User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-11227630-20190803082138/@comment-31580368-20190803173354

You can use the idea "correctness" from TON (4.4). Every α-n-stable is Lα≺nLβ and have Σn-correctness. Then structure Σn-correctness on gap ordnials will be similar to structure Πn-reflecting on stable ordnials.