Covariant Completion of Phase Gravity Logarithmic Background Dynamics, Relativistic Motion, and the Einstein Limit

We present a covariant completion of the phase-topological framework in which mass arises from topological closure and gravity emerges as screening of a scalar background field. Motivated by the structure of the probe equation of motion, we introduce the logarithmic variable φ = ln ρ4 as the fundamental degree of freedom of the background. We show that the static screening law admits a minimal relativistic covariant extension given by a wave-type equation □φ = κT . In the homogeneous limit this reproduces special relativity, while in the weak-field static limit it reduces to the Poisson sector obtained earlier. We further demonstrate that the effective metric structure arises as an emergent encoding of φ and its gradients, and that general relativity appears as a macroscopic geometric representation of the underlying scalar dynamics rather than a fundamental postulate.