A Conformal Scalar–Tensor Extension of General Relativity and the Energy–Distance Relation

We propose an action-based scalar–tensor extension of General Relativity in which gravitational dynamics are modified through a conformal scalar degree of freedom universally coupled to matter. The theory is formulated explicitly at the level of an effective effective action, preserving diffeomorphism invariance and introducing a single additional propagating scalar mode beyond the tensor degrees of freedom of General Relativity. The framework is conceptually motivated by a reassessment of the relationship between energy localization and spatial separation. Rather than postulating new matter components, deviations from standard gravity arise from a redistribution of gravitational response mediated by the scalar field. When the scalar dynamically relaxes to a constant configuration, the theory reduces smoothly to General Relativity with a ΛCDM cosmological background, ensuring consistency with local gravitational tests and standard cosmological expansion. We derive the field equations, analyze linear cosmological perturbations, and examine gravitational lensing within this framework. Scalar fluctuations generically induce scale-dependent modifications to structure growth and light propagation, leading to effective dark-matter-like phenomenology at galactic and cluster scales without invoking additional particle species. The theory is intentionally constructed to be falsifiable. It predicts scale-dependent gravitational slip, modified relations between lensing and dynamical masses, and characteristic signatures in late-time cosmology. The absence of these effects in future observations would decisively rule out the model within its regime of validity.