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

In previous works within the Origin Geometry (OG) framework, inertial mass was shown to arise from geometric resistance in a discrete four-dimensional H₄ lattice. Bulk-supported excitations acquire mass through volumetric metric deformation, while boundary-supported phase solitons remain massless in the continuum but gain a small inertial mass through discretization-induced pinning. Building upon these results, this Part investigates whether boundary phase excitations can support arbitrarily large masses, or whether discrete geometry imposes an intrinsic upper bound. Using analytical arguments and geometric analogies with known discrete soliton systems, we demonstrate the existence of a boundary mass saturation limit: beyond a critical localization scale, boundary-supported phase solitons can no longer remain purely boundary-localized excitations. This saturation emerges from geometric discreteness and does not rely on phenomenological lepton-mass fitting, interaction-specific assumptions, or particle labeling. Within the allowed boundary regime, we further show that distinct boundary phase excitations organize into discrete geometric localization classes rather than a continuous spectrum. These classes correspond to stable equilibrium configurations of the boundary phase structure and are determined by geometric stability conditions on the discrete substrate. Importantly, because inertial mass depends exponentially on localization width through the Peierls–Nabarro mechanism, comparatively small geometric differences can generate extremely large fermionic mass hierarchies without fine-tuning. We argue that boundary mass saturation provides a natural geometric stratification of fermionic excitations, separating light, long-lived boundary-dominated modes from heavier states that necessarily involve bulk participation. This work extends the OG mass program by identifying a previously unrecognized geometric constraint on boundary excitations and clarifying the structural origin of fermion hierarchy in discrete spacetime.