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.

tests/chat-flow.test.ts

@@ -35,7 +35,7 @@ import { buildSubagentSteer } from "../src/core/contracts.ts";
 import { abortTurn, runningTurn, settledTurnResponse } from "../src/server/turn-runs.ts";
 import { chatStore } from "../src/store/chats.ts";
 import { jobStore } from "../src/store/jobs.ts";
-import { buildAnswer, coerceCorrectAnswer, dismissedAnswer, normalizeOptions, resolveCorrect, type QuestionData } from "../src/core/question-core.ts";
+import { buildAnswer, dismissedAnswer, normalizeOptions, resolveCorrect, type QuestionData } from "../src/core/question-core.ts";
 import { danglingQuestionOf, endsInDismissal, needsContinuation, newAnswerResult } from "../src/core/native-messages.ts";
 import { projectView } from "../src/core/view.ts";
 import type { PiEngine } from "../src/agent/models.ts";
@@ -113,7 +113,7 @@ function gradedAnswer(input: typeof QUIZ_INPUT, values: string[]) {
 		question: input.question,
 		mode: "single-select",
 		options,
-		correctIndices: resolveCorrect(coerceCorrectAnswer(input.correctAnswer), options).indices,
+		correctIndices: resolveCorrect(input.correctAnswer, options).indices,
 		explanation: input.explanation,
 		createdAt: new Date().toISOString(),
 	};