Teach it back: the AI becomes the curious student

A new mode where the user explains a topic in their own words and the AI
plays a curious classmate: short reactions, exactly one follow-up question
at a time, never "correct"/"wrong", never lecturing, and everything it knows
comes from the user's own decks and notes.

- Entry points: a sidebar row next to Exam mode and a "Teach it back" action
  in every deck's menu (which prefills the topic and scope).
- Start screen: deck or typed topic, friendly/skeptical student, the same
  context and model chips as Chat (/teach?deck=<id> pre-selects the deck).
- Session: the student prompt is a mode of the existing /api/chat route, so
  model resolution, allowlists, rate limits and the streaming client are the
  same code; the personality lives in lib/prompts.ts (TEACH_*). The deck's
  cards travel as a numbered checklist and the student is told to ask about
  the ideas that were skipped; it must say so rather than guess when the
  notes cover nothing. One constant (TEACH_MAX_EXCHANGES = 10) ends the
  session, or the Finish button does.
- Recap: a teach_summary call (JSON, one repair retry) yields well / gaps
  with card references / next steps; each gap links to the card it means.
  "Add missed cards to review" schedules those activities due now; "Turn
  gaps into cards" generates one flashcard per gap from the deck's notes and
  shows an editable review list before saving.
- Sessions are chat conversations with a mode:"teach" marker + session
  record: same IndexedDB history, backup and sync, filtered out of the Chat
  screens; the recent list reopens a recap or continues a session.
- Reused pieces: MessageRow, Composer (now optional attach/scope chips and a
  mic), Markdown/KaTeX, AiDisclaimer, ContextPicker/ModelPicker, plus a
  shared useChatModels hook and useKeyboardInset extracted from Chat.
- Dictation: Web Speech API recognition (lib/speech-input.ts); the mic only
  renders where the browser supports it, and dictated phrases land in the
  draft.
- All strings in en + cs. teach-test: 44 checks (API prompts and guards,
  both entry points, the session, recap actions, history, exchange limit,
  360/390px layouts, mic on/off). Version 0.8.1, SW v103.

README.md

@@ -1492,6 +1492,31 @@ Things I chose while building, with the reasoning:
     into view as you type. No strings, so no i18n; the additive change sits on top of the merged
     scene-heat rework unchanged.
 
+109. **Teach it back: the AI stops being the tutor and becomes the student.** The mode where the user
+     explains a topic and an AI classmate probes the gaps — never a grader, never lecturing.
+     `/teach` (a sidebar row next to Exam mode, plus *Teach it back* in every deck's actions)
+     starts from a deck — which prefills the topic and checks against that deck's own cards — or
+     from a typed topic with the same context and model chips Chat uses. The student is a prompt
+     variant of the chat route (`mode: "teach"`), so model resolution, allowlists, rate limits and
+     the streaming client are literally the same code; the personality lives in `lib/prompts.ts`
+     (`TEACH_*`): friendly or skeptical, one short reaction plus exactly one follow-up question,
+     never "correct"/"wrong", never its own facts, and it says so instead of guessing when the
+     notes cover nothing. It receives the deck's cards as a numbered checklist ("[3] Why …? → …")
+     and is told to ask about the ideas that were skipped. After `TEACH_MAX_EXCHANGES` (10, one
+     constant) replies — or whenever Finish is pressed — a second `teach_summary` call turns the
+     transcript into the recap JSON (`well` / `gaps` with card references / `next`), with one
+     repair retry, and each gap links back to the exact card it refers to. The recap's two
+     follow-ups: *Add missed cards to review* puts those activities on the SRS schedule due now,
+     and *Turn gaps into cards* generates one flashcard per gap from the chosen deck's own notes
+     and shows them in an editable review list (checkbox, front, back) before anything is saved.
+     Sessions are chat conversations with a `mode: "teach"` marker plus the session record, so they
+     inherit the IndexedDB history, backup and encrypted sync for free while staying out of the
+     Chat screens; the recent list on the start screen reopens a recap or continues an unfinished
+     session. The screen reuses MessageRow, the Composer (now able to drop its attach/scope chips
+     and take a mic), Markdown + KaTeX, AiDisclaimer and the pickers; dictation uses the Web Speech
+     API's recognition side, and the mic simply does not render where the browser cannot listen.
+     Verified by the teach suite (44 checks).
+
 108. **The clock that learned the engine's pace.** "The tts dispatcher thing and the text highlight is
     completely out of sync on desktop" — Firefox + speech-dispatcher (like iOS) reports **no
     `onboundary` events**, so the highlight rides the estimated clock, and that estimate drifts