Lepton Mass Hierarchy from the Quaternionic Dirac Equation in Warped Internal Space: A Z₂ Parity Derivation

We derive the charged lepton mass hierarchy from a quaternionic Dirac equation defined on a warped internal space S³. The Higgs condensate VEV selects a preferred direction in the quaternionic internal space, defining a Z₂ parity symmetry under which spinor spherical harmonics split into even-parity (j=1/2 and j=5/2) and odd-parity (j=3/2) sectors. The condensate interaction conserves this parity, forcing the Yukawa matrix to be block-diagonal: a 2×2 seesaw block coupling the electron and tau (even parity), and a decoupled 1×1 entry for the muon (odd parity). In a Randall-Sundrum warped geometry identified with the vacuum condensate's bulk-to-shear modulus ratio R = kappa/mu = 10.9026, the overlap integrals of fermion profiles with the IR-localized Higgs yield matrix entries R and sqrt(R) for the even-parity block, with the muon entry given exactly by 1/sqrt(2) (proven analytically from the bulk mass parameter c₂ = 1/2 + ln2/(4 ln R)). The resulting eigenvalue condition uniquely determines the remaining small parameter delta = 0.003737 with no additional freedom. All three charged lepton mass ratios are reproduced: m_tau/m_e = 3477.4 (PDG: 3477.2, 0.006% error) and m_mu/m_e = 206.6 (PDG: 206.8, 0.1% error), from the single parameter R derived in Paper I from the fine structure constant. This derivation answers the critique that the mass formula of Papers I-III was empirical numerology: the Yukawa matrix structure emerges from the Lagrangian through the Z₂ selection rule, and the hierarchy emerges from the warped geometry. We extend the framework to quarks via the color Casimir and derive the Cabibbo angle as sqrt(m_d/m_s) = 0.2236 (0.31% match). The Z₂ parity insight was developed in collaboration with the DeepSeek AI system.