大语言模型的词向量嵌入：词被放进高维空间，"国王 − 男人 + 女人 ≈ 女王"之所以成立，是因为模型学出的基让"性别""王权"等语义恰好对齐成了独立坐标轴。同理，一个卡住的产品决策——若你在"成本 vs 体验"这组纠缠的轴上反复权衡，不妨换一组基（如"可逆性 vs 学习价值"），原本的两难常常就解耦成两个能分别拍板的问题。
English Summary
Basis & Transformation — an object is invariant, but the basis (coordinate axes) you describe it in decides how complex it looks. A change of basis is a change of perspective; the right one makes a tangled problem decouple. Eigenvectors are the directions a transformation only stretches, never rotates — a system's natural axes; finding them (diagonalization) turns coupled dynamics into independent scalings. Fourier transforms, PCA, and word embeddings are all one move: re-express the data in a basis where the problem becomes linear and separable. In deep learning, "representation" just means "choice of basis." Many problems are hard only because you're standing in the wrong basis.
AI Prompts
中文提示词
我正卡在一个权衡里：[描述问题，及我当前用来衡量的两三个维度]。请用「换基」的视角帮我：
① 指出我现在隐含使用的「坐标轴」是哪几条，它们之间是否纠缠（动一个就牵动另一个）；
② 提出 2 组不同的「基」（替代维度），在哪一组下这个两难会解耦成可分别决策的子问题；
③ 找出这个问题的「特征方向」——哪个方向上系统其实保持不变、最值得我下注。
English Prompt
I'm stuck in a tradeoff: [describe the problem and the 2-3 dimensions I currently weigh it on]. Use the "change of basis" lens:
1. Name the coordinate axes I'm implicitly using, and whether they're entangled (moving one drags the other).
2. Propose 2 alternative bases (sets of dimensions); say which one makes the dilemma decouple into separately decidable subproblems.
3. Identify the "eigen-direction" of this problem — the axis along which the system stays invariant and is most worth betting on.
分布式系统里的"网络分区"本质是一个拓扑事件：连通性不是慢慢变差，而是在某一刻不连续地断裂，于是 CAP 的取舍被迫上演——一致性与可用性只能保一个。再看育儿：孩子的核心气质是一个拓扑不变量，它会随成长被连续地拉伸、塑形，但你无法靠外力"撕"掉它。聪明的做法不是对抗这个不变量，而是顺着它做连续形变——把同一股能量引向不同的出口。
English Summary
Continuity & Invariants — topology studies what survives continuous deformation (stretching, bending — no tearing or gluing). A coffee mug equals a donut: both have one hole. The power is in dropping mutable details (distance, size, angle) to ask what can't change without a tear. Topological invariants (connectedness, number of holes, boundaries) jump only at a tear, so they're robust to noise — they're the math of phase transitions, identity persistence, and critical points. "Can you get from A to B" is purely topological, independent of distance. The discipline: separate the deformable details from the skeleton that only breaks when torn.
AI Prompts
中文提示词
我在分析这个系统／关系／计划：[描述]。请用「拓扑不变量」的视角帮我分离骨架与细节：
① 哪些是可连续形变的「细节」（数字、规模、形态，变了也不伤本质）；
② 哪些是「不变量」（撕开才会断的核心——连通性、依赖、身份、底层结构）；
③ 当前有没有逼近某个「撕裂点」（会让不变量发生不连续跳变／质变）？该如何提前察觉。
English Prompt
I'm analyzing this system / relationship / plan: [describe]. Use the "topological invariant" lens to separate skeleton from detail:
1. Which parts are continuously deformable details (numbers, scale, shape — can change without harming the essence)?
2. Which are invariants (the core that only breaks when torn — connectivity, dependencies, identity, underlying structure)?
3. Is anything approaching a "tearing point" (a discontinuous jump / phase change in an invariant)? How can I detect it early?
设计 AI agent 流水线：每个 agent 就是一支箭头（输入类型→输出类型），系统能否跑通，取决于上一个的输出类型能否干净接上下一个的输入——设计 agent 即设计可组合的态射。而当你发现"团队的任务调度难题"与"图着色问题"同构，就能把后者成熟的算法整套搬过来，省下从头思考的成本。认出同构，是跨学科迁移最锋利的一招。
English Summary
Composition & Isomorphism — category theory's radical move: don't study an object's internals, study the arrows (morphisms) between objects and how they compose. A thing's essence is how it relates and chains, not what it's made of — the math behind "program to the interface, not the implementation." Composability is the whole game: if A→B and B→C reliably compose into A→C, you can stack arbitrarily complex systems — the mathematics of modularity. Isomorphism is the rigorous form of analogy: when an invertible, composition-preserving map exists between two structures, every theorem transfers for free. Recognizing your scheduling problem is isomorphic to graph coloring inherits the whole solved theory.
AI Prompts
中文提示词
我在设计／拆解这个系统：[描述]。请用「组合与同构」的视角分析：
① 把它拆成一组「箭头」（每个组件的输入类型→输出类型），指出哪两个接口拼接处类型对不齐、会成为脆弱点；
② 哪些环节的「内部实现」我其实不该关心，只需盯住接口契约；
③ 这个问题是否与某个已被充分研究的经典问题「同构」？若是，能直接搬过来的成熟解法是什么。
English Prompt
I'm designing / decomposing this system: [describe]. Use the "composition & isomorphism" lens:
1. Break it into arrows (each component's input type → output type); flag the two interfaces where types don't line up cleanly and will be fragile.
2. Which parts' internal implementation should I deliberately ignore, watching only the interface contract?
3. Is this problem isomorphic to a well-studied classic one? If so, what mature solution can I import wholesale?
Dimensionality & Similarity — high-dimensional data usually lives on a low-dimensional manifold; intelligence is finding the few axes that matter and projecting away the rest (compression = abstraction). A hidden trap: "similar" is meaningless without specifying the space and metric — two things near in one projection can be opposite in another. Good reduction keeps signal and drops noise (the flip side of choosing a basis, e.g. PCA). The curse of dimensionality: in high dimensions, distances concentrate and nearest-neighbor loses meaning — low-dim intuitions break. Embeddings, semantic search, and vector DBs all define similarity in a learned low-dim space, so retrieval quality equals reduction quality. But reduction is lossy — a map, not the territory; the dimension you projected away is often the one that mattered most.
AI Prompts
中文提示词
我要评估／比较这些选项：[描述对象]，目标是 [描述我想要的结果]。请用「降维与相似」的视角检查我：
① 我正隐含地把它们压到哪两三条「轴」上打分？这组轴是否漏掉了对目标最关键的一维；
② 在我选的这个空间里，"相似／更好"成立；换一个合理的空间，结论会不会反转？
③ 给出一个被我「投影掉」但可能致命的维度，并说明怎样把它重新纳入考量。
English Prompt
I'm evaluating / comparing these options: [describe], aiming for [desired outcome]. Use the "dimensionality & similarity" lens to check me:
1. Which 2-3 axes am I implicitly scoring them on? Does this set omit the dimension most critical to the goal?
2. In the space I chose, "similar / better" holds — would the conclusion flip in another reasonable space?
3. Name one dimension I've projected away that could be decisive, and how to fold it back into the decision.