Logics Containing wK4: Selection à la Fine

We generalize Fine's Iterative Selection Method to the weakly transitive setting. In particular, this provides a transparent frame-theoretic proof of the finite model property for (strongly) cofinal subframe logics extending wK4, which was previously established using algebraic methods. Using the same construction, we generalize Fine's Finite Width Theorem to the weakly transitive setting, connecting these two celebrated theorems of Fine.