` block | May be followed by plain‑language unpacking. |
| B | `## Secondary Definitions` | Extra `[definition:id]` blocks | For auxiliary notions (e.g. Sobolev norm). |
| 3 | `## Examples` | `[example:id]` blocks | At least one non‑trivial example. |
| 4 | `## Key Results` | `[theorem:id]` + `[proof]` pairs | Reference others via `[quotetheorem:id]`. |
| C | `## Problems` | `[problem]`↔`[solution]` pairs | Statements students can solve. |
| D | Follow‑up sections | `## Applications`, `## History`, etc. | Any depth ≥ `##`. |
| E | `## References` | Bibliography | Markdown list; use standard citation style. |
> Tip: Narrative paragraphs (or images) may appear between any two headings. Keep each logical chunk short; avoid walls of text.
---
All tags are lowercase, never nested, and always closed. IDs must be unique within the page.
| Tag | Purpose | Typical Location | Rules |
| ------------------ | -------------------------- | ----------------------------------------- | ---------------------------------------------------------------------- |
| `[definition:ID]` | Formal statement | Formal Definition / Secondary Definitions | Exactly one primary definition (first in page). |
| `[example:ID]` | Worked example | Examples | Use verbs/nouns for ID (`LinearFunction`, `FailureOfClassicalSpaces`). |
| `[theorem:Name]` | Result heading | Key Results | Name in Title Case. |
| `[proof]` | Proof body | Immediately after its theorem | One per theorem. |
| `[problem]` | Exercise/problem | Problems | Must be closed by a `[solution]` block. |
| `[solution]` | Solution/explanation | Problems | Follows its `[problem]`. |
| `[motivation]` | Inline detailed motivation | Anywhere | Optional (pairs with `[/motivation]`). |
| `[quotetheorem:ID]` | Cite external theorem | Anywhere prose | Renders a hyperlink or reference number. |
> Future‑proof: introduce new tags sparingly; first register them in this table so parsers stay in sync.
---
1. Display math – only `align`:
```
\begin{align}
...
\end{align}
```
2. Inline math – single dollars: `$f'(x)$`.
3. Forbidden – `$$…$$`, `\[…\]`, `equation`/`gather`, etc.
These constraints simplify downstream LaTeX‑to‑HTML conversion and prevent delimiter clashes inside block tags.
---
Images – standard Markdown syntax: `!alt‑text{width=35%}`.
Always supply alt‑text; optional attribute braces may set width.
Code/CLI – wrap snippets in back‑ticks or triple‑back‑tick code fences (` ```bash `) as usual.
Lists – prefer `-` bullets; sub‑bullets >= two spaces indent.
Heading levels – never skip: `##`, then `###`, etc.
---
Page slug – lower‑kebab‑case: `sobolev-space`, `weak-derivative`.
Tag IDs – `CamelCase` or `TitleCase`, no spaces: `SobolevNorm`.
Theorem numbers – handled by engine; you may supply a human name (`Rolle`) or numeric label (`49`).
---
Forbidden phrasing — remove vacuous words that obscure logic: "one can show", "clearly", "it is obvious", "we can just work in some coordinate basis/chart", etc. State arguments explicitly.
No hidden steps — every deduction required for the result must be written; nothing may be “left as an exercise.” If you prefer the reader to attempt a sub-result, wrap it in `[problem]` and follow with a complete `[solution]`.
Index conventions
Superscripts are reserved for powers or derivative order ($x^2$, $D^3 u$).
Components, enumeration indices, or labels use subscripts only ($v_1$, $A_{ij}$). Never use superscripts for enumeration.
Symbol hygiene — declare every symbol at first appearance within a section; if multiple standards exist, pick one and note alternatives in parentheses.
Coordinate statements — if a proof depends on a specific chart or basis, name it and keep it fixed; avoid hand‑waving phrases.
---
```markdown
A derivative measures instantaneous rate of change — the slope of a curve at an infinitesimal scale.
Why slopes matter in physics, optimisation, and geometry.
[definition:Derivative]
\begin{align}
f'(x_0) := \lim_{h\to 0} \frac{f(x_0+h)-f(x_0)}{h}
\end{align*}