Characterizing the nature of the rigidity transition

Particulate matter, such as foams, emulsions, and granular materials, attain rigidity in a dense regime: the rigid phase can yield when a threshold force is applied. The rigidity transition in particulate matter exhibits {\it bona fide} scaling behavior near the transition point. However, a precise determination of exponents describing the rigidity transition has raised much controversy. Here, we pinpoint the causes of the controversies. We then establish a conceptual framework to quantify the critical nature of the yielding transition. Our results demonstrate that there is a spectrum of possible values for the critical exponents for which, without a robust framework, one cannot distinguish the genuine values of the exponents. Our approach is two-fold: ({\it i}) a precise determination of the transition density using rheological measurements and ({\it ii}) a matching rule that selects the critical exponents and rules out all other possibilities from the spectrum. This enables us to determine exponents with unprecedented accuracy and resolve the long-standing controversy over exponents of jamming. The generality of the approach paves the way to quantify the critical nature of many other types of rheological phase transitions such as those in oscillatory shearing.