We consider a revenue-maximizing seller with k heterogeneous items for sale to a single additive buyer, whose values are drawn from a known, possibly correlated prior D. It is known that there exist priors D such that simple mechanisms - those with bounded menu complexity - extract an arbitrarily small fraction of the optimal revenue ([BCKW15, HN19]). This paper considers the opposite direction: given a correlated distribution D witnessing an infinite separation between simple and optimal mechanisms, what can be said about D? [HN19] provides a framework for constructing such D: it takes as input a sequence of k-dimensional vectors satisfying some geometric property, and produces a D witnessing an infinite gap. Our first main result establishes that this framework is without loss: every D witnessing an infinite separation could have resulted from this framework. An earlier version of their work provided a more streamlined framework [HN13]. Our second main result establishes that this restrictive framework is not tight. That is, we provide an instance D witnessing an infinite gap, but which provably could not have resulted from the restrictive framework. As a corollary, we discover a new kind of mechanism which can witness these infinite separations on instances where the previous “aligned” mechanisms do not.