Disjoint Stationary Sequences on an Interval of Cardinals

We answer a question of Krueger by obtaining – from countably many Mahlo cardinals – a model where there is a disjoint stationary sequence on ${\mathrm{\aleph }}_{n+2}$ for every $n\in \omega$. In that same model, the notions of being internally stationary and internally club are distinct on a stationary subset of $[H(\mathrm{\Theta }){]}^{{\mathrm{\aleph }}_{n+1}}$ for every $n\in \omega$ and $\mathrm{\Theta }\ge {\mathrm{\aleph }}_{n+2}$, answering another of Krueger’s questions. This is obtained by employing a product of variants of Mitchell forcing which uses finite support for the Cohen reals and full support for the countably many collapses.