Generations as Fiber Winding: Counting the families, the pairing theorem, and the selection of their number

SQG Generations, version 4.6.

The framework hosts three generations but has not derived them. This note reduces the number of chiral zero modes of the wound condensate to one topological integer, proves the structural symmetry of those zero modes, asks what fixes the integer, and states precisely under which reading that integer would be the family number.

(1) Counting: a vortex of winding w of the condensate phase on the compact time fiber binds exactly w chiral zero modes, each carrying one 16 of Spin(10). For the framework's Thomas–Fermi profile the counting is an elementary theorem: inside the hole the channel system decouples exactly, the interior solution is a pure power, and normalizability delimits the window l ∈ {0, …, w−1}.
(2) The pairing theorem: the component swap intertwines the channels l and w−1−l exactly, so every density-coupled insertion satisfies I_l = I_(w−1−l) exactly, upgrading the 2+1 degeneracy from a numerical observation to a theorem. This is the load-bearing symmetry of the companion flavour analysis, and it also makes the wall model Spin(10)-consistent: in Spin(10) the mass kernel is of Hankel type, and by the pairing it gives the same masses and mixing as the Toeplitz kernel.
(3) Selecting: pure boson energetics selects w = 1; a fiber holonomy is shown not to be an independent datum, so three generations remain hosted, not derived. A single winding-3 core is dynamically unstable for a Lorentz-invariant condensate; it is held together only by a type-I dark gauge sector (m_A > m_h).

A dimensional obstruction bounds the reading of the zero modes as four-dimensional families: the winding plane contains the Euclidean time, so the defect is centred on the horizon bifurcation surface. The counting, pairing and selection theorems stand; their identification with the observed families is open. The three-family count passes the comparison with data, the Z width excluding w = 4, 5.