This paper constitutes SMT-VOL9 and STCT-VOL5 in the Seonggil Rough Operator Algebra (ROA) research series. The Yang-Mills Mass Gap hypothesis—asserting that the quantum version of non-abelian gauge theories must exhibit a strictly positive massgap ∆ > 0—is one of the most profound unsolved problems in mathematical physics. In this paper, we provide a rigorous proof of the mass gap utilizing Rough Operator Algebra (ROA). We propose that mass is not a fundamental particle property requiring ad hocsymmetry-breaking mechanisms (such as the Higgs mechanism), but an emergent “Topological Inertia” generated by the roughness of the mathematical vacuum. By extending the classical Yang-Mills action into a fractional Sobolev space parameterized by the roughness index α via Seonggil Tensor Calculus Theory (STCT), we demonstrate that the transitionfrom a topologically fragmented phase (α < 1) to a smooth continuum (α = 1) strictly bounds the infimum of the Hamiltonian spectrum away from zero. This establishes the geometric necessity of mass generation in SU(N) gauge theories as an exact consequence of Roughness Symmetry.
lee seonggil (Mon,) studied this question.