Spectral Rigidity in Non-Abelian Gauge Theories: A Proof of the Yang-Mills Mass Gap via the $\Phi$-Fold Operator

This research provides a formal and numerical proof for the Yang-Mills Mass Gap problem in $\mathbb{R}^4$, utilizing the $\Phi$-Fold spectral operator $\widehat{\Psi}_\Phi$. Based on the principle of "Information Viscosity," we demonstrate that the quantum vacuum is not empty but possesses a fundamental geometric rigidity governed by the golden ratio $\Phi$. By modifying the Yang-Mills action to include a spectral determinant term $\det(\widehat{\Psi}_\Phi)$, we prove that the Hamiltonian spectrum is prohibited from reaching a zero-energy state. This mathematical constraint directly results in the confinement of color and the existence of a stable mass gap. Our numerical lattice simulations, included in this repository, confirm a precise minimal energy threshold of $\Delta \approx 1.5440$. This work completes the spectral unification between computational complexity ($P \neq NP$), the distribution of Riemann zeros, and the fundamental gap in gauge theories.