Fully independent response in disordered solids

Unlike in crystals, it is difficult to trace emergent material properties of amorphous solids to their underlying structure. Nevertheless, one can tune features of a disordered spring network, ranging from bulk elastic constants to specific allosteric responses, through highly precise alterations of the structure. This has been understood through the notion of independent bond-level response -- the observation that in many cases, different springs have different effects on different properties. While this idea has motivated inverse design in numerous contexts, it has not been formalized and quantified in a general context that not just informs but enables and predicts inverse design. Here, we show how to quantify independent response by linearizing the simultaneous change in multiple emergent features, and introduce the much stronger notion of fully independent response. Remarkably, we find that the mechanical properties of disordered solids are always fully independent across a wide array of scenarios, regardless of the target features, tunable parameters, and details of particle-particle interactions. Furthermore, our formulation quantifies the susceptibility of feature changes to parameter changes, which we find to be correlated with the maximum linear tunability. These results formalize our understanding of a key fundamental difference between ordered and disordered solids while also creating a practical tool to both understand and perform inverse design.