Discrete Non-Abelian Gauge Connections, Yang--Mills Curvature, and Octonionic Color Sector: The Hurwitz Ladder of Fundamental Interactions

We complete the trilogy of papers formulating the discrete theory of Rational Wave Numbers (RWN) as an autonomous foundation for fundamental physics by constructing non-Abelian gauge connections, discrete Yang--Mills curvature, and the strong color sector on the non-associative octonionic sequence space $\mathbb{O}^\mathbb{Z}$. Grounded in the primitive basis $\mathcal{P}_0 = \{E_Z, E_{\bar{1}}\}$ and the classical Hurwitz normed division ladder $\mathbb{R} \to \mathbb{C} \to \mathbb{H} \to \mathbb{O}$, we demonstrate how the complete Standard Model gauge structure $\mathrm{U}(1)_Y \times \mathrm{SU}(2)_L \times \mathrm{SU}(3)_C$ arises constructively from successive algebraic doublings without continuous manifold assumptions. Coupling discrete parallel transport link operators with modular sequence lattices yields four primary results: (1) The 14-dimensional exceptional automorphism group $G_2 = \Aut(\mathbb{O})$ is broken by a longitudinal phase axis $e_7$ to its exact stabilizer subgroup $\Stab_{G_2}(e_7) \cong \mathrm{SU}(3)$, under which the transverse imaginary units decompose into $\mathbf{3} \oplus \bar{\mathbf{3}}$, identifying the geometric origin of quark colors; (2) Discrete Yang--Mills field strengths are formulated as modular sequence holonomies around elementary lattice plaquettes, yielding strictly finite action sums that eliminate ultraviolet Landau poles; (3) By Artin's Theorem, color-singlet hadrons correspond to closed, associative Fano cycles, whereas isolated color triplets generate non-vanishing associators and destructive Catalan tree phase jitter, establishing color confinement as an exact topological theorem of alternative sequence algebras; and (4) The $\operatorname{Spin}(8)$ triality automorphism naturally selects exactly three generations of chiral fermions through the three quaternionic subalgebras intersecting along the phase axis.