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/Feed.tsx

@@ -1,5 +1,5 @@
 import { useEffect, useRef, useState } from "react";
-import { useInfiniteQuery, useQueryClient } from "@tanstack/react-query";
+import { useInfiniteQuery, useQuery, useQueryClient } from "@tanstack/react-query";
 import { api, type FeedFilters, type Video } from "../lib/api";
 import { toast } from "../lib/toast";
 import VideoCard from "./VideoCard";
@@ -42,6 +42,12 @@ export default function Feed({
 
   useEffect(() => setOverrides({}), [filters]);
 
+  const countQuery = useQuery({
+    queryKey: ["feed-count", filters],
+    queryFn: () => api.feedCount(filters),
+    staleTime: 30_000,
+  });
+
   const sentinel = useRef<HTMLDivElement | null>(null);
   const { hasNextPage, isFetchingNextPage, fetchNextPage } = query;
   useEffect(() => {
@@ -87,6 +93,11 @@ export default function Feed({
 
   return (
     <div className="p-4">
+      <div className="pb-3 text-sm text-muted">
+        {countQuery.data
+          ? `${countQuery.data.count.toLocaleString()} video${countQuery.data.count === 1 ? "" : "s"}`
+          : " "}
+      </div>
       {view === "grid" ? (
         <div className="grid gap-4 grid-cols-[repeat(auto-fill,minmax(260px,1fr))]">
           {items.map((v) => (

frontend/src/components/Header.tsx

@@ -10,6 +10,7 @@ import {
 } from "lucide-react";
 import { SCHEMES, type Scheme, type ThemePrefs } from "../lib/theme";
 import type { FeedFilters, Me } from "../lib/api";
+import SyncStatus from "./SyncStatus";
 
 function IconBtn(props: React.ButtonHTMLAttributes<HTMLButtonElement>) {
   const { className = "", ...rest } = props;
@@ -91,6 +92,8 @@ export default function Header({
         Sub<span className="text-accent">feed</span>
       </div>
 
+      <SyncStatus />
+
       <div className="flex-1 max-w-xl mx-auto relative">
         <Search className="w-4 h-4 absolute left-3 top-1/2 -translate-y-1/2 text-muted" />
         <input

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>
+  );
+}