User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35470197-20191029224813/@comment-35470197-20191031092611

> Do we have any reason to assume that L(n) doesn't grow much faster than E(n) in a typical case?

No, because we do not have any evidence that E(n) is not broken even if we consider typical cases, i.e. the cases where we describe the PTO by an OCF. At least, as I explained, L(n) for the case where T is PA or NT is perhaps effectively estimated by experts, because Gentzen's proof and Pohlers' arguments are also doing them, and I guess that L(n) does not grow much faster than E(n) in this case.

I recall that we (not experts proof-theorists, but googologists) just believe that E(n) does not break for a typical case, and that X(n) and L(n) do not much stronger than it. Since we do not have any natural relation between the typical fundamental sequences and proof-theory, it seems really difficult in the current community in my opinion.