So, to begin, we have the asymptotic solutions,

Where,

. In which,

Now, we begin by creating a sequence,

In which,

uniformly for

when

. In which, this function will satisfy the equation,

These tetrations will have a plethora of singularities; making a grid of some shape of essential singularities/branchcuts/zeroes. What we want to do from here, is map away from these singularities. The bad points are,

So if we "move" lambda while we move s, we can effectively dodge the singularities. This I did with

. So we take the function,

Which is holomorphic for

. This function still satisfies the asymptotic relation,

So we can expect it to look closer and closer like tetration. We insert an error term similarly (this time its more difficult to do), but we get a,

And now we pull back with logarithms; which can be done without causing singularities.

Now, as to your statements with theta mappings, I'm going to point out William Paulsen and Samuel Cowgill's theorem:

I am simply removing the condition that,

It is absolutely necessary that you have this condition in Kneser's uniqueness--as per following William Paulsen and Samuel Cowgill. I think it would be quite remarkable if we could remove this condition and still have Kneser--I am not convinced though. Especially considering that their proof of this uses a similar theta mapping argument.

If we look at

I am not very sure what this will look like. It looks like a good approach though to sus out if there are singularities. I'm personally of the feeling that this will produce branch cuts. or would at least have singularities. This I would feel happen because,

Where this infinity is interpreted as "non-normality", or divergence. And because of this, we will get arbitrarily close to every fixed point of exp, which will cause slog_K to hit infinity infinitely often. So I imagine there are a plethora of singularities/ branch cuts--and then we cannot apply your argument of getting arbitrarily close to singularities at the naturals (at least as far as I can see--we won't have entire function theory at our disposal).

This is to say that, in the words of Samuel Cowgill and William Paulsen:

If it did, we'd be back in their case--and their uniqueness condition would say it must be kneser. Instead, we have that,

Now, the simple answer to when we have a singularity... we're limiting towards

where

a fixed point/cyclic point of exp where

. This would return to your discussion of

being the ONLY real super logarithm upto a theta mapping. Here is where I'm saying that may not be true. Specifically because of how William Paulsen and Samuel Cowgill made their paper.

The other way I'd approach describing this, is more topological in nature. First of all, going back to our periodic tetrations, we have the following functions. I'll refer to these as

for

and,

There will be essential singularities/branchcuts/zeroes all along the lines

. Each of these functions look something like this; but stretched wider and wider with smaller and smaller

:

[

attachment=1583]

Then, the goal of the beta method, is just to stretch this cylinder to infinity. I cannot see any reason this would be impossible.

As to your comments on iterated exponentials not having a well placed structure. I point towards the papers:

M. Lyubich. (1987). The measurable dynamics of the exponential map. J.

Math. 28, 111-127.

And

M. Rees. (1986). The exponential map is not recurrent. Math. Zeit. 191,

593-598.

These papers show first of all that, for almost all

(upto a set of measure zero in the lebesgue measure), the orbits

get arbitrarily close to the orbit

. The missing points are the periodic points and cycles. And also that the julia set of the exponential is the entire complex plane.

Now, the importance of this with beta, is that we can use this, not to show that tetration grows towards infinity, but that

does. However, slowly, however many dips to (almost) zero; which is because there are no cycles in the beta function. This allows us to say (through quite a lot of work) that,

is a contraction mapping on the beta function. This will ensure convergence on a sector including the real line.

From here, we enter the problem you are talking about; that we are not guaranteed that pulling back with logarithms won't arise in a singularity. To which I had a sketch of a theorem that I posted here

https://math.eretrandre.org/tetrationfor...p?tid=1348 --and also in my paper where I explained it a tad quicker.

Very happy to have you back, Sheldon! I apologize if the post is pretty long--I wanted to cover all the bases, and the things you have missed. There's still a large amount of things to explain and figure out about this function. Your criticisms and commentaries always help though! Thanks again for showing an interest.

Regards, James