Derivation of the Sector-Dependent Koide Parameter QK(r) from Face-Axis Geometry in Mo Theory

This record contains the manuscript for a derivation of the sector-dependent Koide parameter Q_K(r) from face-axis geometry within Mo Theory. The paper develops four results. First, a winner-take-one face potential V_face is constructed on the 4-vertex square face of the motic cube, and its ground-state manifold is shown to consist of exactly 8 points (Theorem 1). Second, the face-to-axis projection amplitude g_face = 1/2 is computed from the A_1 representation overlap on the face. Third, the doublet coupling factor g_doublet = 1/√2 is derived from Schur's Lemma applied to the Z_3 irreducible doublet representation, upgrading the phenomenological identification ε = 1/√8 reported in version 3 to a geometric result. Fourth, a Dyson-type boundary-resolvent resummation of the transit-face feedback loop yields B²(r) = 2/(1 − εr), from which the closed-form Koide parameter Q_K(r) = (2 − εr)/[3(1 − εr)] follows algebraically. The formula contains no fitted numerical parameters within the stated boundary-resolvent model. Comparison with PDG 2024 mass data gives relative errors of 0.001% (leptons), 0.93% (down quarks), and 0.001% (up quarks). A structural theorem proves that Q_K is independent of the Brannen phase δ, separating the mass-ratio geometry from the individual mass hierarchy. Appendices present numerical verification, the edge-difference Laplacian proof, unification with the fine-structure constant (ε² = f_vertex = 1/8), the coherence number N_coh = 2 + q², and a motivated (not derived) neutrino-sector prediction. The analysis is confined to the Koide parameter Q_K(r). The paper does not derive: neutrino masses from first principles, quark Brannen phases δ(r > 0), CKM mixing, gauge coupling running, confinement, (g−2), or the absolute electron mass. One dimensionful scale M_0 remains external. The reader is referred to prior versions for the full Koide derivation, the Brannen phase δ(0) = 2/9, the cube-graph action principle, and the fine-structure constant model. This deposition is intended as a transparent research record of the current state of the argument, including its assumptions, derivation steps, and remaining limitations. Previous work: https://doi.org/10.5281/zenodo.19534450