feat: 메인 현재 시각을 최근 통신 기준으로 갱신

완료 턴 시각 대신 lobby serverTime을 분 경계에 맞춰 표시한다. 게임 API 응답과 SSE 및 같은 계정 탭 전달을 최근 통신으로 기록하고, 45초 공백에는 시계를 멈춘다. 공통 시계 투영과 단위 및 Chromium 회귀 검증을 추가한다.

app/game-frontend/src/utils/serverClockProjection.ts

@@ -0,0 +1,67 @@
+export type ServerClockProjectionInput = {
+    serverTime?: string;
+    serverWallTime?: string;
+    clockMode?: 'realtime' | 'manual';
+    clockRunning?: boolean;
+    clockStartsAt?: string | null;
+};
+
+export type SampledServerClock = {
+    serverTimeMs: number;
+    sampledClientTimeMs: number;
+    clockMode: 'realtime' | 'manual';
+    startDelayMs: number | null;
+};
+
+const parseInstant = (value?: string | null): number | null => {
+    if (!value) return null;
+    const parsed = new Date(value).getTime();
+    return Number.isFinite(parsed) ? parsed : null;
+};
+
+export const sampleServerClock = (
+    input: ServerClockProjectionInput,
+    sampledClientTimeMs = Date.now()
+): SampledServerClock | null => {
+    const serverTimeMs = parseInstant(input.serverTime);
+    if (serverTimeMs === null) return null;
+
+    let startDelayMs: number | null;
+    if (input.clockMode === 'manual') {
+        startDelayMs = null;
+    } else if (input.clockRunning !== false) {
+        startDelayMs = 0;
+    } else {
+        const serverWallTimeMs = parseInstant(input.serverWallTime);
+        const clockStartsAtMs = parseInstant(input.clockStartsAt);
+        startDelayMs =
+            serverWallTimeMs !== null && clockStartsAtMs !== null
+                ? Math.max(0, clockStartsAtMs - serverWallTimeMs)
+                : null;
+    }
+
+    return {
+        serverTimeMs,
+        sampledClientTimeMs,
+        clockMode: input.clockMode ?? 'realtime',
+        startDelayMs,
+    };
+};
+
+export const projectServerClock = (sample: SampledServerClock, clientTimeMs = Date.now()) => {
+    const clientElapsedMs = Math.max(0, clientTimeMs - sample.sampledClientTimeMs);
+    const elapsedGameMs =
+        sample.clockMode === 'manual' || sample.startDelayMs === null
+            ? 0
+            : Math.max(0, clientElapsedMs - sample.startDelayMs);
+
+    return {
+        clientElapsedMs,
+        time: new Date(sample.serverTimeMs + elapsedGameMs),
+    };
+};
+
+export const millisecondsUntilNextMinute = (time: Date): number => {
+    const remainder = ((time.getTime() % 60_000) + 60_000) % 60_000;
+    return remainder === 0 ? 60_000 : 60_000 - remainder;
+};