constant applied field of 0.08 T. The [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples show a clear progression in wait time effect that correlates strongly with their relative densities of nuclear spin states. Thus the monopoles recombine during the wait period much more effectively the larger the nuclear spin: that is, the larger the nuclear spin the higher the monopole mobility, the faster the recombination, and the fewer the monopoles at the start of the measurement. In Fig. 3e, f, higher mobility means the relaxation curves ($M(t = 400~{\rm s})$ and $J_{\rm m}(t = 0)$) shift both up and to the left, so a crossover in curves is expected – and this is indeed observed at the longer times. Near to equilibrium a second crossover would be expected (i.e. the equilibrium current density is higher for the highest mobility), but this crossover is clearly very far outside our time window. Hence our [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples are always far from equilibrium.

The effects observed for [Dy$_{2}$Ti$_{2}$O$_{7}$]{} are yet more dramatic in [Ho$_{2}$Ti$_{2}$O$_{7}$]{}, consistent with the Ho$^{3+}$ non-Kramers character, large nuclear spin, and large hyperfine coupling. Relaxation at 200 mK covers more than two orders of magnitude but is practically extinguished for long wait times, showing that excess 