x↑↑x = -1

[tommy1729]

I assume the theta singularities come from the D(z).
I see no other reason.

A function A with singularity at 0 and a function B with singularity at 0 implies :

A(B(z)) and B(A(z)) have a singularity at 0.

Hence singularities do not cancel under composition with other singularities or analytic functions.


Multiple complex base tetrations ??

\( \alpha_1(\text{tet}_b(z)) = z + \theta_1(z) \)

\( \alpha_2(\text{tet}_b(z)) = z + \theta_2(z) \)

Now I had the idea

\( \alpha_1(\text{tet}_b(z + \theta_3)) = z + \theta_3(z) + \theta_1(z + \theta_3(z)) \)

\( \alpha_2(\text{tet}_b(z + \theta_3)) = z + \theta_3(z) + \theta_2(z + \theta_3(z)) \)

And that becomes :

\( \alpha_1(\text{tet}_b(z + \theta_3)) = z + \theta_3(z) + \theta_4(z) \)

\( \alpha_2(\text{tet}_b(z + \theta_3)) = z + \theta_3(z) + \theta_5(z) \)

Further simplify :

\( \alpha_1(\text{tet}_b(z + \theta_3)) = z + \theta_6(z) \)

\( \alpha_2(\text{tet}_b(z + \theta_3)) = z + \theta_7(z) \)

Notice \( \text{tet}_b(z + \theta_3)) = \text{tet_2}_b(z) \) , in other words another tetration function.

SO we do not have uniqueness ??

Or are there branch issues again ?

And how weird would it be to have a proof of non-uniqueness without a proof of existance.

These equations are weird man !

In fact Im having doubts about the carleman matrices approach due to issues such as singularities and possible non-uniqueness.

Still thinking ...


regards

tommy1729