The specific problems Astra solved

According to OpenAI's report, the results spanned eight fields and included proofs, counterexamples, and improved bounds :

• Non-sofic groups — First explicit construction proving the existence of non-sofic groups, closing a central open question in group theory that had resisted progress since the concept of soficity was introduced by Mikhail Gromov in 1999 .
• Connes's rigidity conjecture — Disproved a long-standing conjecture in von Neumann algebras .
• High-dimensional sphere packing — New upper bounds for sphere packing in dimensions beyond those already settled .
• Three problems posed by Paul Erdős — Resolved three open combinatorial problems from the Erdős canon, including Erdős problem No. 183 on multicolored Ramsey numbers .
• Additional results — New results in quantum parallel repetition, the closest vector problem in lattice cryptography, arithmetic circuit complexity, binary and spherical codes, and Ehrhart's volume conjecture .

How OpenAI verified the solutions

OpenAI did not rely solely on the model's output. Every result was accompanied by a Lean 4 formal proof certificate — a computer-checkable proof file that any reader can independently verify on a laptop . Lean is a proof assistant that checks every logical step from axioms to conclusion, catching gaps, type errors, and inconsistencies . OpenAI published all Lean files publicly on GitHub, and the repository's "sorry" count — indicating unproven steps — stands at zero for all ten proofs .

However, multiple commentators noted that formal verification has important limits: Lean can confirm that a formal statement follows from its formal definitions, but it cannot certify that a press-release summary accurately reflects the formal theorem, that the formal definitions match the mathematical community's intended problem, or that the result's novelty and historical framing are correct . Experts still need to audit statement fidelity, definitions, reductions, and the informal-to-formal bridge .

The cost: $2,000 in tokens

OpenAI disclosed that the total compute cost for generating all ten solutions was roughly $2,000 in token costs at the Sol API rate . This figure is an API-equivalent price for the solution-finding tokens only — it excludes failed attempts, parallel exploration runs, and the internal compute spent searching before landing on a certificate . By comparison, a single human mathematics PhD student's annual stipend at a top university can exceed $50,000, making the cost efficiency the most striking aspect of the announcement for many observers .

Criticisms and skepticism

The mathematics community and AI researchers have raised several points of caution:

• Mathematics community divided — Assessment of the results remains mixed. Some researchers caution that while the Lean certificates prove formal correctness, they do not guarantee the results are significant, surprising, or correctly framed as solutions to the intended open problems .
• Gary Marcus (cognitive scientist) argued that "success on math is a special case" and that expertise in one domain does not imply general reasoning ability. He listed eight common misconceptions, warning against overgeneralizing the result to all of science .
• "Who verifies AI proofs?" — Observers raised the open question of how to audit AI-generated mathematics when the reasoning is opaque. There is no agreed method for evaluating whether a black-box model's result is an uncredited reconstruction of known work .
• Lack of peer review and transparency — The announcement came just after the "Leiden Declaration" in June 2026, in which mathematicians criticized AI companies for bypassing peer review with press releases . OpenAI has not released the Astra model publicly, provided a model card, or disclosed the failure rate behind the $2,000 figure .
• Comparison to Google DeepMind — Skeptics noted that earlier AI math claims (e.g., DeepMind's AlphaGeometry) later faced scrutiny, urging caution before declaring a paradigm shift .

Broader implications

• Democratization of research — The $2,000 cost suggests a future where small teams or even individuals can use AI to make genuine research contributions that previously required years of human expertise .
• Crisis of verification culture — The field now faces an urgent need for agreed standards on how to evaluate, attribute, and peer-review AI-generated proofs when the model is a proprietary black box .
• Commercial control of mathematics — If frontier mathematical discovery becomes dependent on API access to proprietary models, core research infrastructure shifts from public institutions to private companies .
• New benchmark for AI capability — Unlike solving fixed competition problems (e.g., IMO or MATH), Astra produced genuinely novel open-problem results that advanced the frontier of knowledge, raising the bar for what "AI doing math" means . Thomas Bloom, a University of Manchester mathematician, called the results "big news" while also rejecting the idea that AI is replacing mathematicians, noting that the AI draws on more than a century of mathematical theory and was built by mathematicians .

The specific problems Astra solved

According to OpenAI's report, the results spanned eight fields and included proofs, counterexamples, and improved bounds :

• Non-sofic groups — First explicit construction proving the existence of non-sofic groups, closing a central open question in group theory that had resisted progress since the concept of soficity was introduced by Mikhail Gromov in 1999 .
• Connes's rigidity conjecture — Disproved a long-standing conjecture in von Neumann algebras .
• High-dimensional sphere packing — New upper bounds for sphere packing in dimensions beyond those already settled .
• Three problems posed by Paul Erdős — Resolved three open combinatorial problems from the Erdős canon, including Erdős problem No. 183 on multicolored Ramsey numbers .
• Additional results — New results in quantum parallel repetition, the closest vector problem in lattice cryptography, arithmetic circuit complexity, binary and spherical codes, and Ehrhart's volume conjecture .

How OpenAI verified the solutions

OpenAI did not rely solely on the model's output. Every result was accompanied by a Lean 4 formal proof certificate — a computer-checkable proof file that any reader can independently verify on a laptop . Lean is a proof assistant that checks every logical step from axioms to conclusion, catching gaps, type errors, and inconsistencies . OpenAI published all Lean files publicly on GitHub, and the repository's "sorry" count — indicating unproven steps — stands at zero for all ten proofs .

However, multiple commentators noted that formal verification has important limits: Lean can confirm that a formal statement follows from its formal definitions, but it cannot certify that a press-release summary accurately reflects the formal theorem, that the formal definitions match the mathematical community's intended problem, or that the result's novelty and historical framing are correct . Experts still need to audit statement fidelity, definitions, reductions, and the informal-to-formal bridge .

The cost: $2,000 in tokens

OpenAI disclosed that the total compute cost for generating all ten solutions was roughly $2,000 in token costs at the Sol API rate . This figure is an API-equivalent price for the solution-finding tokens only — it excludes failed attempts, parallel exploration runs, and the internal compute spent searching before landing on a certificate . By comparison, a single human mathematics PhD student's annual stipend at a top university can exceed $50,000, making the cost efficiency the most striking aspect of the announcement for many observers .

Criticisms and skepticism

The mathematics community and AI researchers have raised several points of caution:

• Mathematics community divided — Assessment of the results remains mixed. Some researchers caution that while the Lean certificates prove formal correctness, they do not guarantee the results are significant, surprising, or correctly framed as solutions to the intended open problems .
• Gary Marcus (cognitive scientist) argued that "success on math is a special case" and that expertise in one domain does not imply general reasoning ability. He listed eight common misconceptions, warning against overgeneralizing the result to all of science .
• "Who verifies AI proofs?" — Observers raised the open question of how to audit AI-generated mathematics when the reasoning is opaque. There is no agreed method for evaluating whether a black-box model's result is an uncredited reconstruction of known work .
• Lack of peer review and transparency — The announcement came just after the "Leiden Declaration" in June 2026, in which mathematicians criticized AI companies for bypassing peer review with press releases . OpenAI has not released the Astra model publicly, provided a model card, or disclosed the failure rate behind the $2,000 figure .
• Comparison to Google DeepMind — Skeptics noted that earlier AI math claims (e.g., DeepMind's AlphaGeometry) later faced scrutiny, urging caution before declaring a paradigm shift .

Broader implications

• Democratization of research — The $2,000 cost suggests a future where small teams or even individuals can use AI to make genuine research contributions that previously required years of human expertise .
• Crisis of verification culture — The field now faces an urgent need for agreed standards on how to evaluate, attribute, and peer-review AI-generated proofs when the model is a proprietary black box .
• Commercial control of mathematics — If frontier mathematical discovery becomes dependent on API access to proprietary models, core research infrastructure shifts from public institutions to private companies .
• New benchmark for AI capability — Unlike solving fixed competition problems (e.g., IMO or MATH), Astra produced genuinely novel open-problem results that advanced the frontier of knowledge, raising the bar for what "AI doing math" means . Thomas Bloom, a University of Manchester mathematician, called the results "big news" while also rejecting the idea that AI is replacing mathematicians, noting that the AI draws on more than a century of mathematical theory and was built by mathematicians .