Some combinatorial properties of Ultimate L and V

This paper establishes a number of constraints on the structure of large cardinals under strong compactness assumptions. These constraints coincide with those imposed by the Ultrapower Axiom, a principle that is expected to hold in Woodin's hypothesized Ultimate $L$, providing some evidence for the Ultimate $L$ Conjecture. We show that every regular cardinal above the first strongly compact that carries an indecomposable ultrafilter is measurable, answering a question of Silver for large enough cardinals. We show that any successor almost strongly compact cardinal of uncountable cofinality is strongly compact, making progress on a question of Boney, Unger, and Brooke-Taylor. We show that if there is a proper class of strongly compact cardinals then there is no nontrivial cardinal preserving elementary embedding from the universe of sets into an inner model, answering a question of Caicedo granting large cardinals. Finally, we show that if $\kappa$ is strongly compact, then $V$ is a set forcing extension of the inner model $\kappa\text{-HOD}$ consisting of sets that are hereditarily ordinal definable from a $\kappa$-complete ultrafilter over an ordinal; $\kappa\text{-HOD}$ seems to be the first nontrivial example of a ground of $V$ whose definition does not involve forcing.