The main result of this paper is the following theorem: Let M be a premouse with a top extender, F. Suppose that (a) M is linearly coarsely iterable via hitting F and its images, and (b) if M * is a linear iterate of M as in (a), then M * is coarsely iterable with respect to iteration trees which do not use the top extender of M * and its images. Then M is coarsely iterable.

By forcing with Pmax over strong models of determinacy, we obtain models where different square principles at ω2 and ω3 fail. In particular, we obtain a model of 2ℵ0=2ℵ1=ℵ2+¬◻(ω2)+¬◻(ω3).