refactor(watch-together): share host peer IDs

lib/watch_together/primitives.dart

@@ -1,5 +1,7 @@
 int watchTogetherSystemNowMs() => DateTime.now().millisecondsSinceEpoch;
 
+String watchTogetherHostPeerId(String sessionId) => 'wt-${sessionId.toUpperCase()}';
+
 bool orderedStringListsEqual(List<String> first, List<String> second) {
   if (first.length != second.length) return false;
   for (var i = 0; i < first.length; i++) {

lib/watch_together/providers/watch_together_provider.dart

@@ -10,6 +10,7 @@ import '../../utils/app_logger.dart';
 import '../models/playback_state.dart';
 import '../models/sync_message.dart';
 import '../models/watch_session.dart';
+import '../primitives.dart';
 import '../services/current_playback_dispatcher.dart';
 import '../services/watch_together_controller.dart';
 import '../services/watch_together_peer_service.dart';
@@ -368,7 +369,7 @@ class WatchTogetherProvider with ChangeNotifier {
       await _peerService!.joinSession(sessionId);
 
       // Session will be fully configured when we receive sessionConfig from host
-      _session = _session!.copyWith(state: SessionState.connected, hostPeerId: 'wt-${sessionId.toUpperCase()}');
+      _session = _session!.copyWith(state: SessionState.connected, hostPeerId: watchTogetherHostPeerId(sessionId));
 
       _displayName = displayName ?? _generateDisplayName();
 

lib/watch_together/services/watch_together_peer_service.dart

@@ -10,6 +10,7 @@ import '../../i18n/strings.g.dart';
 import '../../services/base_peer_service.dart';
 import '../../utils/app_logger.dart';
 import '../models/sync_message.dart';
+import '../primitives.dart';
 
 // Re-export so existing callers that import from here keep working.
 export '../../services/base_peer_service.dart' show PeerError, PeerErrorType;
@@ -352,7 +353,7 @@ class WatchTogetherPeerService with KeepaliveMixin {
 
     _isHost = true;
     _sessionId = sessionId?.toUpperCase() ?? _generateSessionId();
-    _myPeerId = 'wt-$_sessionId';
+    _myPeerId = watchTogetherHostPeerId(_sessionId!);
     _reconnectAttempts = 0;
 
     try {