User blog comment:Deedlit11/A rigorous definition for pentational arrays/@comment-5150073-20130315231954/@comment-5529393-20130321012348

Unfortunately, the "nice" rules don't always work. The most important rule is that the "fundamental sequences" of a structure must go all the way up to the original structure. Due to the peculiarities of our structures, the largest structures less than X^^(f(X)+1) are structures of the form (X^^(f(X))^n, even though X^^(f(X)+1) is not equal to (X^^(f(X))^X.

Unfortunately, 3^(X^^(f(X)-1)) doesn't work because we don't have structures of the form 3^f(X) in our system - adding arbitrary exponentiation would add huge complexity to our system, and I don't want to deal with that. So we have X^^(f(X)) = X^(X^^(f(X)-1)), but it's not clear how to use that to define a fundamental sequence. I think we have to go with (X^^(f(X)-1))^n.