User blog comment:Deedlit11/A rigorous definition for pentational arrays/@comment-5150073-20130315231954/@comment-5150073-20130320184407

Wait, I found some issues: after expansion 3^^^3 & 3 to P_3(X^^f(X)) you get P_3(X^^f(X-1)^3), but tetrational rule tells us that a^^b = a^(a^^(b-1)), not (a^^(b-1))^a. So I believe that it should be expanded as P_3(3^(X^^f(X-1))). That's not equal since exponentiation isn't commutative.