Part 12: Discretization-Induced Mass of Boundary Phase Modes in Discrete H4 Geometry

Within the Origin Geometry (OG) framework, spacetime is modeled as a fundamentally discrete four-dimensional lattice based on H₄ (600-cell) symmetry. Previous works have established that bulk metric deformations generate large inertial mass, while boundary-supported phase modes correspond to exact isometries and are therefore massless in the continuum limit. In this paper, we establish a no-go result: within discrete H₄ geometry, any boundary phase excitation that propagates nontrivially must acquire a finite inertial mass, even in the absence of metric deformation, gauge dynamics, or external symmetry-breaking fields. This mass arises unavoidably from discretization-induced symmetry breaking, which lifts the exact Goldstone degeneracy present in the continuum. We demonstrate that continuous phase translation symmetry is explicitly broken by lattice granularity, leading to a Peierls–Nabarro-type pinning potential for boundary solitons. As a consequence, phase transport becomes energetically nontrivial, and a finite mass gap emerges. Crucially, this mass generation mechanism is inherently non-perturbative: it vanishes identically in any continuous or perturbative expansion and exists only due to the discrete nature of the geometric substrate. This explains why perturbative quantum field theories must treat fermion mass as an external input rather than a derived quantity. The result is pre-dynamical, parameter-free, and falsifiable: in the exact continuum limit, the mass must vanish identically. This paper provides a logically closed foundation for subsequent scaling and hierarchy analyses, while remaining fully independent of phenomenological fitting.