The charged-lepton hierarchy remains one of the most persistent unexplained numerical structures in particle physics. Although the Standard Model successfully parameterizes charged-lepton masses through Yukawa couplings, it does not explain why three charged leptons exist, why their masses follow a strongly hierarchical structure, or why the hierarchy assumes its particular numerical form. Earlier Parts in the Origin Geometry (OG) program established that light fermion-like excitations may emerge as boundary-supported phase defects in discrete H₄ geometry. In particular, discreteness-induced Peierls–Nabarro (PN) pinning naturally generates finite inertial mass for otherwise massless boundary phase solitons. This mechanism provides a geometric explanation for the existence of a light electron-like excitation. However, previous attempts to derive the full charged-lepton hierarchy from spectral cutoffs, Laplacian eigenvalue clusters, or simple pinning-barrier scaling were ultimately insufficient. In the present work, we propose a revised interpretation. The electron corresponds to the minimal pinned boundary defect, while the muon and tau correspond not to ordinary electromagnetic excited states, but to distinct metastable topological winding sectors of the same underlying boundary defect family. The central geometric ingredient is the interface invariant introduced previously in Part 6: Iᵢnt = 20φ⁴ ≈ 137. 0820393 We investigate the phenomenological hierarchy: mₙ = mₑ 1 + (3/2) Iᵢnt · ∑ (k=1 to n) k⁴ with n = 0, 1, 2 corresponding respectively to the electron, muon, and tau sectors. Unlike earlier interpretations, the quartic structure is not interpreted as a literal bulk-volume law. Instead, it is interpreted as an effective fourth-order topological excitation burden associated with nested winding structure in projected H₄ boundary geometry. We further propose that the local icosahedral shell multiplicity associated with the H₄ interface defines a finite coherence-support capacity for metastable boundary winding sectors. In this interpretation, the tau approaches the maximal coherent excitation regime, while higher winding sectors lose coherent boundary support and become unstable against bulk-assisted reconfiguration. Direct radiative transitions such as μ → e + γ are proposed to be strongly suppressed because electromagnetic boundary fluctuations preserve winding-sector topology. Finally, the multiplicative structure of the hierarchy suggests a natural form of chiral protection: in the continuum limit where PN pinning vanishes, all winding sectors become massless simultaneously. This Part does not claim a complete derivation of the Standard Model or electroweak gauge theory. It presents a phenomenological topological-defect framework linking PN pinning, interface invariants, winding-sector structure, metastable coherence, and charged-lepton hierarchy.
The Duy Tan Truong (Mon,) studied this question.