General construction of square-root lattices with arbitrarily degenerate edge states in double bulk gaps

Square-root topological insulators can inherit edge states from an underlying topological parent system, but existing realizations are typically limited to low degeneracy and small winding numbers. Here, we introduce a general design strategy for extending square-root topological phases to an arbitrary winding number N by combining a fixed structural-subdivision framework with long-range couplings, enabling the number of inherited edge states to increase systematically without raising the root order. We reveal a parity-dependent winding-number phase diagram, where odd and even maximum winding numbers exhibit distinct phase-boundary structures and topological transition sequences. The resulting square-root lattices host N-fold degenerate topological edge states within each of two bulk band gaps, with the corresponding modes exhibiting characteristic fixed-position spatial localization. We also discuss experimentally accessible implementations in photonic waveguide arrays and topolectrical circuits. Our approach provides a scalable route to highly degenerate topological modes and multi-gap wave control in artificial lattices.