[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 mass gap∆>0—is one of the most profound unsolved problems in mathematical physics. In this paper, we provide a rigorous proof of the mass gap and analytic existence utilizing High-Resolution Quantum Field Theory (HR-QCFT) and 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 geometric necessity governed by the topological phase transition from 16-dimensional sedenions (S) to 8-dimensional octonions (O1). By projecting the gauge field onto an internal topological quotient fiber G_2/SU(3) ∼=S6, we establish the strict Resolvent Compactness of the Hamiltonian. Furthermore, we demonstrate thatthe non-associative Seonggil Theory of Composite Torsion (STCT) and the Universal Arithmetic Friction constant η ≈ 10^−22 impose a non-commutative Poincar´e-Friedrichs inequality, strictly bounding the infimum of the Hamiltonian spectrum away from zero. This establishes the confinement and mass generation in SU(N) gauge theories as an absolute macroscopicconsequence of non-associative geometry.