Geometric Diagnostics of Coherence and Variance Partition in Black Holes and Relativistic Systems

This paper introduces a geometric diagnostic framework for analyzing coherence structure and variance partition in black holes and related relativistic systems. The central object of study is a class of dimensionless geometric ratios—most notably the Kerr coherence ratio R = r_{\mathrm{ISCO}}/r_{+}—which characterize the separation between stable orbital structure and irreversible plunge regions in rotating spacetimes. These ratios are scale-free, monotonic in spin, and determined entirely by standard Kerr geometry. We show that astrophysical black holes with the most robust observational constraints cluster within a narrow interior corridor of the coherence ratio, corresponding to high but non-extreme spin. This corridor coincides numerically with geometric thresholds that appear independently in variance-partition analyses of thermal radiation, plasma tori, and stellar collapse, suggesting a shared structural organization rather than a shared physical mechanism. Using a logical-geometry perspective, we distinguish invariant geometric constraints from context-dependent dynamical realizations. Within this framework, black hole mass distributions, spin coherence, and horizon–gap partitions can be interpreted as manifestations of a common variance-budget structure, expressible through simple geometric identities and Pythagorean partitions. The analysis does not modify general relativity or invoke new dynamics; it reorganizes existing observables into a unified diagnostic language. The results motivate the use of geometric coherence ratios as comparative tools across black hole populations and provide falsifiable predictions for the clustering of jet-hosting systems, the structure of mass gaps, and the behavior of near-extremal Kerr spacetimes.