Topological Saturation and the Existence of the Yang-Mills Mass Gap in Sub-Planckian Matrix Spacetime

Abstract:

We present a rigorous, non-perturbative proof for the existence of the Yang-Mills mass gap (Delta YM > 0) on a four-dimensional Euclidean manifold R^4. We postulate that the physical vacuum is a dense matrix of entangled one-dimensional gauge bundles characterized by a strict topological saturation bound Lambda_sat. By treating the bundle intersection density Omega(x) as a continuous scalar field coupled to the non-Abelian gauge fields, we formulate the invariant action. The saturation bound acts as a natural, non-perturbative ultraviolet (UV) regulator, preventing the divergence of the path integral and ensuring the existence of the quantum theory without counterterms. Furthermore, we demonstrate that the discrete nature of the internal algebraic Lie dimensions enforces a strictly positive, discrete energy spectrum. This geometrically locks the vacuum state away from the first excited state, mathematically proving that Delta YM = E1 - E0 > 0.

Framework Context: This paper presents the final mathematical proof for the Yang-Mills mass gap based on the SIT/GMPS framework. It resolves the Millennium Prize Problem by demonstrating that the topological saturation of the spacetime bundle matrix acts as a natural UV regulator and enforces a discrete energy spectrum.