify on a laptop.

• Ten open problems solved, spanning group theory, von Neumann algebras, high-dimensional geometry, quantum complexity, lattice cryptography, and extremal combinatorics
• The headline result is an explicit construction of a non-sofic group, open since Mikhail Gromov introduced soficity in 1999
• All Lean 4 certificate files are published on GitHub under Apache 2.0, with a sorry count of zero
• Total compute cost was roughly $2,000 at GPT-5.6 Sol API rates

Why Do Lean 4 Certificates Change the Argument?

The usual difficulty with AI-generated mathematics is that evaluating it requires an expert, and experts are scarce and busy. A Lean 4 proof removes that bottleneck. Lean is a proof assistant: it mechanically checks every inference step against a formal foundation, and it will not accept a proof with a gap.

The detail that matters most in OpenAI's repository is the sorry count. In Lean, sorry is the placeholder that lets you assert something without proving it — a legitimate tool while drafting, and a fatal hole in a finished proof. A count of zero across all ten formalizations means no step anywhere is taken on faith. You do not have to trust OpenAI, or Astra, or a peer reviewer. You run the checker.

What Are the Actual Results?

The non-sofic group construction is the one specialists will care about most. Soficity is a property Gromov described in 1999, and whether every group is sofic has been open ever since; an explicit example that is not sofic settles it.

Beyond that, Astra disproved Connes's rigidity conjecture on von Neumann algebras, proved Ehrhart's volume conjecture, and resolved three problems from Paul Erdős's famous catalog of open questions. Sebastien Bubeck, OpenAI's head of mathematics research, confirmed the results publicly and called them beautiful.

The breadth is worth noting on its own. These are not ten variations on one technique — they sit in six distinct areas, which suggests something more general than a system tuned to a single problem class.

What Does $2,000 in Compute Tell Us?

Roughly $2,000 for ten results is the number that will get repeated, and it is genuinely striking. But it is worth reading precisely: that figure covers the compute for the solutions, priced at GPT-5.6 Sol API rates. It is not the cost of training Astra, and it is not a claim that any $2,000 of inference produces comparable output.

What it does suggest is that once a capable enough model exists, formally verified mathematical discovery is not compute-bound in the way one might assume. The expensive part has moved upstream into training. Astra itself remains unreleased, so the ten certificates are what the outside world has to work with — which is precisely why publishing them in a machine-checkable form was the right call.

For a field that has spent three years arguing about whether language models reason, a repository of independently verifiable proofs is a considerably more useful contribution than another benchmark. We cover the rest of the frontier in our AI section, including recent open-weight releases like Kimi K3.

Sources: SiliconANGLE — August 2, 2026; TNW — August 2026; Forbes — August 3, 2026.