Covariant Gravitational Field Equations of Self-Variation Theory

We develop the gravitational sector of Self-Variation Theory (SVT), a covariant framework in which gravitational dynamics emerge from the self-variation of intrinsic physical quantities and their associated spacetime equilibration momentum. Starting from the self-variation gravitational potential, we derive the corresponding field intensity, propagation velocity, and an effective static spherically symmetric spacetime metric. In the infinite-range regime (k=0), the theory yields a relativistic propagation structure compatible with causal invariance and reproduces the standard weak-field parametrized post-Newtonian limit with parameters (γ=1) and (β=1). Consequently, the theory recovers the classical predictions of General Relativity for perihelion precession, gravitational light deflection, and Shapiro time delay in the weak-field approximation. We further investigate the rotational dynamics generated by the SVT gravitational field. For small negative values of the parameter (k), the theory naturally predicts extended quasi-flat galactic rotation curves without introducing an explicit dark matter halo component. The flattened rotational velocities emerge dynamically from the structure of the self-variation gravitational field itself, while characteristic galactic acceleration scales of order () arise naturally at observationally relevant distances. These results suggest that Self-Variation Theory provides a covariant geometric framework capable of reproducing both local relativistic gravitational phenomena and large-scale galactic dynamics within a unified description.