User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-11227630-20181121071343/@comment-31580368-20181121122653

1) Ohhh..  It is very good. I figure it looked like an offset problem, when S[σ](ω + 1) is actually ω-th admisible. Then, need to save two S's in S[S[σω]]. And I will stick with this option "S[S[σω](1)] is nonprojectable limit of nonprojectables; S[S[σω](2)] is nonprojectable limit of nonprojectable limits of nonprojectables; S[S[σω](3)] is nonprojectable limit of nonprojectable limits of nonprojectable limits of nonprojectables; etc. then S[S[σω](ω)] is the level ω of them." And keep S[σω'1] is nonprojectable and admissible.

5) This means, if remove  "the ordinal x that is x-ply-stable" structure S[S2[σσ'n]] be saved. Just need to be careful with the definitions.