feat: door state persistence + multiplayer sync

[nickmeinhold]

Context

Phase 5 of prompt spell doors (#213). Doors stay open across sessions and sync to all players in real-time.

Scope

Firestore Persistence

• Door state per room: rooms/{roomId}/doors/{doorId} → {unlockedBy, unlockedAt, challengeCompletions}
• Challenge progress per user: users/{uid}/spellProgress → attempts, completions, per-school stats
• Doors stay open permanently once unlocked (first wizard opens the path for everyone)

Multiplayer Sync (LiveKit)

• Door unlock event broadcast on data channel (topic: door-unlock)
• All connected players see door open animation in real-time
• Late-join: door states loaded from Firestore on room join (already-opened doors render as open)

Edge Cases

• Collaborative doors: N players must each solve a different challenge before door opens
• Progress counter sync: "2/3 incantations spoken" updates for all players
• Offline unlock: if Firestore write fails, optimistic local unlock with retry

Depends on

• feat: door component for map editor and game world #252 (door component)
• feat: wire terminal → prompt challenge → door unlock #254 (terminal → door wiring)

Part of

Prompt spell doors (#213)