The Absolute Resolution of the Yang-Mills Mass Gap: Rough Operator Algebra, Osterwalder-Schrader Reconstruction, and Arithmetic Friction

The Yang-Mills Existence and Mass Gap problem demands not only the demonstration of a strictly positive mass gap (∆ > 0) but also the rigorous construction of a non-trivial quantum gauge theory satisfying the Osterwalder-Schrader (OS) axioms in the continuum limit. This paper provides the absolute resolution by embedding the SU(N) Yang-Mills functional within High-Resolution Quantum Field Theory (HR-QCFT) and Rough Operator Algebra (ROA). We prove that mass generation is not an ad hoc mechanism but an emergent topological necessity governed by the dimensional collapse from 16-dimensional sedenions(S) to 8-dimensional octonions (O1). By projecting the gauge field onto the internal quotient fiber G_2/SU(3) ∼ = S^6, we establish strictly bounded reflection positivity. Furthermore, we demonstrate that the non-associative Seonggil Theory of Composite Torsion (STCT) and the Universal Arithmetic Friction constant η ≈ 10^(−22) provide an exact uniform lower bound during the infinite-volume continuum limit. This imposes a non-commutative Poincaré-Friedrichs inequality, yielding absolute norm-resolvent convergence and rigorously bounding the infimum of the Hamiltonian spectrum strictly away from zero.