Systems governed by a free energy landscape can be trapped forever in a local minimum, unable to find more favorable configurations. However, the introduction of transient degrees of freedom can connect otherwise distant regions of the energy landscape and allow for the discovery of rare minima with unusual properties. Using conventionally prepared over-jammed disk and sphere packings as a starting point, we show that the introduction of particle radius as a transient degree of freedom allows for the construction of “ideal” disk packings in 2d and exactly hyperuniform sphere packings in dimensions 3 and higher. Ideal disk packings are the highest density non-overlapping configurations of disks possible for a given set of particle radii. Such structures have nearly all of the thermal and mechanical properties of crystals while entirely devoid of crystalline order.