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 $\kappa_g$ 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.