fix(backend): 생성 장수 첫 턴을 접수 시각에 맞춤

daemon 완료 시각이 뒤처져도 신규 장수 턴을 접수된 논리 게임 시각부터 한 턴 안에 배정한다. 로비 응답에는 보정 시계가 사용할 서버 게임 시각과 clock mode를 추가한다.

app/game-api/test/lobbyRouter.test.ts

@@ -3,7 +3,15 @@ import { describe, expect, it, vi } from 'vitest';
 import type { DatabaseClient, GameApiContext } from '../src/context.js';
 import { appRouter } from '../src/router.js';
 
-const buildContext = (meta: Record<string, unknown>): GameApiContext =>
+const buildContext = (
+    meta: Record<string, unknown>,
+    clock: {
+        baseTime?: Date;
+        tick?: bigint;
+        mode?: string;
+        wallAnchor?: Date;
+    } = {}
+): GameApiContext =>
     ({
         auth: null,
         db: {
@@ -16,6 +24,10 @@ const buildContext = (meta: Record<string, unknown>): GameApiContext =>
                     tickSeconds: 3_600,
                     config: {},
                     meta,
+                    clockBaseTime: clock.baseTime ?? null,
+                    clockTick: clock.tick ?? null,
+                    clockMode: clock.mode ?? 'realtime',
+                    clockWallAnchor: clock.wallAnchor ?? null,
                     updatedAt: new Date('2026-07-31T00:00:00.000Z'),
                 })),
             },
@@ -36,4 +48,23 @@ describe('lobby season state', () => {
 
         expect(result.isUnited).toBe(isunited);
     });
+
+    it('returns the projected server game time and whether the clock is running', async () => {
+        const result = await appRouter
+            .createCaller(
+                buildContext(
+                    {},
+                    {
+                        baseTime: new Date('2026-08-15T00:00:00.000Z'),
+                        tick: 72_000_000n,
+                        mode: 'manual',
+                        wallAnchor: new Date('2026-08-15T17:00:00.000Z'),
+                    }
+                )
+            )
+            .lobby.info();
+
+        expect(result.serverTime).toBe('2026-08-15T02:00:00.000Z');
+        expect(result.clockMode).toBe('manual');
+    });
 });