fix: clean up Roundtable UI state after QA/Discussion request failure

[YizukiAme]

Summary

Fixes #127 — When /api/chat fails mid-stream during a QA or Discussion session, the Roundtable live UI (bubble, thinking animation, engine mode) is never cleaned up, leaving the interface stuck until page refresh.

Problem

The catch blocks in resumeSession(), sendMessage(), and startDiscussion() only append an error message to the chat panel. They don't notify Stage to clean up live UI state, nor do they transition the PlaybackEngine out of live mode.

This affects users with unstable networks or when the LLM provider returns server errors mid-stream.

Changes

File	Change
use-chat-sessions.ts	Add clearLiveSessionAfterError() helper that cleans current buffer, appends error message, and notifies Stage via onLiveSessionError callback
chat-area.tsx	Pass through onLiveSessionError callback
stage.tsx	Add handleLiveSessionError() to reset all live UI state + call engine.handleDiscussionError(). Also add engine.resume() in doResumeTopic() to sync engine mode after soft-pause
engine.ts	Add handleDiscussionError() to exit live mode gracefully without triggering the normal discussion-end flash sequence

How it works

Request fails in catch block
  → clearLiveSessionAfterError()
    → append error message to chat
    → onLiveSessionError callback
      → Stage: resetLiveState() (clear bubble, thinking, streaming flags)
      → Engine: handleDiscussionError() (live → idle, restore lecture cursor)

The session is not hard-closed — it remains retryable.

Verification

• ✅ ESLint: 0 errors on all 4 files
• ✅ TypeScript: only pre-existing latex-to-omml.ts error (unrelated)

[wyuc]

Clean refactor. Consolidates the duplicated error handling into clearLiveSessionAfterError and adds the missing Stage/Engine notification. restoreSavedLectureState extraction is a nice cleanup too. LGTM.

[wyuc]

Approved. There are merge conflicts with main — could you rebase when you get a chance?

[YizukiAme]

Approved. There are merge conflicts with main — could you rebase when you get a chance?

Rebased onto latest main — resolved the conflict with #129's livePausedRef (both additions kept, no logical overlap).
Ready for re-review when you get a chance~