Part 16: Boundary Mass Saturation and Geometric Stratification of Fermionic Boundary Modes in Discrete H₄ Geometry

Previous Parts of the Origin Geometry program developed a structural account of inertial mass in discrete H4 geometry [1–7]. Bulk-supported excitations acquire inertial cost through volumetric metric deformation, while boundary-supported phase solitons remain massless in the ideal continuum limit but acquire a small nonzero mass through discretization-induced Peierls–Nabarro-type pinning [10–15]. Parts 13–15 further showed that boundary mass is exponentially suppressed by soliton delocalization and that light fermion-like states may be interpreted as members of a broader class of Boundary Phase Descendants. Boundary Mass Saturation The present Part investigates a new structural question: can boundary-supported phase solitons support arbitrarily large masses by becoming increasingly localized, or does discrete H4 geometry impose a finite boundary mass capacity? We argue that pure boundary mass cannot grow without limit. As the effective localization width σ decreases, the boundary phase gradient becomes increasingly concentrated. Below a critical localization scale σ_crit, the boundary sector can no longer support the excitation as a purely boundary-localized phase configuration. Bulk metric participation then becomes unavoidable. This defines a boundary mass saturation scale [5–7, 16–20]: m_max_boundary ~ A_boundary exp(−C σ_crit) where A_boundary is a non-exponential geometric prefactor and C > 0 is a geometry-dependent constant. Localization Classes and Fermion Hierarchy Within the allowed boundary regime, stable boundary excitations are not expected to form an arbitrary continuum of localization widths. Instead, they are expected to organize into preferred geometric localization classes, corresponding to stabilized phase configurations of the discrete boundary substrate [7, 18–23]. Because the inertial mass depends exponentially on localization width, comparatively small geometric differences among such classes can generate large fermion-like mass separations without fine-tuning. Scope and Limitations This Part does not derive numerical lepton masses, prove the number of generations, solve the Standard Model flavor problem, or derive the Koide relation. Its purpose is narrower: to identify a geometric saturation constraint on purely boundary-supported mass and to clarify how boundary mass saturation, localization stratification, and boundary–bulk transition may organize fermion-like hierarchy within the stated assumptions of Origin Geometry.