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.