Uniqueness of tetration. Small lemma.

[Kouznetsov]

..
In the tetration curve, the imaginary value for f(z) at the real axis is equal to zero for all z>=-2. For the solution based on the fixed point, is the imaginary value of f(z) nonzero everywhere outside of the real axis?
..

What does it mean, "based on the fixed point"? Do you mean the case of other base?
For base \( 1<b<\exp(1/e) \), there are two real fixed points.
We may use any of them to build-up the "real" super-exponential.
Both resulting superexponentials have imaginary periods.
If such a superexponential has zero somewhere (for example, at minus unity),
then it has countable set of zeros.

..
Otherwise it would seem likely that there would be other singularities, outside of the real axis.

Yes, at \( 1<b<\exp(1/e) \), the superexponential F such that F(0)=1 is periodic; hence, just translate the singularity at -2 for the period, and you get the singularity outside the real axis. See the picture for b=sqrt(2) at
http://en.citizendium.org/wiki/Image:Hol...rt2v01.jpg
It shows one additional singularity and the corresponding cut line.