Ringdown comparison bound for two resonator modes ($Q_1<{\omega }_1/[{\omega }_2/Q_2+\ln R_0/t]$)

For two weakly damped normal modes with initial intensity ratio $R_0>1$, independent exponential energy decay gives $I_1(t)/I_2(t)=R_0{e}^{-t({\omega }_1/Q_1-{\omega }_2/Q_2)}$. An observed reversal to a ratio below one implies the displayed upper bound on $Q_1$, provided the denominator is positive. A strict intensity comparison does not determine a unique quality factor; equality fixes only the crossing threshold.