Repulsive Gravity Theory Based on Scalar Fields and the Effective $\Lambda$

In this study, we propose an extension of the Einstein field equations by introducing a repulsive gravity framework based on the scale-responsive geometric repulsion effect of space, derived from a scalar field. This effect is induced by the geometric configuration of the scalar field, from which a scale-dependent effective cosmological term, $\Lambda_\text{eff}$, emerges. The impact of $\Lambda_\text{eff}$ depends on spatial scale, potentially redefining the structural properties of gravity, especially at galactic scales and within black hole (BH) interiors. We demonstrate that two key observational and theoretical challenges—the flattening of galactic rotation curves in their outer regions and the avoidance of black hole singularities—can be addressed in a unified and quantitative manner through the common scalar-field-derived term $\Lambda_\text{eff}$.