Electroweak Mixing from the Three-Plane Scaffold: sin2 θW = 1/(Nc + 1) and the W/Z Mass Ratio

This paper derives the electroweak mixing angle and the W/Z mass ratio from the three-plane scaffold — a geometric model based on three mutually orthogonal scalar-field domain walls meeting at a Borromean junction. The SU(2)_L gauge sector is identified with the three orthogonal walls and the U(1)_Y hypercharge sector with the single compact rotor. At energies above the wall vacuum mass, the octahedral symmetry of the junction core is restored and each generator carries equal topological weight, fixing the Weinberg angle via a simple degree-of-freedom count: one rotor mode against three wall modes gives sin²θ_W = 1/4 at the scaffold UV scale. The consequent Z mass prediction agrees with observation to 1.8%, and the custodial parameter ρ = 1 at tree level. The residual discrepancy is consistent with standard renormalisation-group running over six decades from the scaffold UV scale to the Z pole, identifying this emergence scale as a derived prediction of the framework rather than a free parameter. The mechanical stiffness ratio of the rotor to the wall modes is shown to be strongly disfavoured as a gauge input — introducing it predicts a Z mass error of 20% — establishing a clean separation between the UV gauge sector, governed by topology, and the IR instanton sector, governed by mechanical stiffness. A structural coincidence connects the electroweak mixing to the quark mass spectrum: the cosine-squared of the mixing angle equals the quark generation step factor, both taking the value 3/4, interpreted as two projections of the same scaffold geometric invariant.