tetration base conversion, and sexp/slog limit equations

[bo198214]

\( \lim_{n\to\infty}\log_b^{\circ n}({\exp_a}^{\circ n}(y)) =\text{sexp}_b(\text{slog}(y)-c_b) \)

I was able to follow up to this point, but shouldn't it be this, where slog(y) substitutes for x?
\( {\exp_a}^{\circ n}(\text{slog}y)))= \)

More detailed: we have
\( \lim_{n\to\infty}\log_b^{\circ n}(\text{sexp}_a(x+n))=\text{sexp}_b(x-c_{a,b}) \)
this translates into
\( \lim_{n\to\infty}\log_b^{\circ n}(\exp_a^{\circ n}(\text{sexp}_a(x)))=\text{sexp}_b(x-c_{a,b}) \)
then I substitute \( y=\text{sexp}_a(x) \), \( x=\text{slog}_a(y) \).

\( \kappa_{b,a}(y) = \lim_{n\to\infty} {\log_b}^{\circ n}({\exp_a}^{\circ n}(y)) \), \( \kappa_{a,b}={\kappa_{b,a}}^{-1} \)

sadly, I didn't understand this step.

Thats just the definition of an ancillary function. You can verify the property \( \kappa_{a,b}^{-1} = \kappa_{b,a} \). The limit has to exist if our paradigm works.

I introduced this function because now we can separate the limits (which went into \( \kappa \)) from the sexps/slogs.

.....
So we need to show in your case that the limit:
\( \lim_{b\to\eta^+}\text{slog}_b(\kappa_{b,a}(x)) \) exists for \( x \) in some initial range, where \( \text{slog}_b \) is your linear approximation.

To show this we could use the Cauchy criterion....

How is this different from:
\( \lim_{b\to\eta^+}\text{slog}_b(\text{sexp}_a(x)) \) exists ...
Is it because we don't know if sexp is an analytic function, but there are better reasons to believe that \( \kappa_{b,a} \) is an analytic function?

Apart from you omitted the \( -x \) the difference is that we have no \( \text{sexp}_a \) at that moment. All we have are linear (or higher order) approximations of an \( \text{slog}_b \). But what we have is \( \kappa \).

\( \kappa_{a,b} \) is a function which is bigger then \( x\mapsto x \) exactly if \( a<b \). In this case it decreases extremely slowly towards \( -\infty \) (so slowly that one could think its converging to a constant) and extremely fast towards \( +\infty \). The more \( a \) towards \( \eta \) the bigger is \( \kappa_{a,b} \) (and infinity for \( a=\eta \)).
Thoug I am not sure how to prove analyticity, \( \kappa \) has straight algebraic properties, like

\( {\kappa_{a,b}}^{-1} = \kappa_{b,a} \)
\( {\kappa_{a,b}\circ \kappa_{b,c} = \kappa_{a,c} \)