Plectonemic Fiber Bundles, Arithmetic Cutoffs, and the Mass Gap in Non-Abelian Gauge Theories — An Algorithmic and Geometric Roadmap to Yang-Mills Quantum Existence —

This document establishes the rigorous mathematical foundation for the non-Abeliangauge field quantization induced by an arithmetic lattice configuration, known as the BellTower Architecture. By projecting the gauge field onto a 3D helical bundle embedding, wenative-define an ultraviolet (UV) geometric cutoff that eliminates trans-Planckian divergencesa priori. Furthermore, utilizing a dimensional reduction over a compact fiber modulated bya stable Bernoulli regularized potential, we formally derive the exact existence of a strictlypositive Mass Gap (∆Mass = e > 0) for the fundamental glueball excitation state. Finally,the same geometric constraint is mapped onto cosmological metrics, leading to a non-singularBig Bounce that regularizes early-universe gravitational infinities.