Discrete Non-Abelian Gauge Connections, Yang--Mills Curvature, and Octonionic Color Sector: The Hurwitz Ladder of Fundamental Interactions
Year: 2026
Type: article
Abstract: 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.
Author: Smith, Terence R.
Institution: University of California, Santa Barbara
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225637039
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