The maximality of the core model

Our main results are: 1) every countably certified extender that coheres with the core model $K$ is on the extender sequence of $K$, 2) $K$ computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of $K$, 4) (joint with W. J. Mitchell) $K\|\kappa$ is universal for mice of height $\le\kappa$ whenever $\kappa\geq\aleph _2$, 5) if there is a $\kappa$ such that $\kappa$ is either a singular countably closed cardinal or a weakly compact cardinal, and $\square _\kappa^{<\omega}$ fails, then there are inner models with Woodin cardinals, and 6) an $\omega$-Erdös cardinal suffices to develop the basic theory of $K$.