Derivation of the Standard Model from Admissibility Mechanics on the Allen Orbital Lattice

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”