x↑↑x = -1

[jaydfox]

SO we do not have uniqueness ??

Or are there branch issues again ?

And how weird would it be to have a proof of non-uniqueness without a proof of existance.

So for complex base tetration, uniqueness would stem from the analytic continuation from Kneser's solution, which would also answer Tommy's question about uniqueness, since Henryk Trapmann has a published proof for the uniqueness of slog for Kneser's solution, but there are multiple branches in the pseudo period of tetration, and you can get these equations, but they should all be equivalent, since real Tetration is unique.

Real tetration is not unique since we also have a 1periodic wave.

So the question of uniqueness is imho not resolved.

regards

tommy1729

For a real base greater than eta, the Kneser solution is unique. Henryk established a condition somewhere, roughly equivalent to Kneser's solution, that the tetration function should be bounded as the imaginary part goes to +/- infinity. This is equivalent to solving for an slog that asymptotically goes to a logarithm at the primary fixed points (which, coincidentally, is how I accelerate convergence of the intuitive/matrix solution). So there is a simple uniqueness criterion, for real bases greater than eta.

But I'm not convinced that the Kneser solution works for complex bases. I can't imagine how to construct it, because I can't help but imagine a discontinuous first derivative at the endpoints of our initial section of the real line. (For reference, I've constructed the Kneser solution explicitly for base e, so I know how to imagine it for a real base greater than eta.)