User blog comment:Mh314159/Help me understand a natural number recursion/@comment-35470197-20191014142325/@comment-35470197-20191016131300

Thank you for the reference. But as far as I understand, the FGH is not referring to the fundamental sequence in Veblen hierarchy. Namely, the original statement is just "TREE(n) is bounded by f_φ(ω@ω)(n) if the fundamental sequences are directly given by the structural induction on the computation algorithm of TREE". Then it is not an effective upperbound because the fundamental sequence directly uses TREE. Say, f_ω(n) with respect to the fundamental sequence ω[n] = TREE(n) trivially surpasses TREE. The statement is doing essentially the same for φ(ω@ω) if I am correct.

Or do their arguments imply that the estimation is also valid for the canonical fundamental sequences, i.e. those in Veblen hierarchy, which are irrelevant to TREE? Otherwise, f_φ(ω@ω) with respect to the fundamental sequences in Veblen hierarchy is not a rough upperbound, if you do not allow f_ω with respect to the usual fundamental sequence to be a rough upperbound of TREE(n).