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.

components/TabBar.tsx

@@ -27,9 +27,9 @@ export function TabBar() {
   const reduceMotion = useReducedMotion();
 
   /* The wizards keep their own bottom action bar (Cancel / Next / Freeze) and
-     chat keeps its own composer — the flow is deliberately focused, so the tab
-     bar steps aside there. */
-  const inWizard = pathname === "/create" || pathname === "/homework" || pathname === "/chat";
+     the chat and teach screens keep their own composer — those flows are
+     deliberately focused, so the tab bar steps aside there. */
+  const inWizard = pathname === "/create" || pathname === "/homework" || pathname === "/chat" || pathname === "/teach";
 
   const item =
     "relative flex min-h-[52px] min-w-0 flex-1 flex-col items-center justify-center gap-1 py-2 text-[10.5px] font-medium leading-none transition-colors";