Part 13: Scaling Law of Discretization-Induced Mass for Boundary Phase Solitons in Discrete H4 Geometry

Overview Part 12 established a structural necessity result within Origin Geometry: boundary-supported phase modes that are massless in the continuum limit acquire a nonzero inertial cost once the underlying H4 substrate is treated as discrete [4, 5]. This mass generation arises from discretization-induced breaking of continuous phase translation symmetry and may be interpreted through a Peierls–Nabarro-type pinning mechanism. Soliton Width and Mass Scaling The present Part investigates the next structural question: what determines the magnitude of this induced boundary mass? We argue that the electron-like boundary mass scale is not a fixed geometric invariant assigned directly to a particle identity, but an emergent quantity controlled by the effective spatial extent of the boundary phase soliton relative to the underlying lattice. In discrete soliton systems, the Peierls–Nabarro barrier is generically suppressed as the soliton becomes increasingly delocalized. Translating this principle into the Origin Geometry framework yields a universal scaling relation of the form: m_boundary(σ) ∝ ΔE_PN(σ) ∝ exp(−Cσ) where σ denotes the effective soliton width measured in lattice units and C > 0 is a geometry-dependent constant [6–13]. Geometric Stabilization Crucially, the soliton width is not treated as an adjustable phenomenological parameter. Within the stated geometric assumptions, it is interpreted as a geometrically selected equilibrium width determined by the competition between phase delocalization, boundary stiffness, topological winding, and geometric tension. The resulting mass scale is therefore structurally stabilized rather than freely tunable. Scope and Limitations This Part establishes a scaling principle for light boundary-supported fermion-like masses within Origin Geometry. It does not derive a numerical value for the electron mass, does not fit observed lepton masses, and does not invoke the empirical proton–electron mass ratio. Instead, it identifies the structural mechanism by which extremely small but nonzero boundary masses may arise naturally from discrete spacetime geometry.