feat(search): live YouTube search backend

Add a live YouTube search that materialises results into the shared catalog so
they render with the normal feed cards + in-app player and gain per-user state.

- YouTubeClient.search_videos(): search.list (100 units), embeddable-only, returns
  flat stubs + nextPageToken; surfaces liveBroadcastContent for live filtering.
- routes/search.py GET /api/search/youtube: require_human + per-user daily cap
  (search_daily_limit_per_user, default 70) + can_spend pre-check (429 on either);
  drops live/upcoming, upserts channel stubs (channels.list) + video stubs, enriches
  (videos.list), runs the youtube.com/shorts probe, then excludes Shorts/live and
  returns feed cards in relevance order with the YouTube pageToken as the cursor.
- Provenance: videos.via_search / channels.from_search (migration 0028) flag
  search-discovered rows; the feed hides them from the Library (scope=all) by default
  via exclude_search_discovered, leaving the Mine feed untouched.
- quota.actions_today() counts a user's per-action events today for the cap; only the
  search.list call is attributed VIDEOS_SEARCH so the counter is exactly 1 per search.

backend/app/routes/feed.py

@@ -124,6 +124,7 @@ def _filtered_query(
     include_live: bool,
     show: str,
     scope: str = "my",
+    exclude_search_discovered: bool = True,
     exclude_tag_category: str | None = None,
 ) -> tuple[Select, object]:
     """Build the feed query (joins + all WHERE filters), shared by /feed and /feed/count.
@@ -171,6 +172,12 @@ def _filtered_query(
     # either way they shouldn't show in any feed view meanwhile.
     query = query.where(Video.unavailable_since.is_(None))
 
+    # Hide videos that only entered the catalog via a live YouTube search (search-discovered
+    # "noise") from the Library by default. Only meaningful in "all" scope: "my" already
+    # restricts to subscribed channels, where a once-searched video is legitimately the user's.
+    if scope == "all" and exclude_search_discovered:
+        query = query.where(Video.via_search.is_(False))
+
     if channel_id:
         query = query.where(Video.channel_id == channel_id)
     if min_duration is not None:
@@ -271,6 +278,7 @@ def _feed_params(
     include_live: bool = False,
     show: str = "unwatched",
     scope: str = "my",
+    exclude_search_discovered: bool = True,
 ) -> dict:
     return {
         "tags": tags,
@@ -287,6 +295,7 @@ def _feed_params(
         "include_live": include_live,
         "show": show,
         "scope": scope,
+        "exclude_search_discovered": exclude_search_discovered,
     }