feat(api): add game:ready message for client state sync

- WsGameReadySchema added to shared schemas and WsClientMessageSchema
- handleGameReady sends current game:question directly to requesting
  client socket (not broadcast) — foundation for reconnection slice
- router dispatches game:ready to handleGameReady handler

apps/api/src/ws/handlers/gameHandlers.ts

@@ -1,6 +1,6 @@
 import type { WebSocket } from "ws";
 import type { User } from "better-auth";
-import type { WsGameAnswer } from "@lila/shared";
+import type { WsGameAnswer, WsGameReady } from "@lila/shared";
 import { finishGame, getLobbyByCodeWithPlayers } from "@lila/db";
 import { broadcastToLobby, getConnections } from "../connections.js";
 import { lobbyGameStore, timers } from "../gameState.js";
@@ -52,6 +52,32 @@ export const handleGameAnswer = async (
   }
 };
 
+export const handleGameReady = async (
+  ws: WebSocket,
+  msg: WsGameReady,
+  _user: User,
+): Promise<void> => {
+  const state = await lobbyGameStore.get(msg.lobbyId);
+  if (!state) throw new NotFoundError("Game not found");
+
+  const currentQuestion = state.questions[state.currentIndex];
+  if (!currentQuestion) throw new NotFoundError("No active question");
+
+  ws.send(
+    JSON.stringify({
+      type: "game:question",
+      question: {
+        questionId: currentQuestion.questionId,
+        prompt: currentQuestion.prompt,
+        gloss: currentQuestion.gloss,
+        options: currentQuestion.options,
+      },
+      questionNumber: state.currentIndex + 1,
+      totalQuestions: state.questions.length,
+    }),
+  );
+};
+
 export const resolveRound = async (
   lobbyId: string,
   questionIndex: number,

apps/api/src/ws/router.ts

@@ -6,7 +6,7 @@ import {
   handleLobbyLeave,
   handleLobbyStart,
 } from "./handlers/lobbyHandlers.js";
-import { handleGameAnswer } from "./handlers/gameHandlers.js";
+import { handleGameAnswer, handleGameReady } from "./handlers/gameHandlers.js";
 import { AppError } from "../errors/AppError.js";
 
 export type AuthenticatedUser = { session: Session; user: User };
@@ -60,6 +60,9 @@ export const handleMessage = async (
       case "game:answer":
         await handleGameAnswer(ws, msg, auth.user);
         break;
+      case "game:ready":
+        await handleGameReady(ws, msg, auth.user);
+        break;
       default:
         assertExhaustive(msg);
     }

packages/shared/src/schemas/lobby.ts

@@ -52,6 +52,12 @@ export const WsLobbyStartSchema = z.object({
 
 export type WsLobbyStart = z.infer<typeof WsLobbyStartSchema>;
 
+export const WsGameReadySchema = z.object({
+  type: z.literal("game:ready"),
+  lobbyId: z.uuid(),
+});
+export type WsGameReady = z.infer<typeof WsGameReadySchema>;
+
 export const WsGameAnswerSchema = z.object({
   type: z.literal("game:answer"),
   lobbyId: z.uuid(),
@@ -66,6 +72,7 @@ export const WsClientMessageSchema = z.discriminatedUnion("type", [
   WsLobbyLeaveSchema,
   WsLobbyStartSchema,
   WsGameAnswerSchema,
+  WsGameReadySchema,
 ]);
 export type WsClientMessage = z.infer<typeof WsClientMessageSchema>;