On extension to "other" iteration roots

[Leo.W]

Actually your expertise and technical arsenal in creating superfunctions is a bit wasted on me.

I am still puzzled how a superfunction can not come out of a iteration (semi)group.

In fact all superfunctions can come out of an iteration group, but the price is that it almost always is multivalued, we mostly speak and talk about a superfunction and automatically take it as singlevalued and avoid talking about branch cuts, but the natural way is to consider the superfunction as a multivalued function, or almost equivalently a riemann surface.
I don't think it's expertise or something or even "wasted", it's very nice to have a simplication as you did.
I pointed out that your \(tet_{e^{-2e}}\) is also generalized by P, only to remind you that it's different from a tetration. It's because
if we have a superfunction \(F(z+1)=f(F(z))\), then F(-z), is superfunction of the inverse of f, that is, \(F(-(z+1))=f^{-1}(F(-z))\), so this time it's not tetration, since it oscillate and eventually converges at 1 limit at negative infinity.

The name of the method is in your hand, you even called it re-construction method once.
I would say to choose a name wisely in the beginning and don't say things like "call it what you want".
We have enough confusion about naming, so that would really improve things to start with good names in the first place.

Sure. I just personally don't have a brain to memorize all different names. I think it's pretty good to call it P method.

James: that the nearest fixed points are the conjugate pairs (which are still there for b=η−+δ, but aren't the "primary fixed point/points"). But once you go outside of the shell thron region, you get (when forcing real valued solutions) two complex conjugate fixed points.

Bo: We really have a problem here. I thought from the picture I gave it was clear that there is no split at eta minor - it is a totally continuous behaviour regarding the fixed points. The inf{|ℑ(z0)||bz0=z0,z0∈C} is 0 because of the fixed point on the real line (in and out of the STR). I really have no clue what split you are referring to or what essentially changes in the fixed points when passing eta minor. Inside and outside we have one real fixed point and many conjugated fixed point pairs.

I think it's my fault I didnt clarify the "2 limits" or "2 fixed points".
The 2 fixed points refers to the 3 real fixed points of \(f(z)=b^{b^z}\), and excluded the one real fixed point of \(f^{1/2}(z)=b^z\). It's exactly the bifurcation in main branch of chaos theory.
When b fits \(0<b<e^{-e}=\eta_-\), the function \(f(z)\) will always have 3 real fixed points, one of them are the real fixed point of \(b^z\). Because the excluded one fixed point is repelling, the tetration of b (to integer heights) won't converge and will oscillate between the other 2 values, or have 2 limits at infinity.
For example, when \(b=0.06\), we have 3 fixed points of \(b^{b^z}\), approx are 0.216898, 0.36158, 0.54323
where 0.36158 is also the fixed point of \(b^z\), this time the tetration to integer heights will oscillate between 0.216898 and 0.54323 like the pic shown below.

By James, it's correct about the splitting choices, when you take the split 2 fixed points and merge their superfunction it's liable to have a real tetration. It's just too hard to compute and make it converge. I'm trying to do this. It makes me feel that we should take a different contour integral as double-dagger method or others do, we should take a broader contour in real-axis-direction.