新智元报道
人类历史上规模最大的数学证明，现已通过计算机形式化验证。
有限单群分类定理（CFSG，Classification of Finite Simple Groups），刚刚被AI彻底形式化！
百余名全球顶尖数学家历经大半个世纪、写下长达15000页的「数学天书」，被初创团队Axiom研发的多智能体系统在短短24天内全部跑通。
更让人头皮发麻的是，AI构建了人类史上首个超4000万行的Lean代码库。
按代码行数计算，这现在是史上最大的开源软件项目。它的规模超过了 Linux 内核、Chromium、Firefox、LLVM、GCC、FFmpeg 等。
它甚至还在原有的证明中揪出了人类的漏洞，并亲自证明了新的数学定理来完成修补！
知名数学家Ken Ono动情发文，悼念今年1月去世的数学家父亲。
他回忆起童年时父亲引荐自己走近群论巨匠的岁月，在见证人机协作跨过这条历史终点线时，这位数学大师忍不住当场泪崩。
在数学界，这个定理有一个如雷贯耳的绰号——「巨型定理」（The Enormous Theorem）。
在此之前，世界上没有任何一个人能够从头到尾完整通读它的所有证明。
到底什么是「群」？
简单来说，群是描述「对称性」的数学语言。
比如，描述正六边形的对称性，对应一个二面体群群：
比如，用图论看魔方，每个状态是一个顶点，每一步移动是这张巨大图中的一条边。
这段动画展示了排列如何穿越 43 百亿亿种配置，最终抵达解法。
而所有的有限群，都可以像化学分子被分解为原子一样，被分解为不可再分的「基本构件」——这就是有限单群（Finite Simple Groups）。
分类定理的伟大之处就在于，它为全宇宙的对称性开列了一张「元素周期表」。
它明确宣告：
所有的有限单群，要么属于素数阶循环群，要么属于至少5个符号的交错群，要么属于16个无限家族的李型群，要么，就属于26个极度特殊、无法归类的「零散群」（Sporadic groups）。
这26个零散群中，最大的一个被数学家敬畏地称为「魔群」（Monster Group） 。
顺便提一句，1998年，Richard Borcherds因证明著名的「怪兽月光猜想」（Monstrous Moonshine），荣获菲尔兹奖 。这一猜想揭示了数学中一个令人震惊的联系：最大的散在单群「魔群」，竟然与数论中的模函数存在深刻的数学关联。
2007年，物理学巨擘Edward Witten更是提出，这套包含魔群的理论，可能被用来描述一个简化宇宙中的引力，用来编码黑洞在不同能级上的量子态分布，从而计算黑洞的熵！
从1950年代开始，为了证明巨型定理，大约100位数学家在500多篇学术论文中，拼凑出了长达15000页的证明，约14部《红楼梦》。
它太庞大、太分散了，以至于1983年数学家Gorenstein宣布证明完成时，其实还漏掉了一个巨大的缺口，直到2004年才由Aschbacher和Smith用1221页的论文补上。
因为证明实在太长，人类极容易出错。正如Aschbacher所言：「要写出一个非常长且复杂的论证而不犯错，是不可能的。」
既然人类的大脑已经逼近处理复杂逻辑的极限，那么，是时候让AI登场了。
面对这座1.5万页的「数学珠穆朗玛峰」，Axiom团队没有选择传统的蛮力计算，而是动用了多智能体蜂群——AxiomProver。
把人类的自然语言证明翻译成Lean代码（即「形式化」），就等同于让一台铁面无私的机器，用放大镜去检验那15000页纸上的每一个标点符号。
2026年10月1日，Axiom团队在一个Slack频道里敲下了启动的指令。
人类工程师和数学家将庞大的「巨型」拆解成108个里程碑和约380个小任务，生成了一张巨大的「证明地图」。
10月2日，第一批AI「蜂群」正式上线。
这群AI智能体像一支训练有素的军队一样开始分工协作。领头的AI负责拆解高难度的论证，其他AI则疯狂搜索现有结果或埋头编写Lean代码。如果一个AI遇到了还没证明的先决条件，它会自动把这个任务外包给另一个AI。所有的结果都会汇聚到一个共享的代码库中。
整个过程并非一帆风顺。由于AI编写代码的速度实在太快，Lean系统在重新检查依赖项时甚至出现了算力瓶颈。
Axiom的工程师不得不临时重构底层的工程基础设施，开发了一套分布式缓存系统，才勉强跟上AI那恐怖的产出速度。
10月10日，Slack频道里弹出了一条让所有人沸腾的消息：
我们证明了魔群是单群。
AI不仅从结构信息中识别出了魔群，还通过代码彻底形式化了它的存在！
仅仅24天！
在24个日日夜夜里，这群不知疲倦的AI智能体，产出了惊人的4310万行Lean代码，构建了人类目前最大的Lean代码库。
人类的智慧，经过了机器无情的检验，而且发现了其中的逻辑漏洞！
人类的语言总是充满省略。
有网友做词一首，调侃数学证明中的思维跳跃：
西江月·证明
即得易见平凡，仿照上例显然
留作习题答案略，读者自证不难
反之亦然同理，推论自然成立
略去过程 Q.E.D.
由上可知证毕
「显然可得」、「同理可证」、「易证」……这些数学论文里常见的词语，对AI来说却是行不通的。
形式化证明要求每一步都必须有明确的逻辑支撑。
在逐行扫描、强迫推理的过程中，AI智能体记录下了数百处异常。其中大部分是人类的笔误、表格错误或遗漏了前提条件，但有34处是重大的逻辑偏差！
而其中最惊险、最核心的一个「Bug」，出在至关重要的不平衡群定理（Unbalanced Group Theorem, UGT） 上。
UGT是整个分类定理中极为关键的一环，它的作用是「通过局部看整体」——通过研究一个群的子群结构，来推断出整个大群的真面目。
Axiom团队最初选择了John Walter在1986年发表的一篇权威文献作为形式化的基础。
然而，在翻译这段人类自以为完美无缺的论证时，AI智能体突然发出了红色警报！
AI发现，Walter证明中的一个关键闭包断言，在逻辑上是站不住脚的。
AI不仅指出了错误，还给出了一个明确的反例：PSL₂(49)。在这个反例中，一个操作会产生一个被明确排除在目标家族之外的群（因为7是梅森素数）。
人类引以为傲的证明，在这一刻断链了。
长达1.5万页的逻辑大厦，随时面临崩塌的危险。
怎么办？如果去借用后人写的「第二代证明」，不仅会引入海量的全新依赖，还会让这24天的努力前功尽弃。
此时，人机协作的巅峰一幕上演了。
Axiom团队紧急连线了当年参与CFSG原始工作的活化石——群论泰斗Richard Lyons和Ronald Solomon。
在听取了AI发现的漏洞后，人类大师的直觉与AI的算力展开了完美共振。基于Solomon在1976年提出的理论，人类数学家构思出了一条新的证明路线。
随后，AxiomProver（AI系统）亲自上阵，根据新思路，严丝合缝地证明了一个全新的群论结果——Preservation of simple tops。
这个由AI主导证明的新定理，像一块完美的定制补丁，精准地填补了人类大师当年留下的窟窿，成功推导出了分类定理所需的UGT形式！
定理依然成立，但证明已经不完全属于人类了。
这是人类历史上第一次，AI在最顶级的纯数学研究中，不仅充当了「验算器」，更成为了发现新定理、修补核心漏洞的「合作研究员」！
这不禁让人想起数学史上类似的一幕：2002年菲尔兹奖得主Vladimir Voevodsky，发现自己的证明是错的，而且救不回来。
1999–2000年，在普林斯顿高等研究院讲课时，他发现早年关键引理有误，原命题无法按原样挽救。
2013年又确认1989年另一项结果也错了——反例早在1998年就有人提出。
他不再相信人类能可靠地检查复杂证明，转向Coq、同伦类型论与单值基础：让计算机验数学。
在追求数学真理的道路上，人类用血肉之躯和有限的大脑，进行了长达近两百年的悲壮远征。
如今，人类有了新的合作伙伴，一起探索知识疆域的星辰大海。
参考资料：
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New Ji-won.
The largest mathematical evidence in human history has now been validated in computer format.
CFSG, Classification of Finite Simple Groups, just completely formalized by AI!
More than a hundred top mathematicians from around the world, who have spent more than half a century writing 15,000 pages of "mathematic books", have been running through the multi-AI agent system developed by the original team, Axiom, in just 24 days.
And even more so, AI built the first more than 40 million lines of the Lean Code Bank in human history.
This is now the largest open source software item in history, calculated by code line numbers. It is larger than the Linux kernel, Chromium, Firefox, LLVM, GCC, FFmpeg, etc.
It even found a human loophole in the original proof and personally proved the new mathematical theorem to complete the repair!
A well-known mathematician, Ken Ono, issued an emotional message in memory of the father of the mathematician who died last January.
He recalled the years of childhood when his father introduced himself to the cosmopolitan giants, and the mathematician couldn't help but shed tears at the moment when the witness machine worked together to cross the end of history.
In mathematics, the theorem has a resonant nickname, The Enormous Therem.
Until then, no one in the world had been able to read all its proofs from beginning to end.
What the hell is "group"?
In short, the group is a mathematical language that describes "symmetry".
For example, the symmetrical description of a positive hexagon corresponds to a diaphragm group:
For example, in graphic terms, each state is a peak, and each move is one side of this giant picture.
This animation shows how the sequence travels through 4,30 billion configurations and eventually reaches the solution.
And all limited groups can be broken down into irretrievable "basic components" as chemical molecules are decomposed into atoms — the limited single groups.
The great thing about the taxonomy is that it sets out a “cycle table of elements” for the symmetry of the universe.
It clearly declares:
All limited single groups, either of prime stratification, or of at least five staggered symbols, or of 16 infinite families, or of 26 extremely special, uncategorized "Sporadic groups".
The largest of these 26 scattered groups is known as the Monster Group by mathematicians with awe.
By the way, Richard Borcherds won the Fields Award in 1998 for proving the famous Monstrous Moonshine. This guess reveals an alarming connection in mathematics: the largest one is scattered in a single herd, and there is a profound mathematical connection to the role of a mathematical model.
In 2007, Edward Witten, the physics giant, put forward the theory of a herd, which could be used to describe a simplified gravitational force in the universe, to codify the quantum pattern distribution of black holes at different levels of power, and to calculate the entropy of a black hole!
Starting in the 1950s, in order to prove the giant theorem, about 100 mathematicians, in more than 500 academic papers, put together certificates of 15,000 pages, about 14 Red House Dreams.
It was too large and dispersed to have been completed by Mathematician Gorenstein in 1983, when he announced that a huge gap had been missed, and was not supplemented by 1221 pages of paper by Aschbacher and Smith until 2004.
Because it's too long to prove that human beings are vulnerable to mistakes. As Aschbacher said, "It is impossible to write a very long and complex argument without making a mistake."
Now that the human brain is nearing the limits of complex logic, it's time for AI to be here.
In the face of this 15,000-page Mathematics Mount Everest, the Axiom team did not choose traditional force calculations, but instead used the multi-AI agent hive, AxiomProver.
Translating the human language into the Lean code (i.e., "formatization") is tantamount to allowing an iron-faced machine to test every symbol on the 15,000 pages with a magnifying glass.
On 1 October 2026, the Axiom team knocked down the launch order on a Slack channel.
Human engineers and mathematicians disassembled the huge "mega" into 108 milestones and about 380 small missions, producing a huge "maps of proof".
On October 2, the first group of AI "bees" officially went online.
This group of AZAI agent began working together like a trained army. The lead AI is responsible for defusing difficult arguments, while the other AI is crazy about searching for existing results or burying the Lean code. If an AI encountered unproven preconditions, it would automatically outsource the task to another AI. All results will be brought together in a shared code library.
The process was not smooth. Because of the speed with which AI prepares the code, the Lean system has even experienced a credit bottleneck when rechecking the dependencies.
Axiom ' s engineers had to temporarily reconstruct the engineering infrastructure on the ground floor and developed a distributed cache system to keep pace with AI ' s horrific output.
On October 10, the Slack channel picked up a message that would boil everyone:
We have proved that the herd is a single herd.
Not only did AI identify the herd from structural information, but it completely shaped its existence through code!
Just 24 days!
In 24 days and nights, the tireless AZAI agent produced an amazing 43.1 million lines of Lean code and built the world ' s largest currently known code repository.
The wisdom of mankind, having undergone the machine's relentless tests, has discovered its logical flaws!
Human language is always filled with omissions.
There's a guy on the Internet who's doing the words, and he's doing the thinking leap in the math certificate:
Saiji Moon.
It's easy to see, as usual.
It's easy for the reader to identify himself.
It's the opposite. It's natural.
Q.E.D.
I'm sure it's from the top.
"Absolutely available", "same evidence", "probable evidence", and "probable evidence". It's not gonna work for AI.
Formalization requires clear logic at every step.
During line-by-line scanning and forced reasoning, AZAI agent recorded hundreds of anomalies. Most of them were human errors, table errors or missing preconditions, but 34 were major logical deviations!
One of the most dangerous, central “Bug” is on the vitally important unbalanced Group Therem, UGT.
The UGT is a crucial link in the whole taxonomy, and its function is to "see the whole through a partial look" — to deduce the trueness of a large group by looking at the sub-cluster structure.
The Axiom team initially selected an authoritative document published by John Walter in 1986 as a basis for formalization.
However, in translating this self-perceptional argument, AZAI agent suddenly issued a red alert!
AI found that one of the key closed assertions in Walter ' s certificate was logically untenable.
Not only did AI point to errors, it also gave a clear counter-reference: PSL2(49). In this retrospect, an operation creates a group that is explicitly excluded from the target family (because 7 is Masonian).
The evidence of human pride is broken at this point.
The 15,000-page Logic Building is in imminent danger of collapse.
What do we do? To borrow the "second-generation certificate" written by a later generation would not only introduce a new mass dependency, but would also make these 24 days of hard work impossible.
At this moment, the culmination of human collaboration takes place.
The Axiom team had an emergency connection to the living fossils that had been involved in the original work of the CFSG - Richard Lyons and Ronald Solomon.
Having listened to the loopholes discovered by AI, the instincts of the human master resonated perfectly with the calculation of AI. Based on the theory put forward by Solomon in 1976, human mathematicians have devised a new line of proof.
Subsequently, the AxiomProver (AI system) personally came up and, in the light of the new ideas, provided a seamless proof of an entirely new ensemble of results.
This new aphrodisiac of AI, like a perfect custom patch, accurately filled the holes left behind by the master of man, successfully deduced the UGT form required for the classification theorem!
Theorem is still valid, but it proves that it is not entirely human.
This is the first time in human history that AI, in its best pure mathematics studies, not only acts as an "accommodator", but also as a "cooperative researcher" who discovers new theorems and fixes core loopholes!
This reminds us of a similar event in the history of mathematics: Vladimir Voevodsky, the 2002 Fields winner, found his proof wrong and could not be saved.
In 1999-2000, when he was teaching at the Princeton Institute of Higher Studies, he found that the key arguments of the early years were incorrect and that the original proposition could not be saved as it stood.
It was also wrong to confirm another 1989 result in 2013 — an example of which was raised as early as 1998.
He no longer believes that humans can reliably examine complex proofs and turn to Coq, the same-style theory and the single-value base: to allow computer mathematics.
In the quest for mathematical truth, human beings have marched for almost 200 years with flesh and blood and limited brains.
Today, humankind has new partners to explore the stars and oceans of the knowledge frontier.
References:
Editor: David
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