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

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, $m_a\propto \mathrm{\mid }Z(2\pi a\mathrm{/}3){\mathrm{\mid }}^2$. 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 ${\theta }_{\mathrm{\ell }}=-2\mathrm{/}9$. Using $m_e$ and $m_{\mu }$ as inputs, the model predicts $m_{\tau }=1776.97\mathrm{M}\mathrm{e}\mathrm{V}$.