Failure of Approachability at the Successor of the first Singular for any Cofinality

We solve two long-standing open problems regarding the combinatorics of ${\mathrm{\aleph }}_{\omega +1}$. We answer a question of Shelah by showing that it is consistent for any $n\ge 1$ that $\mathsf{G}\mathsf{C}\mathsf{H}$ holds and there is a stationary set of points of cofinality ${\mathrm{\aleph }}_n$ which is not in the approachability ideal. As a corollary, we obtain a model where the notions of goodness and approachability are distinct for stationarily many points of cofinality ${\mathrm{\aleph }}_1$, answering an open question of Cummings, Foreman, and Magidor.