Octonions, G2, and Knot Topology in the Arithmetic of Order Framework: A Topological–Algebraic Reinterpretation of the Higgs Mechanism within the Standard Model

We present a unified framework connecting knot theory, exceptional algebra, and particle physics that provides a topological–algebraic reinterpretation of the Higgs mechanism within the Standard Model (SM). The SM gauge groups U(1), SU(2), and SU(3) arise naturally from the three Reidemeister moves on knots and braids, with a rigorous Lie-algebraic identification via Schiller’s strand–tangle model. These gauge symmetries embed into the exceptional Lie group G2, the automorphism group of the octonions, and are realized geometrically through coset spaces such as G2/(SU(2)L × SU(2)R). Within the Arithmetic of Order (AoO) framework, the Higgs field is reinterpreted not as a fundamental scalar particle but as an internal algebraic phase transition on this coset manifold. This phase transition triggers symmetry breaking, producing W and Z boson excitations consistent with collider data. The framework yields a fully computable, discrete physical model that recasts the SM as an inevitable consequence of topology, exceptional algebra, and discrete geometry.