An Empirical Outer-Halo Scaling Law in Galaxy Clusters and its Interpretation in Temporal Synchrony Gradient Gravity

We present an empirical analysis of the radial structure of the mass discrepancy in galaxy clusters using published dynamical mass measurements drawn from the literature. Rather than modeling individual clusters or fitting parametric halo profiles, we adopt an ensemble approach in which cluster radii are normalized to the virial scale r_{200} and mass ratios are compared at fixed fractional radius. Using median statistics to mitigate heterogeneity across studies, we find that dynamical and inner mass determinations are broadly consistent within approximately 0.5\,r_{200}. Beyond this radius, a systematic amplification of the inferred dynamical mass emerges, reaching factors of \sim 3\!-\!4 near the virial radius. This behavior is not consistent with a uniform rescaling or a global power-law trend. The observed radial dependence is well described by a minimal empirical scaling relation featuring a finite turn-on radius near x_0 \simeq 0.5 (where x \equiv r/r_{200}) and a shallow, square-root–like growth beyond this scale. The scaling is extracted directly from ensemble median behavior and does not rely on assumptions about halo profiles, equilibrium conditions, or cluster morphology. Independent of theoretical interpretation, the results establish a new empirical benchmark for the outer regions of galaxy clusters that any successful model of cluster dynamics or gravity must reproduce. We discuss the implications of this scaling in the context of Temporal Synchrony Gradient Gravity (TSGG), while emphasizing that the primary contribution of this work is empirical and model-independent. All quantities analysed are derived from previously published cluster mass measurements, and the analysis is fully reproducible from the cited literature.