WOW! This is even easier than I expected. These values line up really really well!
If I take:
\[
2 \oplus_{0.1, \sqrt{2}} 4 = 6.1242048680713775690019097380798646675079851431412...\\
\]
It's already correct to two digits of
\[
2 \oplus_{1.1,\sqrt[3]{3}} 3 = 6.1408969266137790131714605868589172539150029963802...
\]
So in this instance \(\varphi_{1},\varphi_2\), are going to be very very small. This tends to hold for all these values. It's easy to work with this equation because we can assume that \(2 <s> 2 = 4\) (this will definitely screw up for \(s \approx -1\); but I'm already assuming we will have branching problems on the negative real axis). So we get a solution of values \(\varphi_1,\varphi_2\), which solve this equation exactly. The trouble is rectifying it to a single function as we move the other variables.
\[
2<s>_{\varphi_1} 4 = 2<s+1>_{\varphi_2} 3
\]
The trouble now is figuring out how to properly construct a function:
\[
\varphi(\alpha,y,s)\\
\]
such that:
\[
\begin{align}
\varphi(2,4,s) &= \varphi_1\\
\varphi(2,3,s+1) &= \varphi_2\\
\end{align}
\]
That will be holomorphic in all three variables.
----okay..., I think I can do this for this scenario only. Everywhere else it will fuck up. But since this is a two variable equation, I think I know how. This would solve:
\[
2+u <s> 4 + v = 2 + u <s+1> 3+w\\
\]
for \(u,v,w \approx 0\).
And at least do so locally. God I need to go to bed -_-......
Here is a graph of
\[
\begin{align}
f(s) &= 2 \oplus_{s, \sqrt{2}} 4\\
g(s) &= 2 \oplus_{1+s,\sqrt[3]{3}} 3\\
\end{align}
\]
For \(0 \le s \le 1\). We're essentially trying to zipper this solution together. We're trying to close this small gap.
If I take:
\[
2 \oplus_{0.1, \sqrt{2}} 4 = 6.1242048680713775690019097380798646675079851431412...\\
\]
It's already correct to two digits of
\[
2 \oplus_{1.1,\sqrt[3]{3}} 3 = 6.1408969266137790131714605868589172539150029963802...
\]
So in this instance \(\varphi_{1},\varphi_2\), are going to be very very small. This tends to hold for all these values. It's easy to work with this equation because we can assume that \(2 <s> 2 = 4\) (this will definitely screw up for \(s \approx -1\); but I'm already assuming we will have branching problems on the negative real axis). So we get a solution of values \(\varphi_1,\varphi_2\), which solve this equation exactly. The trouble is rectifying it to a single function as we move the other variables.
\[
2<s>_{\varphi_1} 4 = 2<s+1>_{\varphi_2} 3
\]
The trouble now is figuring out how to properly construct a function:
\[
\varphi(\alpha,y,s)\\
\]
such that:
\[
\begin{align}
\varphi(2,4,s) &= \varphi_1\\
\varphi(2,3,s+1) &= \varphi_2\\
\end{align}
\]
That will be holomorphic in all three variables.
----okay..., I think I can do this for this scenario only. Everywhere else it will fuck up. But since this is a two variable equation, I think I know how. This would solve:
\[
2+u <s> 4 + v = 2 + u <s+1> 3+w\\
\]
for \(u,v,w \approx 0\).
And at least do so locally. God I need to go to bed -_-......
Here is a graph of
\[
\begin{align}
f(s) &= 2 \oplus_{s, \sqrt{2}} 4\\
g(s) &= 2 \oplus_{1+s,\sqrt[3]{3}} 3\\
\end{align}
\]
For \(0 \le s \le 1\). We're essentially trying to zipper this solution together. We're trying to close this small gap.