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/components/SyncStatus.tsx

@@ -0,0 +1,49 @@
+import { useMutation, useQuery, useQueryClient } from "@tanstack/react-query";
+import { Database, Loader2, Pause, Play } from "lucide-react";
+import { api } from "../lib/api";
+import { formatViews } from "../lib/format";
+
+export default function SyncStatus() {
+  const qc = useQueryClient();
+  const { data } = useQuery({
+    queryKey: ["sync-status"],
+    queryFn: api.status,
+    refetchInterval: 30_000,
+    staleTime: 25_000,
+  });
+
+  const toggle = useMutation({
+    mutationFn: () => (data?.paused ? api.resumeSync() : api.pauseSync()),
+    onSuccess: () => qc.invalidateQueries({ queryKey: ["sync-status"] }),
+  });
+
+  if (!data) return null;
+
+  return (
+    <div className="hidden lg:flex items-center gap-2 text-xs text-muted">
+      <Database className="w-3.5 h-3.5" />
+      <span>{formatViews(data.videos_total)} videos</span>
+      <span className="opacity-40">·</span>
+      {data.paused ? (
+        <span className="text-accent font-medium">paused</span>
+      ) : data.channels_backfilling > 0 ? (
+        <span className="flex items-center gap-1">
+          <Loader2 className="w-3.5 h-3.5 animate-spin" />
+          {data.channels_backfilling} syncing
+        </span>
+      ) : (
+        <span>all synced</span>
+      )}
+      {data.is_admin && (
+        <button
+          onClick={() => toggle.mutate()}
+          disabled={toggle.isPending}
+          title={data.paused ? "Resume background sync" : "Pause background sync"}
+          className="ml-1 p-1 rounded-md hover:bg-card hover:text-fg transition"
+        >
+          {data.paused ? <Play className="w-3.5 h-3.5" /> : <Pause className="w-3.5 h-3.5" />}
+        </button>
+      )}
+    </div>
+  );
+}