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/quota.py

@@ -36,6 +36,7 @@ class QuotaAction:
     VIDEOS_BACKFILL_FULL = "videos_backfill_full"
     VIDEOS_ENRICH = "videos_enrich"
     VIDEOS_LOOKUP = "videos_lookup"
+    VIDEOS_SEARCH = "videos_search"
     CHANNELS_DISCOVER = "channels_discover"
     CHANNELS_SUBSCRIBE = "channels_subscribe"
     CHANNELS_UNSUBSCRIBE = "channels_unsubscribe"
@@ -109,6 +110,26 @@ def can_spend(db: Session, units: int) -> bool:
     return remaining_today(db) >= units
 
 
+def actions_today(db: Session, user_id: int, action: str) -> int:
+    """How many quota events of `action` this user has logged so far in the current Pacific
+    day — for per-user, per-action daily caps (e.g. the live-search limit). Counts events,
+    not units, so it only works for actions charged exactly once per user action."""
+    from sqlalchemy import func, select  # local import: keep the module's import head lean
+
+    return (
+        db.scalar(
+            select(func.count())
+            .select_from(QuotaEvent)
+            .where(
+                QuotaEvent.user_id == user_id,
+                QuotaEvent.action == action,
+                QuotaEvent.created_at >= pacific_day_start_utc(),
+            )
+        )
+        or 0
+    )
+
+
 def record_usage(db: Session, units: int) -> None:
     """Atomically add `units` to today's counter (upsert) and log an attribution event."""
     if units <= 0: