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

Overview Within the Origin Geometry (OG) framework, effective spacetime and inertial behavior are modeled as emergent descriptions of an underlying discrete geometric substrate. Earlier Parts introduced H4-derived geometry, identified mass-like inertia with metric deformation cost, and proposed that mass hierarchy arises from sector-dependent geometric participation. Part 11 then developed a boundary-sector mechanism in which an electron-like phase excitation can acquire a small mass scale through discretization-induced Peierls–Nabarro pinning. The Conditional No-Go Result The present Part isolates the logical core of that mechanism. We establish a conditional no-go result for exact boundary masslessness in finite discrete H4-derived geometry. Under the minimal OG boundary-sector assumptions, a nontrivially propagating boundary phase mode cannot remain exactly massless on a finite discrete substrate unless an exact residual symmetry protects the zero mode or the continuum limit is restored. The result follows from the fact that continuous phase translation, which produces an exact zero mode in the ideal continuum boundary sector, is explicitly reduced by lattice granularity. Once the phase coordinate moves across inequivalent discrete configurations, the continuum degeneracy is lifted and a finite pinning landscape appears. Geometric Mass Scale and Non-Perturbative Nature The resulting mass scale is geometric. It does not require bulk metric deformation, gauge dynamics, scalar fields, or phenomenological fitting. It is also non-perturbative with respect to a smooth continuum expansion: the effect vanishes when the discrete boundary is replaced by an exact continuum and is therefore invisible to any description that assumes continuous phase degeneracy from the outset. In the discrete theory, however, the residual Peierls–Nabarro-type pinning potential supplies a finite boundary energy scale. Scope and Falsifiability This Part does not compute the observed electron mass. It does not replace the Higgs mechanism or derive the full fermion spectrum. Its result is more specific and more structural: exact masslessness of an electron-like boundary phase mode is incompatible with finite discrete H4-derived geometry under the stated assumptions. The mechanism is falsifiable. A finite discrete H4 boundary sector admitting a nontrivially propagating phase soliton with strictly zero inertial cost, without exact residual symmetry protection, would invalidate the proposed no-go result. Conversely, a pinning scale that vanishes in the continuum limit and remains finite on the discrete substrate would support the mechanism.