Geometric Origin of the Koide Angle theta_K: A Pure-Geometry Reconstruction Proposal

A sister paper to "Geometric Origin of CP Violation" v15 (DOI: 10.5281/zenodo.19545893). The companion paper reports a CKM matrix reconstruction with zero free parameters that reproduces all eleven CKM observables within 0.52 sigma, built through a test-driven development (TDD) program. A single empirical input remains: the Koide phase theta_K, which parameterizes the quark mass spectrum. This paper attempts a pure-geometry reconstruction of theta_K itself, using only geometric constants already established in the companion paper (Fano rapidities, hierarchy r = pi/8, Fano point count N_Fano = 7, generation integers ell = 1, 2, 3), and proposes: theta_K_up = r * mu_k = (pi/8)(ln2/ln3) ~ 0.24784 (empirical 0.247, 0.34% difference) Delta theta_K = 1/(2 Sigma ell) = 1/12 ~ 0.08333 (empirical 0.083, 0.40% difference) theta_K_down = theta_K_up - Delta theta_K ~ 0.16451 (back-solved 0.164, 0.31% difference) The denominator 2 Sigma ell = 12 coincides with the numerator of the exact k-edge EM coefficient 12/7 = 2 Sigma ell / N_Fano derived at Step 33 of the companion paper. This multiple appearance of the Fano global algebraic quantity 2 Sigma ell in two logically independent constructions (EM phase correction and Koide angle difference) is, within the present framework, the strongest structural argument against accidental coincidence. This paper is NOT a first-principles proof. The mapping between the Koide harmonic sphere and the Fano rapidity simplex is not constructed. The proposal is a structural candidate audited against four criteria (numerical precision, uniqueness, multiple appearance, derivability), with honest [C] labels on remaining open gaps. Published as a separate sister paper to preserve the methodological distinction: the companion paper achieves precision through TDD against PDG measurements; the present paper achieves structural consistency through pure-geometry reconstruction.