Rank-Wise Approximations of Hyper-Operations

[JmsNxn]

Could similar uniqueness criterion work for pentation and higher hyper-operations?

They work for the iteration of any function, so yes.

So how do pentation and higher hyper-operations behave?

Let's start with pentation, and consider \(\sqrt{2}\). For, \(\sqrt{2}\uparrow^3 z\) we have a fixed point \(\omega_2\) such that:

\[
\sqrt{2}\uparrow^3 \infty = \omega_2
\]

This is \(\infty\) in the sense of the closing point of \(\Re(z) > 0\). If we choose tetration as the Schroder iteration about \(2\):

\[
\sqrt{2} \uparrow \uparrow \omega_2 = \omega_2\\
\]

While additionally, this is a geometrically attracting fixed point. Therefore we just rinse and repeat the construction of tetration.

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If we talk about the repelling fixed points (discussions of \(4\), and the subsequent iterations)--it gets very chaotic. But the answer to your question is that it continues to hold.