perf(feed): collapse count + facets into one /feed/summary pass (E2 S4, PERF-13)

The FE fired /feed/count (1 pass) + /facets (3 passes) on every filter change =
4 full scans of the subscribed set. New /api/feed/summary does ONE grouped query
(the base filter with tags stripped, GROUP BY channel_id -> per-channel video
counts) and derives the total count + every tag's distinct-channel facet count in
Python on the ~few-hundred-row channel map, faithfully mirroring filtered_query's
channel-based tag logic + the /facets drill-down (own category dropped, unless a
conjunctive topic-AND). Returns {count, facets} — the same numbers the two old
endpoints returned.

FE: a single shared query (Feed reads .count, Sidebar reads .facets); same key +
filters, so React Query dedupes to ONE request. Invalidations repointed to
qk.feedSummary. The old /feed/count + /facets endpoints (and their now-unused FE
api/qk) are kept as the equivalence-test oracle + safety net; a later cleanup can
drop them.

New equivalence test (test_feed_views, 8 filter combos incl. cross-category,
disjunctive/conjunctive topic modes, search): /feed/summary count == /feed/count
and facets == /facets, byte-identical. Structural win: 2 requests / 4-6 queries
-> 1 request / 2-3 queries (the review measured 530 -> 144 ms). Gate green:
344 pytest, frontend tsc/eslint/knip/prettier/222 vitest.

backend/app/feedquery.py

@@ -340,6 +340,82 @@ def filtered_query(
     return query, rank_expr
 
 
+def feed_summary(db: Session, user: User, **params) -> dict:
+    """Single-pass replacement for /feed/count + 3x /facets (PERF-13): the sidebar's total video count
+    and every tag's distinct-channel facet count, derived in Python from ONE grouped query instead of
+    four full passes over the subscribed set on every filter change.
+
+    The one query is the base filter with the TAG predicate removed, grouped by channel: it yields every
+    channel that has a matching video, with its video count. From that ~few-hundred-row map we compute:
+      - count  = sum of the video counts of the channels that pass the SELECTED tag filter;
+      - facets = per tag, the number of distinct channels that carry it AND pass the drill-down filter
+                 (the tag's own category dropped, unless it's a conjunctive topic-AND — mirroring the
+                 old /facets exclude_tag_category logic exactly).
+    Tag matching is channel-based, identical to filtered_query's tag block (AND across categories, OR
+    within, topic-AND requires ALL selected topics)."""
+    tag_mode = params.get("tag_mode", "or")
+
+    # ONE pass: per-channel video counts for the base filter, tags stripped so every candidate channel
+    # is present to facet over.
+    base, _rank = filtered_query(db, user, **{**params, "tags": []})
+    ch_col = base.with_only_columns(Video.channel_id).subquery()
+    channel_counts: dict[str, int] = dict(
+        db.execute(select(ch_col.c.channel_id, func.count()).group_by(ch_col.c.channel_id)).all()
+    )
+    if not channel_counts:
+        return {"count": 0, "facets": {}}
+
+    # channel -> its visible tag ids, plus each tag's category (for grouping the facet output).
+    visible = or_(ChannelTag.user_id.is_(None), ChannelTag.user_id == user.id)
+    channel_tags: dict[str, set[int]] = {}
+    tag_category: dict[int, str] = {}
+    for cid, tid, cat in db.execute(
+        select(ChannelTag.channel_id, ChannelTag.tag_id, Tag.category)
+        .join(Tag, Tag.id == ChannelTag.tag_id)
+        .where(visible, ChannelTag.channel_id.in_(channel_counts.keys()))
+    ).all():
+        channel_tags.setdefault(cid, set()).add(tid)
+        tag_category[tid] = cat
+
+    # Selected tags grouped by category (read from Tag, so a selected tag that no base channel carries
+    # still filters everything out for its category — matching the SQL `IN (empty)` behaviour).
+    selected_by_cat: dict[str, set[int]] = {}
+    selected = params.get("tags") or []
+    if selected:
+        for tid, cat in db.execute(
+            select(Tag.id, Tag.category).where(Tag.id.in_(selected))
+        ).all():
+            selected_by_cat.setdefault(cat, set()).add(tid)
+
+    def passes(ch_tags: set[int], exclude_cat: str | None) -> bool:
+        for cat, sel in selected_by_cat.items():
+            if cat == exclude_cat:
+                continue
+            if cat == "topic" and tag_mode == "and" and len(sel) > 1:
+                if not sel <= ch_tags:  # conjunctive: channel must carry ALL selected topics
+                    return False
+            elif not (sel & ch_tags):  # disjunctive: channel must carry ANY selected tag
+                return False
+        return True
+
+    count = sum(
+        n for cid, n in channel_counts.items() if passes(channel_tags.get(cid, set()), None)
+    )
+
+    facets: dict[int, int] = {}
+    for category in ("language", "topic", "other"):
+        # Drop this category's own selections so its chips keep independent counts; a conjunctive
+        # topic-AND keeps them (each remaining chip narrows to channels that ALSO have the selected).
+        exclude = None if (category == "topic" and tag_mode == "and") else category
+        for cid, ch_tags in channel_tags.items():
+            if not passes(ch_tags, exclude):
+                continue
+            for tid in ch_tags:
+                if tag_category.get(tid) == category:
+                    facets[tid] = facets.get(tid, 0) + 1
+    return {"count": count, "facets": facets}
+
+
 # --- Keyset (cursor) pagination ---------------------------------------------
 #
 # Each sort is described as an ordered list of key columns (expression + direction