Curvature–Memory Coupling as a Phenomenological Extension of Classical Gravity with Testable Interferometric Signatures

This version presents a substantially revised and focused formulation of the curvature–memory framework developed in earlier versions. The manuscript has been reorganized as a minimal phenomenological effective-field-theory (EFT) extension of classical gravity designed for experimental testability. The new version introduces a covariant curvature–memory response sector implemented through admissible Volterra-type causal kernels added to the Einstein–Hilbert action. The framework preserves diffeomorphism invariance, causal structure, and exact recovery of Einstein’s equations in the vanishing-memory limit. Relative to previous versions, the manuscript: • Reformulates the theory explicitly within an effective-field-theory response framework.• Provides a full variational derivation of the curvature–memory action and modified field equations.• Develops a controlled linearized reduction isolating the matter-coupled scalar curvature sector.• Establishes a complete kernel → curvature response → interferometric observable pipeline.• Introduces a worked near-field atom-interferometer benchmark and representative exclusion curves in the effective coupling and relaxation-time parameter space. The resulting formulation provides a controlled and experimentally falsifiable phenomenological parameterization of possible history-dependent curvature response in matter-coupled gravitational dynamics. No new propagating tensor degrees of freedom or modifications to vacuum gravitational-wave propagation are introduced. Deviations from general relativity arise only in sourced configurations through a small causal curvature–memory response controlled by phenomenological parameters. Version 5 (15 March 2026) This version incorporates final revisions prior to journal submission.