math: dollar delimiters end-to-end (exact answers, escaping examples, context rendering)

- switch the math authoring/rendering contract from \(...\)/\[...\] to the
  standard $...$ inline / $$...$$ display: prompts (20-question-tools,
  40-formatting), singleDollarTextMath on the renderer, legacy reader kept,
  and escape-debris recovery extended to dollar spans (mirroring micromark
  delimiter-run rules)
- exact answer membership now precedes debris cleanup in
  coerceCorrectAnswer (calculus incident: an exact sentence-final-period
  label was stripped as model debris, then rejected); option keys are
  passed at the boundary/grading seams, the dock no longer pre-coerces
- a complete, phase-correct JSON-escaping example rides the probe/quiz
  tool descriptions (serialized from one canonical body; quiz leads with
  type+nodeId), plus the one-sentence JSON-escaping rule in 40-formatting
- ask option descriptions render markdown/LaTeX in the pending dock;
  settled cards keep the details line and option descriptions
  retroactively, read from the persisted call arguments (chat passes
  part.input to AnsweredCard)

1566/1566 tests on both navigator configs; typecheck, lint, production
build, standalone packaging, and Electron-runtime smoke all green.

src/agent/prompts/shared/20-question-tools.md

@@ -7,7 +7,7 @@ Two instruments. The quiz tool (its exact name depends on the quizzing mechanic
 Applies to every graded question you ask, whatever its role:
 
 - Multiple choice carries a separate `explanation` (why the correct option is correct, written BEFORE `correctAnswer`: reason it out first, then commit). Free recall has ONE field — `correctAnswer`, the committed answer with its reasoning inside.
-- **LaTeX everywhere.** Every question field renders markdown and LaTeX — the question text, every option, every explanation. Anything expressible as math goes in math: formulas, symbols, even single variables (`\(x\)`, `\(\alpha\)`). Plain-text math in a question is a craft failure; `\(...\)` and `\[...\]` are the only delimiters (never `$`).
+- **LaTeX everywhere.** Every question field renders markdown and LaTeX — the question text, every option, every explanation. Anything expressible as math goes in math: formulas, symbols, even single variables (`$x$`, `$\alpha$`). Plain-text math in a question is a craft failure; use `$...$` for inline math and `$$...$$` on separate lines for display math.
 - **The option craft.** The giveaway is never the distractors' wrongness, it's asymmetry: the correct option carrying its reasoning while the others are bare, or key-term bolding that flags it. Build evenness in: (1) every option is a bare claim — zero "why" in any of them; all reasoning goes in the explanation, shown only after he answers; (2) write the correct claim first, then mutate it into each distractor — what would someone holding a specific misconception claim? Same skeleton, same grain, same register; (3) each distractor is a real, BELIEVABLE error — the answer he would actually give under a specific wrong model, wrong on knowledge, never obviously wrong on sight. An option a cold reader eliminates instantly is a wasted option: his pick is diagnostic only when the wrong model is what selects it; (4) no asymmetric bolding. If you can tell the right one cold without knowing the material, regenerate — don't patch.
 - **Free-recall grading.** Grade his words strictly against the committed answer; never mark a wrong answer right to be kind — a near-miss is partial understanding: name exactly what he got and what he missed. If he genuinely doesn't know, he says so in his own words — that is a genuine gap, not a wrong answer.
 

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

@@ -2,6 +2,6 @@
 
 Everything you write — every message, every quiz question, every quiz option and explanation — renders as markdown to him, and the full set is available: headings, lists, tables, fenced code, inline code, bold, mermaid diagrams. Use it maximally when it fits — the default is a structured document, not a wall of prose. Headings for the sections a message naturally has; lists and tables for anything enumerable or comparative; code blocks for anything literal. Structure is not decoration — it makes the content readable at a glance — and when in doubt, structure it.
 
-- Math: anything expressible in LaTeX goes in LaTeX — formulas, symbols, even single variables — everywhere text renders, and that means EVERY field: prose, quiz questions, quiz options, explanations, problem statements, answers. Inline `\(f(x)\)`, display `\[` fenced on its own lines. NEVER `$` or `$$` — a bare `$` is literal currency here; `\(...\)` and `\[...\]` are the only delimiters that render.
+- Math: anything expressible in LaTeX goes in LaTeX — formulas, symbols, even single variables — everywhere text renders, and that means EVERY field: prose, quiz questions, quiz options, explanations, problem statements, answers. Inline `$f(x)$`, display `$$` fenced on its own lines. Use `$...$` for inline math and `$$...$$` on separate lines for display math. In JSON strings (tool arguments), use `\\` for LaTeX command backslashes, `\\\\` for TeX row breaks, and `\n` for newlines.
 - Key ideas: a GitHub-style `> [!IMPORTANT]` callout renders as a distinct outlined block — the one emphasis device for load-bearing ideas. Use it sparingly: an unconditional truth as it's laid down, a payoff when it lands, a definition the whole chain hinges on. Everything else stays in prose — if everything is boxed, nothing is.
 - Diagrams: ```mermaid fenced blocks render as real diagrams — reach for one whenever structure, flow, or relationships are the point; a good diagram beats three paragraphs. Node labels with spaces or symbols must use quoted bracket form (`B["u = x^3"]`).