The stationarity of the collection of the locally regulars

Abstract

I analyze various natural assumptions which imply that the set $${\{\omega_1^{L[x]} \mid x \subseteq \omega\}}$${?1L[x]?x⊆?} is stationary in ?1. The focal questions are which implications hold between them, what their consistency strengths are, and which large cardinal assumptions outright imply them.