{"data":{"id":2249,"lang":"en","title":"OpenAI's Unreleased Astra Model Solves 10 Open Math Problems for $2,000","summary":"An internal OpenAI model called Astra produced ten machine-verified proofs of decade-old open math problems for about $2,000 in compute.","category":"ai","category_label":"AI","published_at":"2026-08-25T02:01:16Z","updated_at":"2026-08-25T02:01:16Z","url":"https://techtools.cz/tools/news/article/2249-openai-s-unreleased-astra-model-solves-10-open-math-problems-for-2-000","api_url":"https://techtools.cz/news-api/v1/articles/2249?lang=en","markdown_url":"https://techtools.cz/news-api/v1/articles/2249.md?lang=en","image_url":"https://media.thenextweb.com/2026/08/openai-astra-model-ten-math-proofs-non-sofic-groups.avif","word_count":194,"reading_time_minutes":1,"sources":[{"url":"https://www.implicator.ai/openai-astra-10-math-problems-lean-proofs/","domain":"implicator.ai"},{"url":"https://thenextweb.com/news/openai-astra-model-ten-math-proofs-non-sofic-groups","domain":"thenextweb.com"}],"translation":{"lang":"cs","id":2322,"title":"Nezveřejněný model Astra od OpenAI vyřešil 10 otevřených matematických problémů za 2000 dolarů","url":"https://techtools.cz/tools/news/clanek/2322-nezverejneny-model-astra-od-openai-vyresil-10-otevrenych-matematickych-problemu","api_url":"https://techtools.cz/news-api/v1/articles/2322?lang=cs"},"content":"OpenAI says an internal, still-unreleased model called Astra has produced verified solutions to ten mathematics and theoretical computer science problems that had gone unsolved for a decade or more. The results include the first explicit construction of a non-sofic group, a question open since 1999, a disproof of Connes's rigidity conjecture, and the first improvement to sphere-packing density bounds since 1978.\n\nEach solution shipped with a machine-checkable Lean 4 proof certificate published on GitHub, so mathematicians can verify the logic step by step without trusting OpenAI's claims. Researchers used Astra to draft the arguments, then formalized them in Lean; the accompanying manuscript runs 249 pages and the GitHub repository's \"sorry\" count — Lean's marker for unproven gaps — is zero.\n\nOpenAI's math research head Sebastien Bubeck called the results \"beautiful,\" while researcher Noam Brown noted the lab hadn't spent much per problem, joking there were \"no Millennium Prize Problems (yet).\" Total compute cost across all ten solutions was roughly $2,000 at current API rates. Astra itself remains unavailable for public testing, and the announcement lands amid friction with mathematicians over AI labs publishing results via blog post rather than peer review, per June's Leiden Declaration."}}