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

TL;DR

This paper presents a geometric model for charged-lepton masses based on a compact-cycle framework, reproducing Koide's relation and predicting the tau mass with high precision.

Contribution

It introduces a novel geometric construction using a Green-dressed compact cycle to explain Koide's relation and predict the tau mass.

Findings

Reproduces Koide's relation exactly.

Predicts tau mass as 1776.97 MeV.

Derives the geometric origin of lepton mass relations.

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 $m_\tau=1776.97\,\mathrm{MeV}$.