A minimization theorem for the Koide ratio and its Standard Model calibration

Abstract

The charged-lepton Koide relation remains a striking empirical regularity in Standard-Model flavor data. We prove that for any positive mass set with Koide ratio $Q_0$, the one-particle extension $Q(m_1,\ldots,m_N,x)$ has a unique global minimum $Q_\text{min}=Q_0/(1+Q_0)$ at $m^*=\bigl[(\sum_i m_i)/(\sum_i \sqrt{m_i})\bigr]^2$. This exact kinematic result defines a unique extension benchmark. For the measured charged leptons it gives $m_*^\ell = 1.255\,34(16)\,\text{GeV}$ and $Q_{4,\min}^{\mathrm{exp}} = 0.399\,997\,8(43)$; in the ideal Koide limit $Q_\ell^{\mathrm{K}}=2/3$, the corresponding minimum is exactly $2/5$. In the effective-participant language $N_{\mathrm{eff}}\equiv 1/Q$, the optimal one-particle extension increases $N_{\mathrm{eff}}$ by one, while the equal-$k$ multiplet extension increases it by $k$. The one-particle $N_{\mathrm{eff}}$ profile is exactly Lorentzian in a dimensionless share-mismatch coordinate $u$, which we interpret kinematically rather than dynamically. Using charged-lepton pole masses with the PDG~2024 own-scale $\overline{\text{MS}}$ charm mass gives $Q(e,\mu,\tau,c)=0.400\,002\,5(64)$, i.e. $11.7\,\text{ppm}$ above the measured-input benchmark and $6.2\,\text{ppm}$ above $2/5$. This intentionally mixed-definition comparison is treated only as a phenomenological coincidence. To calibrate it within a stated benchmark class, we perform an exhaustive common-scale scan over non-neutrino Standard Model 2-body and 3-body seeds with one added mass. The charged-lepton-plus-charm continuation ranks $33/12{,}720$ in the raw trial set, $24/2{,}640$ after collapsing repeated scale realizations, and $6/756$ within the fermion-only collapsed subset. We present the charm case as an empirically calibrated example of the theorem, not as a dynamical flavor model.