Part 14: Geometric Mass Hierarchy from Bulk–Boundary Energy Separation in Discrete H4 Geometry

Overview Parts 12 and 13 established two key structural results within the Origin Geometry mass program. Part 12 showed that boundary-supported phase solitons, although massless in the continuum limit, acquire a nonzero inertial cost once continuous phase translation symmetry is broken by discrete H4 geometry. Part 13 then showed that this induced boundary mass is exponentially suppressed with increasing soliton delocalization [4, 5]. The present Part investigates the next structural question: why should a bulk-dominated excitation and a boundary-supported phase excitation possess widely separated mass scales? We argue that this hierarchy is a structurally expected consequence of bulk–boundary energy separation within discrete H4 geometry. Bulk vs. Boundary Mechanisms Bulk excitations and boundary phase excitations couple to different geometric stiffness mechanisms: volumetric metric deformation in the four-dimensional interior versus discretization-induced phase pinning at the effective boundary. Bulk deformation is supported by the geometric inertia and lattice stiffness framework [1, 2, 6–10]. Boundary pinning follows from the boundary soliton and Peierls–Nabarro mechanism [3–5, 11–16]. Structural Incommensurability and Mass Hierarchies These two mechanisms are not merely quantitatively different. They are structurally incommensurate. Bulk metric deformation produces an algebraic or order-unity geometric energy scale associated with volumetric participation. Boundary phase pinning produces an exponentially suppressed energy scale controlled by soliton width. Their ratio therefore naturally generates large bulk–boundary mass hierarchies without numerical fitting, particle-specific assumptions, or phenomenological mass input. Scope and Limitations This Part does not derive the numerical proton–electron mass ratio, does not fit observed particle masses, and does not replace effective low-energy field theory. Instead, it establishes a structural explanation for why proton-like bulk-dominated excitations and electron-like boundary-supported excitations are expected to occupy parametrically different inertial regimes. Within the stated geometric assumptions, large mass hierarchy is therefore not an anomalous feature requiring external tuning, but a natural consequence of dimensional energy separation in discrete higher-dimensional geometry.