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 model in which this vector is sampled from one real amplitude $Z(\phi)$ on an internal circle, while the masses are quadratic overlaps, $m_a\propto |Z(2\pi a/3)|^2$. The amplitude is built from the two lowest antiperiodic modes on the circle; their symmetric square is periodic and gives the minimal three-harmonic family space $e^{i\phi},1,e^{-i\phi}$. A reality condition together with the requirement that the amplitude comes from the square of one two-component spinor fixes the relative weights required by Koide's $45^\circ$ geometry. The remaining orientation angle is fixed by matching one $C_3$ family shift to transport on the full circle: integrating out the higher Fourier harmonics gives the Berry dressing that enters the determinant term and selects $\theta_\ell=-2/9$. Using $m_e$ and $m_\mu$ as inputs, the model predicts $m_\tau=1776.97\,\mathrm{MeV}$.