From X^^X to X^^(X+1)... X^^X * X = X^(X^^(-1+X)+1)
Explanation. The tower power of X X's doesn't expand as in ordinals. The first X is only placeholder, and there is only X-1 (formally!) X's in the power tower above first X.
X^^X * X^X = X^(X^^(-1+X)+X)
X^^X * X^X^X = X^(X^^(-1+X)+X^X)
X^^X * X^^X = X^(X^^(-1+X)*2)
X^(X^^(-1+X)*X) = X^X^(X^^(-2+X)+1)
...
the 1 at climbing joins to 1 on the top, we have (X^)^X (X^^(-X+X)+1) = (X^)^X) (1+1) = (X^)^X 2.
Formalization:
The prime block is formed from cutting all blocks to p size. (to be changed)
To define prime block of X^(X^^(-1+X)+1), we must go to the top and collect all + and *. We only have +1 in the second level. After this we must define some sequence of actions which add 1 to the top.
From X^X to X^(X+1):
X^(X+1) [p] = X^X * p
From X^^X to X^(X^^(-1+X)+1):
X^(X^^(-1+X)+1)) [p] = X^^X * p
From X^X to X^(X2):
X^(X2) [p] = X^(X+p)
From X^^X to X^(X^^(-1+X)*2):
X^(X^^(-1+X)*2) = X^(X^^(-1+X)+X^^(-1+p))
From X^X^X to X^X^(X+1):
X^X^(X+1) [p] = X^(X^X * p)
From X^^X to X^X^(X^^(-2+X)+1):
X^X^(X^^(-2+X)+1) [p] = X^(X^^(-1+X) * p))
From X^X^X to X^X^(X2):
X^X^(X2) [p] = X^X^(X+p)
From X^^X to X^X^(X^^(-2+X)*2):
X^X^(X^^(-2+X)*2) [p] = X^(X^(X^^(-2+X)+X^^(-2+p)))
and so on. (to be changed, add recursive rules)