``Addendum to Paper~III: Sol-Monodromy Regularisation of the Cross-Wall Quantum Coherence Dilaton, the Charged-Lepton Mass Ratio m_ / m_ from () and 12, and the Closure of Y₅₈₁^ (e) under the Dimensional-Shell Hierarchy. ''Paper~III Addendum in MEF Series

Charged leptons and neutrinos are fermions, but their physical properties differ sharply from those of quarks: leptons carry no QCD colour, the neutrino mass scale lies six orders of magnitude below the charged-lepton scale, and the structure of the PMNS mixing matrix bears no apparent resemblance to the CKM hierarchy. The Standard Model accommodates each of these features by assigning independent Yukawa couplings or quantum numbers, but provides no first-principles derivation of any of them. This Addendum derives the lepton sector from the topology of the Master Equation Framework (MEF) twelve-manifold M₁₂ = (Sₒ₎₋ S³) K₈. The compact internal eight-manifold K₈ has Euler characteristic (K₈) = 12 = 4 3, with the factor-of-four spinor multiplicity and the factor-of-three generation count fixed in Paper I 1 from the Spinᶜ structure on the orbifold pillowcase T²/Z₂ K₈ via Reading corner localisation. The framework carries zero continuous moduli: every parameter is a topological invariant or a 2, 3-arithmetic integer, in contrast to string-theoretic compactifications whose Calabi–Yau and flux moduli require independent stabilisation. The derivations are based on the constructive QFT axioms of Paper XX 2, under which four-dimensional singularities of physical observables are projection artefacts of smooth higher-dimensional geometry — the Shadow Principle, invoked as Axiom M-2 in the neutrino mass-scale derivation below. We identify the geometric origin of lepton colourlessness. Under the MEF identification of CP² K₈ with the strong sector — whose SU (3) isometry is QCD colour — leptons cannot propagate as bulk modes on CP². The Dolbeault cohomology supports this directly: quarks sit in h^0, 2 (CP², O (-4) ) = 3, whose three bulk representatives carry the three generations × three colours; leptons sit in h^0, 2 (CP², O (-3) ) = 1, a single bulk representative insufficient for three generations. The three lepton generations therefore arise from orbifold fixed-point localisation on T²/Z₂, whose corners carry no SU (3) representation content. Rigorous. We derive the PMNS mixing matrix from M₁₂ topology, with zero free parameters. The four PMNS observables ₁₂ = 33. 15° (NuFIT 6. 0 3: 33. 68 ± 0. 71°; -1. 6%), ₁₃ = 8. 71° (8. 56 ± 0. 11°; +1. 8%, ~1. 4σ), ₂₃ = 49. 22° (48. 5° +0. 7°/-0. 9°, upper octant; +1. 5%), and ₂₏^ = 165° (177° +19°/-20°; -6. 8%) all lie within 1σ–2σ of the NuFIT 6. 0 global best fit, and the small mass-squared splitting m²₂₁ = 7. 39 × 10⁻⁵ eV² ( (7. 49 ± 0. 19) × 10⁻⁵ eV²; -1. 3%) within 1σ. To our knowledge, no competing parameter-free derivation of the PMNS angles in the literature reaches comparable agreement; representative parameter-free predictions such as tribimaximal mixing miss ₁₃ by orders of magnitude (predicting 0 against the measured 8. 5°). Subsequent work on the Bismut canonical-rescaling expansion are expected to tighten the residual deviations on the large mass-squared splitting and its downstream consequences, disclosed in the lepton-sector closure summary table (Section 4C). We derive the absolute neutrino mass scale from the K₈ warp budget. Bismut canonical rescaling on the warped Spinᶜ bundle fixes the warp power w = 2 from the warped-fibre dimension ₆ = 6, giving the parameter-free baseline: m_^ (baseline) = V_^1/4 \, e^-24 which is approximately 4 meV, satisfying the strict equality m_^ (baseline) = ₄₅₅^1/4. This equality is a structural identity under Axiom M-2 (Shadow Completion) of Paper XX 2, specialised to the M₁₂ vacuum sector. Both quantities are 4D-shadow projections of the same K₈ warp-budget object, sharing prefactor V_^1/4 and exponent e^-24. The exponential suppression by the warp factor e^-24 is the geometric origin of neutrino lightness: the Standard Model accommodates the charged-lepton-to-neutrino mass ratio of ~10⁶ by parameter fitting via a seesaw scale or a tuned Yukawa; the MEF derives it from a single warp integral over K₈ with no such tuning nor parameter freedom. The mechanism is one geometric object, used across eight independent observables.