Part 6: Geometric Hierarchy Structures from Discrete H4 Geometry

Overview Previous Parts of the Origin Geometry (OG) program established a sequence of pre-dynamical structures: a discrete geometric substrate, dimensionless geometric baselines, a taxonomy of interaction regimes, attractor formation, and controlled deviation through screening. The present Part investigates whether discrete H4-derived geometry can naturally generate large dimensionless hierarchy scales before phenomenological fitting, gauge dynamics, or particle-specific assumptions are introduced. Core Geometric Baselines We identify two candidate geometric hierarchy baselines arising from the combinatorial and metric organization of the H4 Coxeter geometry and its associated 600-cell structure: Local Interaction-Connectivity Baseline: I_int = 20φ⁴. The value 20 is associated with the icosahedral local face structure of the 600-cell vertex figure. The symbol φ represents the golden ratio. Numerically, this quantity lies close to the inverse fine-structure constant. Bulk–Boundary Hierarchy Baseline: R_bb = 120√5φ⁴. The value 120 is associated with the global vertex count of the 600-cell. The term √5φ⁴ encodes a candidate golden-ratio-based volumetric amplification factor. Numerically, this quantity lies close to the proton–electron mass hierarchy. Scope and Limitations The present work does not interpret these numerical proximities as exact derivations of Standard Model parameters. No quantum electrodynamics, QCD confinement, hadronic structure, renormalization-group calculation, or particle-mass derivation is performed. Instead, the quantities above are proposed as candidate geometric hierarchy baselines: fixed structural scales arising from intrinsic H4 organization. Conclusion The central claim is deliberately limited: Discrete higher-dimensional geometry may naturally generate large dimensionless hierarchy structures prior to detailed physical dynamics. Observed physical values may later differ from these geometric baselines through screening, coarse-graining, collective relaxation, infrared dressing, and dynamical interpretation. Part 6 therefore provides an initial structural exploration of hierarchy emergence, not a completed theory of constants or particle physics.