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

For every uncountable regular cardinal \( \alpha \), \( S^{*}(\alpha) \) is an explicitly typed four-coordinate forcing motivated by the ladder-system construction of Hayut and Magidor. The forcing is \( \sigma \)-closed and, after adjoining a formal maximum, \( \alpha \)-strategically closed. For regular \( \alpha\geq\omega_2 \), every nonempty family of fewer than \( \alpha \) designated generic branches has a stationary and costationary common trace in \( P_{\omega_1}\alpha \) with respect to the generic ladder-coordinate set \( L_\alpha \). For these cardinals, the minimum number of cofinal branches of the generic tree needed to generate \( L_\alpha \) is exactly \( \alpha \). The lower bound follows from a uniform diagonal argument, whereas the upper bound uses one designated branch per nonempty ladder coordinate. For the Kurepa-bounded companion forcing at the same cardinals, the bistationary-trace conclusion holds, and the corresponding branch-covering number is also exactly \( \alpha \). In the unrestricted forcing at such an \( \alpha \), for every infinite cardinal \( \mu<\alpha \) in the ground model, some level of the generic tree contains a copy of \( ({}^\mu2)^V \). Consequently, the endpoint-corrected restriction family indexed by \( P_{\omega_2}\alpha \) is too wide, whereas the scaled restriction system indexed by \( P_\alpha\alpha \) has all levels of size less than \( \alpha \) exactly when \( \alpha \) is strongly inaccessible in the ground model. When these equivalent conditions hold, the branch-covering number of \( L_\alpha \) relative to the scaled system equals \( \alpha \). For regular \( \alpha\geq\omega_2 \), the low-cofinality empty-value convention also implies that the set of domains of \( L_\alpha \) contains no club in \( P_\alpha\alpha \). As in the displayed tree clause of the motivating presentation, no explicit level-size bound is imposed. No forcing equivalence is asserted.