## Free-then-freeze : transient learning degrees of freedom for introducing function in materials

Many protocols used in material design and training have a common theme : they introduce new “learning'' degrees of freedom, often by relaxing away existing constraints, and then evolve these degrees of freedom based on a rule that leads the material to a desired state at which point these learning degrees of freedom are frozen out. Using this conceptual framework, we first introduce and subsequently remove different sets of degrees of freedom from the energy minimization process of athermal particle packings. By doing so, we are able to create stable jammed packings that exist in exceptionally deep energy minima marked by the absence of low-frequency quasilocalized modes ; this added stability persists in the thermodynamic limit. We show that the inclusion of particle radii as learning degrees of freedom leads to deeper and much more stable minima than does the inclusion of particle stiffnesses. This is because particle radii couple to the jamming transition whereas stiffnesses do not. Thus different choices for the learning degrees of freedom can lead to very different training outcomes.