--- title: OpenAI's Astra model claims to have solved 10 open... description: OpenAI claims an internal frontier model codenamed Astra has produced machine-verified proofs for 10 open problems in mathematics, quantum complexity theory,... canonical: https://daily.dev/posts/openai-s-astra-model-claims-to-have-solved-10-open-problems-in-math-and-quantum-complexity-uwqok2qbn twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: OpenAI's Astra model claims to have solved 10 open problems in math and quantum complexity | daily.dev og:description: OpenAI claims an internal frontier model codenamed Astra has produced machine-verified proofs for 10 open problems in mathematics, quantum complexity theory,... og:url: https://daily.dev/posts/openai-s-astra-model-claims-to-have-solved-10-open-problems-in-math-and-quantum-complexity-uwqok2qbn og:image: https://api.daily.dev/og/posts/uwQOk2qbN.png og:image:alt: OpenAI's Astra model claims to have solved 10 open problems in math and quantum complexity og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # OpenAI's Astra model claims to have solved 10 open problems in math and quantum complexity **[Collections](https://daily.dev/sources/collections)** · 2 min read · 0 upvotes · 0 comments ## Summary OpenAI claims an internal frontier model codenamed Astra has produced machine-verified proofs for 10 open problems in mathematics, quantum complexity theory, and theoretical computer science. Results include a proof of quantum parallel repetition for all finite two-player entangled games, hardness results for the closest vector problem relevant to post-quantum cryptography, and advances in sphere packing, Ramsey numbers, and arithmetic circuit complexity. Four of the ten were counterexamples rather than proofs. The entire compute run reportedly cost around $2,000 in API tokens. None of the proofs have been peer-reviewed or publicly released yet, so independent verification is pending. If validated, this would signal that frontier-level mathematical reasoning could become accessible to smaller teams at budgetable inference costs. ## Content OpenAI says an internal version of its next frontier model, codenamed Astra, has produced machine-verified proofs for 10 longstanding open problems across mathematics, quantum complexity theory, and theoretical computer science. The reported results include a proof of quantum parallel repetition for all finite two-player entangled games, stronger hardness results for the closest vector problem (with implications for post-quantum cryptography), and advances in sphere packing, Ramsey numbers, group theory, and arithmetic circuit complexity. Four of the ten results were apparently counterexamples rather than proofs — meaning Astra didn't just confirm existing conjectures, it disproved some of them. None of this has been peer-reviewed yet. OpenAI hasn't released the full proofs publicly, so independent verification is still pending. That's a significant caveat — "machine-verified" means the proofs check out in a formal system like Lean, but it doesn't mean the mathematical community has had a chance to scrutinize the underlying reasoning or the problem formulations. One detail worth noting: the compute cost for the entire run was estimated at roughly $2,000 in API tokens. That's a strange number to lead with, but the implication is real — if frontier-level mathematical reasoning can be framed as a budgetable inference expense rather than a years-long research program, smaller teams could potentially access capabilities that previously required large institutional resources. The same week, Chinese labs reportedly pushed the cost of comparable AI work close to zero, which adds some context to the competitive pressure OpenAI is operating under. If the results hold up under scrutiny, this would be a meaningful milestone in AI-assisted research. But "if" is doing a lot of work in that sentence until the proofs are out in the open. ## Questions this post answers ### What open problems did OpenAI's Astra model reportedly solve? OpenAI reported machine-verified proofs for 10 longstanding open problems, including quantum parallel repetition for all finite two-player entangled games, stronger hardness results for the closest vector problem with implications for post-quantum cryptography, and advances in sphere packing, Ramsey numbers, group theory, and arithmetic circuit complexity. Four of the ten results were counterexamples disproving existing conjectures rather than confirming them. _Follow how AI-assisted proof generation evolves before trusting unverified research claims, tracked on daily.dev._ ### Has OpenAI's Astra model math results been peer reviewed? No, none of the claimed results have been peer-reviewed, and OpenAI has not publicly released the full proofs, so independent mathematical verification is still pending. Machine-verified in this context means the proofs check out formally in a system like Lean, not that the mathematical community has scrutinized the underlying reasoning or problem formulations. _Developers weighing AI research claims can track verification status of tools like this on daily.dev._ ### How much did it cost to run the AI model that produced these math proofs? The entire run was estimated to cost roughly $2,000 in API tokens, a figure notable because it suggests frontier-level mathematical reasoning could become a budgetable inference expense rather than a years-long institutional research program, potentially giving smaller teams access to capabilities once reserved for large labs. _Teams evaluating AI compute economics for research can follow cost-performance shifts on daily.dev._ ## Similar posts on daily.dev - [Ten advances in mathematics and theoretical computer science](https://daily.dev/posts/ten-advances-in-mathematics-and-theoretical-computer-science-ngbzjedvn) · OpenAI · 1 upvotes · 0 comments - [OpenAI teases Astra, its next major AI model, after it solves 10 long-standing math problems](https://daily.dev/posts/openai-teases-astra-its-next-major-ai-model-after-it-solves-10-long-standing-math-problems-jifwxasvg) · BleepingComputer · 1 upvotes · 0 comments - [OpenAI Publishes 10 AI-Generated Mathematical Breakthroughs Using Its Internal Model Astra](https://daily.dev/posts/openai-publishes-10-ai-generated-mathematical-breakthroughs-using-its-internal-model-astra-v6xjxofws) · Medium · 1 upvotes · 0 comments --- Tags: [#ai](https://daily.dev/tags/ai), [#llm](https://daily.dev/tags/llm), [#openai](https://daily.dev/tags/openai), [#quantum-computing](https://daily.dev/tags/quantum-computing) [View this post on daily.dev](https://daily.dev/posts/openai-s-astra-model-claims-to-have-solved-10-open-problems-in-math-and-quantum-complexity-uwqok2qbn) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"OpenAI's Astra model claims to have solved 10 open problems in math and quantum complexity","url":"https://daily.dev/posts/openai-s-astra-model-claims-to-have-solved-10-open-problems-in-math-and-quantum-complexity-uwqok2qbn","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/openai-s-astra-model-claims-to-have-solved-10-open-problems-in-math-and-quantum-complexity-uwqok2qbn"},"datePublished":"2026-08-10T20:13:59.050Z","dateModified":"2026-09-13T19:23:25.482Z","description":"OpenAI claims an internal frontier model codenamed Astra has produced machine-verified proofs for 10 open problems in mathematics, quantum complexity theory,...","isAccessibleForFree":true,"articleSection":"Collections","inLanguage":"en","publisher":{"@type":"Organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180}},"author":{"@type":"Organization","name":"Collections","logo":"https://media.daily.dev/image/upload/s--fk_6ycEi--/f_auto,q_auto/v1780996001/logos/collections?_a=BAMAMiWQ0","url":"https://daily.dev/sources/collections"},"commentCount":0,"discussionUrl":"https://daily.dev/posts/openai-s-astra-model-claims-to-have-solved-10-open-problems-in-math-and-quantum-complexity-uwqok2qbn","interactionStatistic":[{"@type":"InteractionCounter","interactionType":{"@type":"LikeAction"},"userInteractionCount":0},{"@type":"InteractionCounter","interactionType":{"@type":"CommentAction"},"userInteractionCount":0}],"keywords":"ai,llm,openai,quantum-computing","timeRequired":"PT2M"} {"@context":"https://schema.org","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://daily.dev"},{"@type":"ListItem","position":2,"name":"Collections","item":"https://daily.dev/sources/collections"},{"@type":"ListItem","position":3,"name":"OpenAI's Astra model claims to have solved 10 open problems in math and quantum complexity"}]} {"@context":"https://schema.org","@type":"FAQPage","@id":"https://daily.dev/posts/openai-s-astra-model-claims-to-have-solved-10-open-problems-in-math-and-quantum-complexity-uwqok2qbn#faq","mainEntity":[{"@type":"Question","name":"What open problems did OpenAI's Astra model reportedly solve?","acceptedAnswer":{"@type":"Answer","text":"OpenAI reported machine-verified proofs for 10 longstanding open problems, including quantum parallel repetition for all finite two-player entangled games, stronger hardness results for the closest vector problem with implications for post-quantum cryptography, and advances in sphere packing, Ramsey numbers, group theory, and arithmetic circuit complexity. Four of the ten results were counterexamples disproving existing conjectures rather than confirming them. Follow how AI-assisted proof generation evolves before trusting unverified research claims, tracked on daily.dev."}},{"@type":"Question","name":"Has OpenAI's Astra model math results been peer reviewed?","acceptedAnswer":{"@type":"Answer","text":"No, none of the claimed results have been peer-reviewed, and OpenAI has not publicly released the full proofs, so independent mathematical verification is still pending. Machine-verified in this context means the proofs check out formally in a system like Lean, not that the mathematical community has scrutinized the underlying reasoning or problem formulations. Developers weighing AI research claims can track verification status of tools like this on daily.dev."}},{"@type":"Question","name":"How much did it cost to run the AI model that produced these math proofs?","acceptedAnswer":{"@type":"Answer","text":"The entire run was estimated to cost roughly $2,000 in API tokens, a figure notable because it suggests frontier-level mathematical reasoning could become a budgetable inference expense rather than a years-long institutional research program, potentially giving smaller teams access to capabilities once reserved for large labs. Teams evaluating AI compute economics for research can follow cost-performance shifts on daily.dev."}}]} ```