The mechanical stability of a physical system plays a crucial role in determining its excitations and response to strain. Recent advances have led to protocols that can create particularly stable amorphous solids. Such systems, whether they be physical systems created using vapor-deposition or numerical model systems created using swap or breathing algorithms, exist in exceptionally deep energy minima marked by the absence of low-frequency quasilocalized modes. We introduce new numerical protocols for creating stable jammed packings that first introduce and subsequently remove degrees of freedom such as particle sizes or particle stiffnesses. We find that different choices for the degrees of freedom can lead to very different results. For jammed packings, degrees of freedom that couple to the jamming transition, e.g., particle sizes, push the system to much more stable and deeper energy minima than those that only couple to interaction stiffnesses.
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