The Twin Dilemma: When is a Jammed Packing not Actually Jammed?

Periodic boundary conditions are used in studies of jamming to lessen finite size effects. However, the use of these boundaries can lead to strange and unintuitive results. We show that when critically jammed sphere packings are replicated they unjam through the creation of new zero modes. In fact, systems with fewer than the dimension, d, states of self-stress will always unjam upon replication. We draw links between this result and the constraints needed for shear stability, as well as the Guest-Hutchinson modes found in crystals.