Exact screening constraints for theories of inhomogeneous electrolytes: Generalized Stillinger–Lovett relations and a nonlinear Ward hierarchy

Exact theories of electrolyte structure must satisfy the nonlocal constraints imposed by electrostatic screening, yet these constraints are most commonly formulated only for homogeneous bulk systems and at the level of two-point correlations. Here, we develop a unified field-theoretic framework that extends them to general inhomogeneous ionic fluids and nonlinear charge correlations. Using a standard Hubbard–Stratonovich representation of the partition function, we derive an exact Ward identity relating charge-density correlations to the full electrostatic-field propagator. In homogeneous conducting electrolytes, its long-wavelength limit recovers the classical Stillinger–Lovett screening conditions, whereas in inhomogeneous systems, it remains an exact local kernel relation. Repeated functional differentiation generates a nonlinear Ward hierarchy that constrains higher-order charge correlations. We show explicitly how these exact relations are represented in classical density functional theory, Ornstein–Zernike formulations, the random-phase approximation, and the mean-spherical approximation. The resulting hierarchy provides a common bridge between field-theoretic and liquid-state descriptions of electrolytes and supplies rigorous, model-independent consistency tests for approximate theories of screening, response, and correlations in spatially inhomogeneous classical Coulomb fluids.