What question did this study set out to answer?

This paper aims to reconstruct the Koide angle, focusing on its geometric origins using established geometric constants.

April 14, 2026Open Access

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

Key Points

This paper aims to reconstruct the Koide angle, focusing on its geometric origins using established geometric constants.
Developed a geometric reconstruction proposal for the Koide angle
Utilized geometric constants from a companion paper
Analyzed numerical precision against empirical values
Verified structural consistency through established criteria
Koide angle estimated at 0.24784 with a 0.34% difference from empirical value
The down value of Koide angle estimated around 0.16451 with a 0.31% difference
Demonstrated structural consistency through multiple appearances of geometric quantities

Abstract

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 thetaK, which parameterizes the quark mass spectrum. This paper attempts a pure-geometry reconstruction of thetaK itself, using only geometric constants already established in the companion paper (Fano rapidities, hierarchy r = pi/8, Fano point count NFano = 7, generation integers ell = 1, 2, 3), and proposes: thetaKᵤp = r * muₖ = (pi/8) (ln2/ln3) ~ 0. 24784 (empirical 0. 247, 0. 34% difference) Delta thetaK = 1/ (2 Sigma ell) = 1/12 ~ 0. 08333 (empirical 0. 083, 0. 40% difference) thetaKdown = thetaKᵤp - Delta thetaK ~ 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 / NFano 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. Keywords: CKM matrix, Koide mass formula, Fano plane, Kerr geometry, quark masses, pure geometric reconstruction.

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Kuniyuki Hayashi (Mon,) studied this question.

synapsesocial.com/papers/69ddd9e1e195c95cdefd751f https://doi.org/https://doi.org/10.5281/zenodo.19546004

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