feat: multiplayer slice — end to end working

WebSocket server:
- WS auth via Better Auth session on upgrade request
- Router with discriminated union dispatch and two-layer error handling
- In-memory connections map with broadcastToLobby
- Lobby handlers: join, leave, start
- Game handlers: answer, resolve round, end game, game:ready for state sync
- Shared game state store (LobbyGameStore interface + InMemory impl)
- Timer map separate from store for Valkey-readiness

REST API:
- POST /api/v1/lobbies — create lobby + add host as first player
- POST /api/v1/lobbies/:code/join — atomic join with capacity/status checks
- getLobbyWithPlayers added to model for id-based lookup

Frontend:
- WsClient class with typed on/off, connect/disconnect, isConnected
- WsProvider owns connection lifecycle (connect/disconnect/isConnected state)
- WsConnector component triggers connection at multiplayer layout mount
- Lobby waiting room: live player list, copyable code, host Start button
- Game view: reuses QuestionCard, game:ready on mount, round results
- MultiplayerScoreScreen: sorted scores, winner highlight, tie handling
- Vite proxy: /ws and /api proxied to localhost:3000 for dev cookie fix

Tests:
- lobbyService.test.ts: create, join, retry, idempotency, full lobby
- auth.test.ts: 401 reject, upgrade success, 500 on error
- router.test.ts: dispatch all message types, error handling
- vitest.config.ts: exclude dist folder

Fixes:
- server.ts: server.listen() instead of app.listen() for WS support
- StrictMode removed from main.tsx (incompatible with WS lifecycle)
- getLobbyWithPlayers(id) added for handleLobbyStart lookup

apps/web/src/lib/ws-client.ts

@@ -25,25 +25,27 @@ export class WsClient {
   public onClose: ((event: CloseEvent) => void) | null = null;
 
   connect(apiUrl: string): Promise<void> {
-    // If already connected or connecting, resolve immediately
-    if (
-      this.ws &&
-      (this.ws.readyState === WebSocket.OPEN ||
-        this.ws.readyState === WebSocket.CONNECTING)
-    ) {
-      return Promise.resolve();
-    }
     return new Promise((resolve, reject) => {
-      if (this.ws) {
-        this.ws.close();
-        this.ws = null;
+      if (
+        this.ws &&
+        (this.ws.readyState === WebSocket.OPEN ||
+          this.ws.readyState === WebSocket.CONNECTING)
+      ) {
+        resolve();
+        return;
       }
 
-      const wsUrl = apiUrl
-        .replace(/^https:\/\//, "wss://")
-        .replace(/^http:\/\//, "ws://");
+      let wsUrl: string;
+      if (!apiUrl) {
+        wsUrl = "/ws";
+      } else {
+        wsUrl =
+          apiUrl
+            .replace(/^https:\/\//, "wss://")
+            .replace(/^http:\/\//, "ws://") + "/ws";
+      }
 
-      this.ws = new WebSocket(`${wsUrl}/ws`);
+      this.ws = new WebSocket(wsUrl);
 
       this.ws.onopen = () => {
         resolve();