Three Families from One Winding: Exact zero-mode counting, an exact pairing theorem, and a two-sided bound on the winding

Elementary counting [proved]. For a condensate that vanishes inside a core, the mode equations decouple exactly in the hole, and normalisability selects exactly the w channels l = 0, …, w−1. Exact pairing [proved]. Swapping the two spinor components maps channel l onto channel w−1−l exactly. Paired channels therefore have identical radial densities, and every density-coupled overlap satisfies I_l = I_(w−1−l). For w = 3 this makes the 2+1 structure of the families a theorem, with a middle channel whose profile is fixed by the condensate alone. Two-sided bound. From below, w ≥ 3: only odd w ≥ 3 contains a paired and a self-paired channel, and three is the smallest family number with CP violation. From above: every zero mode carries a light active neutrino. The Z width, N_ν = 2.984 ± 0.008, agrees with w = 3 and excludes w = 4 and w = 5 at 127σ and 252σ. What is confirmed. Against data: three light neutrino species. Within the model, by predictions fixed before the computation: the tunnelling rate of the middle channel, and the r² suppression of the lightest neutrino. By consistency: the counting and pairing theorems, verified numerically. Open. The reading of the zero modes as four-dimensional families is open and stated as such. The mechanism class has close prior art in the vortex programme of Libanov, Troitsky and Frère, which is cited and delimited. Derivations, verification code and drafting were carried out in collaboration with Claude (Anthropic).