Structural Differentiation Gravity (SDG) v1.8: Nonlocal Structural Field, Scale-Dependent Dynamics, and Emergent Motion

We present Structural Differentiation Gravity (SDG) v1.8, a minimal and fully reproducible framework in which gravitational dynamics emerge from a nonlocal structural density field rather than local mass or spacetime curvature. The structural field is defined through a kernel-based convolution: C(x) = ∫ K(|x - x'|) ρ(x') dx' where K encodes the interaction scale of structural relations. In this formulation, motion is not caused by local sources but selected by the global structural configuration. We demonstrate that:- Local gradient-based dynamics are recovered as a limiting case- Large-scale behavior is governed by nonlocal smoothing- Effective dynamics become intrinsically scale-dependent The commonly used expression g = -∇C(x) is interpreted as a coarse-grained representation of the full nonlocal convolution, ensuring consistency between theoretical formulation and illustrative implementations. We further introduce an effective scale parameter k, representing the ratio between observation scale and kernel interaction scale, which governs the transition between local and nonlocal regimes. This framework provides a unified structural interpretation of gravitational phenomena and suggests that dark energy and dark matter effects may emerge from scale-dependent structural dynamics rather than new physical components. All figures are fully reproducible using the provided Python implementation, ensuring transparency and testability. SDG offers a minimal, falsifiable, and observationally relevant alternative to conventional gravity frameworks.