The "cheta" function

[jaydfox]

Can you refresh my memory why change-of-base only makes sense to bases \( >\eta \)?

Well, it can actually be applied to any base greater than 1. I assume you're referring to where I said:

And at any rate, for base b>eta, it definitely does not converge on the way "up", even if by some miracle it manages to converge on the way back "down"

For bases less than or equal to eta, it will converge on the lower fixed point as we go "up". For bases larger than eta, there aren't any real fixed points, so convergence never happens.

Also, my change of base formula works from infinity down, so to speak, so for bases less than or equal to eta, we can't properly get to 0, so it doesn't give tetration per se (though it is a form of super-exponentiation, just not centered at 0).