feat(nav): deep 1:1 zibll search-box + navbar-search + newposts unify

Port aksrt_get_search_box/posts for menu_search and layout 1/3 slide;
unify publish switch; expand user-box admin row; CSS/JS remote-box parity.

inc/helpers/header/aksrt-header-render.php

@@ -183,8 +183,16 @@ if (!function_exists('aksrt_header')) :
 			aksrt_nav_mobile_shell($ctx);
 		}
 
-		if (function_exists('aksrt_render_signout_modal')) {
-			aksrt_render_signout_modal();
-		}
+					// Layout 1/3 search slide (zibll .navbar-search). Layout-2 uses inline menu_search.
+			if (!empty($ctx['search_enabled'])) {
+				$navbar_search = locate_template('templates/parts/navbar-search.php');
+				if ($navbar_search) {
+					include $navbar_search;
+				}
+			}
+
+			if (function_exists('aksrt_render_signout_modal')) {
+				aksrt_render_signout_modal();
+			}
 	}
 endif;