Discrete Gauge-Theoretic Framework for Baryonic Topological Solitons

This work presents a discrete gauge-theoretic framework in which the nucleon is realized as an intrinsically stable topological soliton on a simplicial complex governed by the Johnson association scheme and the arithmetic geometry of the maximal class-number-one Heegner field. By replacing the continuous spacetime manifold with a discrete, countable combinatorial substrate, the model eliminates ultraviolet divergences and non-perturbative Landau singularities without relying on heuristic cutoffs or ad-hoc renormalization procedures. Using discrete exterior calculus, the invariant baryonic constituent count is derived from the bulk graph structure and harmonic layering, closed at the boundary by Borromean bridge triplets dictated by the non-trivial spectral gap of the discrete geometry. Evaluating the discrete hydrodynamic helicity budget demonstrates that the valence core currents account for approximately one-quarter of the total angular momentum, while the remaining three-quarters reside in the orbital precession of the confining boundary bridges, providing a purely geometric resolution to the long-standing proton spin puzzle. Exact topological boundary nilpotency preserves discrete gauge invariance across all simplicial chains. The resulting gauge Hessian possesses a strictly positive spectral gap that eliminates tachyonic instabilities and guarantees mechanical ground-state stability against spontaneous decay. The remaining zero modes correspond identically to the physical collective coordinates of spatial translation, three-dimensional rotation, baryonic phase, and flavor isospin, establishing that the structural rigidity of the nucleon is an exact consequence of discrete topological invariants.