feat: sync status indicator, admin pause/resume, filtered video count

- Header shows a live sync status (total videos + channels still backfilling, or
  "paused"), polled every 30s
- Admins can pause/resume the background sync; a paused flag in app_state makes
  scheduled jobs skip (migration 0006)
- GET /api/feed/count returns the number of videos matching the current filters;
  shared filter builder keeps it in sync with /api/feed; shown above the feed
- /api/sync/status reports backfill progress, pending enrichment and paused state

backend/app/models.py

@@ -237,3 +237,14 @@ class ApiQuotaUsage(Base):
 
     day: Mapped[date] = mapped_column(Date, primary_key=True)
     units_used: Mapped[int] = mapped_column(Integer, default=0, server_default="0")
+
+
+class AppState(Base):
+    """Single-row global app state (admin-controlled)."""
+
+    __tablename__ = "app_state"
+
+    id: Mapped[int] = mapped_column(primary_key=True, default=1)
+    sync_paused: Mapped[bool] = mapped_column(
+        Boolean, default=False, server_default="false"
+    )