Strongest Transformations

We continue our study of maps transforming high-dimensional complicated objects into squares stationary sets. Previously, we proved that many such transformations exist in ZFC, and here address the consistency strongest conceivable transformations. Along way, obtain new results on Shelah's coloring principle $Pr_1$. For $\kappa$ inaccessible, prove $Pr_1(\kappa,\kappa,\kappa,\kappa)$. successors regulars, a full lifting Galvin's 1980 theorem. In contrast, theorem to singulars is shown be inconsistent.