This paper presents a topological bound on the static structure factor of many-body systems with U(1) symmetry, determined solely by the ground state Chern number. The bound is derived from causality and non-negative energy dissipation, and applies to a wide range of systems, including fractional Chern insulators, fractional quantum spin Hall insulators, topological superconductors, and chiral spin liquids. The static structure factor, which measures equal-time density-density correlations, has a lower bound determined by the many-body Chern number, which governs the quantized Hall response. This bound is universal and applies to general chiral topological phases with quantized Hall response. The derivation involves considering the response of the system to an external U(1) gauge field and using the dissipation-fluctuation theorem to relate the static structure factor to the density response function. The result shows that the static structure factor vanishes quadratically at small wavevectors, with the quadratic coefficient (quantum weight) related to the negative-first moment of the optical conductivity. The quantum weight is shown to have a universal lower bound determined by the quantized Hall conductivity, or the many-body Chern number. The bound is applied to various systems, including Chern insulators, topological superconductors, and chiral spin liquids. For Chern insulators, the bound is verified using non-interacting models and is consistent with the quantum metric and Berry curvature of the system. In twisted transition metal dichalcogenide bilayers, the bound is shown to be tight and consistent with experimental observations. The results also apply to spin systems, where the spin structure factor is bounded by the spin Hall conductivity and the Chern number. The topological bound on the structure factor provides a universal feature of topological phases beyond quantized response. It has implications for identifying topological superconductors and quantum spin liquids from the spin structure factor. The bound is derived using fundamental physical principles and holds for a broad range of systems, including strongly interacting systems such as fractional Chern insulators. The results highlight the deep connection between topology and quantum fluctuations in many-body systems.This paper presents a topological bound on the static structure factor of many-body systems with U(1) symmetry, determined solely by the ground state Chern number. The bound is derived from causality and non-negative energy dissipation, and applies to a wide range of systems, including fractional Chern insulators, fractional quantum spin Hall insulators, topological superconductors, and chiral spin liquids. The static structure factor, which measures equal-time density-density correlations, has a lower bound determined by the many-body Chern number, which governs the quantized Hall response. This bound is universal and applies to general chiral topological phases with quantized Hall response. The derivation involves considering the response of the system to an external U(1) gauge field and using the dissipation-fluctuation theorem to relate the static structure factor to the density response function. The result shows that the static structure factor vanishes quadratically at small wavevectors, with the quadratic coefficient (quantum weight) related to the negative-first moment of the optical conductivity. The quantum weight is shown to have a universal lower bound determined by the quantized Hall conductivity, or the many-body Chern number. The bound is applied to various systems, including Chern insulators, topological superconductors, and chiral spin liquids. For Chern insulators, the bound is verified using non-interacting models and is consistent with the quantum metric and Berry curvature of the system. In twisted transition metal dichalcogenide bilayers, the bound is shown to be tight and consistent with experimental observations. The results also apply to spin systems, where the spin structure factor is bounded by the spin Hall conductivity and the Chern number. The topological bound on the structure factor provides a universal feature of topological phases beyond quantized response. It has implications for identifying topological superconductors and quantum spin liquids from the spin structure factor. The bound is derived using fundamental physical principles and holds for a broad range of systems, including strongly interacting systems such as fractional Chern insulators. The results highlight the deep connection between topology and quantum fluctuations in many-body systems.