A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. The stable regularity lemma of Malliaris and Shelah, along with folklore results about half-graphs, imply the following: a hereditary graph property P is stable if and only if for all $\epsilon>0$, every sufficiently large element in P has an $\epsilon$-regular partition with no irregular pairs. It is well known there are several distinct generalizations of Szemer\'edi's regularity lemma to the setting of hypergraphs. Consequently, there are several potential ways to generalize the correspondence between stability and the absence of irregular pairs to higher arities, which gives rise to a collection of new open questions. In this talk we present work, joint with J. Wolf, in which we resolve some of these questions in full, and make partial progress on others, all in the setting of 3-uniform hypergraphs. This work was motivated by companion results in the arithmetic setting, which will be presented in part two.