Elastic Spacetime with Scale-Dependent Coupling (ESSC) v22: Unified Structural Coordinates Across Galaxies and Clusters

This work presents a unified structural analysis of galaxies and galaxy clusters within the framework of Elastic Spacetime with Scale-Dependent Coupling (ESSC).We introduce normalized structural coordinates (u_X, u_pi) that capture the relative placement of characteristic radii associated with slope transitions and pi-related boundary features. Using rotation curve data for galaxies and X-ray profile data for clusters, we construct a common representation that enables direct comparison across scales.The analysis reveals a clear structural separation:Galaxies occupy an internal regime where u_pi < 1 and u_X > u_pi, indicating internally resolved structural ordering.Clusters collapse toward u_pi approximately equal to 1, showing compression of the boundary scale toward a terminal limit rather than a shared internal configuration.This distinction emerges without introducing new dynamical laws, particles, or modifications to gravity. Instead, it reflects a structural constraint on how observational regimes map onto each other.The results support the ESSC perspective that pi appears not as a fixed constant in dynamics, but as a boundary manifestation associated with structural termination and observational closure.This work does not propose new physical interactions or predictive models. It provides a structural filter for interpreting cross-scale gravitational systems and highlights a fundamental asymmetry between internally structured systems and boundary-compressed regimes.