陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

Matthew Bolan, Joachim Breitner, Jose Brox, Nicholas Carlini, Mario Carneiro, Floris van Doorn, Martin Dvorak, Andrés Goens, Aaron Hill, Harald Husum, Hernán Ibarra Mejia, Zoltan Kocsis, Bruno Le Floch, Amir Livne Bar-on, Lorenzo Luccioli, Douglas McNeil, Alex Meiburg, Pietro Monticone, Pace P. Nielsen, Emmanuel Osalotioman Osazuwa, Giovanni Paolini, Marco Petracci, Bernhard Reinke, David Renshaw, Marcus Rossel, Cody Roux, Jérémy Scanvic, Shreyas Srinivas, Anand Rao Tadipatri

The aim of the project was to work out which of these laws imply which others. For instance, all laws imply the trivial law , and conversely the singleton law implies all the others. On the other hand, the commutative law does not imply the associative law (because there exist magmas that are commutative but not associative), nor is the converse true. All in all, there are implications of this type to settle; most of these are relatively easy and could be resolved in a matter

已知结果和反例

After a rather hectic setup process (documented in this personal log ), progress came in various waves. Initially, huge swathes of implications could be resolved first by very simple-minded techniques, such as brute-force searching all small finite magmas to refute implications; then, automated theorem provers such as Vampire or Mace4 / Prover9 were deployed to handle a large fraction of the remainder. A few equations had existing literature that allowed for many implications

Even after applying all these general techniques, though, there were about a dozen particularly difficult implications that resisted even these more powerful methods. Several ad hoc constructions were needed in order to understand the behavior of magmas obeying such equations as E854 , E906 , E1323 , E1516 , and E1729 , with the latter taking months of effort to finally solve and then formalize.

证明或构造的主线

A variety of GUI interfaces were also developed to facilitate the collaboration (most notably the Equation Explorer tool mentioned above), and several side projects were also created within the project, such as the exploration of the implication graph when the magma was also restricted to be finite. In this case, we resolved all of the implications except for one (and its dual):

See this blueprint page for some partial results on this problem, which we were unable to resolve even after months of effort.

阅读时建议盯住的点

Interestingly, modern AI tools did not play a major role in this project (but it was largely completed in 2024, before the most recent advanced models became available); while they could resolve many implications, the older “good old-fashioned AI” of automated theorem provers were far cheaper to run and already handled the overwhelming majority of the implications that the advanced AI tools could. But I could imagine that such tools would play a more prominent role in future

值得单独记下的条目

• ( Central groupoid law ):
• Linear models, in which the carrier was a (commutative or noncommutative) ring and the magma operation took the form for some coefficients , turned out to resolve many cases.
• Smarter ways to utilize automated theorem provers, such as strategically adding in additional axioms to the magma to help narrow the search space, were developed over the course of the project.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源