Charged-Lepton Mass Predictions from Golden-Ratio Scaling

Abstract

Starting from the approximate formula $y_e \approx (1 - 1/(2\pi))\,e^{-4\pi}$ for the electron Yukawa coupling [Zhang 2026e], we observe that the logarithms of the charged lepton Yukawa couplings satisfy golden-ratio scaling:

where $\varphi = (1+\sqrt{5})/2$ is the golden ratio. Combined with the electron formula, these two rules predict all three charged lepton masses from $\pi$, $\varphi$, and the Higgs vacuum expectation value $v$: $m_\mu = 105.4$ MeV (measured: 105.66 MeV, 0.24%) and $m_\tau = 1786.5$ MeV (measured: 1777.0 MeV, 0.54%). An independent check confirms: $m_\tau/m_\mu = e^{\pi - 1/\pi}$ to 0.08%. The golden ratio connects to the Fibonacci chain structure of the mass sector in the super-flag framework on $\mathbb{CP}^{3|4}$ [Zhang 2026h]. The scaling does not extend to quarks: down-type quarks scale as ~2 rather than $\varphi$, and up-type quarks are disrupted by the top quark ($y_t \approx 1$). We report the lepton predictions honestly and the quark non-extension equally honestly.

The empirical relation reported here is also reviewed in the companion DAEDALUS review [Zhang 2026 review], where it is classified as a Cabibbo-scenario regularity; in the broader Twistor Configuration Geometry (TCG) framework [Zhang 2026 TCG], the Perron eigenvalue $\varphi$ of the Fibonacci transfer matrix appears as a structural pattern, and the precise map to the lepton mass operator (resolving the sign-inversion $m_\mu/m_\tau < 1$ versus $\varphi > 1$) remains an open task.

DOI

https://doi.org/10.5281/zenodo.19981197