HUZZAH!
I've managed to show that \(x [s] y\) is indeed analytic at \(y =e\), and the above artifacts are my code failing. When in doubt, return to basics. All I have to do is show that \(x [s] y\) is complex differentiable at \(y = e\). This can be done pretty simply, because \(\frac{d}{dy}y^{1/y} = 0\) when \(y = e\). So the derivative has a bunch of cancellations. By which, a closed form expression for the derivative at \(e\) is given as:
\[
\frac{d}{dy}\Big{|}_{y=e} x[s]y = \frac{d}{du}\Big{|}_{u=e} \exp^{\circ s}_{\eta}\left( \log^{\circ s}_\eta(x) + u\right)\\
\]
EDIT: I should probably prove this claim to be thorough.
Recall:
\[
x[s]y = \exp^{\circ s}_{y^{1/y}}\left(\log^{\circ s}_{y^{1/y}}(x) + y\right)\\
\]
Differentiate \(\exp_{y^{1/y}}^{\circ s}(U)\) first, which looks like zero because \(y^{1/y}\) has a critical point at \(y=e\). Now differentiate through \(U\):
\[
\frac{d}{dU} \exp^{\circ s}_{y^{1/y}}(U) \frac{dU}{dy}\\
\]
The derivative of \(U = \log^{\circ s}_{y^{1/y}}(x) + y\) at \(y=e\) is just \(1\), because we have the same critical point argument.
Therefore:
\[
\frac{d}{dy}\Big{|}_{y=e} x[s]y = \frac{d}{dU}\exp^{\circ s}_{y^{1/y}}(U)\Big{|}_{y=e}\\
\]
This is always a finite value for \(x > e\) and \(0 \le s \le 2\). OH YA!!!
So yes, the above dips are artifacts in the program because they aren't even approaching this limit, which is a must. YES!
So all that's left now, is to make an adaptive algorithm which works as \(y \to e\). I just need to be able to run the standard Ecalle construction of the abel function about a parabolic point for the boundary of the Shell-Thron region, while simultaneously running Schroder iteration about the repelling fixed point for the interior. This is going to be tricky.
You can see the error more clearly here, here is:
\[
\exp^{\circ 0.5}_{y^{1/y}}(3)\\
\]
graphed across \(y \in (2,3)\).
This dip is clearly artificial, because it's not a local maximum, which the math prerequisites it being. Also it's happening in the second decimal point, near \(e\), where Schroder becomes untenable. I'll try to see if I can make this more accurate, but without constructing an Abel function that is analytic as \(y \to e\), which is easier said than done. We might have to suffice with a program that fails near the Shell thron boundary but works everywhere else. Luckily, the math needed to construct \(\varphi\) doesn't depend on being holomorphic near \(e\), so we should still be okay theory/computation wise--just have to avoid when \(|\log(y)| = 1\).
So, as per the theory of this post. If I were to run a much much higher precision/deeper recursive version of the above graph, the blip should be smaller. And voila! I had to fucking eat ram like it was a fat kid with a bag of cheetos, but the blip is a tiny bit smaller here. Use the tickers to weigh the difference:
I've managed to show that \(x [s] y\) is indeed analytic at \(y =e\), and the above artifacts are my code failing. When in doubt, return to basics. All I have to do is show that \(x [s] y\) is complex differentiable at \(y = e\). This can be done pretty simply, because \(\frac{d}{dy}y^{1/y} = 0\) when \(y = e\). So the derivative has a bunch of cancellations. By which, a closed form expression for the derivative at \(e\) is given as:
\[
\frac{d}{dy}\Big{|}_{y=e} x[s]y = \frac{d}{du}\Big{|}_{u=e} \exp^{\circ s}_{\eta}\left( \log^{\circ s}_\eta(x) + u\right)\\
\]
EDIT: I should probably prove this claim to be thorough.
Recall:
\[
x[s]y = \exp^{\circ s}_{y^{1/y}}\left(\log^{\circ s}_{y^{1/y}}(x) + y\right)\\
\]
Differentiate \(\exp_{y^{1/y}}^{\circ s}(U)\) first, which looks like zero because \(y^{1/y}\) has a critical point at \(y=e\). Now differentiate through \(U\):
\[
\frac{d}{dU} \exp^{\circ s}_{y^{1/y}}(U) \frac{dU}{dy}\\
\]
The derivative of \(U = \log^{\circ s}_{y^{1/y}}(x) + y\) at \(y=e\) is just \(1\), because we have the same critical point argument.
Therefore:
\[
\frac{d}{dy}\Big{|}_{y=e} x[s]y = \frac{d}{dU}\exp^{\circ s}_{y^{1/y}}(U)\Big{|}_{y=e}\\
\]
This is always a finite value for \(x > e\) and \(0 \le s \le 2\). OH YA!!!
So yes, the above dips are artifacts in the program because they aren't even approaching this limit, which is a must. YES!
So all that's left now, is to make an adaptive algorithm which works as \(y \to e\). I just need to be able to run the standard Ecalle construction of the abel function about a parabolic point for the boundary of the Shell-Thron region, while simultaneously running Schroder iteration about the repelling fixed point for the interior. This is going to be tricky.
You can see the error more clearly here, here is:
\[
\exp^{\circ 0.5}_{y^{1/y}}(3)\\
\]
graphed across \(y \in (2,3)\).
This dip is clearly artificial, because it's not a local maximum, which the math prerequisites it being. Also it's happening in the second decimal point, near \(e\), where Schroder becomes untenable. I'll try to see if I can make this more accurate, but without constructing an Abel function that is analytic as \(y \to e\), which is easier said than done. We might have to suffice with a program that fails near the Shell thron boundary but works everywhere else. Luckily, the math needed to construct \(\varphi\) doesn't depend on being holomorphic near \(e\), so we should still be okay theory/computation wise--just have to avoid when \(|\log(y)| = 1\).
So, as per the theory of this post. If I were to run a much much higher precision/deeper recursive version of the above graph, the blip should be smaller. And voila! I had to fucking eat ram like it was a fat kid with a bag of cheetos, but the blip is a tiny bit smaller here. Use the tickers to weigh the difference: