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Morning Edition · Tuesday, August 4, 2026Published at 2:03 AM EDT · New York
OpenAI Publishes Lean-Verified Proofs From Its Astra Model for Ten Long-Open Mathematics Problems
Each result ships with a machine-checkable Lean 4 formalization, and OpenAI says the tokens behind all ten would have cost roughly two thousand dollars at its own API rates.
OpenAI published ten results on August 1 that it says an internal version of its next model family, named Astra, produced in mathematics, operator algebras, quantum complexity and theoretical computer science. The release includes a 249-page technical manuscript, a 62-page account of how the arguments came together, and a formalization of each result in the proof-checking language Lean 4, posted on GitHub. That last element is the one that matters. A Lean proof either type-checks or it does not, which moves these claims out of the category of assertions a lab makes about its own model and into the category of artifacts anyone can run.
The named results are substantial. According to OpenAI's writeup, Astra produced the first explicit construction of a non-sofic group, a question open since the mathematician Mikhail Gromov introduced the notion of soficity in 1999. It also disproved Connes's rigidity conjecture on von Neumann algebras, proved Ehrhart's volume conjecture, and settled three problems from the catalogue of the mathematician Paul Erdős, including the entry on multicoloured Ramsey numbers. OpenAI says each of the ten problems had been open for at least a decade.
The economics attached to the claim are as notable as the mathematics. OpenAI states that the tokens used to generate all ten solutions would have cost about two thousand dollars at its Sol API rates. If that figure holds, the marginal cost of a research-grade proof attempt has fallen to an amount a single graduate student could expense, which changes who can afford to run a search at this scale.
Separate and much weaker claims are circulating alongside this one. Emad Mostaque, the former chief executive of Stability AI, said in remarks relayed on Telegram that a billion-parameter model trained on nothing published after 1911 independently reconstructed general relativity, and that artificial intelligence had recovered more than a century of missing algebra in Einstein's equations. No paper, weights or replication accompany that assertion, and it should be read as a claim by an interested party rather than as a result. The distinction between the two is exactly the distinction Lean certificates are designed to enforce.
Veracity: Corroborated
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