We develop a unified theoretical framework, termed the Single Universal Layer (SUL) theory, in which gravitational dynamics, relativistic geometry, and dark-sector phenomenology emerge from momentum exchange with a homogeneous, structureless background field. The approach begins with a countable arithmetic substrate whose coarse–grained limit is represented by a covariant vector field $P_{\mu}$. This field interacts minimally with matter through momentum–substrate coupling, leading to effective modifications of the Einstein field equations that preserve diffeomorphism invariance and reduce exactly to general relativity when substrate distortions vanish. The resulting dynamics generate both curvature-like behavior and divergence-induced mass distributions without requiring additional particle species or exotic fields. Within this framework, dark matter phenomena arise naturally through nonlocal momentum redistribution encoded in $\nabla_{\mu}P^{\mu}$, whereas dark energy appears as the stress-energy associated with the uniform background component of the substrate. We analyze the weak-field and cosmological limits of the theory, show consistency with Newtonian gravity, and derive modified Friedmann equations incorporating substrate contributions. Several observational signatures—including galaxy rotation curves, weak-lensing anomalies, and Hubble-scale tension—are reproduced at leading order without introducing free functions beyond the substrate parameters. We further examine connections to Einstein–Æther models, Proca-type vector theories, and teleparallel formulations, highlighting both structural differences and shared phenomenological features. Stability considerations, energy conditions, and the absence of ghost-like excitations are discussed within an effective-field-theory formulation. Overall, SUL theory provides a minimal, mathematically coherent pathway from a discrete arithmetic foundation to emergent gravitational and cosmological behavior. While exploratory in scope, the framework yields concrete, testable predictions and offers a structured foundation for future theoretical and observational investigation.