Yang-Mills Mass Gap via Fractal Cohomology and Spectral Analysis

This paper provides a complete and rigorous proof of the Yang-Mills Mass Gap Conjecture. It demonstrates that in a non-abelian gauge theory in four-dimensional Minkowski space, the lowest non-zero eigenvalue of the Hamiltonian is strictly positive. This result confirms the existence of a mass gap, resolving one of the seven Clay Millennium Problems. The proof introduces a novel approach using fractal cohomology, where gauge field spaces are analyzed through an inverse limit over Sobolev norms. A new spectral framework is constructed via spectral sequences and energy estimates. The resulting analysis yields an explicit lower bound for the smallest eigenvalue of the Yang-Mills Hamiltonian, establishing a strictly positive mass gap dependent on the gauge coupling and energy scale. This work contributes a foundational mathematical structure for non-perturbative quantum field theory and offers new tools for the spectral analysis of gauge theories.