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We report on the measurement of the two-neutrino double-$\beta$ decay of$^{82}$Se performed for the first time with cryogenic calorimeters, in theframework of the CUPID-0~experiment. With an exposure of 9.95 kg$\times$yr ofZn$^{82}$Se, we determine the two-neutrino double-$\beta$ decay half-life of$^{82}$Se with an unprecedented precision level, $T_{1/2}^{2\nu} = [8.60 \pm0.03 \textrm{(stat.)}~^{+0.17}_{-0.10} \textrm{(syst.)}] \times10^{19}~\textrm{yr}$. The very high signal-to-background ratio, along with thedetailed reconstruction of the background sources allowed us to identify thesingle state dominance as the underlying mechanism of such process,demonstrating that the higher state dominance hypothesis is disfavored at thelevel of 5.5 $\sigma$.

Evidence of Single State Dominance in the Two-Neutrino Double-$\beta$ Decay of Se-82 with CUPID-0