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

Building upon the necessity result established in Part 12—that boundary phase solitons in discrete H₄ geometry must acquire a nonzero inertial mass due to discretization-induced symmetry breaking—we investigate how the magnitude of this mass is determined.

We demonstrate that the electron mass is not a fixed geometric invariant, but an emergent quantity controlled by the effective spatial extent of the boundary phase soliton relative to the underlying lattice. Using general results from lattice soliton theory and Peierls–Nabarro pinning, we show that the discretization-induced mass exhibits exponential suppression with increasing soliton delocalization.

Crucially, we show that this soliton width is not a free parameter, but stabilizes dynamically at a unique geometric equilibrium determined by the balance between phase delocalization and boundary geometric tension. As a result, the electron mass is fixed, universal, and structurally stable.

This establishes a universal scaling law for light fermionic masses in Origin Geometry, independent of particle identity, interaction dynamics, or phenomenological fitting, and explains why extremely small but nonzero masses are natural consequences of discrete spacetime geometry.