Phase 5: Redis pub/sub for horizontal scaling

Splits the WebSocket layer into three pieces so one app instance and many
behave identically: ConnectionManager stays a purely local socket registry;
RoomBroadcaster publishes chat messages to a per-room Redis channel and
every instance (including the publisher) forwards received messages to its
own local sockets via a single psubscribe("room:*") listener started in
main.py's lifespan; Presence is a Redis-backed refcounted hash per room
tracking who's connected across all instances.

Presence replaces the old process-local connected_user_ids check that
Phase 4's offline-push logic used -- without it, a user connected on a
different instance would look offline and get a redundant push. Fixing
this was scoped in beyond the issue's literal ask (message fan-out only)
since it's a real correctness gap in a phase specifically about running
more than one instance; a known limitation (no heartbeat/TTL, so a hard
crash leaks a presence increment) is documented in the README instead of
solved here.

New tests/test_broadcast.py spins up two independent app instances sharing
one Postgres + Redis to prove delivery and presence both actually cross
the Redis boundary, not just work in-process. Manually verified the same
thing against two real uvicorn processes on different ports.

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

backend/app/ws/broadcaster.py

@@ -0,0 +1,41 @@
+import json
+import uuid
+
+from redis.asyncio import Redis
+
+from app.ws.connection_manager import ConnectionManager
+
+ROOM_CHANNEL_PREFIX = "room:"
+
+
+class RoomBroadcaster:
+    """Cross-instance message fan-out (ARCHITECTURE.md phase 5).
+
+    Publishes to a per-room Redis channel; every app instance -- including
+    the one that published -- subscribes via a single pattern subscription
+    and forwards to its own locally connected WebSocket clients via
+    ConnectionManager. A single instance just talks to itself through Redis,
+    so there's no separate code path for the 1-instance vs N-instance case.
+    """
+
+    def __init__(self, redis: Redis, manager: ConnectionManager) -> None:
+        self._redis = redis
+        self._manager = manager
+
+    async def publish(self, room_id: uuid.UUID, payload: dict) -> None:
+        await self._redis.publish(f"{ROOM_CHANNEL_PREFIX}{room_id}", json.dumps(payload))
+
+    async def listen(self) -> None:
+        pubsub = self._redis.pubsub()
+        await pubsub.psubscribe(f"{ROOM_CHANNEL_PREFIX}*")
+        try:
+            async for message in pubsub.listen():
+                if message["type"] != "pmessage":
+                    continue
+                channel = message["channel"]
+                room_id = uuid.UUID(channel.removeprefix(ROOM_CHANNEL_PREFIX))
+                payload = json.loads(message["data"])
+                await self._manager.broadcast(room_id, payload)
+        finally:
+            await pubsub.punsubscribe(f"{ROOM_CHANNEL_PREFIX}*")
+            await pubsub.aclose()