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/api/src/controllers/gameController.test.ts

@@ -7,7 +7,46 @@ type ErrorResponse = { success: false; error: string };
 type GameStartResponse = SuccessResponse<GameSession>;
 type GameAnswerResponse = SuccessResponse<AnswerResult>;
 
-vi.mock("@lila/db", () => ({ getGameTerms: vi.fn(), getDistractors: vi.fn() }));
+vi.mock("@lila/db", async (importOriginal) => {
+  const actual = await importOriginal<typeof import("@lila/db")>();
+  return { ...actual, getGameTerms: vi.fn(), getDistractors: vi.fn() };
+});
+
+vi.mock("../lib/auth.js", () => ({
+  auth: {
+    api: {
+      getSession: vi
+        .fn()
+        .mockResolvedValue({
+          session: {
+            id: "session-1",
+            userId: "user-1",
+            token: "fake-token",
+            expiresAt: new Date(Date.now() + 1000 * 60 * 60),
+            createdAt: new Date(),
+            updatedAt: new Date(),
+            ipAddress: null,
+            userAgent: null,
+          },
+          user: {
+            id: "user-1",
+            name: "Test User",
+            email: "test@test.com",
+            emailVerified: false,
+            image: null,
+            createdAt: new Date(),
+            updatedAt: new Date(),
+          },
+        }),
+    },
+    handler: vi.fn(),
+  },
+}));
+
+vi.mock("better-auth/node", () => ({
+  fromNodeHeaders: vi.fn().mockReturnValue({}),
+  toNodeHandler: vi.fn().mockReturnValue(vi.fn()),
+}));
 
 import { getGameTerms, getDistractors } from "@lila/db";
 import { createApp } from "../app.js";