feat: 메인 화면에 정식 기수 표시

app/game-api/src/context.ts

@@ -44,6 +44,7 @@ export type WorldStateConfig = z.infer<typeof zWorldStateConfig>;
 
 export const zWorldStateMeta = z.object({
     serverId: z.string().optional(),
+    gameIdx: z.number().int().positive().optional(),
     starttime: z.string().optional(),
     opentime: z.string().optional(),
     preopenAt: z.string().optional(),

app/game-api/src/router/lobby/index.ts

@@ -53,6 +53,8 @@ export const lobbyRouter = router({
 
         return {
             serverId: worldState.meta.serverId?.trim() || ctx.profile?.name || 'game',
+            profile: ctx.profile.id,
+            gameIdx: worldState.meta.gameIdx ?? 1,
             year: worldState.currentYear,
             month: worldState.currentMonth,
             userCnt,

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

@@ -15,6 +15,7 @@ const buildContext = (
 ): GameApiContext =>
     ({
         auth: null,
+        profile: { id: 'che', scenario: 'default', name: 'che:default' },
         db: {
             worldState: {
                 findFirst: vi.fn(async () => ({
@@ -75,6 +76,7 @@ describe('lobby season state', () => {
                 buildContext(
                     {
                         serverId: 'che_260819_season',
+                        gameIdx: 101,
                         preopenAt: '2026-08-19 22:00:00',
                         opentime: '2026-08-19 23:00:00',
                         scenarioMeta: { title: '【가상모드27-b】 아시아 명장전(비급)' },
@@ -103,6 +105,8 @@ describe('lobby season state', () => {
 
         expect(result).toMatchObject({
             serverId: 'che_260819_season',
+            profile: 'che',
+            gameIdx: 101,
             preopenAt: '2026-08-19 22:00:00',
             opentime: '2026-08-19 23:00:00',
             scenarioTitle: '【가상모드27-b】 아시아 명장전(비급)',