Part 15: Universality of Light Fermions as Boundary Phase Descendants in Discrete H4 Geometry

Overview Parts 12–14 established the structural basis of boundary-supported mass generation within Origin Geometry. Part 12 showed that boundary phase solitons acquire a nonzero inertial cost when continuous phase translation symmetry is broken by discrete H4 geometry. Part 13 showed that this induced boundary mass is exponentially suppressed by soliton delocalization. Part 14 then showed that large bulk–boundary mass hierarchy is structurally favored because bulk metric deformation and boundary phase pinning belong to incommensurate geometric energy sectors [4–6].

Boundary Phase Descendants (BPDs) The present Part addresses the next structural question: is the electron-like boundary excitation an isolated object, or is lightness a universal property of a broader geometric class? We introduce the concept of Boundary Phase Descendants (BPDs): excitations that are primarily boundary-supported, propagate through phase reorientation rather than bulk metric deformation in the continuum limit, and acquire finite inertial cost only through discretization-induced phase pinning.

Within the stated assumptions of the OG framework, any such excitation is expected to acquire a small but nonzero mass governed by the same exponential suppression mechanism:

m_BPD(σ) ∝ exp(−Cσ)

where σ denotes the effective boundary soliton width and C > 0 is a geometry-dependent constant [3–5, 9–14].

Topological Winding and Fermion Families The Part further argues that multiple light fermion-like boundary modes may be represented by distinct topological winding sectors of boundary phase solitons [15–19]. Differences in internal phase complexity can modify the stabilized soliton width, thereby generating mass differences without introducing independent coupling constants or phenomenological fitting. This provides a structural explanation for why light fermion-like states may form a family rather than appearing as isolated exceptions.

Scope and Limitations This Part does not identify specific Standard Model fermions, derive charged-lepton masses, compute flavor spectra, or solve the neutral fermion sector. Its purpose is narrower: to establish Boundary Phase Descendants as a universal geometric class of light fermion-like excitations within discrete H4 geometry, and to prepare the ground for later work on fermionic hierarchy, boundary mass saturation, and physical particle mapping.