OpenAI Models Solve Open Sphere-Packing, Group Theory Conjectures

OpenAI模型攻克球堆积与群论未解猜想 随着自动化推理模型接手技术性执行工作来解决长期悬而未决的公开难题，人类数学家的核心价值将从证明引理转向发挥审美鉴赏力与建立群体理解。 OpenAI数学家Mehtaab Sawhney与Mark Sellke探讨了前沿推理模型如何从基准测试解答者，蜕变为能够解决球堆积、编码理论和群论领域未解研究猜想的工具。这些模型并非依赖暴力破解，而是展现出类似于人类专家同事的行为——提出假设、在陷入死胡同时回溯、综合不同领域的数学文献，并且不知疲倦地执行精妙的epsilon-delta论证。研究人员详细解析了具体的突破性成果，包括确立Cohn–Elkies高维线性规划界，以及用精炼的15页证明构造出非sofic群。随着人工智能承担起证明生成中艰巨的机械性重担，与会者总结认为，数学家的主要角色将从技术性执行工作，转向高层次的鉴赏力、问题界定以及培育学术群体的共同理解。 OpenAI推理模型已开始独立解决未解的研究难题，包括证明高维球堆积的渐近线性规划界，以及构造出首个非sofic群。 (24:30) 模型的推理过程并未产生难以理解的内容或暴力穷举的计算，而是极度类似于人类专家的意识流笔记，展现出审慎的回溯、假设检验以及跨领域的文献综合能力。 (13:00) 该AI对非sofic群的构造仅用了15页直接的群论与组合数学推理，绕过了此前用于反驳相关Aldous–Lyons猜想的长达数百页的量子复杂性理论。 (50:00) 随着模型接手证明中繁琐的技术性执行工作，研究人员预测人类数学将脱离机械推导，转向学术策划、审美鉴赏以及群体的阐释与交流。 (60:00)

OpenAI researchers Mehtaab Sawhney and Mark Sellke argue that frontier reasoning models are evolving from search tools into genuine collaborators capable of solving open problems in pure mathematics. Rather than rendering human mathematicians obsolete, this shift decouples proof execution from comprehension, refocusing the profession on taste, synthesis, and communal understanding. Execution Doggedness and Bypassing Human Context Pollution (4:45) Human research often stalls because mathematicians face mental fatigue, become emotionally discouraged after early failures, or suffer from what Sellke describes as mental context pollution—an inability to clear cognitive baggage once an idea becomes tangled with an unsuccessful technical approach. Models, by contrast, navigate delicate technical hurdles like verifying epsilon-delta bounds without clerical error, and they can systematically evaluate alternative paths without emotional bias. Sawhney explains that this capability was initially apparent in literature discovery—such as surfacing buried solutions to open problems from Paul Erdős's archive—and extends to executing notoriously difficult existing strategies, such as approaches to the unit distance problem. The researchers note that because general-purpose reasoning models naturally acquire behaviors like backtracking and reformulation, they read like the unvarnished scratch notes of an expert collaborator. Resolving Conjectures in Sphere Packing and Error Codes (22:15) Exact sphere packing densities in Euclidean space remain known only for dimensions 1, 2, 3, 8, and 24. For higher dimensions, Henry Cohn and collaborators had used numerical experiments to conjecture an asymptotic linear programming (LP) bound scaling as roughly (e / 2π)^d ≈ 2^(-0.601d), but a structural proof remained elusive. An OpenAI reasoning model resolved the high-dimensional Cohn–Elkies LP bound by using complex analysis to construct the optimal asymptotic function, proving both that the bound achieves this value and that the linear program can do no better. When tasked with examining spherical codes and binary error-correcting codes on hypercubes, the model bypassed standard Kabatiansky–Levenshtein bounds by deploying representation theory to exploit geometric symmetries. When prompted by the researchers to push further, the model connected its representation-theoretic formulas back to the Euclidean sphere packing limit, revealing deep structural continuity across discrete and continuous spaces. Compact Disproofs: Constructing a Non-Sofic Group (43:30) Group theory studies symmetry through associative binary operations with identity and inverses, with sofic groups defined as those capable of being approximated by finite groups. While mathematicians historically hoped every group would be sofic—which would facilitate lifting properties like surjunctivity from finite to infinite structures—OpenAI's model proved the existence of a non-sofic group. While the related, broader Aldous–Lyons conjecture on unimodular random networks was disproved around 2024 through a 250-page paper drawing on quantum complexity theory, the model disproved that all groups are sofic in a direct, roughly 15-page argument. Sellke and Sawhney note that prior literature by mathematicians such as Gábor Kun and Andreas Thom had circled a particular combinatorial obstruction without being able to rule it out. Interrogating the model's reasoning traces revealed that it successfully neutralized this conspiratorial substructure by introducing a specific algebraic relation, demonstrating that machine proofs can be concise and mathematically elegant rather than opaque brute-force outputs. Redefining Mathematical Taste and Communal Absorption (40:00) Historically, generating a proof was so grueling that understanding, contextualization, and exposition were secondary byproducts of the author's individual struggle. With AI models absorbing the burden of long-horizon execution, the researchers suggest separating execution from supervision: a worker model can tackle technical execution while a supervisor maintains strategic focus and prevents tunnel vision. Sawhney defines mathematical taste in utilitarian terms, arguing that sound judgment is demonstrated by the ability to select the right paths and solve harder problems faster. Furthermore, as models drastically increase the volume of mathematical literature, they also accelerate its absorption. Sawhney notes that uploading complex arXiv papers to models to extract high-level strategies significantly accelerates literature review, lowering barriers for applied scientists in physics or engineering who need advanced mathematical machinery without decades of hyper-specialized training. Even with sustained exponential improvements in automated reasoning, Sellke cautions that the discipline's ceiling remains exceptionally high, meaning grand challenges like P versus NP may still remain out of reach. Ultimately, the researchers expect AI to elevate human mathematics by replacing mechanical derivation with framework building, curation, and the communal sharing of understanding.

OpenAI 研究员 Mehtaab Sawhney 与 Mark Sellke 提出，前沿推理模型已超越单纯的文献检索，展现出回溯、剪枝与严谨执行技术细节等类似人类专家的数学推理能力；随着 AI 接管繁重琐碎的证明推演，数学研究的核心瓶颈将转向人类的学术品味、框架构建与群体理解。 从繁琐执行到避免认知上下文污染 (4:45) 人类数学家在探索复杂猜想时，常因精力有限、惧怕冗长的“epsilon-delta”等精细技术验证，或在遭遇死胡同后产生认知“上下文污染”而过早放弃潜在路径。人类大脑容易将整体思路与失败的技术细节绑定，产生挫败感并偏见性地否定该方向。Sawhney 与 Sellke 指出，模型具备极强的执行韧性，既不因繁复推导而疲倦，也不会受到情绪干扰；它能客观评估障碍是根本性阻碍还是可绕过的代数细节，并随时重置上下文以探索新分支。这种机制并非纯粹的算力暴力穷举，而是展现了提出假设、回溯尝试与通过良好判断力剪枝的综合推理行为。 攻克高维球堆积与纠错码上界猜想 (20:00) 在高维空间中，球体体积随维度增加呈指数级收缩，最佳堆积密度问题困扰学界已久。此前数学家仅对极少数维度（如 1、2、3、8、24 维）获得了精确解，而由 Kabatiansky 与 Levenshtein 在 1970 年代建立的高维渐近上界约为 2^(-0.599d)。针对 Henry Cohn 与 Noam Elkies 提出的无限维线性规划（LP）框架，数学界长期通过数值模拟猜想高维极限应为 (e / 2π)^d ≈ 2^(-0.601...d)，却始终缺乏理论解释。OpenAI 的推理模型构造出了渐近最优检验函数，并利用仅数页优雅的复分析证明了该 LP 上界不仅能够达到、而且无法超越这一常数，解决了这一悬案。在球面码与超立方体上的二进制纠错码问题上，模型亦通过深入运用表示理论挖掘几何代数对称性，成功改善了长期停滞的经典理论上界。 以极简组合论证明非 Sofic 群的存在 (43:30) 关于是否所有群都是 Sofic 群（即可在精确技术意义上由有限群逼近的群）是群论中长达数十年的核心问题，该问题直接关涉动力系统中的 Gottschalk 猜想与 surjunctivity 性质。相比于大约两年前通过长达 250 页且极度依赖量子复杂性理论的论文才得以证伪的更广泛网络猜想（Aldous–Lyons 猜想），AI 模型构造非 Sofic 群的证明仅有约 15 页，完全限定在经典群论和组合数学框架内。模型精确识别出以往人类学者怀疑却未能排除的特定组合“阴谋”子结构，并通过引入恰当的代数关系彻底消解了该结构，其推导的极简性甚至引起了 Gábor Kun 和 Andreas Thom 等领域专家的迅速跟进与拓展研究。 数学研究范式向品味与群体理解演进 (55:00) 以往数学界的研究流程中，定理证明与理解完全捆绑：学者唯有亲自完成艰难推导，才能建立起对概念的深刻把握，而论文写作往往洗净了探索中的摩擦与直觉。Sawhney 认为，随着模型接管大规模推导任务，“品味”在实用主义层面上体现为凭借更优的判断力挑选高产路径并更快解决问题；未来研究流程或将演变为“负责品味与宏观方向的监督者”配合“长程推演执行者”的协作形态。同时，模型不仅成倍提高新定理的产出速度，也被学者用于快速解析 arXiv 预印本的核心结构与证明策略，从而缓解学术信息过载。数学研究的评价重心正从单一的证明占有权，转向跨领域阐明、框架构建以及促进人类社群共同理解的显性贡献。 Sellke 提醒，数学问题的难度上限极高，即便 AI 推理能力呈指数级增长，诸如 P vs NP 这类宏大谜题在相当长时期内仍可能无法企及；但模型普及将显著降低前沿数学工具的使用门槛，使理论物理与工程等应用学科无需依赖狭窄领域的顶尖专家，即可直接调用高深理论推动科研进展。

## Background 本期节目由 a16z 基础架构合伙人 Lisha Li 与 OpenAI 研究员 Mehtaab Sawhney、Mark Sellke 共同录制并发布于音频播客平台（2026年9月8日）。节目简介显示，对话探讨了 AI 数学推理能力的迅猛发展、近期模型在数学前沿取得的突破（如球堆积与非索菲克群的构造），以及当 AI 开始攻关数十年未解之难题时，对数学研究范式、人类品味与知识消化的深远影响。 ## TL;DR 两位从纯数学学术界转型加入 OpenAI 的青年数学家，分享了模型如何从跨领域文献检索演进到展现出严密的端到端推演与回溯能力。他们重点复盘了 AI 在超球堆积渐近线性规划界、纠错码优化以及构造非索菲克群（Non-sofic group）等前沿课题上的突破，打破了“机器只会暴力穷举或输出天书证明”的刻板印象。对谈强调，机器并未剥夺人类的乐趣，而是承担了耗竭精力的技术推导细节，同时通过辅助论文精读极大加速了数学吸收效率。未来数学家的核心价值将从单纯寻找证明，转向确立认知框架、概念整合、审美判断以及跨学科的沟通与赋能。 ## Atmosphere & style 对谈整体充满学者式的智性严谨与由衷的兴奋感，毫无浮夸的公关辞令，宛如两位顶尖数学青年在白板前与同行复盘震撼的实验笔记。两人以生动的比喻拆解晦涩的高等代数与分析学，讲述自己读研时苦思数月无果的问题被模型用数页优雅证明打通时的欣慰与释怀。整体节奏既有对机器推理本质的冷静审视，又洋溢着一种见证数学研究迎来“复兴”与认知解放的轻快与乐观。 ## Bulletin - `25:00` OpenAI 凭借模型突破独立攻克了超球堆积（Sphere Packing）的高维线性规划渐近上界，将渐近指数严格推导至约 2^(-0.601d)，并给出了极简短的复分析证明。 - `50:00` 在代数拓扑与群论领域，模型以仅约 15 页的精炼组合学证明直接构造出“非索菲克群（Non-sofic group）”，摆脱了人类以往依赖 450 页量子复杂性理论的沉重框架。 - `07:30` 模型展现出人类数学家般的推导与主动回溯（Backtracking）能力，同时消除了人类在研究中因沉没成本和思维定势导致的“上下文污染”。 - `01:00:00` 数学研究范式正在经历从“寻找证明”向“宏观框架理解与成果吸收”转移，AI 不仅呈指数级生成数学成果，还大幅降低了跨学科调用和理解前沿数学的门槛。 ## Key takeaways - `00:00` Mark Sellke 与 Mehtaab Sawhney 在见证 AI 达到国际数学奥林匹克（IMO）金牌水平及 GPT-5 快速检索并解决学术难题的能力后，从纯数学界转型加入 OpenAI。 - `05:00` AI 在数学推演中具备极高的微观执行力，能高精度稳定推进繁琐的 ε-δ 分析细节，有效克服了人类注意力耗尽的痛点。 - `07:30` 人类数学家在失败后容易产生认知偏差并放弃有潜力的路径，而模型在推理树展开与回溯中没有心理负担，不会产生上下文污染。 - `10:00` 数学论文和教科书仅展示高度精炼的公理化证明，抹去了中间的挫折与直觉，但强推理模型通过通用强化学习自然涌现出了试错与全局策略重构能力。 - `15:00` 公开的思维链表明，AI 的推理方式与资深数学家的非正式草稿极其相似，并非暴力穷举或随机猜测，而是具有清晰的数学审美与剪枝逻辑。 - `18:00` 在超球堆积问题中，除 1、2、3、8、24 等少数特殊维度外，通用维度的精确密度仍是未解之谜，贪心极大填充给出的下界为 2^(-d)。 - `25:00` 利用 Cohn-Elkies 线性规划框架，OpenAI 模型 Astra 严格给出了高维极限下的精确等式，攻克了长期悬而未决的渐近猜想。 - `30:00` 纠错码、超球面码与欧氏全空间球填充在几何与分析极限下呈现深层统一，模型通过深入挖掘表示论中的代数对称性推导出了超越以往文献的新界。 - `34:55` 模型表现出较强的“任务导向性”，达到最初设定的指数界目标后即停止，人类数学家的提示词“你能把它推向更深处吗”促成了表示论到全空间球填充的彻底闭环。 - `36:45` 所谓的“数学品味”在实用主义视角下本质上是在巨大搜索空间中做出高胜率决断并协同解决多个复合子问题的能力。 - `40:00` 宏观品味与底层蛮力执行在上下文结构上适合进行解耦分工，类似于监督模型与执行下属模型的协同机制。 - `43:24` 针对 1970 年代哥特沙尔克猜想延伸出的“是否存在非索菲克群”这一重大问题，模型利用纯粹群论和精致的组合学论证给出了确凿证明。 - `50:00` 相比于人类学者推翻奥尔德斯-莱昂斯猜想所耗费的 450 页量子复杂性论证，AI 生成的非索菲克群证明出人意料地短小精炼，仅占约 15 页。 - `53:00` 模型构造非索菲克群的核心在于通过巧妙施加代数事实，严格排除了一种前人无法解决的隐秘组合学子结构（即“排除了一场阴谋”）。 - `56:30` AI 带来的不仅是数学成果产出的指数级增长，更带来了“吸收数学”的效率革命，论文的快速解构极大加速了人类对新知识的消化。 - `01:00:00` 寻找证明不再是核心瓶颈后，数学家的角色正向宏观框架搭建、跨领域直觉融合以及社群共同理解等方向深刻演变。 - `01:02:18` 像 P vs NP 这样的超级难题构成了数学的极高天花板，而常规数学谜团的快速破解正在推动应用数学和工程学科迎来广泛复兴。 ## Detailed walkthrough ### `00:00` 数学研究的转折点与两位数学家的转型 数学研究往往伴随着漫长而充满挫败感的探索过程：数学家灵光一闪产生一个直觉，尝试数小时、数天乃至数周，最终因为细节卡壳或精力耗尽而放弃。然而，随着 AI 模型的突破，这种范式正经历前所未有的剧变。本期节目中，a16z 基础架构合伙人 Lisha Li 与 OpenAI 的两位数学家 Mehtaab Sawhney 和 Mark Sellke 展开对谈。Mark 和 Mehtaab 在进入工业界前均在纯数学领域深耕多年（两人曾师从同一个导师团队并在学术界发表过合著论文），他们的转向正是由大语言模型在数学推理上的突飞猛进所催化的。 Mark 提到，他的转折点源于去年夏天看到 AI 在国际数学奥林匹克（IMO）中斩获金牌水平的表现，这让他萌生了“一定要弄清楚底层机制”的冲动，并随即加入了 OpenAI。到了秋天，Mark 给还在学术界的 Mehtaab 开通了一个内部的 GPT-5 账号。Mehtaab 回忆，当时他一上手体验便迅速被彻底说服。作为组合数学领域的学者，Mehtaab 经常会浏览收集保罗·埃尔德什（Paul Erdős）猜想的知名公开数学问题库。在过去，最令他苦恼的事情莫过于看到一个标为“未解决（Open）”的问题，却无法确定它在海量、分散的数学文献中是否真的尚未被破解。 在一次尝试中，Mehtaab 和几位朋友正在讨论 Erdős 网站上的一个难题，众人琢磨了数小时仍毫无头绪，无法断定该问题是否在现有技术可触及的范围内。Mehtaab 随手将问题输入 GPT-5，短短五分钟后，模型便在茫茫文献中定位到了准确的参考文献，并直接给出了解决方案：“And it was just very nice, okay, to be told, yes, this is in reach.” 随后两人顺藤摸瓜，又在网站上接连找出了约 10 个类似被误标为未解、实则已有文献破解的案例。这一经历让 Mehtaab 彻底意识到了大模型在跨领域文献检索与知识连接上的惊人威力。 ### `05:00` 从全景检索到硬核推演：模型在复杂细节中的执行力 Lisha Li 指出，学术检索虽然是机器的天然强项，但过去一年来 AI 展现出的突破早已超越了单纯的“跨领域文献搬运”，而是迈入了更深层次的数学推理，其工作方式越来越贴近职业数学家的工作笔记。Mark 对此表示赞同，他分析道，除了熟悉全局文献的“搜索”优势外，AI 目前展现出的另一个决定性长板是极高精度的方案执行力（execution）。在真实的数学研究中，当提出一个直觉性思路后，往往需要处理大量繁琐严苛的推导，例如确保各项分析不等式中“$\epsilon$ 是否比 $\delta$ 还要小”。人类面对海量技术性细节时极易疲倦或迷失方向，而 AI 却能以极高的精度始终稳健地推进这类技术论证。 谈及 OpenAI 在 Astra 系列成果中所攻克的几何与组合难题（如单位距离问题及球堆积相关突破），Mehtaab 解释说，很多时候模型的思路并不是空穴来风的奇思妙想，其构造路径可能与过去数学大师们尝试过的方向非常接近。然而在数学研究中，把一个构想“真正做通”往往构成了整个攻关的核心战役： > “A not so uncommon experience is that you find out a year or two later that somebody else got the idea to work that you thought that didn't work.” —— Mehtaab Sawhney 在单位距离猜想等难题中充斥着极度棘手、吹毛求疵的微观细节。人类在面对这种问题时是在“和概率对赌”——明知一条路希望渺茫，由于时间精力有限，往往会理性地选择止损放弃；而模型不仅极为顽强，更展现出了一种优秀的“数学品味（mathematical judgment）”。在公开的思维链（chain of thought）总结中可以清晰看到，模型绝非盲目尝试所有分支的暴力穷举，而是凭借精准的数学知识对庞大的搜索树进行有效剪枝，最终锁定极难打通的正确推导路径。 ```mermaid flowchart TD accTitle: Human vs Model Mathematical Reasoning accDescr: 比较人类数学家与大模型在面对难题时的不同认知与搜索路径机制 subgraph Human["人类数学家模式"] H1["提出数学直觉与构想"] --> H2["陷入微观细节推演"] H2 --> H3{"遇到阻碍/失败"} H3 -->|"精力耗尽/降低信心"| H4["沉没成本与思维定势<br>（上下文污染，放弃尝试）"] end subgraph Model["大语言模型推理模式"] M1["数学知识储备与策略构思"] --> M2["严密执行细节分析<br>(ε-δ 对齐、符号推导)"] M2 --> M3{"路径受阻"} M3 -->|"无心理负担回溯"| M4["修剪搜索树分支"] M4 -->|"重构直觉与探索路径"| M2 M4 --> M5["打通完整定理证明"] end ``` ### `07:30` 认知偏差的消除：摆脱“被污染的上下文窗口” 两位研究员进一步探讨了人类研究者与模型在“试错心理学”上的本质区别。Mark 提出了一个关于人类大脑“上下文窗口”的生动比喻：当一个数学家沿着某条错误路径钻研很久后，整个人会陷入思维定势，最初的核心灵感会与那些行不通的错误分支紧密纠缠在一起。此时人类无法像对待机器那样直接把大脑回滚，重新克隆一个上周的自己并命令道：“不要走这条死胡同，换个方向去建立直觉”。人类在失败过一两次后，往往会本能地大幅调低对该思路可行性的心理预期，甚至因疲劳而彻底封杀这一方向。 与此相反，模型在推理树展开与回溯时表现出近乎冷酷的精确计算。当它在某条分支碰壁时，既能冷静地执行回溯（backtrack），又不会让失败的先入为主偏见污染整体的全局推理。面对 Lisha 关于“这究竟是随机采样的运气还是真正的数学推理”的提问，Mehtaab 强调，模型在多次解题过程中表现得异常精准，它展现出的并非胡乱猜测，而是一种极其审慎且计算周密的决策能力。面对复杂命题，它能够克制住人类常有的情绪化降级偏见，在浩瀚的推导迷宫中做出最符合逻辑的决断。 - 关键事实与论点总结： - Mark Sellke 与 Mehtaab Sawhney 均受去年 IMO 金牌水准模型启发，先后从全职纯数学研究转型加入 OpenAI。 - Erdős 开源问题库存在文献脱节的痛点，GPT-5 展现了数分钟内检索隐蔽文献并纠正“假性未解问题”的强大能力。 - 机器在数学推演中的决定性优势之一是精准执行细碎技术逻辑（如分析学中的 $\epsilon$-$\delta$ 检验），克服了人类容易注意力分散和精力耗损的弱点。 - 人类数学家常因“沉没成本”和“上下文污染”在受挫后放弃潜在可行的方向，而大模型凭借无认知偏差的回溯和搜索树剪枝，成功攻克了前人屡屡折戟的极端精细化问题。 ### `10:00` 数学训练集中的“动机缺失”与通用推理能力的涌现 Lisha Li 与研究员们探讨了模型在解题时展现的探索与回溯机制。虽然人们可以通过并行启动多个模型会话来同时尝试不同的路径，但如果在单一轨迹内模型能够主动反思并回溯（backtracking），这种行为就与人类数学家极其相似。因为人类正是通过不断犯错，才建立起对解题空间的直觉，进而排除那些不可行的死胡同。在人类合作中，一个人提出构想后，往往需要另一个人花费漫长的时间去推演验证；而 AI 模型的大规模推理则极大地压缩了这一迭代周期。 ```mermaid flowchart TD accTitle: Mathematical Training vs Model Reasoning Emergence accDescr: 比较传统数学文本的缺陷与通用推理模型涌现出的思维能力之间的差异。 subgraph DataGap["传统数学资料的缺陷"] D1["数学论文 / 教材"] --> D2["高度精炼的公理化形式<br/>（如 Rudin 实分析）"] D2 --> D3["抹去了直觉推导与失败历程"] end subgraph Emergence["通用推理能力涌现"] R1["通用强化学习 / 纯思考时间扩展"] --> R2["试错与主动回溯<br/>（Backtracking）"] R1 --> R3["全局策略重构<br/>（Starting over）"] end D3 -.->|"无法仅靠字面复现"| Emergence ``` Lisha 提出了一个深刻的认识论问题：从本质上看，现存的数学论文或教科书对于训练数学推理而言是极其匮乏的数据集。以经典教材（例如 Walter Rudin 的《实分析原理》）为例，其呈现的数学证明高度精炼且无可挑剔，但完全掩盖了人类为了构想出这些抽象定义所经历的漫长挣扎——读者根本看不到“为什么必须以这种极度抽象的方式来定义实数”。大多数学术论文撰写时也从未考虑过如何传授数学思维，然而模型输出的思考痕迹却表现出惊人地贴近真实数学思维的探索过程。 OpenAI 研究员对此作出回应，强调 OpenAI 致力于训练通用的推理模型，让模型能够跨领域、长时间地进行深度思考。诸如选择路径、遭遇阻碍时回溯、推倒重来等行为，并非数学领域的专属特例，而是通用推理在达到极高强度后必然展现的核心工具。“很多在数学中观察到的行为（例如回溯或重新开始），其实并不是数学特有的，它们是通用推理工具。如果你在推理方向上下足功夫，最终自然会观察到这些规律。” - 数学文本的局限：学术论文和教材仅记录成功构建的最终形态，遗失了研究过程中探索、碰壁和形成直觉的中间语义，缺乏代码库中天然具备的大上下文与紧密结构。 - 通用推理的泛化：模型并未依赖 Lean 等交互式定理证明器的“自动形式化”（auto-formalization）来作为推理的主导脚手架，其展现出的数学思维更接近人类自然的非形式化直觉。 --- ### `15:00` 思维链的透明度与阅读同行的“邮件草稿” 对于外界普遍抱有的“AI 是否仅仅是在暴力穷举或胡乱猜测”的质疑，研究员解释了 OpenAI 决定公开提炼后思维链（summarized chains of thought）的初衷。如果研究界只看到模型最终吐出的复杂完整证明，往往会陷入怀疑，无法判断模型是否在以某种彻底异化的外星逻辑在做推演。“实际上，它的推理方式与人类专家的思考过程惊人地相似。” 研究员生动地形容了阅读模型推理过程的感受：这非常像在阅读某位数学合作者发来的原始工作笔记或长篇邮件。虽然这些思考过程显得稍微凌乱、缺乏系统排版，但那种在邮件中向你倾倒未经雕琢的思路、层层剖析直觉的感觉，与资深同行的交流别无二致。 - 破除黑盒迷思：公开思维链使人类数学家能够核验其推导脉络，确认其具有合乎逻辑的论证结构与数学审美。 - 协作体验的真实感：模型的内部思考痕迹充满了非线性的草稿感，展现出跨文献组合概念、设定猜想与自我质疑的鲜活性。 --- ### `18:00` 超球堆积问题：从蜜蜂蜂巢到 8 维与 24 维的奇迹 在讨论近期公开发布的测试问题集时，研究员分享了自己最青睐的一个几何学经典课题——超球堆积问题（Sphere Packing Problem）。该问题的表述极其直观：在给定的 $d$ 维欧几里得空间中，放入互不重叠的单位半径超球体，球体所能占据空间的最大比例（密度 $\Delta_d$）是多少？ ```mermaid flowchart LR accTitle: Sphere Packing Problem across Dimensions accDescr: 展示不同空间维度下超球堆积问题的已知成果与代表性方法。 subgraph LowDim["低维空间探索"] D1["d = 1<br/>线段分割（密度 100%）"] D2["d = 2<br/>六边形晶格（蜂巢结构）"] D3["d = 3<br/>面心立方（Hales 计算机辅助证明）"] end subgraph MagicDim["特殊极值维度"] D8["d = 8<br/>E8 晶格"] D24["d = 24<br/>李奇晶格（Leech Lattice）"] end subgraph HighDim["高维未知领域"] DGeneral["d 为任意高维<br/>密度下界 2^(-d) 呈指数衰减"] end LowDim --> MagicDim MagicDim --> HighDim ``` 研究员按维度梳理了这一问题漫长而波折的探索史： 1. **$d=1$ 维**：问题平凡，单位球即单位线段，可以无缝填满整个实数轴，密度为 1。 2. **$d=2$ 维**：解为规则的六角形晶格（hexagonal lattice），类似于蜜蜂建造的蜂巢结构。研究员笑称，关于为何这种结构最优，“我对此最好的直觉就是蜜蜂就是这么做的，如果还有更高效的方式，大自然进化早就驱使蜜蜂换一种方式筑巢了”。尽管其形态显而易见，但人类直到 1960 年代才给出了严格的数学证明。 3. **$d=3$ 维**：解法正如杂货铺中堆叠橙子般的面心立方堆积（开普勒猜想）。托马斯·黑尔斯（Thomas Hales）在 2000 年代初期给出的证明长达数百页，充斥着庞杂的线性规划计算与极其精细脆弱的几何分部讨论，“这是数学界出了名的极丑证明（famously ugly argument）”。 4. **$d=8$ 维与 $d=24$ 维**：在极少数特殊的维度上，数学家迎来了极其优美的突破。2017 年前后被严格证明的最优堆积分别对应于 $E_8$ 根晶格与 24 维的李奇晶格（Leech lattice）。这两种晶格具有极高的对称性与异常紧密的填充密度，其构造与模形式、弦理论及编码理论紧密交织。 除上述 5 个特殊维度（1、2、3、8、24 维）之外，人类对于其余任何维度的精确堆积密度一无所知。针对普遍的 $d$ 维空间，通过贪心策略填装超球（即无法再放入多余球体时的状态）只能得出一个非常宽松的初等下界 $\Delta_d \ge 2^{-d}$。随着维度不断增加，高维单位球的体积相较于外接超立方体呈断崖式指数衰减，空间中绝大部分体积都集中在角落与外壳边缘，导致在极高维度下最优堆积所占据的真实空间比例极其微小，且人类现存的数学工具依然在此步履维艰。 ### `20:00` 高维球体填充与 Kabatiansky-Levenshtein 界 在高维几何中，球体体积的退化行为往往反直觉。由于球体不能相互重叠，高维空间中的最大填充密度（sphere packing density）$\Delta_d$ 会随着维度 $d$ 的增加呈指数级衰减。为了直观理解这一衰减现象，数学家很早就提出了一个极其简洁的贪心覆盖论证： 假设在空间中构建一个“极大填充”（maximal packing）——即无法再放入任何新的互不相交的球体。如果把该填充中的每个球体半径放大为原来的 2 倍（体积随之放大 $2^d$ 倍），这些放大后的球体必定能完全覆盖整个空间。否则，只要还存在任何未被覆盖的空白点，原本就能在该点处直接塞入一个新的球体。由此立即可知，任何极大填充的密度至少为 $2^{-d}$。 然而在高维空间中，普通的网格结构（如高维立方体网格）的性质极其奇特。如果在高维立方体的每个顶点处放置一个球体，这些球体的体积占比实际上极其微小；由于空间对角线方向极其空旷，立方体中心甚至可以塞入一个全新的球体而不会与角上的球体发生碰撞。 ```mermaid flowchart LR accTitle: 贪心球填充下界论证逻辑 accDescr: 展示极大填充如何通过将半径翻倍并覆盖全空间来推出密度至少为 2 的负 d 次方 subgraph GreedyPacking["极大填充构造与空间覆盖"] A["极大球体填充\n(无法再放入新球)"] -->|"半径放大 2 倍\n(体积放大 2^d 倍)"| B["放大球体完全覆盖整个空间"] B -->|"若有空隙则矛盾\n(否则原本可塞入新球)"| C["填充密度下界:至少占 2^(-d) 比例"] end ``` 在寻找高维填充密度的上限（上界）方面，数学界长期以来的最佳记录是 1970 年代苏联数学家 Kabatiansky 和 Levenshtein 证明的一个著名界限：大约为 $2^{-0.599d}$。这个数字并不是凭空出现的巧合，而是源于一个极其复杂的优化问题。虽然他们当年的论文极其精炼且难以搜寻（往往只有一两页长，省略了大量细节），但该上界几十年来一直未被撼动。 - **贪心极大填充下界**：通过半径放大 2 倍覆盖全空间的论证，填充密度至少为 $2^{-d}$。 - **1970 年代苏联经典上界**：Kabatiansky 与 Levenshtein 给出的渐近上界约为 $2^{-0.599d}$。 --- ### `25:00` Cohn-Elkies 线性规划界与 AI 模型的渐近突破 为了进一步改进球填充的界限，数学家 Henry Cohn 与 Noam Elkies 曾提出了一套基于线性规划（Linear Programming, LP）的方法。其核心思想是将寻找高维填充上界松弛为一个在无限维凸集上的优化问题：寻找一个满足特定性质的检验函数 $f: \mathbb{R}^d \to \mathbb{R}$。 具体而言，函数 $f$ 需要满足两条性质： 1. 当欧氏距离 $\|x\| \ge 1$ 时，$f(x) \le 0$。 2. $f$ 的傅里叶变换 $\widehat{f}(x)$ 在所有点上均非负（$\widehat{f}(x) \ge 0$）。 由于傅里叶变换是线性算子，寻找最优函数 $f$ 实际上构成了一个线性规划问题。Cohn-Elkies 证明，填充密度 $\Delta_d$ 受到比值 $\frac{f(0)}{\widehat{f}(0)}$ 乘以半径为 $1/2$ 的高维球体体积的约束。Maryna Viazovska 正是因在 8 维和 24 维中构造出了神奇的显式函数，使得该 LP 界与 $E_8$ 格和 Leech 格完全重合，从而赢得了 2022 年的菲尔兹奖。 然而在高维渐近极限（$d \to \infty$）下，这个 LP 优化问题的真实最优渐近值究竟是多少，一直是一个悬而未决的谜题。Cohn 等人此前曾通过精妙的高精度数值实验提出过渐近猜想，但完全无法给出理论证明。OpenAI 的模型（Astra）仅在被提示“分析高维空间中的这一线性规划”后，便独立攻克了该问题：它不仅构造出了显式函数证明了可达性，还严格证明了匹配的下界，给出了确切的等式： $$\mathrm{LP}(d) = \left( \frac{1}{e^{2\pi}} + o(1) \right)^d \approx 2^{-0.601\dots d}$$ “The model solution, especially for this being like, the LP can't do better than this was quite short. It's a few pages of... like complex analysis, but it's kind of exactly the right approach. Like once you see it, it's kind of, it's like unbelievable. Like why, why hadn't somebody done this before?” Mehtaab Sawhney 赞叹道。他回忆自己读研究生时曾在这个问题上苦思冥想了整整六个月，结果“取得了绝对为零的进展”（made absolutely zero progress），因此看到模型几页优雅的复分析证明时，感受到了极大的震撼与满足。 在此突破之后，讨论顺理成章地延伸至 Astra 解决的另外两个密切相关的问题：球面码（Spherical Codes）与纠错码中的二元码（Binary Codes）。从几何本质上看，球面码无非就是把同样的球填充搬到了高维超球面上；而信息论中的二元码，则是将同样的填充几何离散到了超立方体的顶点集上。 - **Cohn-Elkies LP 框架**：通过满足空间截断与傅里叶非负性的函数 $f$，给出球填充密度的松弛上界。 - **菲尔兹奖级别前作**：Viazovska 利用模形式为 8 维与 24 维构造了精确匹配的函数。 - **AI 取得的渐近严格解**：模型确立了高维极限下的精确等式，将渐近指数推进到了约 $2^{-0.601d}$。 - **从球填充到编码理论**：球填充、超球面上的球面码以及超立方体上的二元纠错码，在数学结构上均可归结为同一种几何填充与距离约束问题。 ### `30:00` 纠错码、高维超立方体与球填充的深层数学统一 研究人员首先深入解释了纠错码（Error-Correcting Codes）的基本直觉与数学定义。通信的本质是发送方传输一串比特，但传输信道难免会出现噪声或比特翻转损坏。优秀的通信协议必须能够容忍一定比例的随机错误，并精准解码还原出原始信息。就像日常英语具备容错能力一样，哪怕句中出现几个打字错误，读者依然能轻松理解其含义，而脆弱的通信方案则会直接崩溃。在数学上，固定长度的二进制字符串可以视作高维超立方体上的顶点。为了构建抗干扰字典，设计者必须选取彼此保持足够距离的合法码字集合——例如在超立方体中挑选所有数字求和为偶数的顶点，确保任意两个码字之间不相邻。当码字之间的汉明距离（Hamming Distance）足够大时，即使发生单比特翻转，接收端也可以根据几何邻近性断定它源自哪一个合法码字。 在高维设定下，信息论的核心实际问题演变为：在固定的错误率（如 1%）下，要容忍该错误，消息长度究竟需要膨胀多少？这构成了通信的信息论极限。这一问题与球面码（Spherical Codes）及欧氏全空间中的球填充（Sphere Packing）展现出深刻的对偶与渐近等价关系：当大球面上的小球半径趋近于零时，大球面的曲率影响逐渐隐去，局部几何便严格等价于全空间球填充。此前，KL（Kabatiansky-Levenshtein）等学者曾分别对球面与超立方体码给出了界，而 OpenAI 的模型在这类问题上均推导出了超越以往文献的新界。 尽管全空间线性规划界分析主要依赖复分析（Complex Analysis），但针对球面与超立方体的高维优化方法却重度依赖表示论（Representation Theory）。由于球面和超立方体具有高度的几何对称性，该证明的核心飞跃在于将已有的对称性利用程度大幅推向极端，使代数对称性以更为精妙复杂的表示论形式介入证明。令人惊奇的是，当研究人员对模型基于表示论给出的球面码公式取小球极限时，它直接自然退化并重现了全空间球填充的新界与特定猜想数值。 ```mermaid flowchart TD accTitle: 纠错码与球填充的等价性与表示论分析 accDescr: 展示从二进制超立方体与球面码出发，通过表示论方法取得突破，最终取极限推导出全空间球填充界的过程。 subgraph ProblemSpace ["高维码与对称空间"] A["高维超立方体二进制码<br/>(汉明距离约束)"] B["高维球面码 (Spherical Codes)<br/>(曲面几何约束)"] end subgraph ProofEngine ["模型数学突破"] C["深度挖掘代数对称性<br/>(高级表示论 Representation Theory)"] D["推导超越已有文献的<br/>全新紧致上界"] end subgraph Convergence ["几何极限退化"] E["小球极限 (Small-Sphere Limit)<br/>曲率趋近于零"] F["全空间球填充界<br/>(重现并证明猜想数值)"] end A --> C B --> C C --> D D --> E E --> F ``` - 几何解释：超立方体上的顶点集合若两两不相邻，即使无法纠错也至少具备检错能力；若码字间距达到相反对角线级别的汉明距离，则发生单比特错误时可 100% 纠错还原。 - 理论联系：球面码问题在曲率消弭的极限状态下，与全空间连续球填充共享相同的变分与分析极限。 --- ### `34:55` 人机交互提示：“你能把它推向更深处吗？” 在球填充与纠错码的系列突破中，绝大多数问题都是模型端到端独立输出的，唯独这一对关联成果的产生经历了一次极为关键的人机交互。研究人员最初仅仅提示模型去改进纠错码现有的指数衰减界。模型迅速给出了改进方案，并且在其证明架构中引入了一定程度的表示论工具。看到这一结果后，数学家们敏锐地意识到该路径未尽的潜力，于是追问了一句极其简短的提示：“你能把这个方法推向更深处吗？会发生什么？”（“Can you push this further? What happens?”）。 模型随即给出了更加深奥晦涩的高阶表示论推导，并穷尽了该方法的极限；令人震惊的是，推导出的终极公式在极限下居然直接给出了全空间球填充的猜想理论值。于是数学家们紧接着让模型直接从复分析和全空间角度补全论证，彻底闭环了整个猜想。谈及为什么模型起初没有一次性做到极致，研究员解释称这并非模型推理能力不足，而是因为模型具有极强的“任务导向性”（Task-Oriented）。 > “In this case what the model was asked to do originally for codes was to improve the bounds by some exponential factor... And, you know, it improved the bounds and it didn't try to push things too much further... It wasn't like a capabilities issue. It just didn't feel like it at the time.” —— Mark Sellke - 任务导向行为模式：只要满足了用户指令中界定的具体目标（例如改进常数或指数阶），模型就会认为任务圆满完成并停止延伸，除非被明确要求探索极限。 - 引导与自主的界限：当前数学研究自动化工作流中，资深数学家的敏锐直觉（辨认出某种工具蕴含更大潜力）依然扮演着阶段性路标的角色。 --- ### `36:45` 数学品味的功利主义定义与研究自主性 针对当前需要依赖外部 Harness（评估与交互套件）或人类提示的现象，Lisha Li 提出了深层疑问：随着模型一代代飞速扩展，AI 是会天然具备这种“自主判断哪条路线更有前景”的高阶研究品味，还是说它始终需要外部框架的介入？在数学发现中，“数学品味”（Mathematical Taste）究竟应该如何界定？ Mark Sellke 给出了坚定的功利主义（Utilitarian）回答。他认为，不必将所谓“品味”玄学化；在科研实践中，品味的本质就是能在巨大的可能性空间中迅速做出高胜率的决断。解决更困难的数学难题，本质上需要同时攻克大量环环相扣的中小型子问题。模型不可能在一瞬间像魔术般凭空抓出全部解法，而是必须在较长的连续上下文里协同推进多个不同想法、评估其交织效果并完成自洽拼装。模型能够攻克越来越难的综合性大题这一事实本身，就证明了它在一个连续推理单元中承载复合工作量的能力在急剧跃升，因而它已经在实质上拥有了越来越优秀的“品味”。 > “I tend to be pretty utilitarian in my view of taste. And if you're able to solve problems faster by making better judgments, I think that's the best general proxy I have for taste. And somehow the fact that solving harder problems means it has, kind of by definition means it has better taste.” —— Mark Sellke - 复合问题拆解：真正的数学突破从来不是单一灵感的闪现，而是多个彼此作用的组件形成合力；模型能自主串联起这些小问题的解决，本身就代表了高阶逻辑能力。 - 任务导向带来的暂时局限：模型偶尔会在已经完成突破的关键节点上“见好就收”，不再自动压榨出全部理论蕴涵，但这只是指令遵循特性的副产物，相较于其底层推理能力的爆炸式跃迁而言只是非常次要的工程细节。 ### `40:00` 数学品味、长程推理与监督机制的设计 在探讨高难度数学任务时，Mark Sellke 提出了一个关于认知分工的设想：如果让同一个模型一边专注于需要长上下文和深度投入的具体长程任务，一边又要展现宏观的“品味”（Taste），它很可能会陷入混乱。他构想道：“如果由一个模型负责品味，另一个模型作为监督 AI 的下属，负责长期投入去解决难题，我觉得目前来看这种模式完全行得通。” ```mermaid flowchart TD accTitle: 监督与执行分离的推理架构 accDescr: 展示负责宏观品味的监督模型与负责长程求解的下属模型之间的交互与协作模式 subgraph DualArchitecture ["双模型分工架构"] Supervisor["监督模型（负责宏观品味与判断）"] Worker["下属模型（负责长程任务与硬核求解）"] end Supervisor -->|"下发方向与修剪搜索空间"| Worker Worker -->|"反馈长程推理轨迹"| Supervisor Supervisor -->|"跳出局部死胡同 / 重新定向"| Worker ``` Lisha Li 对这种将宏观品味与具体求解解耦的观点深感好奇。她指出，这意味着两者的上下文不能互相污染；在应用 AI 和初创企业中，人们也经常面临类似的张力——究竟是模型本身决定了一切，还是外部框架（Harness）更关键？有时极简的 harness 能最大化释放模型的原始能力，但随后的训练往往又需要让 harness 与模型联合训练，以牢牢掌控模型展现出的推理轨迹。 Mehtaab Sawhney 则从做数学研究的日常体验出发，给出了一个非常切合人类经验的比喻：“这就像你做任何任务时，偶尔都会钻进牛角尖，埋头苦干却走不出来；这时只要有个朋友站在你肩膀后面看一眼，问一句‘你到底在干嘛？’——就仅仅是这种让你后退十秒钟抽身审视的瞬间，往往就极其有用。”他强调，人类团队合作通常比单打独斗更强大，模型之间亦是如此。Mark 补充说，所谓的数学品味，很大程度上是对“自己或某种方法究竟擅长解决哪些问题”的直觉嗅觉；这种对可推进方向的判断力，本身也是模型在不断成功完成任务的过程中顺理成章衍生出的副产品。 - 宏观品味与底层蛮力执行在上下文上可能需要一定程度的隔离，避免长程求解时逻辑互相干扰。 - 求解过程不仅是暴力枚举，更是在庞大的路径空间中进行关键抉择与极端剪枝。 - 协作式设计（类似于同伴审视）能帮助模型打破局部死胡同，实现思维的“退后一步”。 ### `43:24` 群论基础与索菲克群（Sofic Groups）的数学意涵 讨论随后转向了模型最新攻克的重大课题之一：非索菲克群（Non-sofic group）的存在性证明。为了建立清晰的数学直觉，Mark Sellke 首先回归群论（Group Theory）的基本定义。在数学中，群是刻画“对称性”（Symmetry）的核心语言。它包含一个元素集合以及一个满足结合律的乘法运算，集合中存在单位元（Identity $e$），并且对每个元素都存在唯一的逆元（Inverse）。它可以抽象地表示各种操作的复合，比如数的相乘、矩阵乘法，或是旋转——后者本质上也是矩阵乘法的特例。 群可以是有限的，也可以是无限的。例如，一个正方形的旋转对称群只包含 4 个元素；而一个圆的连续旋转群则拥有不可数无限多个元素。所谓索菲克群（Sofic groups），通常针对的是有限群或可数无限群（元素个数与整数集相当）。直观而言，一个群如果是索菲克的，就意味着它在某种精确的拓扑或组合意义上能够被“有限群”所逼近。 - 群的定义包含元素集合、乘法运算、结合律、单位元 $e$ 以及逆元。 - 索菲克群（Sofic groups）是一类能够被一系列有限群在局部或统计意义上良好逼近的群。 - 长期以来，数学界并不知道是否存在非索菲克群（Non-sofic group），而最新的突破性进展正是证明了“存在非索菲克群”。 ### `46:08` 哥特沙尔克猜想、奥尔德斯-莱昂斯猜想与凯莱图的逼近思想 对于为什么数学界如此渴望所有群都是索菲克群，Mark 解释道，如果所有群都能被有限群逼近，那将是一种美妙的“数学奇迹”，因为很多定理在有限群上容易证明，随后只需借助索菲克群的逼近性质便能将结论平移至无限群。这一问题最早可以追溯到 1970 年代瓦尔特·哥特沙尔克（Walter Gottschalk）关于群上动力系统的“满射群猜想”（Surjunctive Group Conjecture）。数学家证明了所有索菲克群必然都是满射群，这直接推动了大家去探究到底存不存在“非索菲克群”。 Mark 随后将索菲克问题与概率论中著名的奥尔德斯-莱昂斯猜想（Aldous-Lyons Conjecture）联系起来。该猜想大致表述为：任何具有“幺模性”（Unimodularity）的无限随机图，是否都能被一系列大型有限图所逼近？为了抛开繁琐的技术细节、用最直观的方式解释这种逼近，Mark 以整数集 $\mathbb{Z}$ 的凯莱图（Cayley Graph）为例展开说明： ```mermaid flowchart LR accTitle: 整数凯莱图与其有限模算术图逼近 accDescr: 展示无限整数链条如何通过模 n 整数构成的有限环状图来进行逼近 subgraph InfiniteZ ["无限凯莱图 (整数集 Z)"] Z1["..."] --- Z2["-1"] --- Z3["0"] --- Z4["1"] --- Z5["2"] --- Z6["..."] end subgraph FiniteZn ["有限逼近图 (整数模 n: Z/nZ)"] M0["0"] --- M1["1"] --- M2["2"] --- Mdot["..."] --- Mn["n-1"] --- M0 end InfiniteZ -.->|"局部邻域逼近 (当 n 趋于无穷)"| FiniteZn ``` 如果将整数集画成图，只需将每个相邻整数用边相连，形成一条双向无限延伸的链条。而要用有限图来逼近这个无限结构，最自然的构造就是取模 $n$ 剩余类群（Integers mod $n$）。在模 $n$ 的图结构中，顶点连成了一个包含 $n$ 个结点的闭合环；当 $n$ 变得极大时，对于任何站在某个点附近的观察者来说，局部的几何景象与整条无限的整数直线别无二致。正是这种通过有限对象逼近无限对象的结构特征，构成了探讨索菲克性与相关几何群论猜想的核心直觉。 - 满射群性质（Surjunctivity）：属于群上动力系统的重要性质；已知索菲克群必为满射群。 - 哥特沙尔克猜想（Gottschalk's Conjecture）：源自 1970 年代，询问是否每一个群都是满射群，是促成非索菲克群研究的核心历史动因。 - 奥尔德斯-莱昂斯猜想（Aldous-Lyons Conjecture）：概率论中的著名强猜想，探讨所有幺模随机网络是否都能由大型有限图逼近。 - 凯莱图（Cayley Graph）直观：整数集 $\mathbb{Z}$ 构成一条无限链条，其局部性质可以通过足够大的有限环 $\mathbb{Z}/n\mathbb{Z}$ 实现近乎一致的逼近。 ### `50:00` 索菲克群与奥尔德斯-莱昂斯猜想的几何直观与证明对比 讨论从索菲克群（Sofic group）直观的几何图像展开：想象用一个巨大的圆环来近似一条无限长的直线。在微观尺度上，如果只观察局部的任何一点及其邻域，圆环与直线看起来是完全一样的；只有当你把视野拉得非常远时，才能察觉到这种全局的几何差异——你面对的是一个封闭的圆，而非无限延伸的直线。在代数结构上，整数群 $\mathbb{Z}$ 和模 $n$ 整数群 $\mathbb{Z}/n\mathbb{Z}$ 本质上都是群（运算分别为普通的加法与模 $n$ 加法）。这些有限的模 $n$ 整数群就是完整整数群的索菲克近似，正是这种能被局部逼近的特性使得整数群成为一个索菲克群。 由此推演，“每个群都是索菲克群”这一著名猜想，本质上是将上述逼近性质推广到所有群。而奥尔德斯-莱昂斯猜想（Aldous-Lyons conjecture）则是一个更为广泛的几何/概率猜想：它认为对任意网络都可以进行类似的近似，且对代数结构的要求更少。然而，奥尔德斯-莱昂斯猜想在两年前已被人类学者证伪。那是一篇真正的力作（tour de force）——整整 250 页的论文建立在另外 200 页的前人工作之上，且深度依赖于量子计算复杂性理论（Quantum complexity theory），在纯数学与物理之间搭建了一座极其复杂的桥梁，几乎没有多少人能彻底读懂。 由于奥尔德斯-莱昂斯猜想比“所有群皆为索菲克群”更强，因此推翻强猜想在逻辑上弱于推翻“群皆索菲克”。但出人意料的是，AI 模型直接证明“存在非索菲克群（Non-sofic group）”的过程却出奇地短小精悍。它没有借用复杂的量子复杂性工具，而是完全扎根在纯粹的群论领域，仅使用了包括 Gábor Elek（注：音频中 Kuhn 等为语音识别偏差，实际涉及 Elek-Szabó 或 Thom 类似工作）等数学家现有的合理前人成果，整篇证明只有约 15 页左右。两者在图论视角下的本质区别在于：群对应的是确定的凯莱图（Cayley graph），需要确定性的网络逼近；而奥尔德斯-莱昂斯猜想则允许更宽泛的随机图结构。 ```mermaid flowchart TD accTitle: Sofic group and Aldous-Lyons conjecture comparison accDescr: 比较奥尔德斯-莱昂斯猜想的人类证伪与非索菲克群的AI证明在技术路径和篇幅上的差异 subgraph AL["奥尔德斯-莱昂斯猜想 (更宽泛)"] A1["允许随机图结构"] --> A2["人类证明：跨越量子复杂性理论"] A2 --> A3["体量巨大：约 250 页 + 200 页底座"] end subgraph Sofic["群皆索菲克猜想 (确定性凯莱图)"] B1["基于确定性凯莱图逼近"] --> B2["AI 模型证明：纯粹群论与精致组合学"] B2 --> B3["体量紧凑：仅约 15 页"] end ``` - 几何近似直观：用巨大圆环的局部平坦性来近似无限长直线，对应以有限模 $n$ 循环群局部逼近无限整数群 $\mathbb{Z}$。 - 难度反差：推翻奥尔德斯-莱昂斯猜想需要 450 页跨越量子复杂性的宏篇巨著，而 AI 构造非索菲克群仅用了约 15 页“非常正常、讲道理”的数学证明。 - 技术路径：AI 的证明完全留在群论体系内部，无需借助晦涩的外域跨界桥梁。 --- ### `53:00` AI 证明的短小精炼与其所排除的“代数组合阴谋” AI 所生成的数学成果呈现出一个非常反直觉的鲜明共性：其证明普遍非常简短优雅。在这项成果之前，人们普遍担忧 AI 一旦证明高深猜想，会吐出人类根本无法理解的成千上万页符号垃圾，但现实完全相反——“一年前如果有人告诉我这些 AI 证明全都是非常短小而优雅的，我会极其惊讶……现在只有人类才能写出两百页长的证明。” 构造非索菲克群的核心难点在于，“索菲克”这一属性很难被直接用代数手段抓住，你必须给出一个具象的判据来证明某个特定群“无法被任何有限群局部近似”。AI 找到的证明本质上是一个“极其精致微妙的组合学论证（delicate combinatorics）”。研究团队在完成构造后对模型进行了事后复盘（post-mortem），要求它解释其背后的真实机制。模型解释称，前人研究中其实已经隐约感觉到了某种无法被排除的具象组合子结构（combinatorial substructure），这正是前人止步之处；而模型敏锐地找到并追加了一条巧妙的代数事实，从而严格排除了这种异常情况。Mark Sellke 形容这就像是模型在逻辑上精准破获了一场隐秘的协同破坏：“这非常明显就是在试图排除一种‘阴谋’（trying to rule out a conspiracy）。” - 紧凑性反差：与机器暴力搜索可能带来的冗长推导不同，AI 目前给出的反例证明通常只有十几页，逻辑极其精巧。 - 组合学精妙性：非索菲克性的证明不是纯形式化的大力出奇迹，而是结合文献中已有的关键切入点，通过精准施加代数约束，排除了阻碍证明的特殊子结构。 - 人类数学家跟进：该结果发布后，被引用的原作者（如 Thom 等人）迅速发表了后续研究，进一步扩展了非索菲克群的例子，显示出学术社区已开始主动接纳并构建于 AI 成果之上。 --- ### `56:30` 数学吸收效率革命与未来数学家的角色转变 AI 在数学研究中的渗透引发了关于学术规范与数学家角色的深层思考。Lisha Li 提出，数学界目前对 AI 的接受度呈现出光谱分布：一些学者已经意识到不把 AI 纳入工作流将处于严重劣势，而另一些学者则对署名权、归属（attribution）以及论文审核的压力感到焦虑。 Mehtaab Sawhney 指出，AI 在指数级制造更多待验证数学成果的同时，也提供了一剂完美的解药——它极大加速了人类“吸收数学”的速度。在实际研究中，如果想读懂 arXiv 上一篇高深论文的证明，通常研读导读和引言非常耗时，但现在“直接把 PDF 丢进最喜欢的模型里，让它梳理出大致的证明策略”，效率呈数量级提升。针对 AI 是否会引发数学论文过载的担忧，Mehtaab 强调：“当然，模型会帮我们产出指数级增加的数学，但它们同时也让我们吸收这些数学变得容易得多。” 展望未来，数学研究的分工将经历深刻重构。以往，由于“证明一个猜想”耗费了人类学者 99% 的心血，导致理解、消化以及将结论融入人类知识图谱等工作都只是解题过程的副产品。随着 AI 接管艰深的证明推导，数学家的核心价值将更多转向审美、消化、概念整合，以及开拓不同领域直觉的新参与者如何共同推动数学发展。 > “当然，模型会帮我们产出指数级增加的数学，但它们同时也让我们吸收这些数学变得容易得多。” —— Mehtaab Sawhney - 论文阅读范式转变：研究人员不再逐页硬啃晦涩的预印本，而是利用大模型快速提取宏观证明策略和核心引理骨架。 - 降低数学门槛：强大的模型有望让非传统背景、拥有独特几何或物理直觉的人也能产出严肃数学成果，打破深奥推导技巧设置的壁垒。 - 劳动重心转移：数学活动的重心正从“极其艰苦地完成形式化推导”转向“评判问题价值、理解结构意义以及将其编织进人类认知整体”。 ### `01:00:00` 瓶颈转移与数学研究范式的演变 对谈首先讨论了传统数学与 AI 赋能数学在认知组织结构上的差异。在以往的数学界中，如果你能独自证明出一个定理，你自然会非常深入地理解它，并且在某种意义上，你有责任维护这项成果并向他人解释说明。但现在，过去的重大瓶颈——即推导和寻找证明本身——已经大大削弱，其他维度的限制随之浮现，这可能彻底改变数学知识的最优组织结构。 Lisha Li 提出疑问：这是否会让数学变得更像一门经验科学（empirical）？当原本稀缺的推理工作被机器分担，学科的运作逻辑是否会发生根本改变？毕竟，数学中最具乐趣的部分之一在于“理解”。如果未来的核心转向理解、沟通、整合以及保持人类的审美与品味（human taste），这是否就是当代数学家需要调整自身适应并衡量贡献的新方式？ Mehtaab Sawhney 认为，随着产生出来的数学成果越来越多，理解如何将它们放入恰当的框架，并向其他人类解释清楚以供欣赏，将变得越来越重要。在过去，这种能力被隐含地看重，是因为你正是那个给出证明、为所有人带来理解的人；而未来，数学家的价值将更多体现在担任桥梁的角色。关于群体理解的价值，Mehtaab Sawhney 明确指出： > “I think that more sort of that communal understanding will become much more implicit in how we view Massachusetts, but I think it will be an increasingly more explicit and valuable part of the subject.” - 过去的范式：证明者 = 理解者 = 知识维护者与传播者；寻找证明是绝对瓶颈。 - 未来的范式：机器提供证明与推理，瓶颈转向建立宏观认知框架与人类社群内部的沟通理解。 --- ### `01:02:18` 难题天花板、未知谜题与大众的“数学复兴” 讨论随后转向人类对重大未解之谜的探索。Mark Sellke 指出，数学这门学科的一个美好特质在于难题的难度上限极高。即使 AI 在数学上的能力呈现指数级提升，在现实中可能依然无法解决像 P vs NP 这样极其艰深的终极猜想。未来的数学研究可能会进一步向这些巨大的谜团靠拢，而那些较为常规、程式化的较小谜团则会退居次要。 对于投身研究多年的数学家而言，这一变革带来了极为实际的宽慰与兴奋。Mehtaab Sawhney 感慨道，有些问题他曾花费数月甚至数年苦苦思索却不得其解，而现在人类很快就能知道其中一部分问题的答案，“对此我感到非常开心”。Lisha Li 赞同道，数学本身是一个具有无限空间（“infinite field”）的领域，可以被创造的成果极其丰富，AI 将带来一场成果与理解的复兴（renaissance）。 随着 AI 降低理解门槛，即使那些不以数学为全职工作的人，理解前沿数学并探索心中好奇的能力也将大幅提高。Mark Sellke 形象地调侃道，如果你的工作需要用到高级数学，以往必须去专门寻找一位该领域的顶级学者帮忙；而现在，AI 让应用领域的从业者不再受限于此： > “Also, you know, if you're working on something that requires some math, you know, suddenly you don't need to like find a world expert on this topic to be able to use it in your own work.” - **重大未解之谜的地位**：如 P vs NP 等极高难度的核心问题仍具有坚固的堡垒性质，机器的辅助会促使学科注意力向最顶层的终极难题汇聚。 - **科研效率与满足感**：长年悬而未决的个人研究困惑，正借助模型推理加速找到答案。 - **跨学科外溢效应**：不仅惠及理论物理，还将极大加速应用数学与工程科学的发展，让非数学专业人士也能直接调用顶尖数学能力。 ```mermaid flowchart TD accTitle: AI推动下的数学知识演进流向 accDescr: 展现AI加速推理如何重塑数学难题分类与知识传播 subgraph DifficultyCeiling["数学问题的复杂度分层"] A["常规/较小的数学谜团"] -->|"AI规模化攻克"| B["快速转化为已知结论"] C["超高难度终极猜想\n(如 P vs NP)"] -->|"依然构成极高天花板"| D["人类与高级AI共同聚焦的核心前沿"] end subgraph KnowledgeDistribution["知识的转化与赋能"] B --> E["认知框架整理与人类传播沟通"] E --> F["应用数学与理论物理等跨学科赋能"] E --> G["非专职研究者也能理解与调用数学工具"] end ``` --- ### `01:04:14` 结语与节目声明 对谈在对应用数学加速发展的期待中走向尾声。Mark Sellke 提到，若应用数学的发展速度能大幅提升，这对整个世界而言将是一件极好的事。Lisha Li 对两位嘉宾 Mehtaab Sawhney 和 Mark Sellke 的分享表达了感谢，赞赏这场讨论充满趣味，并对 OpenAI 模型的飞速进步感到由衷振奋，期待不久后再次邀请两位做客。 随后播出了节目标准结语与免责声明： - 节目由 a16z 制作出品，收听平台涵盖 YouTube、Apple Podcasts 与 Spotify，社交平台推特账号为 `@a16z`，Substack 专栏为 `a16z.substack.com`。 - 免责声明明确指出，播客内容仅供信息参考，不构成任何法律、商业、税务或投资建议，亦不得用于评估任何投资或证券价值，且不针对任何 a16z 基金的现有或潜在投资者。a16z 及其附属机构可能持有节目中所讨论公司的投资头寸。详情见 `a16z.com/disclosures`。 ## Notable quotes > `56:30` 模型会帮我们产出指数级增加的数学，但它们同时也让我们吸收这些数学变得容易得多。 — Mehtaab Sawhney > `40:00` 如果由一个模型负责品味，另一个模型作为监督 AI 的下属，负责长期投入去解决难题，我觉得目前来看这种模式完全行得通。 — Mark Sellke ## Mentioned Lisha Li, Mehtaab Sawhney, Mark Sellke, OpenAI, GPT-5, Astra, Paul Erdős, Thomas Hales, Henry Cohn, Noam Elkies, Maryna Viazovska, Walter Rudin, Walter Gottschalk, Gábor Elek, Kabatiansky, Levenshtein, a16z ## Video description a16z Infra Partner Lisha Li sits down with OpenAI mathematicians Mehtaab Sawhney and Mark Sellke to discuss how quickly AI’s mathematical capabilities are advancing, what recent results reveal about model reasoning, and what happens when AI begins making progress on problems mathematicians have struggled with for decades. Mehtaab and Mark unpack several recent results from OpenAI’s models, including advances in sphere packing and the construction of a non-sofic group. They explain why the surprising part isn’t simply that models can search more possibilities or work longer than humans: in many cases, the reasoning traces look remarkably similar to the work of an expert mathematician, including choosing promising approaches, backtracking when they fail, and combining ideas from across the literature. They also explore what this means for mathematics itself: how the role of human taste and judgment may change, whether AI could produce far more mathematics than humans can absorb, and why models that accelerate discovery may also make sophisticated results easier to understand. [Source](https://mgln.ai/e/1344/afp-848985-injected.calisto.simplecastaudio.com/3f86df7b-51c6-4101-88a2-550dba782de8/episodes/0936ea4f-0124-43ff-b2b1-641cc247a0c3/audio/128/default.mp3?aid=rss_feed&awCollectionId=3f86df7b-51c6-4101-88a2-550dba782de8&awEpisodeId=0936ea4f-0124-43ff-b2b1-641cc247a0c3&feed=JGE3yC0V)

[56:30] Mehtaab Sawhney: 模型会帮我们产出指数级增加的数学，但它们同时也让我们吸收这些数学变得容易得多。 [40:00] Mark Sellke: 如果由一个模型负责品味，另一个模型作为监督 AI 的下属，负责长期投入去解决难题，我觉得目前来看这种模式完全行得通。

[55:00] Mehtaab Sawhney: I mean, of course, models are going to help us produce exponentially more mathematics, but they also make it much easier to absorb it. [35:00] Mehtaab Sawhney: I tend to be pretty utilitarian in my view of taste. And if you're able to solve problems faster by making better judgments, I think that's the best general proxy I have for taste.

[0:00] Often as a practicing mathematician, you have an idea, and then you kind of think it might work. Then you try for a few hours, a few weeks, and at some point you give up. Whereas for GPT, like, okay, a human told me to do this, like, let's just do this. And so that's why we're sort of in this renaissance of, like, reachable results. This is the best part about this problem, which is really nobody had any idea. Is the model just guessing in some insane way? Doesn't seem like there's a limit so far, but it doesn't have that context yet. a lot faster. The ceiling for difficulty of a math problem is pretty high. Even if AI continues getting exponentially better at math, Plausible will never solve something like P versus N. What's the ideal way that this is being taken up by the math community? Probably most at this point are like, okay, AI is obviously doing some non-trivial stuff. So AI isn't just getting better at math benchmarks. It's beginning to make progress on mathematical problems that have Leisha Lee sits down with OpenAI mathematicians, Metab Swani and Mark Selke to understand what's actually changing. They walk through recent results in sphere packing, coding theory, and group theory and explain why these advances can't be reduced to brute force. The models tried different approaches, abandoned dead ends, connect ideas across fields, and in some cases produce reasoning that reads surprisingly like the notes of a human mathematician. They also tackled the bigger question. becomes less of a bottleneck. From mathematical taste and human judgment to understanding an explosion of new results, Leisha, Metab, and Mark explore how AI could change not just what problems get solved, but what it means to practice mathematics. Well, thank you guys for coming. This is really exciting because I think math has been moving so fast. With AI, I just love to get to both practicing mathematicians and who work at OpenAI some of these results. So we have with us Mark Selke and Mithab Swani. We're connected actually because you, Faye, was actually your advisor. And so both of you guys have worked much more deeply in math since I've quit many, many, like over a decade ago. So this is very exciting to kind of hear a download of your thoughts on how OpenAI has been sort of approaching this and also just like where you think math is going with the incredibly rapid advance of how AI has been helping We can start off with some very basic questions. What do you do to the extent that you can, of course, share? And how did you come from being a practicing mathematician to working at OpenAI? Yeah, I mean, I guess we both... Bradley got excited... last year when the models started to really take off in math. So I joined a little bit before Metab. I saw the IMO gold medal last summer, basically. I thought, this is amazing. I want to see what the heck they did. Let me go see. And then, yeah, I guess in the fall, Mark gave me a GPT-5 account, and then I started playing with the models and very quickly became convinced that, yeah, it was extremely exciting to play with them. You two were collaborating before then. Yeah, we've known each other for a while. We have one paper we actually wrote jointly. Yeah. Yeah. So 2PG5 was your conversion? Yeah, yeah. What was the magic that sort of... What question do you throw at it? What process? Yeah, so I think actually, yeah, so I think how this started was, at least for me, the starting moment was something like, There's a collection of problems called So Paul Erdős is a very famous mathematician. He posed a bunch of problems, and so they've now all been collected on this site. And, So I specifically worked in combinatorics and A lot of these questions are among the most important, so it's always fun to flick through the slight, but One thing that often happened to me that was extremely frustrating was I would look at a question, see that it's marked as open, and then not actually know if it's correct, not actually know if it was still unsolved, because the literature is often quite hard to search. One instance, I just plugged it into GPT-5, and five minutes later, it found a reference. And this was a case where a few of my friends actually started thinking about the problem on the site. I was talking with them. And, I mean, we had spent a few hours. It wasn't clear if the problem was within reach. And it was just very nice, okay, to be told, yes, this is in reach. Here's how you do it. And, yeah, GPT-5 told me this. And then I told Mark about this. Yeah, this was sort of, yeah, this for me was quite a surprising moment. Yeah. And then we looked more into it and we found 10 more cases sort of like this. At the time, I feel like, being better at maybe making connections between people, as you're saying, like the search for whether there's been a result or a related thing earlier is just kind of humanely hard, but maybe better for machine. But I imagine as a the progress that's happened in the last year, what has been impressive has kind of reached beyond that. And maybe through talking about it more abstractly or if it's more natural to talk about it through one of the problems that has been recently announced through Astra, [5:00] enlightened me as to how the recent progress has been a lot more than just searching through more areas, making these connections between the field and perhaps just actually deeper more mathematical reasoning that's similar to it. a working mathematician. Yeah, I mean, I think... this like... search point of being familiar with everything is still definitely like a relative strength that maybe informs like, the types of problems that AI is solving now. I think there are some other relative strengths and weaknesses. Another relative strength that's pretty noticeable is just Like, it's very good at executing on some, like, idea once it has it. Whenever you have an idea, there's usually some amount of getting everything lined up. Is epsilon smaller than delta? This kind of thing. You have to get everything correct. And, Like for a human, it's easy to get lost in these kinds of details. And the AIs just kind of always nail these kinds of arguments, I find. Yeah, I feel like you guys will know more detail on this, but like for the unit distance problem, it was just like the approach, there was definitely contributions from OpenAI, but like the approach perhaps was suggested even earlier. originally by Erdős, and then... It's just that the actual reasoning was a very, very momentous feat. And so for a human, you're like, well, I only have a limited amount of time. And if after so many steps, it is still not clear. I mean, maybe you're like Andrew Wiles and you actually spent 10 years alone and do something. But it's not clear that the risk reward is not good enough, whereas for a human, GPT, okay, a human told me to do this, let's just do this. And so that's why we're sort of in this renaissance of reachable results. Does that track? And did you feel like with the ASTRO results, is that sort of like where the strengths have been primarily or there's an extra... ingredient or magic here? I feel like I think the unit distance example is quite telling in the sense of, Maybe the exact construction, you can make it look very similar to what people had tried before, but I think, I mean, often as a practicing mathematician, you have an idea and then you kind of think it might work. Then you try for a few hours, a few days, a few weeks. And... At some point you give up, and then A not so uncommon experience is that you find out a year or two later that somebody else got the idea to work that you thought that didn't work. So somehow getting an idea to work can even be a large portion of the battle. And I think, especially in the case of the unit distance conjecture, there's just a lot of extraordinarily finicky details. And very often when you're doing math, you're kind of gambling against the problem. You're like, maybe I should try this approach, but it seems really unlikely and just not worth my time. And the model, I think, In several of these cases, both by combining what it knew and having good taste, kind of made the correct math. And you can kind of see this in the summarized chain of thought we released. You can sort of look at it. It's reasonably like a mathematician, and because it knows a few very correct bits, it makes the right decisions and eventually able to prune the search tree. It's not really trying everything. It tries a lot of different things. It's extremely dogged. I mean, it can't try every idea. It has to try a limited set of ideas. And it's able... to kind of use its knowledge, plus good mathematical judgment and find the right path to go along. So, I mean, that for me was like, because this was a problem which a lot of people have thought about. And I mean, the fact that the idea is not so foreign probably indicates that a lot of people have tried it, or at least a few very serious mathematicians have tried it. And I think that's what made it really interesting to see. I think something else that I feel when I see these proofs is like, If I have an idea and I'm trying to execute, It might be that I have some wrong plan for like how to get things to work. And, As a human, if you have some wrong path you go down for a while, it can be hard to rewire your brain to start over and try a different path. The initial idea is kind of linked in your brain with these other things that ended up not working. It's sort of like your context window is like a little polluted and you can't just make another clone of yourself from last week and say, don't do this. Try something else. Build your intuition in another direction. But it's very easy to do this with an AI. So I think this is another reason that it's like, Getting the details right once you have some good general direction is much less of a barrier all of a sudden. And when you say it's much easier to do with AI, it's like it's not actually being directed with human interference to it. As you were saying, the reasoning traces, it's like making these choices. Maybe it backtracks, but then it's able to not be distracted by maybe the context in which it's thinking about the problem, like by these machinery. And it's like, do you see it kind of go back as well? Or is it just making good choices? Like, is it a lucky sample or is it actually reasoning like a mathematician or, OK, it doesn't do well in this path, but it goes back, but then it doesn't let that pollute? and thinks about it. I think it's somehow very calculating, very correct. I mean, as human mathematicians, you're not always perfect at making these decisions. The first time something doesn't work, you automatically kind of downgrade how likely this approach is to work, and you keep doing this a few times. The model somehow is much better able to, like... It seems... for several of the solutions we've seen, somehow, [10:00] It seems much better. that are able to update how likely the path is to work versus rejecting a path versus a human doing it. But I think even if it weren't, the fact that you could just start another model session over means like, you know, it's always going to be the case that, Got it, got it. So in some sense, it is like still leveraging the fact that you could like run kind of parallel, you know, agents on the problem. But if it were kind of backtracking, then does make it seem much more like a human, you know, mathematician. And perhaps it is kind of doing some of that stuff, too. Because like, obviously, like, we have to like make mistakes in order to like even gain intuition for like, why that solution space is like, not, you know, not in the set of paths that it could be in. approach, you might tell another human your kind of general idea, and then they'll come back and figure out how to get it to work. And, you know, it's just, Yeah. It takes more time to do this like with humans. I wonder, I mean, you know, maybe this gets to the extent that you can actually talk about sort of like, obviously don't talk about the training recipes or whatever. But like, it's interesting that if you're just studying, for instance, for math papers, it's like a very poor training set, like a priority for math. Because, I mean, maybe math textbooks are even a pure example of this. It's like really bad at actually reconstructing the motivation for why things were. You know, it's like don't, I mean, maybe some people like it, but don't learn real analysis from Rudin. very clean already and crisp. And I think that's bad because it doesn't show the struggle that made us formulate definitions in a certain way. Like, why do we even need to have real numbers be defined in this super abstract way? way, et cetera. And so, you know, I think paper is also, I mean, unless you're, Most people don't write papers with the... context of I need to educate somebody to be a mathematician. And so like the actual maybe curriculum of like learning math is not inherent in like a lot of our artifacts as mathematicians. So maybe another way to ask this question is if the reasoning tracers are actually producing things that's like, okay, this is actually more close to mathematical thought, like how does that arise? Um... I mean, yeah, I guess OpenAI has been like the pioneer of reasoning models and teaching AI to reason in this way. We're doing a lot of work at kind of kind of all possible directions on, you know, teaching models to reason better and for longer and, you know, in all kinds of different domains. Um, I mean, I think... I think we're training general purpose reasoning models. Yeah. kind of one... A lot of these behaviors that we're describing mathematically, like backtracking or kind of starting again, I mean, these are these are not really specific to math. I mean, we're seeing them specifically in mathematics in these examples, but kind of their general purpose tools for reasoning. And I think. If you work hard at reasoning, you should see these patterns eventually. So it's a submergent because it's, I mean, I do think, That's why the OpenAI approach, um, was so, I mean, it's like it doesn't rely on, you know, doing auto formalization in order to like guide the reasoning. I think that's like obviously more like us. But it's just like so not obvious that if you're just like training on, say, a corpus of like math proofs, maybe auto formalized and lean that you get the sort of like projection of like how to think well. Like put another way, like with maybe if we think about it with code, like code is such a good corpus to train on, It's one of the few... data sets that have such large contacts. You just like, I mean, maybe you see this kind of with books, but they're less structurally interconnected. There's just like less structure there, I think. It's safe to say, like on average, in a book compared to like a piece of code. And so like with math papers, I feel like maybe what... What we're still bad at with coding models is stuff that that data set doesn't contain, which is like kind of the semantics, like the syntax is there, but there's a little bit of like the higher level semantics of what produced like, why do I have to write it this way? I'm kind of getting too much of the philosophical, but it is just like really interesting how it's still emergent that it's doing good mathematics. will push the brute force thing. You clearly are impressed with some of the reasoning traces, and it's just not obvious that's gleaned from... what we would imagine would be the easy training set here. Absolutely. I mean, I think this kind of thing is... one reason we decided it was important to release these summarized chains of thought for these kinds of results. Because if you... If you've never seen these and you just see all these proofs coming out, you're kind of you're not sure what it means. Is the model just guessing in some insane way? [15:00] Is it thinking in some totally foreign, like what's going on? But actually, it's reasoning kind of shockingly like an expert human would. Yeah, yeah. Yeah, I... It's very much like reading a colleague's notes. I mean, it's a little more disorganized in some way, but kind of, especially if you work close enough with the cobbler, sometimes you'll just see them like, spill out their thoughts in an email to you. And it kind of, it feels like reading a lot of those chained together. So it's, Yeah, it's... Sorry. Sponsored. It's quite surprising the first few times. Were you two sort of very involved in choosing the problems to release in this last 10 problem set that Astra is applied to? Which was your favorite? Yeah, we are definitely involved. Do you want to start? Yeah, I mean, yeah, I guess. Yeah, so I guess my personal favorite among these problems is the following. It's an extremely simple question, which is just like, it's just about how efficiently can you put a bunch? my circles are not very good and they're not all the same size. But we're assuming they are. Yeah. So the question is just like, How dense can you place a bunch of... So you have a bunch of spheres of radius one. and D-dimensions. So the question is, how densely can they pack? And so... Yeah, so... So in two dimensions, it's kind of like the... So D equals 1, this is not an interesting question, kind of. It's just the real line, and yeah, you can cut it up. A sphere in dimension one is just a unit segment, so you can cover everything. So in d=2, it's kind of, the picture that you know, that everybody loves. It's just like, uh, It's just a bunch of spheres which sort of form like, a hexagonal lattice. Hopefully I've drawn it well enough that I can draw the hexagon. Kind of betraying my naivete on this problem. Is that like obvious? Is it like a very elegant proof that it's a regular lattice? Yeah, it's not so obvious that this should work. It was only proven in the 60s, I think. There's a short argument, but it's not so easy. where's the intuition? Like, where is kind of like the, machinery of the argument. I mean, it definitely looks like it should work. That's why I'm going to go. Yeah, I think this is the best part about this problem, which is really nobody has any idea of this problem. So, yeah, so... I mean, Honestly, the best situation I have for this is that bees do this. And if there was a more efficient way, then probably bees would pack pottycombs some other way. Evolution is efficient. Yeah. I think beyond that, I don't have a great argument. I mean... And I think how little we know is demonstrated by the fact, so, okay, d equals 3, the answer is just like... it's how you pack oranges in a grocery store. And this was proved by Hales sometime in the 2000s. And like, And we don't have a short proof of this. I think the shortest proof is like a few hundred pages. What area does it draw from? So it's a lot of linear programming arguments and it's very delicate geometry. It's quite ugly actually. This is like a famously ugly argument. And then the two most famous results are d=8 and 24. 8 and 24. There must be some weird subspace thing. Yeah, exactly. So this was done in 2017. Sounds slightly prettier, though. So the reason it works out in these two very special dimensions is that So this is called a lattice packing. So it's kind of very regular. And it turns out in these two dimensions, there are two very special lattices. They're called the E8 and leech lattice. and they're very nice and they're like unusually dense like kind of they're just very very pretty structures coming from other areas of math and it turns out that they're the optimal structures but there's still like Regular. Yeah, they're very regular. But... I mean, beyond this, so we don't know any more exact dimensions. We know these five dimensions and we kind of don't know anything else. And I mean, To give an indication of how little we know, So there are two very surprising things about this. So you can define delta d to be the densest sphere packed in d dimensions. So... there's kind of an easy lower bound of like, 2 to the minus D. Yeah, this is... This is not so hard to show. Basically, um, Any packing where you can't put in another sphere has to have this density. So, okay, it's not ridiculously small. And we know that it has to decay exponentially. So it has to decay, like it grows like 1 minus C for some at least for some quantities. So in large dimensions, you can only cover like a vanishingly small portion. But we know basically nothing else. That's just because of the high-dimensional sphere thing where it occupies... [20:00] the volume behavior is weird. Yeah, so basically, I mean, basically they don't want to touch next to each other. I don't know if, I don't think there's a particularly short way to see that it's like exponentially small, but it's known to be exponentially small, and For a long, long time, the best bound was something like, this funny number, like 2 to the minus 0.599d. And this was proved by two mathematicians in the 70s. Kapitansky. Okay, that's a weird number, where's that? spit out it from? Is it like combinatorial? It's the answer to some extremely ugly optimization problem. There's like a nice underlying strategy. Okay, let's discuss what the music is about. Yeah, these were these two Russian mathematicians in the 70s. Yeah, it's actually very hard to find their paper. Like one page, it's like, two pages long. Yeah, they don't write very many details because the paper was fine. But The two negative d's, there's like a square lattice, like the dumb one. So yeah, it's actually not so easy to... So the argument for this is as follows. Basically, imagine that you have a set of spheres, and I tell... You can construct a set of spheres so that you can't put down another sphere. Because... If you could put down an extra sphere, you just keep putting it down. So you have a set of spheres so that there's no other sphere which you can put down. That's your, it's like almost like a... Just, yeah, just take any such packing. Okay, yeah, yeah. And I claim that this has to cover at least 2 to the minus d fraction. The reason is that, like, if you blew up each of these spheres by a factor of 2, then, like, they have to cover every point in space. And the reason is otherwise you could put down, if there was any empty space, you could put down a sphere there. Yeah, I agree. So I guess if you take the usual lattice, there are actually more places you can put things kind of diagonally. Okay, yeah, yeah. So that's actually not good. It's like a worse bound. Yeah, you can just keep plopping things in. So this is like more... Okay, good. Yeah, this is related to this really funny fact where you put a sphere on every point. If you take a cube in high dimensions, you put a sphere on every point. It's vanishingly small. It's vanishingly small. It's so small that you can put another sphere in the middle. Yeah, yeah. And it fits. Another high dimensional sphere behavior. Yeah, it's very weird. And so Okay. So the great part is that Um, So the model shows the following. I'll write two things. This is... Astra, I guess. Probably the right way to refer to this. So, okay, I'm going to write something called the LP bound. I'm not, I'll explain this in a second. And it shows that it's smaller than This very nice number. Do you want to say equals? Yeah, it's equals actually. Oh. e to the 2pi. was little of 1 to the d. And if you can work out what this number is, it's like... It's roughly something like 2 to the minus 0.6... Close about. Yeah, it's surprising. But this one, you're like, oh, maybe there's some nicer kind of structure there that fell out. But it's MOs. This is roughly something like 2 to the minus 0.6. 01 dot dot dot d. I think that's the numerics. I thought it was six or four. Great. Yeah, this shows mine. Yeah. Okay, so there are a couple of things. So first, what is this LP of D? So via Zasco's work actually builds on some earlier work. It turns out that there's a way to attack sphere packing via what's called a linear programming bound. So LP just stands for linear programming. So Conan Elke's um, gave an approach for sphere pocketing based on linear programming. So it's like a linear optimization problem over a convex set, but it's all kind of infinite dimensional here. And basically what this reduces down to is you try to understand the following. what you try to show is you, Basically, you construct a function f, So this is in D dimensions and it's mapping to R. And it has the following properties. So first... Um, Applebacks. So this is a function in d-dimension, so it's always less than zero. F. Liked. the size of X is bigger than one and you, Second, have the Fourier transform of X. This is always non-negative. So this is just, this is a linear program because the Fourier transform is a linear operator. Uh, Night Race. So you're taking just some arbitrary f that satisfies this property? Yeah, so you can take any f that satisfies these properties. And what they prove is that delta d is bounded by... the ratio of the Fourier transform at zero to its... uh, to the, um, the Fourier transform would zero up. an Apple zero to those. [25:00] over the Fourier transform of zero times the volume of the ball of radius one-half. and d-dimensions. And And so, Okay, this proof is not so short for experience math petition. It's like, it's half a paragraph to prove it, but a little bit tricky. And the point is, so... It turns out, so this is a relaxation problem. There's no guarantee that taking the optimal last will give you a good bound on delta d. But, So what Viasovsk did, and this was sort of the key I mean, a large part of the reasons you want to feel this medal and, um, in 2020 was that, or in 2022, was that she constructed a function in 8 and 24 dimensions such that this upper bound matches exactly these two very special lattices. And these are kind of miracles of nature that both you can construct this function and that it gives you the optimal balance. But you can just, this is a very, very natural problem. I mean, It's a function with two very simple properties, and you just want to understand the value. how this behaves in for large dimensions d and that was a big mystery um There was a numerics paper by Cohn and several others, which conjecture that, just based on doing numerics, that this was the answer. But they had no idea why this would be the answer. And what the model shows is that actually... The linear programming bound in large dimensions has this extremely nice asymptotic behavior. And the proof kind of explains where this is coming from. And because you understand this LP bound perfectly, this actually just gives a better bound on delta D. It turns out that this old bound can be kind of reinterpreted in this framework. And what the model does is it shows you the best possible bound you can get by this framework. So the model sort of made the connection. And what is this sort of like? So the model gives a function f, which so first it constructs a function f, which gives you this bound. better. So it's an equality, which is quite strong. So we now understand this problem in high dimensions very well. and that's Pretty remarkable. And the model was just kind of told, like, analyze this linear program in high dimensions, you know, go have fun. Got it. And to give an indication of how it was known, I think this conjecture was based basically only by doing numerics. Extremely clever numerics, but numeric. Okay. Yeah, you have to kind of figure out why this is the right thing to aim for, and it does. And that was pretty remarkable. Yeah, I mean, I had actually thought about this problem for about six months at some point when I was a graduate student. And yeah, I remember making like absolutely zero progress on it. So it was very nice to be like, explained why it was why it was true. So that was a pleasant experience. I think also in general, it was one of these solutions which I knew several people had tried the problem. It's pretty remarkable because like, The model solution, especially for this being like, the LP can't do better than this was quite short. It's a few pages of... like complex analysis, but it's kind of exactly the right approach. Like once you see it, it's kind of, it's like unbelievable. Like why, why hadn't somebody done this before? It was, it was like, there are many types of good mathematics, but I think one of them is just like, you see it and you're like, oh man, why didn't I think of this? And it was really fun. And I mean, I sort of knew why I didn't think of it, but it was quite nice to see it. And it was fun to see. that's why I like this I'm a lot. Yeah, so this is the first of the 10 problems that Estra solved. But the second is actually closely related. So this was sphere packing. The second one is spherical and binary codes. So, What's your draw code? You drew a packing. It's going to be the same picture. Okay, sure. Otherwise, we're going to have his picture. The hexagonal packing in our minds. Yeah, a spherical code is literally just a sphere packing, but on another sphere. Um, Yeah, so a spherical code is basically just a sphere packing on the surface of another sphere. So... Yeah. Yeah. Yeah, I'll cut the story. Yeah. Okay, so... Same picture as before, except you're kind of on a curved surface. Okay. Okay, so why is it called a code? Well, um... You can... I guess the reason is because of binary codes, which is, again, the same sort of thing, but now it's on a cube. Okay, yeah, fine. Let me draw a picture of a cube and some, like, simplest possible code on it. So, like... When you're sending a... So this is really about error correcting codes. [30:00] So what are error correcting codes? It's like I send you some string of bits, right? I'm, Mike. and I maybe I'm worried that some of the bits I send you get corrupted, right? So maybe just because of some errors in my system, this one gets changed. And we want some communication protocol so that... you can... decode this like small amount of error and like recover what I was trying to tell you. Uh, And you know, like normal English language kind of has this sort of property, right? If I make a few typos, you're going to be able to understand what I'm saying. But if we have some like really brittle communication scheme, it's not going to work. So... Um, Codes are kind of the... where you solve this. And mathematically, it just means like, you know, What's a binary string like this of a fixed length? It's like a point on some hypercube. And we want a dictionary of allowable code words that are separated from each other. So like in this case, if I don't want any two to be adjacent, I would kind of take these four vertices, kind of the even ones, if you sum up the... The digits, Right, and like... Okay. In this case, I guess if I have an error, you can't tell which one it's from, but at least you can tell it's like, Not. At least you can tell there was an error. Oh, I see, I see. Yeah, yeah, yeah, because it's like kind of sparse in the, yeah, it's like it's not too adjacent, so that like, this is like when the hemming distance is not the case? Yeah, yeah, yeah, right, right. You want, yeah, so you want like a large hemming distance between any distinct in your dictionary. And yeah, I guess if you take two opposite corners, then if I have like a single bit error, I can always like recover which point it was coming from. One that it's definitely closest to. very high-dimensional setting, what kind of rate can you get? And, like, for binary codes, it's really, like, you know, an extremely practical question. It's sort of like, if I send you... like an n-bit string, And there's like, you know, 1% error rate. Like, how much longer does my message have to become to tolerate that amount of errors? It's like some fundamental information theoretic limit of communication. Um, And, you know, but you can see, like, certainly this spherical case is, like, it looks very much like sphere packing. For example, if you, like, If you make all these little spheres really small, then the curvature of the big sphere is kind of not going to matter so much, and it looks like just packing spheres in full space. And in fact... Yeah, like these problems turned out to be very related. So for these problems, there were similar bounds coming from the KL authors and like There's like something for the sphere and something for the cube, but it's all kind of the same stuff. And our models found better bounds for these cases as well. And, uh, like, It's. You... I mean, the techniques look pretty different, actually, if you write them out. So this full-space analysis of this linear programming was using, like, Just complex analysis. Um, But if you... The method for these cases, we're using representation theory. Both the sphere and the cube have a lot of symmetry. And basically, the... The idea of the proof was to really leverage this symmetry. I, I, There's some amount of this in the previous existing method. And really, the improvement is to lean into the representation theory really hard and kind of make the algebraic symmetry enter in a more sophisticated way. And then it turns out that from the representation theory formulas, if you kind of take this small sphere limit in the spherical code case, you recover part of this result and you recover this value. So, Like this result isn't a special case. You kind of only went one direction of the bound from looking at it from the code's point of view, but like there's like a very close connection. Okay. Yeah. You guys let this like run in parallel. So it's like kind of discovering or because you're not sort of like feeding it So actually, this was the one case where there was some interactivity involved. Oh, interesting. Okay. So for all of, so except for this pair, it was just, you know, we had some problems, we fed them in, and we, you know, the model came back with some solutions. What happened here is actually pretty interesting. So we first asked it to improve the bounds for the codes. And it came back with an improvement that used some amount of representation theory. And then we kind of [35:00] asked it, hey, can you push this further? What happens? And then it came back with some much more sophisticated representation theory, and it turned out that you got this conjectured value for full space sphere packing out of that method by pushing it as far as it can go. So then we kind of asked to directly analyze this guy and try to complete the picture. Okay. Yeah. So the relationship isn't a coincidence. Yeah, yeah. It's interesting when you're saying the first prompt, which is maybe so basic, which is like, can you push this further? It does require some judgment from mathematicians. But eventually, you would imagine by scaling the models, you don't need to do that. Or there's another view that the harness actually does matter, and this is kind of part of the harness apparatus. Do you guys have any views on that with your working with Astra? Especially generations of models and how much you have to kind of input or how much the harness matters versus not? I mean, yeah, I guess there have been some funny quirks like this that just come from exactly what you asked the model to do, basically. In this case what the model was asked to do originally for codes was to improve the bounds by some some exponential factor So it really like shows up in this like leading constant up here And you know it improved the bounds and it didn't try to push things too much further. Like sometimes you see it do, but sometimes it just doesn't bother. But yeah, you know, you just ask it again and it goes further. So it wasn't like a capabilities issue. It just didn't feel like it at the time. Do you call that judgment or But like, what is the, because there is a, yeah, what do you call that? Models tend to be pretty task oriented. If you tell it to do a task, it accomplishes the task. It's pretty happy. So yeah, the task orientedness, it's like, but do we expect that level to kind of ascend up to, it's not that they will be less good at being task oriented. It's like, they'll ascend to the level of like, okay, no, let's go in this direction. You'll have the judgment too, because you guys have the judgment too. Like, okay, this is pretty promising. looks like you're using a lot of representation theory. It doesn't seem like there's a limit so far, but it doesn't have that context yet. But I guess what I'm trying to say is this one is maybe harder to extrapolate, but from previous generations when you had to give it more maybe prompting more of that harness work but eventually probably had to give it less So it probably gives you some confidence that there this like really fast ascension And do you see yeah like where Somehow solving a harder math problem is like you have to solve many smaller, like somewhat less hard math problems. And the fact that the math problems are getting harder is kind of an indication that you're, the model is able to take on more and more work in like a single continuous unit. And I think that's the thing that looks very promising. Somehow like any of these solutions, it's not like one idea, and you're kind of home free. You need several pieces to kind of interact and talk to each other. The model doesn't come up with all the ideas at once, right? It doesn't pull everything out in an instance. So kind of the fact that it needs to sort of see how this piece interacts with another piece, that's kind of like solving a problem in itself or piecing together many problems in itself. It could just be that, okay, when you're telling it, okay, push this even further, that was of the same order of magnitude as all the smaller things it's solving as well in between. And so you don't think that this is kind of a privileged direction. It's just sort of like, hey, let's give it one more help. Or you actually think that there's... I guess what we'll try to get at a bigger question is, is there a good sense of taste Because when people talk about for instance how well the models are getting at like doing research for instance we want a little bit of RSI and sort of like there surprising things about how that improves And then there's like, oh, you know, maybe right now it's a level of still like a junior researcher. It's like not really asking like the right problems. And so I'm just trying to get maybe a sense of where you're seeing that progress through the model advancements each generation. What is taste even? Yeah, I think somehow I tend to be pretty utilitarian in my view of taste. And if you're able to solve problems faster by making better judgments, I think that's the best general proxy I have for taste. And somehow the fact that solving harder problems means it has, kind of by definition means it has better taste. I think occasionally because they are task-oriented, you do occasionally get these symptoms of like, oh, it clearly has made a breakthrough. It kind of understands it's made a breakthrough, and then it doesn't kind of push all the way to the limit because that's not what you asked. But that seems rather minor compared to the state of progress we've seen so far. Okay, yeah. I think that's a pretty clear... [40:00] I think it's like maybe you're liable to get confused if you're trying to like, do a concrete long horizon task and show taste kind of at the same time. But like... If you... if you have like one model that's responsible for taste and one model that's responsible for going out and like, working for a long time at solving a hard problem, kind of as the like, uh, you know, I don't know. underling of the supervising AI. I feel like that kind of going to be fine. currently. Oh, interesting. Because that is like saying that these two things are somewhat separate, if not separate, at least they shouldn't kind of pollute each other's context, which is a little bit, I mean, it could be potentially like a strong, stronger statement than, I guess, you know, it's just kind of interesting because it's, it might just be like, to your point, it's, you know, let's take the utilitarian answer is solving harder and harder problems. It's doing a lot more than just like, you know, brute forcing something. It's making choices. It's like pruning, you know, a vastly large space of possible paths into something that's like really, you know, small path within that space. But like having like, why would, what would be like a separate model, a separate generation or something that's, a different version of the model that would contribute to taste, or maybe that's Totally, it's too abstract, doesn't make any sense. We should just let the actual... The related question would be like, what is the thing that gets us to... a better version of intelligence, the harness and the model? Is it just the model? And it's like we see this at least in... applied AI or startups where it's a continual battle of you need the harness, but then the harness adapts very poorly to a new model because sometimes a very, very minimal harness is still the best way to expose to the raw power of the model. But then now we also have these like, training regimes require the harness to be trained with them. Maybe part of this is to keep things more proprietary and harder for other people to use it. But I think partially it's maybe actually that it helps have more control on the reasoning traces you care about. It's a long rambling way of saying it's like, yeah, I don't actually. This is so interesting to see how the models have gotten better at math. And maybe something that's like very abstract and hard to describe, like taste, is a way to tease out what is actually necessary here. I think my only, like, non-trimonial thought here is that, like, when you're working, I mean, just when you're doing any task, occasionally you get pigeonholed and you like work really hard and just, having a friend look over your shoulder and be like, what are you doing? And then just like, just having that one bit of like, step back for 10 seconds, like this is often very useful. Yeah, yeah. I mean, I see no reason why humans would be so different than models somehow. Yep, yep. Having, or models would be so different than humans. Having a few humans working together is often more powerful than just having one. Yeah, it's like in this kind of collaborative thing, you actually, you kind of, yeah, artificially created it, but it's very similar and dynamic. But I think a lot of taste is also like having a sense of what problems you or like some method you have in mind are going to be good at solving. Like it's, I mean, certainly there's some amount of absolute aesthetic... point, right? But there's also just like, you know, having a nose for what you might want to pursue because you'll be able to make progress. And I think for that, like, there's, You would expect that as a side product of being good at completing tasks, you would get there sort of, right? Let me know if we still want to do a section on SOPHIC groups, because I think, you know, up to you guys, it's definitely super interesting. So maybe the first question is, what is a group? Let's remind ourselves. Group is... I set of elements. with some multiplication operation. Thank you. And basically, this is... how mathematicians think about symmetry. So, So you're like... Basically, if G and H are elements of your group, then G H. has some, is some other well-defined, uh, element of your group. And you have like... associativity, And you have an inverse. So for every G, there's some inverse. uh, And there's some like specific elements in the group that, uh, is kind of the identity. [45:00] E. I've, Okay. So it's some abstraction of like, composing operations. So these could be like numbers, they could be like multiplying matrices, they could be like like rotating something, which is, you know, a special case of multiplying matrices. Um, And, A group is... Sofic, If, well, there's some, you know, precise definition, uh, But... you know, roughly... It means... it. So I should say, groups that can be finite or infinite. So like, you know, Um, If you have a square, all the rotations of it form a group with four elements. If you have a circle, then the rotations form a group with uncountably many elements. Um, And so sofic groups are either finite or countable. You should think of them as being countably infinite. So there's like the same number of... elements as like integers. And So it's so fake if in some sense it can be, approximated by finite groups. So we didn't know if there was a non-sophic group. So the result that Aster proved is simply, that There exists... A non-sophic. Thank you. Before going to that, I feel like a lot of the programs in math is like, okay, we are such finite creatures. Let's see how well our finite approximations do. And in this case, especially for the countable case, maybe you'll be relating it to the Aldous Leons thing. It helps kind of anchor the picture of what we're doing. It's. seems like such a I mean, it's a nice result if it were true, but it's not. And it seems almost like reasonable. And so, yeah, I actually didn't go. I would love to hear the explanation of how it found a counterexample. Yeah, I mean, I would say that like, the hope that there was no non-Sophic group, so every group has this kind of approximation, like maybe this is... sort of like people hoping that there's a miracle. Because it turns out that groups like this have a lot of nice properties. because you can... run certain proofs for finite groups and then, you know, kind of approximate them and, whatever way the definition of being so thick lets you approximate them and I get the results. So like there's this notion of being a surjunctive group. So... there's, There's some fact-telling, that, uh, Any group which is Sofec, is also surjunctive. uh, So jumptiff is some property of like, um, dynamical systems on the group. And, um, I guess the original question was whether every group is, uh, surjunctive. This is some question of Gottschalk from the seventies. Uh, And this fact that follows this pattern of prove it for finite groups and then do this approximation is what motivated the question about if there's a non-sulfate group. Mm-hmm. Um, Yeah, maybe I'll say a little bit about this Aldous-Lyon's conjecture. Yeah, sure. Yeah, yeah. So I guess I had heard of this a little bit beforehand, because there's a related stronger conjecture in probability that was made popular by Aldous and Lyons. This conjecture, roughly what it says, is like... Any... like infinite graph with some nice property called... modularity. Um, I... Thank you. in modular random graph, can be... approximated... by large finite graphs. So maybe the way to... like, explain what these kinds of things are trying to say without getting into technical weeds is to say what they mean about the integers. So, So, like, how would I draw the integers as a graph? Um, So this is called the Kaley graph. You're just going to connect nearest neighbors. So there's some... kind of canonical way in which this is like the graph that represents the integers. Okay. And There's some sense in which you can approximate this by finite graphs, Why? Well, Um, If you look at integers mod n, then you kind of get the same picture. [50:00] But, uh... Like you have like a big circle instead of infinite line. Um, And the point is, if you look at any point here, and any point here. In nearby, things look the same. You have to go like very far away to kind of see this global geometric structure that you have a circle and not a line. And in fact, the integers and integers mod n are both groups just by like adding numbers or adding numbers mod n. So these integers mod n are like, Sofic approximations for the it full integers. So like this approximation is kind of why the integers are a Sofic group. Mm-hmm. So, um, So the the specificity, The The statement that every group is sofic is sort of a generalization of the fact that you can do this approximation with groups. And this Aldous-Leyne's conjecture, is kind of a broader conjecture that, like any network, you can do this, um, and it, like, you don't require as much algebraic structure, roughly. So it's kind of a broader conjecture. So, so this conjecture was disproved, uh, earlier. like two years ago. And it was kind of a really tour de force work. It was like 250 pages building on another 200 pages. It uses quantum complexity theory. So it really builds this like you know, very complicated bridge. And like, you know, I think not many people could understand this. Right. So since this is a stronger conjecture, the disproof, uh, is weaker than disproving this statement that all groups are so thick. But it turns out that the direct proof that there's a non-Sulfa group was like, much shorter and easier than this really amazing disproof of the Aldous-Leyens conjecture. It's like, like 15 pages maybe. And it doesn't, have any of this like very complicated connection with quantum complexity. It just kind of stays in group theory land. I mean, it uses some important like existing results by other mathematicians like Kuhn and Kuhn and Tom, but it's like, It's like a very reasonable, normal kind of proof. Yeah. And kind of spelling maybe out the obvious, but like the connection between the Sophic group statement is just you take the Cayley graph and that's the one that is like what they use or for the Aldous Leon. Yeah. Yeah. And so that's why it's like a, you know, a subset of. Right. So yeah, basically what happens is, yeah. So for a group, you can take exactly a Cayley graph. So you take some like, elements that like generate the group and you kind of connect elements that are adjacent. So in this case, like this is a Kaley graph of the integers. So when you do that from a group, you get like a deterministic graph. You just get like a single graph. You fix some set of generators. So this conjecture is stronger basically because it allows a broader set of graphs that aren't deterministic. It allows them to be random, but have some extra you know, um you know, modularity property that constrains exactly how it can be random. But yeah, basically that's the difference. Here you kind of have to give a deterministic network instead of a random one. Yeah, anything kind of... interesting, surprising about the results. I mean, you mentioned some things, which is like, it stayed within group theory, the techniques. I mean, I think maybe it's like a nice example of this general pattern that... theorems produced by AI have generally been, like, the proofs are pretty short, generally. They're, like... Um, Like with the counter examples so far. Yeah, but this one, it's like, okay, it's sort of a counter example, but like there's some, you know, there's some like, stuff you have to do to analyze things. The difficult part here is that like, The property of being a Sofit group is not so easy to get your hands on. So you have to find like... a concrete way of saying like, like producing a, producing a way of saying this group cannot be so big. And like, and the proof is actually, it's very short. It's like, it's almost a, it's a combinatorics argument, but it's like a very, very delicate combinatorics. The model is, Somehow you need to both have the right statement and know what piece of the literature and then execute it correctly. And that's very nice. Like, the difficulty of this problem is that, like, It's just a really, really hard, it's like very hard to get your hands on like being approximated by any possible finite group. Yeah, it's like what is happening at that countable infinity that's like resisting this approximation? Like, do you guys kind of, do you give a sense of, like, when you do postmortem and you're like, okay, Asterix, explain to me, like, what is the. Oh, yeah, we did that. Yeah, yeah, yeah. What was a good explanation you got out of it? I think there's some concrete combinatorial instruction. Basically, it's like... It's hard to explain, but there's like some concrete combinatoric substructure, which If you read the previous papers, [55:00] realized that that's what they couldn't rule out. And Astra found a way to kind of say, "Okay, no, no, if you add this one extra algebraic fact, this like, weird conspiracy can't happen. It's like very clearly trying to rule out a conspiracy. Yeah. that sort of previous authors had implicitly written about. And those were the actual suspects, it turned out. So they were sort of on the right track. And then this did the last mile of, well, whatever, however you quantify that. But I think it's like, Like a year ago, I would have been very surprised to learn like, All of these... AI proofs are like, very short and elegant. You're kind of afraid that they're going to generate all these thousand-page things. I'm never going to be able to understand it. But it's been kind of the opposite. Only humans can generate 200-page proofs right now. Yeah. And also, I was asking, if you do that post-mortem, it ends up usually engendering more mathematics. Because when you do that with humans, that's what... breeds new mathematics. So maybe if you kind of alter the prompt a little bit and be like, how would you generalize this or something? Yeah, I don't know if that's been a technique for you guys to have it explore and exploit what it has already developed. Well, there has been some, there has been follow-up on this already, actually, by Kun and Tom, who this result was built on. So math community is coming on. Yeah, which is kind of what we're hoping. We don't want to be But if there's some interesting follow-up, we're very, very happy that there's some follow-up building out these ideas more and giving more examples of non-selfic groups in this case. Yeah. Well, actually, maybe that's a great segue into how... What's the ideal way that this is being taken up by the math community? Because I feel like there's a spectrum of answers from working mathematicians. A sense of like... Probably most at this point are like, okay, AI is obviously doing some non-trivial stuff. It would be a disadvantage not to admit that in my workflow. I've definitely heard some stories where people are kind of, you know, boom would find it hard to either take AI as a co-author or like, how do you even do kind of attribution this way? But I don't know, like what, maybe to paint the more optimistic picture, so you're saying you want the mathematicians to be building on these results. It definitely generates a lot more results to be verified. So it, you know, puts pressure on the community and the profession? Like, how do you kind of expect the evolution of kind of uptake and collaboration with mathematicians? I mean, given that the fact that the models can produce sophisticated mathematics means that they can help you understand like sophisticated mathematics i mean like i don't know occasionally like i enjoy looking at the archive and i want to understand some proof and like i could read the introduction but in practice it's just much faster take the pdf put it into put into my favorite model and then like get a get an output of like what is the rough proof strategy and to have this like I mean, of course, models are going to help us produce exponentially more mathematics, but they also make it much easier to absorb it. Yeah. Yeah. Right now, okay, it's still a bit of a challenge back and forth, but I think it's, for me at least, a... It's much, much faster at understanding. It's much, much faster to understand a piece of mathematics with a model than without it. So, yeah. it's helping solve the problem it creates anyways. Yeah, I feel like that at least. And it's, you know, I don't, I don't view it as like creating much more problem. But again, like I don't have such, you know, high stakes and like, okay, I'm going to get, I'm not going to get tenure, et cetera. So I agree, making it more accessible. If I'm not spending so much time absorbing an area, I can put it into chat GPT and then expect to... I mean, you guys have an even more powerful model, hopefully releasing, for other people to enjoy as well. But it's... I think the positive version of that is actually more people can participate in mathematics. It's like people might be coming with other intuitions and they could actually maybe generate good mathematics. Is that sort of closer to the vision of what you're hoping this is going to be? you know, pushing towards, um, Or like what things do you think mathematicians do? be wary of to kind of adapt fast enough to take advantage of AI. Yeah, I mean, I think Certainly there will be a lot of changes, right? Like, I guess... In math, there are There are a lot of things that are kind of important for a given result, right? You need someone to come up with it, but you also need people to... understand and absorb it and like you know, internalize it enough to do more with it and like figure out where it fits into like humanity's understanding. Right. And like a, a couple of years ago, like, proving the result was like so hard that kind of the other stuff was just kind of coming along for the ride right [60:00] If you manage to prove this thing yourself, you're automatically going to understand it quite well. You're kind of responsible for maintaining it in some sense and explaining it to other people. And yeah, now this kind of... what was the main bottleneck before is kind of, um, much less of a bottleneck. And, you know, these other kind of constraints, uh, come into play. So it yeah, the like optimal... structuring for organizing the knowledge could look rather different. Yeah. How does that look? I mean, did— Does this make the field a lot more kind of empirical? Will people do sort of the hard work? like the first thing that was scarce, which is like all the reasoning and then more. I mean, not that it's like a bad thing to make it empirical, but it's almost like it functions as a very different discipline. Like a lot of the fun stuff is understanding, you know. And so understanding, communicating, maybe assembling, having still the human taste. Does that sort of remain... and that's how, you know, current mathematicians need to adapt and reward contributions or Or is this too much of a caricature? It's like something else. I think certainly understanding how to put as we get more and more mathematics, put it in like a proper framework and sort of how sort of like being able to explain it to other humans so that they can also appreciate it. I mean, So implicitly we valued this, but it was usually because you were the person proving the results that gave everybody else the understanding. But I think increasingly it would be a function of like, you're sort of helping, you're the human who can sort of, give this understanding to other people and sort of help them with it. I think that more sort of that communal understanding will become much more implicit in how we view Massachusetts, but I think it will be an increasingly more explicit and valuable part of the subject. I mean, a nice thing about math is that, um, the ceiling for difficulty of a math problem is pretty high. So even if... you know, kind of, Even if AI continues getting exponentially better at math, it might... you know, plausible will never solve something like P versus NP. And it could be that, like, the field of kind of becomes more... you know, attached to like, like these... big mysteries and less to like smaller mysteries that are more like routine now. Yeah, yeah. I think that's a positive vision. I mean, also, like, I don't know, there are things I spent, like, months or years of my life wondering about, not getting to know. And now we get that. Yeah, some portion of them will get to know the answer to it. I'm pretty happy about that. No, exactly. No, I'm excited about this, like, renaissance of results and understanding. And I feel like, I mean, this is such an infinite, you know, field. Like, no pun intended. But, like, it's just, like, it's just, there's so much that you can actually create here. So, I mean, especially for somebody like me who's not going to have the time to actually, like, practice mathematics. Now there's, like, a lot more that you can actually do in the activity now. So... Yeah, I think the... The ability of someone who's not working on math is like their literal job all the time to like understand what's going on and like, you know, learn about some of the mysteries they might have wondered about will go up quite a lot. Also, you know, if you're if you're like if you're working on something that requires some math, you know, suddenly you don't need to like find a world expert on this topic to be able to, you know, use it in your own work. Sorry, mathematicians. No, it's true. like people who could do that. And so I think this is helpful. Maybe it's helpful for theoretical physics, like we'll see. But a lot of other applied areas as well. It'd be nice for the world if applied mathematics went a lot faster. Yes. I mean, I'm a bad candidate. Well, thank you guys for joining. This is a lot of fun. And I'm, you know, just so excited for how much the models are advancing. So maybe we'll have you guys back soon. Thanks so much for having us. Yeah, thanks for having us. to this episode of the A16Z podcast. If you liked this episode, be sure to like, comment, subscribe, leave us a rating or review, and share it with your friends and family. For more episodes, go to YouTube, Apple Podcasts, and Spotify. Follow us on X at A16Z, and subscribe to our substack at a16z.substack.com. Thanks again for listening, and I'll see you in the next episode. As a reminder, the content here is for informational purposes only. It should not be taken as legal, business, tax, or investment advice, or be used to evaluate any investment or security, and is not directed at any investors or potential investors in any A16Z fund. Please note that A16Z and its affiliates may also maintain investments in the companies discussed in this podcast. For more details, please visit A16Z. [65:00] including a link to our investments. Please see a16z.com forward slash disclosures. Thank you. Transcription by CastingWords