Energy scale independence of Koide’s relation for quark and lepton masses

Koide's mass relation of charged leptons has been extended to quarks and neutrinos, and we prove here that this relation is independent of energy scale in a huge energy range from 1 GeV to 2×10¹⁶ GeV. By using the parameters kᵤ, kd, and k_ν to describe the deviations of quarks and neutrinos from the exact Koide's relation, we also check the quark-lepton complementarity of masses such as kₗ+kd≈k_ν+kᵤ≈2, and show that it is also independent (or insensitive) of energy scale.