Change of base formula for Tetration

[bo198214]

To put this into perspective, think about the iterated multiplication formula (you know, exponentation). Let's say that we know that 2^4 equals 4^2 and 2^6 equals 4^3. In fact, let's say that for all integers k, we know that 2^2k = 4^k.

Wouldn't you expect 2^3 to equal 4^1.5? That's essentially the basis for my initial confidence that my change of base formula was correct. I had found that it was correct for all integer tetrations, so why should I have the slightest concern that it wouldn't be correct for fractional tetrations? It would be as absurd as doubting that 2^3 equals 4^1.5.

Though I did not understand everything you wrote, this is a base problem of tetration (imposed by right-bracketing). For most values is:

\( {}^{xy}b\neq {}^x({}^y b) \)

particularely we cannot define the n-th super root by \( {}^{\frac{1}{n}}x \) because \( {}^{\frac{1}{n}}({}^nx)\neq x \).

Barring a really good reason, I'd be perfectly fine saying that Andrew's solution was in error, not mine. And yet the positive/convex nature of the odd derivatives of his solution is so beautiful as to make all my doubts melt away. I cannot fathom that Andrew's solution is wrong.

Attributes like "wrong" or "right" are completely inappropriate here.
In the realm of mathematics we assign a "right" if we can prove it and a "wrong" if we can disprove it.
Of course much research is a pursue of beauty but this is in the eye of the beholder. I would leave it there.

It'd be like saying that \( 4^x=2^{x\log_2(4,x)} \), where log_2(4,x) is no longer a constant, but a function of x that is cyclic though very nearly constant. Absurd!

I mean the interrelation between \( t \)-th superroot and superpowers \( {}^tx \) for real t, was not yet considered on this forum. How much differs \( {}^{\frac{1}{t}}x \) from the \( t \)-th superroot \( (x\mapsto {}^tx)^{-1} \)?

We must find some underlying reason why having all the odd derivatives be convex is a desirable property, besides the fact that there is (almost certainly) only one such solution per base. Uniqueness alone is insufficient, because my solution is unique in its own way and based on "the" unique solution for base eta.

As far as I know we couldnt prove any uniqueness conditions yet.