User blog comment:Mh314159/Help me understand a natural number recursion/@comment-35470197-20191014142325/@comment-39541634-20191016124937

See here:

https://math.stackexchange.com/questions/1811602/explicit-upper-bound-of-tree3

It looks like I mis-remembered the details of how it was proven (proof theory has absolutely nothing to do with it) but the end result is the same: fφ(ω@ω)(n) is a rough(*) upper bound fo TREE(n).

(*) "rough" in the sense that TREE(n) might concievably grow as fφ(ω@ω)(n^^n) or perhhaps even fφ(ω@ω)+1(n). For an actual numeric upper bound, though, fφ(ω@ω)+2(n) should be more than enough.