  Determining the fate of the Pauling entropy in the classical spin ice
material Dy$_2$Ti$_2$O$_7$ with respect to the third law of thermodynamics has
become an important test case for understanding the existence and stability of
ice-rule states in general. The standard model of spin ice - the dipolar spin
ice model - predicts an ordering transition at $T\approx 0.15$ K, but recent
experiments by Pomaranski $et\ al.$ suggest an entropy recovery over long time
scales at temperatures as high as $0.5$ K, much too high to be compatible with
theory. Using neutron scattering and specific heat measurements at low
temperatures and with long time scales ($0.35$ K$/10^6$ s and $0.5$ K$/10^5$ s
respectively) on several isotopically enriched samples we find no evidence of a
reduction of ice-rule correlations or spin entropy. High-resolution simulations
of the neutron structure factor show that the spin correlations remain well
described by the dipolar spin ice model at all temperatures. Further, by
careful consideration of hyperfine contributions, we conclude that the original
entropy measurements of Ramirez $et\ al.$ are, after all, essentially correct:
the short-time relaxation method used in that study gives a reasonably accurate
estimate of the equilibrium spin ice entropy due to a cancellation of
contributions.
