Part 16A: Topological Winding Sectors, Interface Coherence, and the Charged-Lepton Hierarchy in H₄ Boundary Geometry

Overview The charged-lepton hierarchy remains one of the most persistent unexplained numerical structures in particle physics. The Standard Model successfully parameterizes the electron, muon, and tau masses through Yukawa couplings [12–14], but it does not derive from first principles why the charged leptons appear in three observed generations, why their masses form a strong hierarchy, or why the numerical pattern takes its observed form. Earlier Parts of the Origin Geometry program developed a geometric account of light fermion-like boundary excitations in discrete H4 geometry [5–10]. In that framework, boundary-supported phase solitons are approximately massless in the ideal continuum limit, but acquire finite inertial mass when discrete geometry breaks continuous phase translation symmetry and induces a Peierls–Nabarro-type pinning barrier [18–23]. This mechanism provides a geometric interpretation of an electron-like minimal boundary defect. Topological Winding Sectors and Hierarchy Ansatz Attempts to derive the full charged-lepton hierarchy directly from spectral cutoffs, Laplacian eigenvalue clusters, simple localization scaling, or purely bulk-volume arguments are insufficient. The present Part therefore proposes a revised interpretation. The electron is identified with the minimal pinned boundary defect, while the muon and tau are interpreted not as ordinary electromagnetic excited states of the electron, but as metastable topological winding sectors of the same underlying boundary defect family. The central geometric ingredient is the interface invariant introduced in Part 6: I_int = 20ϕ⁴ ≈ 137.0820393 where ϕ = (1 + √5)/2 [1, 2, 11]. We investigate the phenomenological hierarchy ansatz: m_n = m_e [ 1 + (3/2) I_int ∑ (k=1 to n) k⁴ ] with n = 0, 1, 2 corresponding respectively to the electron, muon, and tau sectors [12, 32]. The quartic structure is not interpreted as a literal four-dimensional bulk-volume law. Instead, it is interpreted as an effective fourth-order topological excitation burden associated with nested winding complexity in projected H4 boundary geometry [24–28]. The coefficient 3/2 is treated as a provisional phenomenological normalization suggested by boundary-interface geometry, not as a derived universal constant. Coherence Saturation and Chiral Protection We further propose that the local icosahedral shell multiplicity associated with the H4 interface defines a finite coherence-support scale for metastable boundary winding sectors. In this interpretation, the tau approaches a near-saturation regime of coherent boundary support, while higher winding sectors are expected to lose stable boundary coherence and become unstable against bulk-assisted reconfiguration. This provides a possible geometric explanation for the observed finite charged-lepton family structure, but not a rigorous theorem proving exactly three generations. Direct radiative transitions such as µ → e+γ are argued to be strongly suppressed because smooth electromagnetic boundary fluctuations preserve winding-sector topology to leading order. The present Part does not compute decay rates; it only identifies a possible topological selection mechanism. Finally, the multiplicative form of the hierarchy suggests a natural chiral-protection structure: in the continuum limit where Peierls–Nabarro pinning vanishes, all winding-sector masses vanish simultaneously [15–17]. Thus, the hierarchy is not generated by independent additive mass terms, but by topological burden anchored to a common pinned boundary baseline. Scope and Limitations This Part does not claim a derivation of the Standard Model, electroweak gauge theory, weak decay rates, or exact charged-lepton masses. It presents a phenomenological topological-defect framework linking Peierls–Nabarro pinning, interface invariants, metastable winding sectors, coherence saturation, chiral protection, and charged-lepton hierarchy within the stated assumptions of Origin Geometry.