User blog comment:Deedlit11/A rigorous definition for pentational arrays/@comment-5150073-20130315231954/@comment-5529393-20130321220809

Actually, X^^X reduces to X^^p so 3^^3 @ 3 will reduce to 3^(3^3) @ 3. (I still hate that notation.) It's X^^(f(X) + 1) that reduces to (X^^f(X))^p.

So my notation isn't really up-arrow or down-arrow. The important thing is that the strength of the notation is phi(2, 0), which is as strong as we can hope for.