User blog comment:Ikosarakt1/Formal definition of tetrational arrays/@comment-5529393-20130329075222/@comment-5150073-20130330084836

X^(A & b[p]) certainly doesn't work because A & b[p] solves to a single number, so we get X^n \&\ b[p], where n = A & b[p] when solved.

As for the rule A7, when X^A_1 is not in the successor structure, we can take A_1 = X^A_2*n+P, where P is the rest of polynomial (see notes). So for X^(X^3+X^2) we have A_1 = X^3+X^2, n=1 (since X^3*1 = X^3), and P = X^2.