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

frontend/src/lib/api.ts

@@ -97,14 +97,30 @@ function feedQuery(f: FeedFilters, offset: number, limit: number): string {
   return p.toString();
 }
 
+export interface SyncStatus {
+  subscriptions: number;
+  channels_total: number;
+  channels_backfilling: number;
+  videos_total: number;
+  pending_enrich: number;
+  quota_used_today: number;
+  quota_remaining_today: number;
+  paused: boolean;
+  is_admin: boolean;
+}
+
 export const api = {
   me: (): Promise<Me> => req("/api/me"),
   tags: (): Promise<Tag[]> => req("/api/tags"),
-  status: (): Promise<any> => req("/api/sync/status"),
+  status: (): Promise<SyncStatus> => req("/api/sync/status"),
   feed: (f: FeedFilters, offset: number, limit: number): Promise<FeedResponse> =>
     req(`/api/feed?${feedQuery(f, offset, limit)}`),
+  feedCount: (f: FeedFilters): Promise<{ count: number }> =>
+    req(`/api/feed/count?${feedQuery(f, 0, 0)}`),
   setState: (id: string, status: string) =>
     req(`/api/videos/${id}/state`, { method: "POST", body: JSON.stringify({ status }) }),
+  pauseSync: () => req("/api/sync/pause", { method: "POST" }),
+  resumeSync: () => req("/api/sync/resume", { method: "POST" }),
   savePrefs: (p: Record<string, any>) =>
     req("/api/me/preferences", { method: "PUT", body: JSON.stringify(p) }),
 };