Topological Derivation of the PMNS Matrix from Spinᶜ Holonomy on T²/Z₂. " Paper III in MEF Series

We derive the structural form of the charged-lepton mass matrix Mₑ of the Standard Model from the Spinᶜ geometry of the internal manifold K₈ = (CP² S²) w (T²/Z₂) ^Spinᶜ of the Master Equation Framework. The lepton zero modes are identified as three-dimensional -localised distributions on a submanifold ₀₁ = S² ₀₁ K₈ disjoint from the colour-bearing factor CP², lifted to four-dimensional smooth tempered distributions via the Zwegers shadow completion of the Spinᶜ-twisted partition function. Four results are established. (i) The geometric origin of the lepton/quark distinction emerges from the broken Liu rigidity decomposition A₀ A₁ A₂ of the Spinᶜ-twisted equivariant elliptic genus on CP²: leptons project onto the colour-singlet component A₀ alone because ₀₁ is geometrically disjoint from CP²; quarks see all three components because they live on bulk Dolbeault modes that traverse CP². (ii) The Zwegers shadow integral g is structurally identified with the constructive-QFT counterterm tower regulating the codimension-5 contact divergence of the 3D -mode. This is, to our knowledge, the first explicit instantiation of the "mock modular forms constructive quantum field theory" correspondence within a physical theory. (iii) The five-factor mass matrix Mₑ (i, j) = Aᵢ^ (L) GH (Cᵢ^ (L), Cⱼ^ (R) ) Aⱼ^ (R) Q₈₉^ (e) Y₈₉^ (e) e^iH₈₉^{ (e) } ₋₈₅ₓ, ₈₉ is closed at the structural level: corner amplitudes are uniform under canonical 4D normalisation; the universal WKB exponent for LL and eR is ₄₃ = ₀ + 1; the geodesic-class magnitudes of GH follow from explicit warped-pillowcase geometry; the corner-pair Clifford matrix Q^ (e) is from Paper XXI; the Spinᶜ Wilson-line phase matrix H^ (e) accumulates the half-period holonomies _ = 5/12 and _ = /2 rigorously derived in Paper V §8; and the Atiyah–Bott -lift sign (-1) ^3mn flips column 3 of Mₑ. The S²-fibre Yukawa overlap Y₈₉^ (e) is identified by the audited corpus (Handoff Audit) as the open factor of the architecture and is not closed in this paper. (iv) A structural floor m_/m_ 0. 30 is conditional on a Y^ (e) closure that approximately preserves the relative magnitudes of the three -independent matrix entries (1, 2), (2, 1), (2, 3). The structural form of the floor — a -independent sub-block whose singular-value ratio is fixed by these three entries' relative magnitudes — is robust to the Y^ (e) closure; the specific numerical value is contingent. The corner-dependence decomposition (§8. 1) identifies a fourth source of corner-dependence missing from the standing Y^ (e) architecture — the radius-dependent S² Clebsch–Gordan modulation W^ (C-G), set by the warped fibre radii Rₒℂ (ₐ) = Rw e^A (ₐ) — as structurally required to break the floor and reproduce the observed lepton mass hierarchy. The framework predicts its own missing computation. The corner-geodesic mechanism that drives this paper is identical, term for term, to the engine of the Yang–Mills mass gap (Paper XVII Theorem B), the localisation of zeros of the Riemann zeta function on the critical line (Paper XXI §5), and the four-dimensional smoothness of Navier–Stokes flow (Paper XVIII). The same architecture solves four classical-mathematics-and-physics problems; the Spinᶜ structure is the structural foundation across all four. Every claim is classified using the four-tier MEF rigour convention.