Charged-Lepton Koide Geometry from a Green-Dressed Compact Family Cycle

Abstract

Koide's charged-lepton relation suggests that $(\sqrt{m_e},\sqrt{m_{\mu }},\sqrt{m_{\tau }})$ is the natural family vector. We construct an effective compact-cycle realization in which this vector is sampled from a real amplitude $Z(\phi )$ on an internal circle, while the masses arise from quadratic chiral-profile overlaps. The coherent branch-real rank-one sector is an explicit assumption of the effective theory and realizes the equal singlet--doublet norm associated with Koide's $45^{\circ }$ cone. The empirical proximity of the $Z_3$ phase to $2/9$ predates the present work. The new ingredient proposed here is an endpoint-dipole Green response for one elementary family-shift link. The compact harmonic sum fixes the geometric factor ${\gamma }_{C_3}=2{\lambda }_B/9$, while ${\lambda }_B$ remains an undetermined matching coefficient, and the determinant anisotropy gives ${\theta }_{\ell }^{(0)}=-2{\lambda }_B/9$. Thus ${\lambda }_B=1$ yields ${\theta }_{\ell }^{(0)}=-2/9$ only as a unit-normalized benchmark, not as an exact first-principles prediction. With the current pole masses, this benchmark requires a relative finite matching correction of $9.83\times 10^{-6}$ in $m_e/m_{\mu }$; no microscopic theoretical uncertainty for that correction is derived here.

arXiv comment19 pages, Published in Journal of Physics G: Nuclear and Particle Physics