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「Snap preview?」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

As you may have noticed, I had turned off the “ Snap preview ” feature on this blog (which previews any external link that the mouse hovers over) for a few weeks as an experiment. After a mixed response, I have decided to re-enable the feature for now, but would be interested in getting feedback on whether this feature makes noticeable differences (both positive and negative) to the viewing experience (in particular, to the loading time of the web page), so that I can decide

This post would also be a good forum to discuss any other formatting and presentation issues with this blog (wordpress blogs are remarkably customisable). For instance, I have been looking out for a way to enable previewing of comments, but as far as I can tell this appears to not be possible on wordpress. But if anyone knows some workaround or substitute for this feature, I would like to hear about it.

已知结果和反例

[ Update , Oct 17: Judging from the comments, the response to Snap preview continues to be mixed. But I have discovered how to turn Snap preview on and off for individual articles or links . So I will keep Snap preview on for Wikipedia links and off for the rest; see the articles currently in the front page of this blog for examples of this. I also note that in the Snap preview options menu, there are options to turn Snap off permanently, or to increase the delay before the p

Also, following suggestions in the comments, I have darkened the text and also changed the blockquote format, in order to increase readability, and also changed the RSS feed from summaries to the full article (or more precisely, the portion of the article before the jump). ]

证明或构造的主线

[ Update , Oct 18: Comments moved to just below the post, as opposed to below the sidebar.]

阅读时建议盯住的点

先写出对象、假设和失败的例子，再进入证明。没有反例的直觉，很容易把局部技巧当成一般定理。

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