feat(admin): live active-work view (5s refresh)

apps/admin/src/lib/elapsed.test.ts

@@ -0,0 +1,14 @@
+import { describe, expect, it } from 'vitest';
+import { formatTime } from './elapsed';
+
+describe('formatTime', () => {
+  it('formats seconds as HH:MM:SS', () => {
+    expect(formatTime(0)).toBe('00:00:00');
+    expect(formatTime(65)).toBe('00:01:05');
+    expect(formatTime(3661)).toBe('01:01:01');
+  });
+
+  it('never returns a negative time', () => {
+    expect(formatTime(-10)).toBe('00:00:00');
+  });
+});

apps/admin/src/lib/elapsed.ts

@@ -0,0 +1,11 @@
+// Pure timing helper — server-authoritative elapsed.
+// Elapsed is computed from the server start_time (wall-clock), not a tick counter,
+// so it survives backgrounding. Ported from the worker's lib/stopwatch.ts.
+
+export function formatTime(totalSeconds: number): string {
+  const s = Math.max(0, Math.floor(totalSeconds));
+  const h = Math.floor(s / 3600);
+  const m = Math.floor((s % 3600) / 60);
+  const sec = s % 60;
+  return `${String(h).padStart(2, '0')}:${String(m).padStart(2, '0')}:${String(sec).padStart(2, '0')}`;
+}