fix: 빈 특기의 획득 예정 나이를 표시한다

메인 장수 read model에 내정·전투 특기 예정 나이를 투영하고, Ref와 같은 현재 나이 다음 해 하한을 적용한다. API 회귀와 Chromium 데스크톱·모바일 검증을 추가한다.

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

@@ -510,6 +510,7 @@ describe('in-game my information ownership', () => {
                 personalCode: 'che_안전',
                 specialCode: 'che_상재',
                 special2Code: 'che_신산',
+                meta: { specage: 31, specage2: 35 },
             }),
         });
 
@@ -520,6 +521,10 @@ describe('in-game my information ownership', () => {
                     specialDomestic: '상재',
                     specialWar: '신산',
                 },
+                traitAges: {
+                    specialDomestic: 31,
+                    specialWar: 35,
+                },
                 traitInfo: {
                     personal: '사기 -5, 징·모병 비용 -20%',
                     specialDomestic: '[내정] 상업 투자 : 기본 보정 +10%, 성공률 +10%p, 비용 -20%',
@@ -543,6 +548,31 @@ describe('in-game my information ownership', () => {
         expect(fixture.db.nation.findUnique).not.toHaveBeenCalled();
     });
 
+    it('returns the scheduled acquisition ages when the owned general has no domestic or war trait', async () => {
+        const fixture = createContext({
+            me: buildGeneral({
+                age: 30,
+                specialCode: 'None',
+                special2Code: 'None',
+                meta: { specage: 35, specage2: 29 },
+            }),
+        });
+
+        await expect(appRouter.createCaller(fixture.context).general.me()).resolves.toMatchObject({
+            general: {
+                age: 30,
+                traits: {
+                    specialDomestic: '-',
+                    specialWar: '-',
+                },
+                traitAges: {
+                    specialDomestic: 35,
+                    specialWar: 29,
+                },
+            },
+        });
+    });
+
     it('reads legacy top-level settings and dispatches only the session-owned general', async () => {
         const requestCommand = vi.fn(async () => ({ type: 'setMySetting', ok: true, generalId: 7 }));
         const fixture = createContext({ requestCommand });