Rank-Wise Approximations of Hyper-Operations

[JmsNxn]

If you used similar uniqueness criterion to the one I proposed for tetration‚ but for higher hyper-operations than for tetration‚ when in the complex plane would a circulated to the infinity converge?

I'd wager a conjecture is in order.

There exists a domain \(a \in \mathcal{D}\) such that:

\[
a \uparrow^\infty z\\
\]

is finite. I'd propose that solely (under any iteration protocol) this domain is \(\mathcal{D} \supseteq \mathcal{S}\) for the Shell-thron region \(\mathcal{S}\). This would be based on the fact that I know \(1 \le \alpha \le \eta\) converges for \(\Re(z) > 0\).

As we sort of move and iterate more exotic areas of \(a \in \mathcal{S}\), the domain \(\Re(z) > 0\) moves in some manner. I do not know how. But, bounded iterations ellicit bounded iterations. So since bounded iterations of the exponential exist in the shell-thron region, and these iterations ellicit bounded iterations themself. We can expect that the Shell thron region \(a \in \mathcal{S}\) always has a value \(z\) such that:

\[
a \uparrow^\infty z = a\\
\]

This is all I'm willing to talk on the matter. When the simple answer is I don't know, and I don't think anyone knows. I think this would be worthy of the greats if you could answer this question entirely.