Grand Unity Conjecture

[bo198214]

Is it already proven that holomorphic tetration is unique?

You mean holomorphic = holomorphic on \( \mathbb{C}\setminus (-\infty,-2] \)?

There is even an entire tetration (regular sexp at a complex fixed point) which though is not real on the real axis.

Dmitrii's approach via lemma about almost identical functions (also somewhere on the forum) has gaps yet.

[Base-Acid Tetration]

There is even an entire tetration (regular sexp at a complex fixed point) which though is not real on the real axis.

does the entire tetration cheat by using different branches of logarithm? like, for the sake of argument, tet(0)=1, but tet(-1)= \( 2k \pi i \) for some integer k

whatevs.
why don't we adopt the entire tetration?

To answer my own question:
It just feels so perverse having nonreal values at INTEGERs but, again, whatevs.

[bo198214]

does the entire tetration cheat by using different branches of logarithm? like, for the sake of argument, tet(0)=1, but tet(-1)= \( 2k \pi i \) for some integer k

No, haha, it doesnt cheat. An entire function takes on every value infinitely often. Particularly the value 0, say sexp0(z_0)=0. Then you can just shift the function letting sexp(z)=sexp0(z0+1+z), then sexp(-1)=0. And by the equation sexp(z+1)=exp(sexp(z)) you get real values on all integers > -2.

why don't we adopt the entire tetration?

Actually Kneser constructed from that entire tetration a real analytic tetration (for details see my post with Kneser in the title).

[Base-Acid Tetration]

No, haha, it doesnt cheat. An entire function takes on every value infinitely often. Particularly the value 0, say sexp0(z_0)=0. Then you can just shift the function letting sexp(z)=sexp0(z0+1+z), then sexp(-1)=0. And by the equation sexp(z+1)=exp(sexp(z)) you get real values on all integers > -2.

i used the word entire to destinguish that tetration from the real ordinary tetration.
i thought entire meant holomorphic everywhere. if the entire tetration is with real values at integers, it can't be differentiable at z=-2

[bo198214]

if the entire tetration is with real values at integers, it can't be differentiable at z=-2

Oh, you are right with that. My mistake. An entire takes on every value at most *except* one, and this seems to be 0 here.

Yes then you are right, you can choose it such that sexp(0)=1, but it then would not satisfy sexp(-1)=0, but that sexp(-1)=2*pi*k for some nonzero k.