feat(lobby): cap simultaneous quick games at 10

Limit a player to 10 active quick games (auto-match + AI); friend games created
by invitation are not counted. At the cap the backend refuses both new-game
entry points — quick enqueue and invitation creation — with 409
game_limit_reached, while accepting an incoming invitation stays allowed, so
friend games are capped from the other end. The lobby disables "New Game" and
shows a low-emphasis notice, driven by a new at_game_limit flag on games.list
(no per-event payload: a turn change does not move the count, and the lobby
already re-fetches games.list on entry and every game event).

- game.MaxActiveQuickGames + Store/Service.CountActiveQuickGames (active/open
  seats, no game_invitations row; hidden games still count -> dedicated count)
- Server.ensureUnderGameLimit gating handleEnqueue + handleCreateInvitation;
  game.ErrGameLimitReached -> 409 game_limit_reached
- FB GameList.at_game_limit (regenerated Go + TS) through the gateway transcode
  and UI codec; gameListDTO + lobbycache snapshot + Lobby.svelte + i18n
- tests: integration count rule + HTTP gate + accept bypass; server error map;
  gateway transcode round-trip; UI codec + lobbycache unit; e2e gamelimit
- docs: PRERELEASE (GL), FUNCTIONAL(+ru), ARCHITECTURE 8, UI_DESIGN, backend README

backend/internal/game/store.go

@@ -450,6 +450,28 @@ func (s *Store) ListGamesForAccount(ctx context.Context, accountID uuid.UUID) ([
 	return out, nil
 }
 
+// CountActiveQuickGames counts the account's in-progress quick games — the ones the
+// simultaneous-game limit (MaxActiveQuickGames) is checked against. It includes both
+// active and still-open (awaiting-opponent) games, the honest-AI ones among them, and
+// excludes friend games (those linked to a game_invitations row) and finished games.
+// A hidden game still occupies a slot, so this is a dedicated count rather than a
+// filter over ListGamesForAccount (which drops hidden games). Joining on the account's
+// own seat counts each game once (an open game's empty opponent seat has no account).
+func (s *Store) CountActiveQuickGames(ctx context.Context, accountID uuid.UUID) (int, error) {
+	// The status literals are game.StatusActive / game.StatusOpen, matching the
+	// games.status CHECK in the baseline migration.
+	const q = `
+SELECT COUNT(*) FROM backend.games g
+JOIN backend.game_players gp ON gp.game_id = g.game_id
+LEFT JOIN backend.game_invitations gi ON gi.game_id = g.game_id
+WHERE gp.account_id = $1 AND g.status IN ('active', 'open') AND gi.game_id IS NULL`
+	var n int
+	if err := s.db.QueryRowContext(ctx, q, accountID).Scan(&n); err != nil {
+		return 0, fmt.Errorf("game: count active quick games: %w", err)
+	}
+	return n, nil
+}
+
 // HideGame hides a game from the account's own lobby list (idempotent). The caller validates the
 // game is finished and the account is a player.
 func (s *Store) HideGame(ctx context.Context, accountID, gameID uuid.UUID) error {