UI iteration 2: toggleable sidebar, floating composer blob, pending-question morph

Sidebar (replaces the modal drawer):
- Inline, toggleable, no backdrop/focus theft — can stay open while working
- Open state persists across pages (provider in root layout) and reloads
  (localStorage; starts closed server-side to avoid hydration mismatch,
  stored value applied post-mount; default open on first run)
- Hamburger in page headers only when closed; close button in sidebar header

Floating composer blob (per reference screenshot):
- Floats over the transcript (absolute overlay), persists at bottom while
  scrolling, content fades behind it (gradient mask); pb-44 scroll clearance
- Concentric corner radii: blob rounded-2xl(16px) − p-1.5(6px) = send
  rounded-[10px] = 25% of the 40px button; CornerDownLeft icon per reference
- Stop moved from header into the blob (busy → square icon)
- prompt-input.tsx + 7 orphan ui files pruned (attachments/command palette
  unused); hand-rolled Composer is ~100 lines and render-tested

Pending-question morph (one text field on screen, never two):
- Quiz/ask-select pending → blob becomes the note field (send hidden, hint
  line above); card's internal note input removed; Submit consumes it
- Ask free-text pending → blob becomes the answer field; its send submits
  via the card's registered submit api (ref bridge); Enter works
- Chat owns the single text state; card clears it on submit
- canSend for answer mode mirrors the card predicate directly — reading the
  ref during render lagged one commit (found in browser testing)

Widths: one shared column (max-w-3xl + px-4) for transcript, cards, and
composer — all three measured pixel-identical (left 416, width 736).
Assistant Message max-w-full/MessageContent w-full (was max-w-[95%]/w-fit —
the 'cards are a bit shorter' bug).

User messages render plain text again — no math pipeline on learner-authored
text (currency hazard accepted for the model only, guarded by the prompt).

Tests: 58 pass (5 new composer pins; card tests updated to controlled
props). Verified live: note flow, answer flow, sidebar persistence, float,
fade, zero console errors — plus the live model spontaneously honoring the
$ currency escape contract in a continuation.

tests/chat-flow.test.ts

@@ -130,7 +130,9 @@ describe("teacher chat flow (scripted model)", () => {
 
 		// The pending-question gate sees it.
 		const pending = findPendingToolCall(saved1.messages);
-		assert.deepEqual(pending, { toolCallId: "call-1", tool: "quiz" });
+		assert.equal(pending?.toolCallId, "call-1");
+		assert.equal(pending?.tool, "quiz");
+		assert.deepEqual(pending?.input, QUIZ_INPUT);
 
 		// ── The learner answers: client grades, output goes back on the part ──
 		const options = normalizeOptions(QUIZ_INPUT.options);