diff --git a/DESIGN.md b/DESIGN.md
index 0f32967..8dca2c9 100644
--- a/DESIGN.md
+++ b/DESIGN.md
@@ -663,6 +663,7 @@ source_assets:
 | `tests/ledger-invalid/bad-toolkit.yaml` | `SP001`（enum） |
 | `tests/ledger-invalid/duplicate-id.yaml` | `SP002` |
 | `tests/notes-loads-sty.tex` + 最小 ledger | `SP003` |
+| `tests/no-teach-block/` | `SP025` |
 
 ---
 
@@ -1333,6 +1334,7 @@ examples/excerpt-toy/
 | `SP022` | 4 | v1（`SUPERPAPER_PHASE` 缺省 2，PR 7 已落地；仅当显式设为 1 时触发）出现 `toolkit: superderive` |
 | `SP023` | 4 | 非 dropped 的新 id ∈ `retired_ids` |
 | `SP024` | 4 | `figures[].claim` / `derivations[].claim` 不是已有 `C*` |
+| `SP025` | 4 | `notes/sections/sec-*.tex` 首个 `\section` 之前缺 `% gap:` / `% takeaway:` / `% jump:` / `% omit:` 任一非空行（`% generated by` 开头的文件豁免） |
 
 stderr 格式**抄** `superfig/scripts/lint.py`：先 `!! {path}`，随后缩进行 `   line N: SP00X message`（无行号则 `   SP00X message`）。
 
diff --git a/SKILL.md b/SKILL.md
index 9413b13..4f41286 100644
--- a/SKILL.md
+++ b/SKILL.md
@@ -47,7 +47,7 @@ or when the source is a lecture video.
 1. `./scripts/preflight.sh`。0 = 全路径，1 = 降级（说出口），2 = 无 LaTeX。
 2. `./scripts/ingest.sh --work <dir> …`（`references/input.md`）。
 3. 从 `assets/ledger.example.yaml` 写 `ledger.yaml`，再写 `outline.md`，再写正文。
-4. 公式三拍：中文动机 → `\[` / `align` → 扁平符号表（`references/pedagogy.md`）。
+4. 写正文：教，不要复述（`references/pedagogy.md`）。公式三拍：直觉 → `\[` / `align` → 扁平符号表。
 5. 按 `references/router.md` 填 figure plan。每张重绘图一份 `F*.request.md`，figure agent 只读它和对应子仓库 `SKILL.md`。
 6. `./scripts/lint.py --work <dir>` 然后 `./scripts/build.sh --work <dir>`。
 7. 改术语或改图：先改 ledger，再改那一处。
@@ -59,9 +59,9 @@ Worked trees: `examples/excerpt-toy/`（`align`）、`examples/pipeline-delegate
 ```
 $superpaper <源> 请 spawn 多 sub agents，隔离上下文：
   - 1 个 outline agent：写 ledger.yaml + outline.md + 节边界
-  - N 个 writer agents：各写 sections/sec-XX.tex，引用 ledger id
+  - N 个 writer agents：先读 references/pedagogy.md，各写 sections/sec-XX.tex，引用 ledger id
   - 每个计划重绘图 1 个 figure agent：只读 F*.request.md，调用对应子仓库 skill
-  - 1 个 consistency agent：符号、术语、claim 覆盖、路由一致性
+  - 1 个 consistency agent：符号、术语、claim 覆盖、路由一致性、是否复述（pedagogy.md）
 ```
 
 必须拆 agent 当且仅当：正文 > 12 页，或顶层节 > 4，或重绘图 ≥ 2，或用户显式要求 spawn。
diff --git a/assets/notes-template.tex b/assets/notes-template.tex
index 1ba4928..8dadaf9 100644
--- a/assets/notes-template.tex
+++ b/assets/notes-template.tex
@@ -13,7 +13,9 @@
 % \section{推导链一览}
 % \section{图表清单}
 %
-% Formula: Chinese motive, then \[ / align, then a flat symbol list.
+% Teach, don't recap. Each writer section starts with
+% % teach: gap / takeaway / jump / omit  (references/pedagogy.md).
+% Formula: intuition off-stage, then \[ / align, then a flat symbol list.
 % Keep figures outside knowledgebox / importantbox / warningbox / quotebox.
 % Do not \usepackage{superfig|supertensor|superderive}.
 
diff --git a/examples/derive-delegate/notes/sections/sec-01.tex b/examples/derive-delegate/notes/sections/sec-01.tex
index 1345846..886e3c7 100644
--- a/examples/derive-delegate/notes/sections/sec-01.tex
+++ b/examples/derive-delegate/notes/sections/sec-01.tex
@@ -1,2 +1,7 @@
+% teach:
+% gap: 见过 $1/\sqrt{d_k}$，但不知道它是装饰性常数还是一次必须看见的改写
+% takeaway: 这一节要判断的是「这步改写值不值得画出来」
+% jump: none
+% omit: 摘录之外的全文结构
 \section{这篇论文在问什么}
 注意力里的 $1/\sqrt{d_k}$ 是装饰，还是一次必须看见的改写？
diff --git a/examples/derive-delegate/notes/sections/sec-02.tex b/examples/derive-delegate/notes/sections/sec-02.tex
index 0b623b7..089bb63 100644
--- a/examples/derive-delegate/notes/sections/sec-02.tex
+++ b/examples/derive-delegate/notes/sections/sec-02.tex
@@ -1,3 +1,8 @@
+% teach:
+% gap: 知道要除 $\sqrt{d_k}$，但看不见方差是从哪一步冒出来的
+% takeaway: 缩放来自点积方差随 $d_k$ 增长，是改写不是新算子
+% jump: fixture 直接给出结论，没展开方差那一步
+% omit: 其他归一化方案的对比
 \section{主张与贡献}
 \splabel{C1}
 点积方差随 $d_k$ 涨。除掉 $\sqrt{d_k}$ 是改写，不是新算子。
diff --git a/examples/derive-delegate/notes/sections/sec-03.tex b/examples/derive-delegate/notes/sections/sec-03.tex
index a1f226b..cc31f94 100644
--- a/examples/derive-delegate/notes/sections/sec-03.tex
+++ b/examples/derive-delegate/notes/sections/sec-03.tex
@@ -1,2 +1,7 @@
+% teach:
+% gap: 不知道什么时候该出 superderive 图、什么时候 align 就够
+% takeaway: $\le 4$ 步且没有消项或代入要看见时，留在 align
+% jump: none
+% omit: superderive 的完整语法（见子仓库 SKILL.md）
 \section{总结与延伸}
 四步以内、没有消项或代入要画出来时，仍然用 \texttt{align}。
diff --git a/examples/excerpt-toy/notes/sections/sec-01.tex b/examples/excerpt-toy/notes/sections/sec-01.tex
index 06abff0..61dd9ef 100644
--- a/examples/excerpt-toy/notes/sections/sec-01.tex
+++ b/examples/excerpt-toy/notes/sections/sec-01.tex
@@ -1,3 +1,8 @@
+% teach:
+% gap: 知道模型要输出预测，但没想过损失算不算前向路径上的一站
+% takeaway: 这份摘录只问一件事——前向到哪里为止
+% jump: none
+% omit: 摘录之外的全文结构与实验
 \section{这篇论文在问什么}
 要把输入变成预测，最简单的机制是什么？损失要不要走在前向主路上？
 这是摘录 fixture，不假装读完全文。
diff --git a/examples/excerpt-toy/notes/sections/sec-02.tex b/examples/excerpt-toy/notes/sections/sec-02.tex
index 7170234..2a35734 100644
--- a/examples/excerpt-toy/notes/sections/sec-02.tex
+++ b/examples/excerpt-toy/notes/sections/sec-02.tex
@@ -1,3 +1,8 @@
+% teach:
+% gap: 还没有一句能离开 PDF 复述的核心主张
+% takeaway: 一次前向是 $x\to f_\theta\to\hat y$；损失在预测之后单独比较
+% jump: none
+% omit: 重复摘要的贡献条目
 \section{主张与贡献}
 \splabel{C1}
 一次前向是 $x \to f_\theta \to \hat y$。损失 $L(\hat y,y)$ 在预测之后单独计算，不是主路上的一站。
diff --git a/examples/excerpt-toy/notes/sections/sec-03.tex b/examples/excerpt-toy/notes/sections/sec-03.tex
index af57f2d..9a73cdd 100644
--- a/examples/excerpt-toy/notes/sections/sec-03.tex
+++ b/examples/excerpt-toy/notes/sections/sec-03.tex
@@ -1,2 +1,7 @@
+% teach:
+% gap: 不知道 $f_\theta$、$\hat y$ 在本讲义里各指什么
+% takeaway: 预测器就是 $\hat y=f_\theta(x)$，其余符号查附录，不在正文重列
+% jump: none
+% omit: 论文式的记号巡游
 \section{预备：定义、假设、符号}
 预测器定义为 $\hat y = f_\theta(x)$。符号见附录。
diff --git a/examples/excerpt-toy/notes/sections/sec-04.tex b/examples/excerpt-toy/notes/sections/sec-04.tex
index 21127ed..7be895b 100644
--- a/examples/excerpt-toy/notes/sections/sec-04.tex
+++ b/examples/excerpt-toy/notes/sections/sec-04.tex
@@ -1,3 +1,8 @@
+% teach:
+% gap: 会算内积，但不知道维数一涨为什么会把后面的非线性弄坏
+% takeaway: 除 $\sqrt{d}$ 是把方差按回 1 的改写，不是新算子
+% jump: 摘录直接写下 $1/\sqrt{d}$，没说维数涨会让点积方差跟着涨
+% omit: 数据集、超参、硬件
 \section{一次前向与损失}
 内积的方差会随维数 $d$ 涨。为了不让后续非线性饱和，要把点积除掉 $\sqrt{d}$。
 
diff --git a/examples/excerpt-toy/notes/sections/sec-05.tex b/examples/excerpt-toy/notes/sections/sec-05.tex
index 522cc36..9198f8f 100644
--- a/examples/excerpt-toy/notes/sections/sec-05.tex
+++ b/examples/excerpt-toy/notes/sections/sec-05.tex
@@ -1,2 +1,7 @@
+% teach:
+% gap: 学完这一条机制，不知道同样的流程怎么套到一篇完整论文
+% takeaway: 先用 ledger 钉住 claim，再按路由决定出不出图
+% jump: none
+% omit: 摘录没覆盖的实验与消融
 \section{总结与延伸}
 摘录只保留一条机制：一次前向加侧路损失。更长的论文用 ledger 把 claim 钉住，再按路由出图。
diff --git a/examples/pipeline-delegate/notes/sections/sec-01.tex b/examples/pipeline-delegate/notes/sections/sec-01.tex
index e6be679..4f21d3a 100644
--- a/examples/pipeline-delegate/notes/sections/sec-01.tex
+++ b/examples/pipeline-delegate/notes/sections/sec-01.tex
@@ -1,2 +1,7 @@
+% teach:
+% gap: 知道有前向和损失两件事，但不知道它们在图上谁接谁
+% takeaway: 这一节问的是数据流拓扑，不是张量轴长
+% jump: none
+% omit: 摘录之外的全文结构
 \section{这篇论文在问什么}
 预测怎么从输入算出来，损失该不该站在前向主路上？
diff --git a/examples/pipeline-delegate/notes/sections/sec-02.tex b/examples/pipeline-delegate/notes/sections/sec-02.tex
index 825a213..7ab3ad5 100644
--- a/examples/pipeline-delegate/notes/sections/sec-02.tex
+++ b/examples/pipeline-delegate/notes/sections/sec-02.tex
@@ -1,3 +1,8 @@
+% teach:
+% gap: 容易把损失画成前向链条上的下一个方块
+% takeaway: 损失是侧路，不是主路上的一站
+% jump: fixture 的框图没说清 $L$ 是挂在 $\hat y$ 之后还是在链条之内
+% omit: 重复摘要的贡献条目
 \section{主张与贡献}
 \splabel{C1}
 一次前向是 $x\to f_\theta\to\hat y$。损失在预测之后单独比较。
diff --git a/examples/pipeline-delegate/notes/sections/sec-03.tex b/examples/pipeline-delegate/notes/sections/sec-03.tex
index c8fd1b8..ef9cbbb 100644
--- a/examples/pipeline-delegate/notes/sections/sec-03.tex
+++ b/examples/pipeline-delegate/notes/sections/sec-03.tex
@@ -1,2 +1,7 @@
+% teach:
+% gap: 不知道这类图为什么交给 superfig 而不是 supertensor
+% takeaway: 主张是「谁吃谁」时走 superfig
+% jump: none
+% omit: 轴长与形状细节（那是 supertensor 的活）
 \section{总结与延伸}
 这张图走 superfig，因为要画的是谁吃谁，不是轴长。
diff --git a/examples/tensor-delegate/notes/sections/sec-01.tex b/examples/tensor-delegate/notes/sections/sec-01.tex
index 1ded60b..719c514 100644
--- a/examples/tensor-delegate/notes/sections/sec-01.tex
+++ b/examples/tensor-delegate/notes/sections/sec-01.tex
@@ -1,2 +1,7 @@
+% teach:
+% gap: 会写 $QK^\top$，但没想过收缩沿哪条边、$K^\top$ 要不要真的换面
+% takeaway: 这一节问的是形状对齐，不是模块拓扑
+% jump: none
+% omit: 摘录之外的全文结构
 \section{这篇论文在问什么}
 每头的 $Q$ 和 $K$ 沿哪一条边收缩，$K^\top$ 要不要真的换面？
diff --git a/examples/tensor-delegate/notes/sections/sec-02.tex b/examples/tensor-delegate/notes/sections/sec-02.tex
index b316723..cb5af2d 100644
--- a/examples/tensor-delegate/notes/sections/sec-02.tex
+++ b/examples/tensor-delegate/notes/sections/sec-02.tex
@@ -1,3 +1,8 @@
+% teach:
+% gap: 把 $K^\top$ 当成记号上的装饰，不当成一次真实的换面
+% takeaway: 打分沿 $d_h$ 收缩，收缩边必须等长
+% jump: fixture 只写 $QK^\top$，没说明转置是物理换面而非书写约定
+% omit: 多头拼接与输出投影
 \section{主张与贡献}
 \splabel{C1}
 打分是 $(T\times d_h)(d_h\times T)\to(T\times T)$。$K^\top$ 必须物理换面，收缩边等长。
diff --git a/examples/tensor-delegate/notes/sections/sec-03.tex b/examples/tensor-delegate/notes/sections/sec-03.tex
index 4390f1d..ed3330c 100644
--- a/examples/tensor-delegate/notes/sections/sec-03.tex
+++ b/examples/tensor-delegate/notes/sections/sec-03.tex
@@ -1,2 +1,7 @@
+% teach:
+% gap: 不知道形状类的图该交给哪个子仓库
+% takeaway: 主张是轴长与收缩边时走 supertensor
+% jump: none
+% omit: 模块拓扑（那是 superfig 的活）
 \section{总结与延伸}
 形状对齐的公式图只交给 supertensor。
diff --git a/references/agents.md b/references/agents.md
index a049465..ae54a01 100644
--- a/references/agents.md
+++ b/references/agents.md
@@ -4,11 +4,13 @@ Must split when pages > 12 **or** top sections > 4 **or** redraws ≥ 2 **or** t
 
 Writer jobs are lecture rows in `outline.md`, not `coverage.sections_in`. Outline writes `ledger.yaml`, `outline.md`, `notes.tex` inputs, and `F*.request.md`. Writers only touch their `sec-XX.tex`. Figure agents only see the request.
 
+Outline, writers, and consistency read `references/pedagogy.md`. Consistency flags missing `% teach:` blocks and section-order recaps.
+
 ```
 $superpaper <源> 请 spawn 多 sub agents，隔离上下文：
   - 1 个 outline agent：ledger.yaml + outline.md
-  - N 个 writer agents：sections/sec-XX.tex
+  - N 个 writer agents：先读 references/pedagogy.md；sections/sec-XX.tex
   - 每个计划重绘图 1 个 figure agent：F*.request.md + sibling skill
-  - 1 个 consistency agent：符号、术语、claim 覆盖
+  - 1 个 consistency agent：符号、术语、claim 覆盖、是否复述（pedagogy.md）
 完成后用 scripts/lint.py 与 build.sh 收口。
 ```
diff --git a/references/antipatterns.md b/references/antipatterns.md
index 0fd758f..72fc277 100644
--- a/references/antipatterns.md
+++ b/references/antipatterns.md
@@ -1,5 +1,6 @@
 # Antipatterns
 
+- Recapping the paper's sections in lecture clothing (see `pedagogy.md`).
 - Drawing a rewrite as `\sfnode` boxes (use `align`).
 - Sending a tensor-shape claim to superfig, or an architecture to supertensor.
 - Loading a figure `.sty` in the notes so sibling warnings leak into the article log.
diff --git a/references/checklist.md b/references/checklist.md
index 0858f23..459d4f3 100644
--- a/references/checklist.md
+++ b/references/checklist.md
@@ -1,6 +1,8 @@
 # Delivery checklist
 
 - Ledger written first; every core claim has `\splabel`.
+- Notes teach from a reader gap, not the paper's section order (`references/pedagogy.md`).
+- Every `sec-*.tex` opens with the four `% teach:` lines (`SP025`).
 - Formula three-beat present wherever display math appears.
 - Each figure toolkit matches `scripts/router.py`; mixed-class rows were split.
 - Notes do not `\usepackage{superfig|supertensor|superderive}`.
diff --git a/references/pedagogy.md b/references/pedagogy.md
index fd0d744..8d5e937 100644
--- a/references/pedagogy.md
+++ b/references/pedagogy.md
@@ -1,15 +1,237 @@
 # Pedagogy
 
-Reuse `youtube-render-pdf` teaching order: motive → idea → mechanism → evidence → takeaway.
+Job: teach the argument thread. The paper is the source of claims, not
+the outline of the notes.
 
-Paper-side changes:
+Section order is still motive → idea → mechanism → evidence → takeaway.
+That order is not a license to recap the PDF in a new sequence.
 
-- Cite `§` / `Eq.(n)` / Figure / Table / page, not timestamps.
-- Front page is a bibliography card, not a PDF cover screenshot.
-- Math is `\[` or `align`, never `$$`.
-- Formula three-beat: Chinese motive, display math, flat symbol list.
-- `quotebox` for a short quotation with a source; no long PDF paste.
-- End major sections with `\subsection{本章小结}`; end the notes with `\section{总结与延伸}`.
-- Figures stay outside boxes.
+## Recite vs teach
 
-Locked first/last titles: 「这篇论文在问什么」…「总结与延伸」, then appendix 符号表 / 推导链一览 / 图表清单.
+The default failure is a **closed-book failure**: a colleague who has
+not read this paper still cannot use the core claim, because the notes
+assume everything the paper assumes.
+
+| | Recite | Teach |
+|---|---|---|
+| Starts from | the paper's next subsection | what the reader still lacks |
+| Formula | "本节给出 Eq.(n)" then symbols | a world without the formula, then the formula |
+| Gap | whatever the paper already wrote | the jump the paper did not write |
+| Stops when | every cited paragraph is covered | the takeaway is usable |
+
+Reader model: a competent colleague in the ambient field, textbook
+level. They have **not** read this paper, do not know its notation, and
+will not reconstruct skipped algebra. Do not reteach the ambient
+textbook (softmax, SGD, …) unless a core claim hangs on a twist.
+
+If deleting the PDF would make the section unreadable as a lecture, it
+was a recap.
+
+## Before each section
+
+At the top of every writer `sec-XX.tex` (not `symbols.tex` / generated
+appendix), answer these four lines **before** the `\section`:
+
+```tex
+% teach:
+% gap: <读者此刻还缺什么，不是论文小节标题>
+% takeaway: <离开本节必须带走的一句>
+% jump: <论文跳过、读者会卡住的那一步；没有则 none>
+% omit: <论文这里有、本讲义故意不写的>
+```
+
+- `gap` — entering deficit, in the reader's words.
+- `takeaway` — one sentence. This is also the `本章小结`.
+- `jump` — the missing why / scale / naive alternative / extreme case.
+- `omit` — ceremony you will not import "for completeness".
+
+Do not start the body until all four are filled. `lint.py` enforces the
+four keys on every `notes/sections/sec-*.tex` as **`SP025`** (presence
+only — it cannot judge whether the answers are honest).
+
+## Coverage density
+
+`coverage.sections_in` is **cite permission**, not a to-do list.
+`ledger_ids` on the outline row are the teaching budget.
+
+- Outline: fewer lecture rows than paper subsections. Merge until each
+  row has **one** takeaway. A row per paper `\subsection` is too dense.
+- Writer: teach this row's ids. A paper paragraph that is not required
+  for a `status: core` claim stays out.
+- Supporting claims: in only if they unblock a core claim.
+- Dropped claims: never.
+
+Default skip (paper ceremony):
+
+- "the rest of this paper is organized as follows"
+- related-work tour (keep the 2–3 contrasts that **define the gap**)
+- contribution bullets that restate the abstract
+- dataset / hyperparameter / hardware laundry lists, unless a core
+  empirical claim hangs on a specific number
+- every ablation, every appendix proof, every lemma not on an
+  `expand: true` derivation
+- notation paragraphs that only duplicate the symbol appendix
+
+## Intuition before formulas
+
+Display math is a three-beat that must not be split:
+
+1. **Intuition in Chinese, with the formula still off-stage.** Use at
+   least one of: analogy, contrast with the naive move, extreme case
+   (0 / 1 / ∞), or tiny numbers. 「下面给出公式」/「本节引入 Eq.(n)」
+   is not a motive.
+2. `\[` or `align` / `aligned`. Never `$$`.
+3. Flat symbol list: one item per symbol — 符号 — 含义 — 定义处.
+
+The section that owns the core mechanism then puts **one**
+`importantbox{如果你只记一件事}` with the takeaway sentence. At most
+one such box per major section. `本章小结` restates that sentence; it
+is not a new inventory.
+
+Pick one intuition tool, not all four. A running numeric example is
+allowed when scale or cancellation is the point; it is not mandatory.
+
+## Paper jumps
+
+Papers skip the why, the scale, and the obvious alternative. Before
+mechanism, name the stall and fill it:
+
+- Why this form, not the naive one?
+- What happens at 0 / 1 / ∞, or if the term is dropped?
+- Which quantity is a count vs a rate vs a scale?
+- Where does the gradient / information / mass actually flow?
+
+Fill in prose or `warningbox`. If the paper is silent and you would
+have to invent a result, say it is silent — do not fabricate.
+
+## Boxes and citations
+
+| box | payload |
+|---|---|
+| `importantbox` | walk-away: core claim, mechanism, 「如果你只记一件事」 |
+| `knowledgebox` | not the main thread: prerequisite, analogy, running example |
+| `warningbox` | naive-vs-correct, hidden assumption, paper-is-silent |
+| `quotebox` | short quotation + `§` / `Eq.(n)` in the title; ≲ 8 lines; no PDF paste |
+
+Routine exposition stays in prose. Figures stay outside every box.
+
+Cite `§` / `Eq.(n)` / Figure / Table / page, not timestamps. No
+`[cite]` placeholders. `\spsource{...}` at the point of use.
+
+Front page is a bibliography card (title, authors, year, venue, arXiv),
+not a PDF-cover screenshot. Default language is Chinese.
+
+End every major section with `\subsection{本章小结}`. End the notes
+with `\section{总结与延伸}` (limitations + compressed takeaways +
+open questions).
+
+Locked first/last titles: 「这篇论文在问什么」…「总结与延伸」, then
+appendix 符号表 / 推导链一览 / 图表清单. Outline may add or drop
+middle sections; it may not rename or drop the lock.
+
+Suggested skeleton (same lock as `assets/notes-template.tex`):
+
+```tex
+\section{这篇论文在问什么}      % questions[] + motive
+\section{主张与贡献}            % claims[status=core]，\splabel{C*}
+\section{预备：定义、假设、符号} % D* / A* / 符号摘要；不是论文 §2 巡游
+% --- outline-chosen mechanism sections ---
+\section{实验与证据}            % only E* that isolate a core claim
+\section{总结与延伸}
+```
+
+## Antipatterns
+
+Each pair is the same lecture beat. Left recites; right teaches.
+
+### 1. Paper order, new wording
+
+```tex
+% recite
+本文第 3 节提出 scaled dot-product attention。
+作者将 $Q$、$K$ 做点积，除以 $\sqrt{d_k}$，再 softmax 乘 $V$。
+```
+
+```tex
+% teach
+% gap: 还以为「对齐再加权」就完了，不知道维数一涨会出什么事
+读者已经会「用相似度加权」。先看极端情况：$d_k$ 很大时，
+点积方差跟着涨，softmax 进饱和，梯度没了。
+除以 $\sqrt{d_k}$ 不是新算子，是把方差按回 1。
+```
+
+### 2. Formula with no off-stage intuition
+
+```tex
+% recite
+本节引入如下公式：
+\[
+  \mathrm{Attention}(Q,K,V)=\mathrm{softmax}(QK^{\top}/\sqrt{d_k})V.
+\]
+\begin{itemize}
+  \item $Q$ — query
+  \item $K$ — key
+  \item $d_k$ — key 维
+\end{itemize}
+```
+
+```tex
+% teach
+两个随机向量的点积有多尖，随维数走。不先压方差，后面的 softmax 是死的。
+\[
+  \mathrm{Attention}(Q,K,V)=\mathrm{softmax}(QK^{\top}/\sqrt{d_k})V.
+\]
+\begin{itemize}
+  \item $Q,K,V$ — query / key / value
+  \item $d_k$ — key 维（shape parameter；缩放跟它走）
+\end{itemize}
+\begin{importantbox}{如果你只记一件事}
+$\sqrt{d_k}$ 是改写，不是新注意力。
+\end{importantbox}
+```
+
+### 3. Paper jump left as 「如式所示」
+
+```tex
+% recite
+如 Eq.(3) 所示，取 $A=d/H_{\mathrm{act}}$。
+```
+
+```tex
+% teach
+% jump: 论文直接写下 A，没说为什么不是 1/N
+走得越宽，越要把输出缩小，否则残差被宽隐层撑爆。
+所以是 $A=d/H_{\mathrm{act}}$：宽 $4096$ 时 $A=1/4$，宽 $1024$ 时 $A=1$。
+不是 $1/N$——专家总数还没进这把尺子。
+```
+
+### 4. Ceremony imported for completeness
+
+```tex
+% recite
+文献三条线：…（半页 related work）
+数据集为 ImageNet / CIFAR / …，硬件为 8$\times$A100，超参见表 7。
+```
+
+```tex
+% teach
+% omit: related-work 巡游、数据集表、硬件
+现成方法把问题当静态参数；缺的是带延迟和互馈的那一截。
+经验数字只在它能隔离这个机制时才进正文。
+```
+
+### 5. One lecture row per paper paragraph
+
+```tex
+% recite
+\subsection{Lemma 2}
+\subsection{Lemma 3}
+\subsection{Ablation on dropout}
+\subsection{Ablation on warmup}
+```
+
+```tex
+% teach
+% omit: 不在 DER1 上的 lemma；不能隔离 C1 的 ablation
+只展开 DER1 用到的那一步。Ablation 只留「去掉这项，C1 是否还成立」。
+```
diff --git a/scripts/lint.py b/scripts/lint.py
index c0784a9..a0fee91 100755
--- a/scripts/lint.py
+++ b/scripts/lint.py
@@ -43,6 +43,13 @@ KIND_ALIASES = {
 }
 STY_RE = re.compile(r"\\usepackage(?:\[[^\]]*\])?\{(superfig|supertensor|superderive)\}")
 LABEL_RE = re.compile(r"\\splabel\{(C[1-9][0-9]*)\}")
+SECTION_RE = re.compile(r"^\s*\\section\b", re.MULTILINE)
+GENERATED_RE = re.compile(r"^\s*%.*generated by", re.IGNORECASE)
+TEACH_KEYS = ("gap", "takeaway", "jump", "omit")
+TEACH_RE = {
+    key: re.compile(rf"^%\s*{key}\s*:\s*\S", re.MULTILINE)
+    for key in TEACH_KEYS
+}
 
 
 def emit(path: Path, messages: list[str]) -> None:
@@ -70,6 +77,19 @@ def collect_ids(data: dict) -> list[str]:
     return ids
 
 
+def is_writer_section(path: Path) -> bool:
+    return path.parent.name == "sections" and path.name.startswith("sec-")
+
+
+def teach_gaps(text: str) -> list[str]:
+    """Keys missing from the `% teach:` block above the first \\section."""
+    if GENERATED_RE.match(text):
+        return []
+    match = SECTION_RE.search(text)
+    head = text[: match.start()] if match else text
+    return [key for key in TEACH_KEYS if not TEACH_RE[key].search(head)]
+
+
 def notes_text(notes_paths: list[Path]) -> str:
     chunks: list[str] = []
     for path in notes_paths:
@@ -155,6 +175,12 @@ def lint(
             text = path.read_text(encoding="utf-8")
             if STY_RE.search(text):
                 errors.append(f"SP003 {path.name} loads a figure package")
+            if is_writer_section(path):
+                missing = teach_gaps(text)
+                if missing:
+                    errors.append(
+                        f"SP025 {path.name} % teach: block missing {', '.join(missing)}"
+                    )
 
     labels = set(LABEL_RE.findall(body)) if notes_scan else set()
 
diff --git a/scripts/test.sh b/scripts/test.sh
index a5b1b03..530871b 100755
--- a/scripts/test.sh
+++ b/scripts/test.sh
@@ -76,6 +76,7 @@ expect_work "$ROOT/tests/loads-sty" SP003
 expect_work "$ROOT/tests/missing-request" SP012
 expect_work "$ROOT/tests/unlabeled-core" SP020
 expect_work "$ROOT/tests/missing-include" SP021
+expect_work "$ROOT/tests/no-teach-block" SP025
 
 echo "==> compile"
 for dir in \
diff --git a/tests/no-teach-block/ledger.yaml b/tests/no-teach-block/ledger.yaml
new file mode 100644
index 0000000..1c25654
--- /dev/null
+++ b/tests/no-teach-block/ledger.yaml
@@ -0,0 +1,20 @@
+schema: superpaper.ledger/v1
+retired_ids: []
+paper:
+  id: toy
+  title: Toy
+  source: {kind: excerpt}
+coverage:
+  mode: excerpt
+questions: []
+claims:
+  - {id: C1, text: core claim, kind: contribution, status: core}
+definitions: []
+assumptions: []
+lemmas: []
+symbols: []
+derivations: []
+figures: []
+evidence: []
+terms: []
+source_assets: []
diff --git a/tests/no-teach-block/notes/notes.tex b/tests/no-teach-block/notes/notes.tex
new file mode 100644
index 0000000..2b526dc
--- /dev/null
+++ b/tests/no-teach-block/notes/notes.tex
@@ -0,0 +1,5 @@
+\documentclass{article}
+\usepackage{amsmath}
+\begin{document}
+\input{sections/sec-01}
+\end{document}
diff --git a/tests/no-teach-block/notes/sections/sec-01.tex b/tests/no-teach-block/notes/sections/sec-01.tex
new file mode 100644
index 0000000..ad1d381
--- /dev/null
+++ b/tests/no-teach-block/notes/sections/sec-01.tex
@@ -0,0 +1,4 @@
+% takeaway: 有 takeaway，但缺 gap / jump / omit —— 闸门没走完
+\section{一次前向与损失}
+\splabel{C1}
+前向只负责预测；损失在预测之后单独比较。