Geometric Origin of the Koide Angle thetaK: Structural Closure via Klein Parent Curve. Version v2 updates the v1 proposal (DOI 10. 5281/zenodo. 19546005) to reflect the completion of the C1 task (torus mother construction) of the companion paper Toward a Geometric Theory of CP Violation v17 (DOI 10. 5281/zenodo. 19582482). In v1, the composite proposal A+B was labeled as a "conditional closure" pending the construction of a common null parent for the Koide harmonic sphere and the Fano rapidity simplex. In v2, this common parent is explicitly identified as the Klein quartic curve (genus 3, automorphism group PSL (2, 7) = Fano plane automorphism group), via the modified axiom A2' (orbifold base sphere quotient S² (2, 3, 7) ). The closure is therefore structural rather than conditional. Key updates: - thetaKᵘp = r * muₖ = (pi/8) (ln 2 / ln 3) is now derived from Lemmas C1-6 (half-speed fixation from the ratio 3) and C1-15 (localization of ln 2 on the outer i-edge only), both proved in the companion paper Appendix C. - Delta thetaK = 1/ (2 Sigmaₑll) = 1/12 is now derived from Lemma C1-10 (Fano global sum 6 preserved across the three outer edges), with C1-8 fixing the minimal marking required for the correspondence to be unique. - The open gaps C-theta 1, C-theta 2, C-theta 3, C-theta 4 from v1 are either closed A or reclassified as engineering verification. - Relative differences against the empirical Koide angles (0. 34%, 0. 40%, 0. 31%) are now interpreted as regularization-level residues within the empirical uncertainty window, not as fitting errors. The remaining truly open gaps at the Perfect CP project level are C2 (Berry holonomy ansatz derivation, currently E) and C3 (three-generation selection rule, currently C), both in the companion paper and not within the scope of this sister paper.
Kuniyuki Hayashi (Wed,) studied this question.