A Constitutive Framework for Weak-Field Gravitational Dynamics in the Low-Acceleration Regime with a Minimal Relativistic Extension

We develop an effective phenomenological framework for weak-field gravitational dynamics motivated by analogy with classical electromagnetism in material media. By modeling the gravitational field with a constitutive response function κg, analogous to permittivity or permeability in electromagnetism, we construct a Maxwell-like field equation applicable in the non-relativistic regime. Under observational constraints from solar-system dynamics and galactic rotation curves, the low-acceleration limit reproduces the phenomenology associated with Modified Newtonian Dynamics (MOND), including the baryonic Tully--Fisher relation, while the high-acceleration limit reduces to Newtonian gravity. The framework incorporates known gravitomagnetic effects and includes a uniform effective phantom density associated with the cosmological constant, representing the gravitational contribution of vacuum energy in the Newtonian limit. We further outline how both the nonlinear gravitational response κg and the background phantom density may arise from a common scalar-field effective action describing the gravitational vacuum. The resulting formulation provides a phenomenological framework relevant to galactic dynamics and large-scale cosmological background effects, including possible connections between low-acceleration gravitational scales and the observed late-time expansion of the Universe. We further construct a minimal relativistic extension by embedding the effective potential in a weak-field spacetime metric, allowing both massive particle dynamics and light propagation to be governed by the same field. This provides a unified description of rotation curves, gravitational lensing, and time dilation within the weak-field regime. The relativistic formulation is presented as a quasi-static limit of a more general covariant theory.