 The paper presents an operationally motivated procedure to identify equivalence classes of states in physics, arguing that the renormalization group arises from inherent ambiguities associated with these classes. The paper justifies the use of low-mementar endpoint functions as statistically relevant observables around a Gaussian hypothesis and enables the calculation of distinguishability in quantum field theory. Additionally, the methods provide a way to extend renormalization techniques to effective models that are not based on the usual quantum field formalism and elucidate relationships between various types of Rg. This article was authored by Cedric Benny and Tobias J. Osborne.