The Coupling Factor $\mathcal{F}$. A Comprehensive, Operational Framework for Collective Systems

This document presents a comprehensive, operational framework for the coupling factor $\mathcal{F}$, a topological property that modulates the strength of quantization constraints in complex collective systems. It integrates empirical observations, theoretical derivations, and practical protocols for calculating $\mathcal{F}$ across domains such as neurobiology, thermodynamics, astrophysics, psychology, and sociology. \inflabel{} Preliminary applications suggest that systems with $\mathcal{F}$ in the approximate range $[0.05, 0.15]$ tend to exhibit stable, coherent dynamics; values outside this range correlate with pathological or unstable states. The framework is designed to be immediately usable by researchers with standard computational tools, while providing a clear path for deeper theoretical exploration and experimental validation.