Inner Model Theory for Supercompact Cardinals and the Consistency of the Proper Forcing Axiom

We construct a canonical inner model L[E] for a super-
compact cardinal, extending the hyper-Woodin hierarchy
through a fully iterable extender sequence E allowing for
long extenders. We establish the existence of a comparison
process for mice exhibiting λ-supercompactness, resolving
the iterability problem for extenders overlapping Woodin
cardinals. Furthermore, we prove that the existence of such
a model Msc provides the exact consistency strength re-
quired for the Proper Forcing Axiom (PFA). Specifically, we
demonstrate that if there exists a supercompact cardinal in
the core model K, then there is a forcing extension satisfying
PFA. Conversely, we show that PFA implies the existence of
an inner model with a supercompact cardinal, thereby estab-
lishing the equiconsistency of ZFC + ∃κ(κ is supercompact)
and ZFC + PFA via canonical inner model theory.