On August 1, 2026, OpenAI announced that Astra, its next model, had solved 10 open problems in mathematics and theoretical computer science — each unsolved for at least 10 years. And for the first time, it made the result verifiable by anyone, not just claimed.
What it solved
The 10 problems span a wide range: group theory, high-dimensional geometry, coding theory, quantum complexity, lattice cryptography, and extremal combinatorics.
The headline result is the first explicit construction of a non-sofic group — a group theory question that had stood open since 1999, when mathematician Mikhail Gromov introduced the concept of soficity. For 27 years, no one had managed to prove or disprove whether non-sofic groups actually existed.
Why these proofs are different
The key point isn't just that Astra found the answers — it's how OpenAI presented them.
Alongside the announcement, OpenAI published a 249-page manuscript and the Lean 4 proof certificates on GitHub (Apache 2.0 license), a formal proof language. The count of "sorry" statements — Lean's placeholder marking an unproven step — is zero across all ten proofs.
In practice: Lean's compiler either accepts or rejects a proof with no room for ambiguity. Verifying that a proof is correct doesn't require trusting OpenAI — it only requires running the certificates through the compiler, something anyone can independently repeat.
The context: why verifiability matters
OpenAI had claimed significant math results before. In 2025, a company claim about solving Erdős problems was partly walked back after independent analysis by mathematician Thomas Bloom, who questioned its actual scope.
Bloom himself, commenting on the Astra results on non-sofic groups, called them "big news" — rating them more significant than OpenAI's earlier Erdős unit-distance result from May 2026.
The difference is structural: when a proof is mechanically verified by a compiler like Lean, there's no room for the interpretive ambiguity that fueled criticism of the 2025 result.
The cost: roughly $2,000
For all 10 problems combined, OpenAI reports a spend of roughly $2,000 in compute — a minimal figure compared to the academic value of problems that had remained unsolved for decades.
This isn't proof that AI is about to replace mathematicians. But it's a concrete signal that pairing increasingly capable models with formal verification tools like Lean can produce results that no longer require an act of trust — just a compiler.