Forcing "NS䃑 is ₁-dense" From Large Cardinals

We answer a question of Woodin by showing that assuming an inaccessible cardinal which is a limit of <-supercompact cardinals exists, there is a stationary set preserving forcing P so that V^ P``NS䃑 is ₁-dense". We also introduce a new forcing axiom QM, show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies Q₌₀ₗ- (*). Consequently, QM implies ``NS䃑 is ₁-dense".