Algebraic Transport and the Gauge-Factor Hierarchy Underlying the Standard Model: Normed Division Algebras and Symmetry Selection in Null-Directional Kinematics

Previous work established that the thermodynamic and kinematic transport of the directional vacuum is governed by Fisher information, uniquely selecting a privileged amplitude variable where $P = |\psi|^2$. Furthermore, gauge connections become kinematically mandatory when these directional amplitudes carry internal phase redundancies. However, local internal covariance alone does not uniquely select the $SU(3) \times SU(2) \times U(1)$ symmetry group of the Standard Model. This paper introduces an algebraic transport axiom: the requirement that local transport updates of directional amplitudes compose multiplicatively while preserving the probability norm selected by Fisher transport. By Hurwitz's Theorem, this stability constraint restricts the internal amplitude spaces strictly to the normed division algebras ($\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$). Mapping these algebraic sectors to their corresponding Hopf fibrations, this paper shows that the required unit-norm redundancies naturally recover the gauge-factor hierarchy underlying the Standard Model. Furthermore, it is shown that the finite informational throughput and directional strain of the substrate generate a subsequent thermodynamic ordering phase. This provides a structural route to an emergent scalar sector, supporting an effective Higgs-like mechanism without invoking primitive scalar fields. Thus, the admissible gauge factors and the mass-generating scalar sector can be understood as constrained emergent features of stable informational transport.