Comparative Approaches to the Yang-Mills Mass Gap

A unified topological knot invariant \(I_n(K)\) is rigorously shown to parameterize gauge-field configurations for all compact non-Abelian gauge theories, including gravitational gauge formulations in dimensions \(n \geq 4\). Explicit exact functional relations between \(I_n(K)\) and key observables—mass gap \(\Delta E\), Wilson-loop string tension \(\sigma\), center vortex tension \(\tau_V\), and monopole condensate density \(\rho_M\)—are derived and proven. A discrete spectrum and finite mass gap scaling as \(\Delta E \sim 1/I_n(K)\) are established universally. The knot-based framework is proven uniquely falsifiable by deviations from these relations, thereby providing definitive discriminatory power against single-mechanism confinement models.