This paper derives the Standard Model gauge structure SU(3) × SU(2) × U(1), the Yukawa sector, fermion mass hierarchy, and the three-generation structure from admissibility mechanics on the Allen Orbital Lattice (AOL) within Pattern Field Theory (PFT). The Allen Orbital Lattice is treated as a prime-indexed admissibility graph on which fermionic degrees of freedom are realized as relay-partitioned transport structures. Within this framework: - chirality asymmetry,- weak participation,- vacuum alignment,- gauge compensation,- and mass generation all emerge from the same closure constraints: - Phase Alignment Lock (PAL),- Equilibrion (EQUI),- Rationic Admissibility,- depth-dependent closure cost. Yukawa couplings are reinterpreted as admissibility coefficients measuring persistent closure burden across chirality partitions rather than arbitrary free parameters. Fermion masses scale with cumulative prime-depth realization, with the muon-electron hierarchy emerging from recursive closure-depth ratios. The paper further establishes that the 137-mode admissibility manifold generated through QUART reduction supports exactly three stable fermion generations and forbids a fourth persistent stable family. The Standard Model Lagrangian is therefore interpreted as an effective closure description of admissibility mechanics on the Allen Orbital Lattice rather than a fundamental primitive structure. Companion foundational ontology paper: https://zenodo.org/records/20101474“The Iniverse: Space as a Consequence of Admissible Depth Complexity”