Bistationary Traces, Wide Levels, and Branch-Cover Rigidity for an Unrestricted Typed Variant of the Hayut-Magidor Forcing

For every uncountable regular cardinal $α$, $\mathbb S^{\ast}(α)$ is an explicitly typed four-coordinate forcing motivated by the ladder-system construction of Hayut and Magidor. The forcing is $σ$-closed and, after adjoining a formal maximum, $α$-strategically closed. For $α\geqω_2$, every nonempty