The paper presents a topological bound on the static structure factor of many-body systems with $U(1)$ symmetry, which is determined solely by the ground state Chern number. This bound holds for a wide range of systems, including fractional Chern insulators, quantum spin Hall insulators, topological superconductors, and chiral spin liquids. The bound is derived using causality and non-negative energy dissipation, and it reveals universal features of topological phases beyond quantized responses. The authors apply their theory to various systems, demonstrating that the bound is saturated for Landau level systems and tight for zero-field Chern insulators in twisted semiconductor bilayers. The bound is useful for identifying topological phases and understanding the energy gap in Chern insulators. The work also discusses the implications for chiral topological superconductors and chiral spin liquids, providing a theoretical framework for experimental verification.The paper presents a topological bound on the static structure factor of many-body systems with $U(1)$ symmetry, which is determined solely by the ground state Chern number. This bound holds for a wide range of systems, including fractional Chern insulators, quantum spin Hall insulators, topological superconductors, and chiral spin liquids. The bound is derived using causality and non-negative energy dissipation, and it reveals universal features of topological phases beyond quantized responses. The authors apply their theory to various systems, demonstrating that the bound is saturated for Landau level systems and tight for zero-field Chern insulators in twisted semiconductor bilayers. The bound is useful for identifying topological phases and understanding the energy gap in Chern insulators. The work also discusses the implications for chiral topological superconductors and chiral spin liquids, providing a theoretical framework for experimental verification.