OpenAI's Astra Model Solves Ten Open Mathematics Problems for $2,000
Internal version of Astra solved previously open problems in mathematics and theoretical computer science, with proofs verified on GitHub.
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OpenAI Announces Astra Solves Ten Open Mathematics Problems
OpenAI announced on August 1, 2026 that an internal version of its next model, Astra, solved ten previously open problems in mathematics and theoretical computer science. The company published formal Lean proofs of the ten solved problems on GitHub, verifiable by any mathematician using the Lean proof-checker.
The problems were solved for roughly $2,000 in compute costs.
Notable Results
One of the ten results is a construction proving the existence of non-sofic groups, described as a central open question in group theory.
The ten results also include new upper bounds on sphere-packing density down to the Cohn-Elkies threshold.
The Astra model family reportedly disproved the unit distance conjecture, associated with mathematician Paul Erdős, concerning the maximum number of pairs of points in a plane that can be exactly one unit apart.
Expert Endorsement
Fields Medal winner Timothy Gowers stated he would recommend one of the Astra model family’s proofs for publication in Annals of Mathematics without hesitation.
Availability
The version of Astra used for the math results is described as internal and unreleased; Astra is not yet publicly available.
Source: Build Fast with AI
Developments since publication
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OpenAI announced that an internal version of its next major model, called Astra, produced ten new results in mathematics and theoretical computer science. Source
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OpenAI published a 249-page manuscript alongside machine-checkable Lean 4 certificates for every result on GitHub. Source
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The total compute cost for all ten Astra solutions was roughly $2,000 at Sol API rates, according to OpenAI. Source
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OpenAI's head of mathematics research, Sebastien Bubeck, confirmed the Astra results on X, calling them 'beautiful'. Source
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OpenAI has not said when Astra will be released publicly, describing it only as its 'next major model'. Source
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OpenAI released a 249-page manuscript alongside the Astra announcement. Source
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OpenAI released Lean 4 proof certificates for all 10 results on GitHub under an Apache 2.0 license. Source
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The GitHub repository for the Astra Lean proofs has a 'sorry' count of zero, indicating that every step across all ten formalized proofs is fully verified. Source
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Astra remains unreleased as of the announcement date; OpenAI describes it as its 'next major model.' Source
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