Stability functions inducing stability data

In this paper, we study when a stability data on an abelian length category is induced by a stability function. We give an intrinsic characterization of such stability data. For at most six phases, an inducing function exists if and only if every semistable subcategory is wide. An example with seven phases shows that this bound is sharp. For finite phase sets and finite rank, an inducing function exists if and only if an inducing red path exists. We also give a maximal green sequence of length eleven in type $D_4$ that no stability function induces. This disproves the red-path conjecture of Brüstle--Smith--Treffinger.