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OpenAI's Astra solved ten decade-old math problems with Lean 4 proofs for $2,000 in compute

· by Pondero Newsdesk

The short version

OpenAI published machine-checkable Lean 4 proofs for ten long-standing problems in mathematics and theoretical computer science, crediting Astra, its unreleased next model, with results spanning group theory to lattice cryptography.

OpenAI's Astra solved ten decade-old math problems with Lean 4 proofs for $2,000 in compute

A 27-year question in group theory fell on August 1 when OpenAI published Lean 4 certificates proving non-sofic groups exist, one of ten long-standing open problems its unreleased Astra model resolved for roughly $2,000 in compute.

What

OpenAI posted a 249-page manuscript and ten machine-checkable Lean 4 proof certificates on August 1, crediting an internal version of Astra, described as the company's "next major model" per the OpenAI announcement. The results span six domains: group theory, von Neumann algebras, high-dimensional geometry, quantum complexity, lattice cryptography, and extremal combinatorics.

The headline result is the first explicit construction of a non-sofic group. Mikhail Gromov introduced the concept of soficity in 1999, and whether non-sofic groups exist had remained one of modern group theory's central open questions per TechTimes. Other results include a disproof of Connes's rigidity conjecture and three resolved Erdos problems. Every certificate file is on the openai/ten-proofs GitHub repository under an Apache 2.0 license, with zero "sorry" placeholders, meaning the Lean 4 kernel verified each step end-to-end against the mathlib foundation.

Greg Brockman reported the work cost approximately $2,000 at current Sol API rates per The Next Web. Astra has no announced release date, API availability, or pricing.

Why it matters

Lean 4 certification is a structural difference from earlier AI mathematics milestones. The proof assistant's trusted kernel gives a binary verdict: a certificate either compiles against the mathlib foundation of 210,000+ formalized theorems or it fails. Prior results, including OpenAI's own Erdos unit distance work from May 2026, relied on human mathematicians reading and signing off on arguments. With the certificates now publicly licensed on GitHub, any operator who installs the Lean compiler can re-check these proofs independently.

Thomas Bloom, the University of Manchester mathematician who curates the Erdos problems catalogue and who publicly dismantled OpenAI's false October 2025 math claim, called these results "big news" on X. He rated them more significant than the unit distance counterexample per TechTimes. That endorsement carries weight specifically because Bloom is the same expert who caught OpenAI's previous fabricated math results.

Astra used a root agent coordinating specialist sub-agents for sustained, multi-domain work across six distinct fields, per OpenAI's description. That multi-agent architecture, not single-call inference, is what produced results at this breadth and depth.

What to watch next

None of the ten results has cleared peer review yet. A journal acceptance would be the next concrete milestone for any of the proofs. Astra's release timeline remains unresolved, with OpenAI describing it only as its "next major model family" without a date. Google DeepMind's AlphaProof Nexus, which resolved nine Erdos problems with Lean-verified proofs in May 2026, is now the direct comparison point as both labs converge on Lean 4 as the shared machine-checkable verification standard.

Sources