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/routes/multiplayer.tsx

@@ -1,6 +1,13 @@
 import { createFileRoute, Outlet, redirect } from "@tanstack/react-router";
-import { WsProvider } from "../lib/ws-provider.js";
+import { useEffect } from "react";
 import { authClient } from "../lib/auth-client.js";
+import { WsProvider } from "../lib/ws-provider.js";
+import { useWsConnect } from "../lib/ws-hooks.js";
+
+const wsBaseUrl =
+  (import.meta.env["VITE_WS_URL"] as string) ||
+  (import.meta.env["VITE_API_URL"] as string) ||
+  "";
 
 export const Route = createFileRoute("/multiplayer")({
   component: MultiplayerLayout,
@@ -13,9 +20,23 @@ export const Route = createFileRoute("/multiplayer")({
   },
 });
 
+function WsConnector() {
+  const connect = useWsConnect();
+
+  useEffect(() => {
+    void connect(wsBaseUrl).catch((err) => {
+      console.error("WebSocket connection failed:", err);
+    });
+    // eslint-disable-next-line react-hooks/exhaustive-deps
+  }, []);
+
+  return null;
+}
+
 function MultiplayerLayout() {
   return (
     <WsProvider>
+      <WsConnector />
       <Outlet />
     </WsProvider>
   );