{"rewrite":{"id":"r_ffce2b0f534d04109a69571e","clusterId":"c_afb70de318c33796bf315310","slug":"openai-model-astra-tackles-10-long-unsolved-math-problems-with-lean-4-proofs","model":"deepseek-v4-flash:free","headline":"OpenAI Model Astra Tackles 10 Long-Unsolved Math Problems With Lean 4 Proofs","summary":"OpenAI announced on August 1, 2026, that an internal version of its next flagship model, Astra, produced new results on 10 unsolved problems in mathematics and theoretical computer science. The targets had no major progress for at least a decade. OpenAI published a 249-page paper and machine-checkable proof data, with arguments formalized in Lean 4 for independent verification.","whyItMatters":"OpenAI is now grading its models on unsolved research questions rather than exam problems, publishing proofs a computer can verify.","webCardHtml":"\u003cp\u003eOpenAI has moved beyond exam benchmarks. An internal version of Astra, its next flagship model, worked on problems that had seen no major progress for at least 10 years, and the company published all 10 results in a 249-page paper with machine-checkable proof data.\u003c/p\u003e\u003cp\u003eThe mathematical arguments were drafted by human staff into paper manuscripts, then Astra formalized each one in Lean 4, software that verifies each reasoning step. The GitHub repository includes Lean-format proof data and tools for independent verification.\u003c/p\u003e\u003cp\u003eThe results include new upper bounds for sphere packing densities, a counterexample to Connes\u0026#39;s rigidity conjecture, and new lower bounds for arithmetic circuits computing the permanent. For binary codes and spherical codes, general upper bounds improved for the first time since 1977 and 1978.\u003c/p\u003e\u003cul\u003e\u003cli\u003e\u003cstrong\u003eSphere packing densities\u003c/strong\u003e: Improved upper bounds in high dimensions to the limits obtained by the Cohn-Elkies method.\u003c/li\u003e\u003cli\u003e\u003cstrong\u003eBinary and spherical codes\u003c/strong\u003e: Exponentially stronger upper bounds for maximum size, first general improvement since 1977 and 1978.\u003c/li\u003e\u003cli\u003e\u003cstrong\u003eNon-sofic group\u003c/strong\u003e: Constructed a group that cannot be approximated by finite permutations.\u003c/li\u003e\u003cli\u003e\u003cstrong\u003eConnes\u0026#39;s rigidity conjecture\u003c/strong\u003e: A counterexample where a group is not uniquely determined by its von Neumann algebra.\u003c/li\u003e\u003cli\u003e\u003cstrong\u003eArithmetic circuit complexity\u003c/strong\u003e: New lower bounds for computing the permanent of a matrix.\u003c/li\u003e\u003cli\u003e\u003cstrong\u003eParallel repetition theorem\u003c/strong\u003e: Extended to any finite two-player game using quantum entanglement.\u003c/li\u003e\u003cli\u003e\u003cstrong\u003eClosest vector problem\u003c/strong\u003e: Approximation is hard even with polynomial factor errors relative to lattice dimension.\u003c/li\u003e\u003cli\u003e\u003cstrong\u003eEhrhart\u0026#39;s volume conjecture\u003c/strong\u003e: Proved for convex bodies where the centroid is the only interior lattice point.\u003c/li\u003e\u003cli\u003e\u003cstrong\u003eMulticolor triangle Ramsey numbers\u003c/strong\u003e: Superexponential lower bounds proved.\u003c/li\u003e\u003cli\u003e\u003cstrong\u003eExtremal graph theory\u003c/strong\u003e: Counterexamples to a compactness conjecture and a degeneracy conjecture.\u003c/li\u003e\u003c/ul\u003e","blueskyPost":"OpenAI says its next flagship model, Astra, produced new results on 10 math problems unsolved for a decade-plus. Proofs are formalized in Lean 4 and published for anyone to verify.","twitterPost":"OpenAI's Astra model produced new results on 10 unsolved math problems, from sphere packing to Connes's rigidity conjecture. Proofs are in Lean 4, published on GitHub.","threadsPost":null,"newsletterBlurb":"OpenAI announced that an internal version of its next flagship model, Astra, achieved new results on 10 problems in math and theoretical computer science. The proofs, formalized in Lean 4, are published for independent verification.","attributionJson":"[{\"source\":\"GIGAZINE\",\"url\":\"https://gigazine.net/news/20260803-openai-astra-lean/\",\"title\":\"OpenAI's Next Flagship AI Model 'Astra' Achieves New Results on 10 Problems in Mathematics and Theoretical Computer Science, with Proofs Formalized in Lean 4 for Machine Verification\"}]","lintFlagsJson":null,"lintHits":0,"costUsd":0,"inputTokens":4646,"outputTokens":859,"status":"published","repairAttempts":0,"nextRepairAt":null,"factsAttemptedAt":1786305003,"createdAt":"2026-08-09T19:44:38.000Z","publishedAt":"2026-08-09T19:46:44.000Z","updatedAt":"2026-08-09T19:44:38.000Z"},"cluster":{"id":"c_afb70de318c33796bf315310","canonicalTitle":"OpenAIの次期主力AIモデル「Astra」が10件の数学・理論計算機科学の課題で新成果、証明をLean 4で形式化し機械検証可能に","representativeArticleId":"a_f413412fdcd91cd6550ad025","sourceCount":1,"writtenSourceCount":1,"writeAttempts":0,"isSolo":true,"entitiesJson":"{\"anime_titles\":[],\"manga_titles\":[],\"work_titles\":[\"Astra\"],\"studios\":[],\"people\":[],\"type\":\"news\",\"domain\":\"other\",\"is_roundup\":false}","contentType":"news","status":"published","firstSeenAt":"2026-08-03T03:40:00.000Z","lastSeenAt":"2026-08-03T03:40:00.000Z","updatedAt":"2026-08-09T19:46:44.000Z"},"attribution":[{"source":"GIGAZINE","url":"https://gigazine.net/news/20260803-openai-astra-lean/","title":"OpenAIの次期主力AIモデル「Astra」が10件の数学・理論計算機科学の課題で新成果、証明をLean 4で形式化し機械検証可能に"}],"entities":{"anime_titles":[],"manga_titles":[],"work_titles":["Astra"],"studios":[],"people":[],"type":"news","domain":"other","is_roundup":false},"keyFacts":null}