model. To assess the impact of differences in the reconstruction of data and MC, for every variable used in the selection, we compute the percent shift of the atmospheric $\nu$ MC distribution necessary to minimize its chi-square against the corresponding data distribution. The cut value in the event selection is then shifted by that percentage and applied to the signal MC to recalculate the efficiency. The total systematic uncertainty due to reconstruction is calculated by summing in quadrature the independent percent changes in signal efficiency due to each percent-shifted cut. For nucleon decay in the SK-IV period only, we also add in quadrature with the other reconstruction uncertainties an additional 10% uncertainty due to neutron tagging, as was done in Ref. [@2016miura]. This is the reason that the reconstruction uncertainties for nucleon decay are $\sim$6% larger than the corresponding uncertainties for dinucleon decay. To estimate the uncertainty in the signal efficiency arising from uncertainties in correlated decay, we assume 100% uncertainty on the correlated decay effect, reweight the correlated decay events accordingly, and recalculate the signal efficiency, taking the overall change in signal efficiency as the systematic uncertainty. The nuclear model uncertainty is estimated as the percent change in signal efficiency when the Fermi gas model is used to compute the true momentum of the protons within the signal MC events instead of the spectral function fit to data described earlier.

The systematic uncertainty on the rate of background events is conservatively taken to be 100% for decay modes where the background events are scarce (all dinucleon decay modes, and the $p \rightarrow e^+ \gamma$ “Low $P_{\text{tot}}$" nucleon decay). For the other nucleon decay signal regions, we use an event-by-event database with uncertainty weights from 73 sources of background systematic uncertainty including uncertainties in flux, cross section and energy calibration, as described in the 2018 SK oscillation a