  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.
