Push a live signal when a user is added to a room (#26)

Previously GET /api/rooms/mine was only ever fetched once at app mount,
so a room added mid-session stayed invisible until a full page reload
-- add_member had no way to reach an already-open client at all.

Backend: ConnectionManager and Broadcaster (renamed from RoomBroadcaster)
now support per-user channels alongside the existing per-room ones, so a
signal can reach a user's socket even for a room they haven't joined
(and by definition can't have, until this fires). add_member publishes
a room_added event on the target user's channel.

Frontend: the WebSocket connection is no longer scoped to whichever
room is open -- ChatShellPage now owns one persistent connection for
the whole session (including while no room is open, which is exactly
when this bug showed), and ChatPane joins/leaves rooms on top of it.
A room_added event triggers a room-list refetch with no reload needed.

Verified end-to-end in the browser: a user sitting on the empty room
list saw a newly-added room appear live, then chatted in it normally.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>

backend/app/ws/chat.py

@@ -78,6 +78,7 @@ async def chat_endpoint(websocket: WebSocket, db: AsyncSession = Depends(get_db)
     presence = websocket.app.state.presence
     broadcaster = websocket.app.state.broadcaster
     joined_rooms: set[uuid.UUID] = set()
+    manager.register_user(user.id, websocket)
 
     try:
         while True:
@@ -225,5 +226,6 @@ async def chat_endpoint(websocket: WebSocket, db: AsyncSession = Depends(get_db)
         pass
     finally:
         manager.leave_all(websocket)
+        manager.unregister_user(user.id, websocket)
         for room_id in joined_rooms:
             await presence.leave(room_id, user.id)