fix(game): clear nudges on game completion

Nudge badges lingered in the lobby on games that ended by turn-timeout,
resignation or forfeit: those paths commit the finish directly, bypassing
the move path's per-mover NudgeClearer, so the awaited seat's nudge was
never marked read. A finished game's nudges are stale, so clear them all.

Add a wired NudgeExpirer (social.ExpireNudges) called from the shared
commit finish block — covering every completion path — and from the
voidGame recovery path. It clears every seat's nudge bits for the game
and leaves chat messages unread; unlike ClearNudges it records no
publish-to-read latency, since a completion is an expiry, not a read.

An integration test reproduces the timeout case (nudge cleared, chat
kept). Docs: ARCHITECTURE §9.1, FUNCTIONAL (+_ru), backend/README.

backend/internal/social/chatread.go

@@ -55,6 +55,25 @@ func (svc *Service) ClearNudges(ctx context.Context, gameID, accountID uuid.UUID
 	return nil
 }
 
+// ExpireNudges marks every pending nudge in the game read, for when the game finishes: a nudge
+// badge is stale once the game is over, so completion clears them all for every seat. Chat
+// messages are left untouched (they stay unread). It satisfies game.NudgeExpirer and is wired
+// into the completion path; failures are the caller's to log (the game has already finished).
+// Unlike ClearNudges it records no publish-to-read latency — a completion is an expiry, not the
+// recipient reading the nudge — so the latency metric is not skewed by games that end hours later.
+func (svc *Service) ExpireNudges(ctx context.Context, gameID uuid.UUID) error {
+	ctx, span := svc.tracer.Start(ctx, "social.ExpireNudges",
+		trace.WithAttributes(attribute.String("game.id", gameID.String())))
+	defer span.End()
+	n, err := svc.store.expireNudges(ctx, gameID)
+	if err != nil {
+		span.RecordError(err)
+		return err
+	}
+	span.SetAttributes(attribute.Int64("expired.count", n))
+	return nil
+}
+
 // UnreadGames returns the set of games in which viewerID has at least one unread chat
 // entry, for seeding the lobby's per-card unread badge in a single query.
 func (svc *Service) UnreadGames(ctx context.Context, viewerID uuid.UUID) (map[uuid.UUID]bool, error) {
@@ -140,6 +159,21 @@ RETURNING m.created_at`
 	return out, rows.Err()
 }
 
+// expireNudges marks every still-unread nudge in the game read for all seats at once, used when
+// the game finishes. It returns the number of nudge entries it cleared. Only 'nudge' rows are
+// touched, so chat messages keep their unread bits; it returns no post times because a completion
+// is an expiry, not a player reading, and must not feed the publish-to-read latency metric.
+func (s *Store) expireNudges(ctx context.Context, gameID uuid.UUID) (int64, error) {
+	const q = `UPDATE backend.chat_messages
+SET unread_seats = 0
+WHERE game_id = $1 AND kind = 'nudge' AND unread_seats <> 0`
+	res, err := s.db.ExecContext(ctx, q, gameID)
+	if err != nil {
+		return 0, fmt.Errorf("social: expire nudges: %w", err)
+	}
+	return res.RowsAffected()
+}
+
 // unreadGames returns the games where viewerID has an unread entry, resolving their
 // seat per game through the game_players join.
 func (s *Store) unreadGames(ctx context.Context, viewerID uuid.UUID) (map[uuid.UUID]bool, error) {