  Understanding the way disordered particle packings transition between jammed
(rigid) and unjammed (fluid) states is of both great practical importance and
strong fundamental interest. The values of critical packing fraction (and other
state variables) at the jamming transition are protocol dependent. Here, we
demonstrate that this variability can be systematically traced to structural
measures of packing, as well as to energy measures inside the jamming regime. A
novel generalized simultaneous particle swap algorithm constructs overjammed
states of desired energy, which upon decompression lead to predictable critical
packing fractions. Thus, for a given set of particle sizes, states with
extraordinarily high critical packing fractions can be found efficiently, which
sustain substantial shear strain and preserve their special structure over the
entire jammed domain. The close relation revealed here between the energy
landscape of overjammed soft-particle packings and the behavior near the
jamming transition points towards new ways of understanding and constructing
disordered materials with exceptional properties.
