SMT-Vol9 & STCT-Vol5: The Resolution of the Yang-Mills Mass Gap: A Geometric Approach via Rough Operator Algebra and Arithmetic Friction

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 hoc symmetry 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 incorporate the Universal Arithmetic Friction constant η ≈ 10−22. We demonstrate that the non-abelian self-interaction term induces a strict non-commutative residual against the underlying arithmetic lattice. The transition from a topologically fragmented phase (α < 1) to a smooth continuum (α = 1) under this friction 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 macroscopic consequence of Arithmetic Friction.