[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"news-slug-openai-astra-ten-math-proofs-2026":3},{"id":4,"title":5,"summary":6,"content":7,"original_url":8,"source_id":9,"tags":10,"translations":27,"news_slug":34,"published_at":35,"created_at":36,"modified_at":37,"is_published":38,"publish_type":39,"image_url":14,"view_count":40},"0fd9ee7a-5b8f-49d2-9032-57f763de40e3","OpenAI 下一代模型 Astra 一口气破解 10 个数学难题:从 27 年未决的非 sofic 群到 46 年未动的高维球体堆积","OpenAI 内部下一代模型 Astra 一次性在 10 个长期未解的数学与理论计算机科学难题上取得重大突破,涵盖高维几何、群论、算子代数、量子复杂度、格密码等。整套发现仅用 2000 美元 Sol API 成本完成,所有证明均经 Lean 4 形式化验证并开源。","# OpenAI Astra 一口气破解 10 个数学难题:AI 当数学家,这次是来真的\n\n## 一次发布,十道题,199→2026 三个时间尺度的突破\n\n8 月 1 日,OpenAI 公布了内部下一代模型 **Astra** 在数学与理论计算机科学领域的最新成果。一次性给出 **10 个**长期未解难题的新结果。这些问题涉及**高维几何、编码理论、算术电路复杂度、群论、算子代数、量子复杂度、格密码学、极值组合学**八个领域,覆盖的每个问题都至少十年没有实质性进展——而其中最久的一个,**已经被数学家们追问了 46 年**。\n\nOpenAI 选择这条路径的动机并不意外:延续 5 月公布 Erdős 单位距离猜想反例之后的同一路线——**在评估未发布模型的过程中,顺手把那些\"卡住人类几十年的问题\"解掉**。差别在于,上次是一道题,这次是十道。\n\n## 三个最值得记住的硬骨头\n\n**第一个:27 年未决的\"非 sofic 群\"问题。** 1999 年,阿贝尔奖得主 Gromov 提出 sofic 群的概念——如果一个无限复杂的群能用有限置换完美逼近,那它就是 sofic 的。问题在于:**是否所有可数群都是 sofic?** 27 年间,无数顶尖数学家尝试构造反例,无一成功。Astra 从「二元 Leavitt 代数的单位群」出发,结合 Kun-Thom 扩展图和 Thompson 群 V,硬生生推出一组逻辑矛盾,首次证明了\"非 sofic 群的存在性\"。\n\n**第二个:46 年冰封的高维球体堆积上界。** 1978 年,两位苏联数学家给出 Cohn–Elkies 阈值后,整整 46 年,任何维度的密度上限都寸步难进。2022 年,Serhiivna Viazovska 因解出 8 维与 24 维的特定维密度拿了菲尔兹奖——但走向无穷维时,人类还是停在 1978 年。Astra 这次直接给出了全新的证明,**精确算出指数衰减率,首次突破 1978 年的边界**。\n\n**第三个:1982 年菲尔兹奖得主 Connes 的\"刚性猜想\"被证伪。** Connes 当年认为:某些特殊的群生成的冯·诺依曼代数是独一无二的。Astra 没有只找一个反例——它构造了一个**可数无限的群家族**,彼此互不同构,但生成的冯·诺依曼代数完全相同。Fable 5 给出的评价是:**\"按菲尔茨奖标准,任何一项都足以获奖\"**。\n\n## 2000 美元,10 道题,Lean 4 全验证\n\n最让数学圈震撼的并不是结果本身,而是**整套发现的成本结构**。OpenAI 透露,Astra 找到这些解决方案消耗的 token,如果按 Sol API 价格计算,**总成本不到 2000 美元**,平均每道题约 200 美元——\"相当于一名研究生一个周末的津贴\"。\n\n更关键的是可信度。所有论证都由 Astra 自己用 **Lean 4 写了形式化证书**,附在每篇手稿之后。Lean 4 的形式化核验意味着:**每一个推理步骤都经过了机器检验,不存在靠\"感觉对\"蒙混过关的空间**。数学家 Elliot Glazer 在第一时间确认消息属实后,称这是\"**迄今最重要的 AI 辅助数学成果**\"。\n\nGitHub 仓库 openai\u002Ften-proofs 已经公开,10 道题对应的 Lean 证书、推理过程的模型自述,全部可独立复核。\n\n## 100,000 名科学家免费用 Astra\n\nOpenAI 在公告里同步推出一项新计划:为 **100,000 名科学家和数学家**免费开放 ChatGPT 的最强模型。同步公开的还有 reasoning-walkthroughs.pdf——10 道题对应的完整模型思维链自述,几乎可以看作 AI 数学家的工作日志。\n\n这与 OpenAI 近期\"模型自我评估 → 评估中意外出新结果 → 公布并开源\"的路线一脉相承:5 月的 Erdős 反例是\"意外之喜\",这次则是**用 Astra 评估窗口有意为之的成果**。数学界对这次发布的态度是分裂的:有人赞这是\"分水岭\",也有人提醒\"独立同行评议是最后一道关\"。\n\n## 这意味着什么:AI 数学家从\"会算\"到\"会构造\"\n\n如果说 5 月的 Erdős 反例是 AI 第一次在数学里\"发现\"——这次 Astra 干的事更远。它**构造**反例、**证明**不可逼近性、**证伪**菲尔兹奖级猜想、**精确化**已知极限。这不是 LLM 在做检索或模仿,这是**机器在执行真正的数学创造**——从选择构造对象、找到证明路径,到写出自洽的形式化证明,全部由模型独立完成。\n\ntest-time compute 远没到天花板。OpenAI 推理模型核心成员 Noam Brown 公开放话:\"**测试时计算远未封顶,连百万美元级别的世界性难题也可能被啃下**\"。\n\n数学还会是人类心智的荣耀吗?这个问题,可能从今天起,要重新回答了。\n\n---\n\n**参考资料:**\n- OpenAI 官方公告:https:\u002F\u002Fopenai.com\u002Findex\u002Ften-advances-in-mathematics\u002F\n- 论文 PDF:https:\u002F\u002Fcdn.openai.com\u002Fpdf\u002Ften-proofs-oai.pdf\n- 推理过程 PDF:https:\u002F\u002Fcdn.openai.com\u002Fpdf\u002Freasoning-walkthroughs.pdf\n- Lean 证明开源:https:\u002F\u002Fgithub.com\u002Fopenai\u002Ften-proofs","https:\u002F\u002Fopenai.com\u002Findex\u002Ften-advances-in-mathematics\u002F","15975962-b5fe-49e5-ae68-687ba6cb7015",[11,15,18,21,24],{"id":12,"name":13,"slug":13,"description":14,"color":14},"5e628969-6d2a-437f-998a-104e4b16cfb1","ai-progress",null,{"id":16,"name":17,"slug":17,"description":14,"color":14},"120fa59a-ff6f-4537-9bf5-f818df636a0e","benchmark",{"id":19,"name":20,"slug":20,"description":14,"color":14},"baf131c1-687a-49f4-87f6-4dd87c1c692f","gpt",{"id":22,"name":23,"slug":23,"description":14,"color":14},"01598627-1ea6-4b27-a5d8-874971571a71","llm",{"id":25,"name":26,"slug":26,"description":14,"color":14},"42e59a88-7795-47dc-a334-ef1e72c24347","openai",[28],{"id":29,"lang":30,"title":31,"summary":32,"content":33},"6d49ada3-b375-4cbf-922f-d41e72e88e70","en","OpenAI's next-generation model Astra cracks 10 open math problems at once: from the 27-year-old non-sofic group question to the 46-year-old high-dimensional sphere packing upper bound","OpenAI's internal next-generation model Astra delivers major advances on 10 long-standing open problems in mathematics and theoretical computer science, spanning high-dimensional geometry, group theory, operator algebras, quantum complexity, and lattice cryptography. Total cost of all the discoveries was about ,000 at Sol API rates, and every proof is formalized in Lean 4 and open-sourced.","# OpenAI's Astra Cracks 10 Open Math Problems at Once: When AI Becomes a Real Mathematician\n\n## One release, ten problems, breakthroughs across three time horizons (1999-2026)\n\nOn August 1, OpenAI published results from its internal next-generation model Astra across mathematics and theoretical computer science, delivering new results on ten long-standing open problems. The problems span eight fields — high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, operator algebras, quantum complexity, lattice cryptography, and extremal combinatorics. Every one of them had seen no meaningful progress on the main question for at least a decade, and the oldest has been stuck for 46 years.\n\nThe motivation is a direct continuation of May's move: OpenAI disproved the Erdős unit-distance conjecture while evaluating an unreleased model. Back then it was a single result. This time it is ten.\n\n## Three highlights worth remembering\n\n**First: the 27-year-old \"non-sofic group\" problem.** In 1999, Abel Prize laureate Gromov introduced the concept of sofic groups: a countable group is sofic if it can be perfectly approximated by finite permutations. Is every countable group sofic? For 27 years, the world's top mathematicians failed to construct a counterexample. Astra started from the unit group of the Leavitt algebra, combined Kun-Thom extension graphs and Thompson's group V, and pushed the logic to a contradiction — the first-ever proof that non-sofic groups exist.\n\n**Second: the 46-year-old high-dimensional sphere packing upper bound.** After two Soviet mathematicians set the Cohn-Elkies threshold in 1978, the density upper bound for general dimensions was frozen for 46 years. In 2022, Serhiivna Viazovska won the Fields Medal for solving 8- and 24-dimensional specific cases — but the infinite-dimensional case was still stuck. Astra delivered a fresh proof and computed the exponential decay rate, breaking through the 1978 boundary for the first time.\n\n**Third: Connes' rigidity conjecture (proposed by a 1982 Fields medalist) is disproved.** Connes claimed that certain groups are uniquely determined by their von Neumann algebras. Astra did not stop at a single counterexample — it constructed a countably infinite family of pairwise non-isomorphic groups that produce exactly the same von Neumann algebra. Fable 5's verdict: \"by Fields Medal standards, any one of these would be enough to win a prize.\"\n\n## ,000, 10 problems, full Lean 4 verification\n\nThe shock is not the results themselves, it is the cost structure. OpenAI reveals that the tokens Astra used to find these solutions cost less than ,000 at Sol API rates — roughly 00 per problem, \"about a graduate student's weekend stipend.\"\n\nThe credibility mechanism is what makes this airtight. Every argument was written by Astra itself in Lean 4, with the formal certificate appended to each manuscript. Lean 4's machine-checked verification means: every step was validated by a theorem prover, with no room for hand-waving. Mathematician Elliot Glazer confirmed the news within hours and called it \"the most important AI-assisted mathematics result to date.\"\n\nThe openai\u002Ften-proofs GitHub repository is now public, with all ten Lean certificates and the model's own narration of its reasoning chain, every step independently auditable.\n\n## 100,000 scientists and mathematicians get free Astra access\n\nOpenAI bundled the announcement with a new initiative: free access to its strongest ChatGPT models for 100,000 scientists and mathematicians. The companion reasoning-walkthroughs.pdf is also public — ten papers of the model thinking out loud while solving the problems, essentially a working log of an AI mathematician.\n\nThis is consistent with OpenAI's recent pattern: model self-evaluation → unexpected new results during evaluation → publish and open-source. May's Erdős disproof was a \"happy accident\"; this time the breakthroughs are the deliberate output of Astra's evaluation window. The mathematical community is split: some call it a watershed moment, others remind everyone that independent peer review is the final gate.\n\n## What this means: AI mathematicians from \"can compute\" to \"can construct\"\n\nIf May's Erdős disproof was the first time AI \"discovered\" something in mathematics, Astra now goes much further. It constructs counterexamples, proves non-approximability, disproves a Fields-medalist conjecture, and tightens a long-standing limit. This is not retrieval or imitation — this is a machine performing genuine mathematical creation, from picking the right construction, to finding the proof path, to writing a self-consistent formal certificate, all done by the model end-to-end.\n\nTest-time compute is nowhere near a ceiling. OpenAI's inference model lead Noam Brown said it publicly: \"test-time compute is far from saturated, even million-dollar-scale world problems could be cracked next.\"\n\nIs mathematics still the glory of the human mind? Starting today, that question probably needs a new answer.\n\n---\n\n**References:**\n- Official announcement: https:\u002F\u002Fopenai.com\u002Findex\u002Ften-advances-in-mathematics\u002F\n- Paper PDF: https:\u002F\u002Fcdn.openai.com\u002Fpdf\u002Ften-proofs-oai.pdf\n- Reasoning walkthroughs: https:\u002F\u002Fcdn.openai.com\u002Fpdf\u002Freasoning-walkthroughs.pdf\n- Lean proofs (open source): https:\u002F\u002Fgithub.com\u002Fopenai\u002Ften-proofs","openai-astra-ten-math-proofs-2026","2026-08-01T10:00:00Z","2026-08-03T00:07:44.630393Z","2026-08-03T00:07:44.630402Z",true,"agent",680]