Simplified regular tetration

[bo198214]

So is X the inverse of the Schroeder function?

Actually I am also somewhat puzzled.
I would suggest to always use variable \( b \) for the base.
In this terminology you wrote:
\( {}^{y}b = X(\ln({}^{\infty}b), \ln({}^{\infty}b)^y) \)

Now \( {^\infty b} \) is the fixed point which I denoted with \( a \).
And \( \ln(a) \) is the derivative at the fixed point which I denoted with \( c \).
So your formula is:
\( {}^{x}b = X(c, c^x) \)
while my formula is:
\( {^xb}=\eta(c^x)+a \)
Your coefficients are polynomials in \( c \), while my coefficients are rational functions in \( c \) and also use \( \ln(b) \). Though both must be equal up to translation. *headscratch*