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.

components/chat.tsx

@@ -1487,7 +1487,9 @@ export function Chat({
 														// Pending/streaming questions live in the dock, not the
 														// transcript; answered ones render as graded records.
 														if (part.state === "output-available") {
-															return <AnsweredCard key={pi} answer={part.output as QuestionAnswerData} />;
+															return (
+																<AnsweredCard key={pi} answer={part.output as QuestionAnswerData} input={part.input} />
+															);
 														}
 														return null;
 													}