feat(admin): show paused sessions in live view; reset to live on logout

- Live cards freeze the worked timer at paused_at and show an amber
  "Gepauzeerd" badge plus a "Pauze H:MM:SS" total when paused.
- AuthContext.signOut resets the path to / so the next admin login lands
  on Live rather than the tab it logged out from.

apps/admin/src/screens/Live.test.tsx

@@ -23,6 +23,8 @@ function makeSession(over: Partial<WorkSession>): WorkSession {
     start_time: new Date(Date.now() - 65_000).toISOString(),
     end_time: null,
     duration_seconds: null,
+    paused_seconds: 0,
+    paused_at: null,
     status: 'active',
     source: 'app',
     notes: null,

apps/admin/src/screens/Live.tsx

@@ -49,7 +49,13 @@ export default function Live() {
 }
 
 function LiveCard({ session, now }: { session: WorkSession; now: number }) {
-  const elapsed = formatTime((now - Date.parse(session.start_time)) / 1000);
+  // While paused the worked timer freezes at the pause moment; otherwise it counts to now.
+  // Worked = (base - start) - paused_seconds, where base is the pause moment when paused.
+  const base = session.paused_at ? Date.parse(session.paused_at) : now;
+  const worked = Math.max(
+    0,
+    Math.floor((base - Date.parse(session.start_time)) / 1000) - session.paused_seconds
+  );
   return (
     <article className="live-card">
       <div className="live-card-head">
@@ -58,7 +64,11 @@ function LiveCard({ session, now }: { session: WorkSession; now: number }) {
       </div>
       <div className="live-activity">{session.activity_name ?? 'Onbekende handeling'}</div>
       <div className="live-meta">{session.pair_count} zolen</div>
-      <div className="live-timer">{elapsed}</div>
+      <div className="live-timer">{formatTime(worked)}</div>
+      {session.paused_at && <span className="live-badge-paused">Gepauzeerd</span>}
+      {session.paused_seconds > 0 && (
+        <span className="live-paused-total">Pauze {formatTime(session.paused_seconds)}</span>
+      )}
     </article>
   );
 }