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 $Q(m_1,\dots ,m_N,x)$ has a unique global minimum $Q_{\text{min}}=Q_0/(1+Q_0)$ at $m^{\ast }=[({\sum }_i{m}_i)/({\sum }_i\sqrt{m_i}){]}^2$. This exact kinematic result defines a unique extension benchmark. For the measured charged leptons it gives $m_{\ast }^{\ell }=1.25534(16)\text{GeV}$ and $Q_{4,\min }^{\exp }=0.3999978(43)$; 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 $Q(e,\mu ,\tau ,c)=0.4000025(64)$, 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.