I read your paper again, and I think I have some more thoughts, but I have more questions. I think I'll formulate a couple of questions and try to explain myself through an air of questioning; and hone the questions better and then ask. First, I thought it warranted to try to talk categorically.

Can we write,

So that is say, a diffeomorphism (I believe that's the word, if not; it's something like that) of . Just so my shallow brain can think of a representative of the category; and it's not all up in the air. Let's additionally assume that:

For some constants . Which will make the exponential convergents behave well. And it would imply it's inverse at worse grows like somethin' somethin'. This would be a perfectly good algebraic space where we could derive,

Now I haven't proven that, not entirely sure how to, but it's manageable--I could probably prove something close enough to continue the discussion.

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With that out of the way, I'm going to keep thinking about this as operations on and functors; but to me they make sense as functors on ; or subgroups, or different versions or whatever. What I mean is, can we think of as an almost IDEAL space. Like the best space possible; where all the algebra is simple. Rather than monsters like we look at simple amoebas like . And build from the bottom up. Because I agree with a lot of what you are saying. But from a categorical perspective, start simple, no?

Unless I'm missing something drastic. You're paper was the most riveting the 3rd time... Maybe I just got over analytical, lmao

Can we write,

So that is say, a diffeomorphism (I believe that's the word, if not; it's something like that) of . Just so my shallow brain can think of a representative of the category; and it's not all up in the air. Let's additionally assume that:

For some constants . Which will make the exponential convergents behave well. And it would imply it's inverse at worse grows like somethin' somethin'. This would be a perfectly good algebraic space where we could derive,

Now I haven't proven that, not entirely sure how to, but it's manageable--I could probably prove something close enough to continue the discussion.

-------------------------

With that out of the way, I'm going to keep thinking about this as operations on and functors; but to me they make sense as functors on ; or subgroups, or different versions or whatever. What I mean is, can we think of as an almost IDEAL space. Like the best space possible; where all the algebra is simple. Rather than monsters like we look at simple amoebas like . And build from the bottom up. Because I agree with a lot of what you are saying. But from a categorical perspective, start simple, no?

Unless I'm missing something drastic. You're paper was the most riveting the 3rd time... Maybe I just got over analytical, lmao