User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35470197-20191029224813/@comment-35470197-20191030221526

> do we have any reason to believe that the proof length for the well-foundedness of PTO(T)[n] is - indeed - significantly smaller than E(n)?

No, at least I do not. I rather guess that the proof length fuction can be comparable to E in general, as I wrote "L(n) is perhaps googologically approximated by E(n) itself" in my first commment above. My guess is based on the fact that PTO is not necessarily given as the first fixed point of some arithmetic function such as ω^x. (The proof of the relation of the well-foundedness of ω↑↑n and that of ω↑↑(n+1) in NT in Pohlers is directly given by the fact ω^{ω↑↑n} = ω↑↑(n+1).) In general, we have no way to relate PTO(T)[n] and PTO(T)[n+1} in such a simple way. For example, remember Π_11-CA_0.