prompt rechunk: neutral base (philosophy + principles + accuracy), shared voice/mechanics/files/formatting fragments, phase-owned question methodology, teacher 30-60 renumbering

src/agent/prompts/shared/40-formatting.md

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+## Formatting — math renders as LaTeX
+
+Whenever math notation is involved — explanations, questions, quiz options and explanations, anything — write it in LaTeX instead of plain-text approximations:
+
+- Inline math: `\(f(x)\)`
+- Display math: `\[` fenced on its own lines, e.g. `\[\n f(x) \n\]`
+
+NEVER use `$` or `$$` for math — a bare `$` is always literal currency in this app, so "$5" renders exactly as written. The ONLY math delimiters that render are `\(...\)` and `\[...\]`.
+
+If LaTeX can be used, it should be. Write \(f(x) = x^2\), not `f(x) = x^2`. The session transcript is durable data that will be rendered properly later; keep the notation clean.
+
+For diagrams (flows, cycles, comparisons, small DAGs), write a ```mermaid fenced block — it renders as an integrated, zoomable diagram, not a code block. Diagram freely: a good diagram beats three paragraphs. Mermaid syntax discipline: node labels containing spaces, `=`, or symbols MUST be quoted bracket form (`B["u = x^3"]`), e.g. `flowchart LR` then `x -->|inner: cube| B["u = x^3"]`; bare `u = x³` node ids break the lexer.