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/prompt-segments.ts

@@ -70,7 +70,7 @@ export function listSourceSegments(agent: PromptAgent): Array<{ path: string; sh
 	// file mechanics stay out (the doctrine contradicts them; 00-base.md
 	// plays the base role), but the FORMATTING guide composes for every
 	// agent (owner, 2026-10-04 late): LaTeX delimiters and markdown apply
-	// in all settings — practice must write \(...\)/\[...\] too.
+	// in all settings — practice shares the $...$/$$...$$ authoring contract.
 	const segments: Array<{ path: string; shared: boolean; file: string }> = agent === "practice" ? [] : [
 		{ path: "system-prompt.md", shared: true, file: PATHS.teacherPrompt },
 	];