Universal uniqueness criterion?

[Kouznetsov]

I asked you som time ago to apply your algorithm to other fixed points, but you somehow did not follow that path.

Assume that for another tetration \( F \), we have some probe function \( E(y)=F(iy) \) for real \( y \); id est, along the imaginary axis.
Assume, Function \( E \) is suposed to go to \( L_m \) at infinity and \( L_m^* \) at minus infinity; \( \log(L_m)=L_m+2\pi i m \), and \( m \) is integer constant.
In order to adjust the probe function to some tetration, we need to evaluate the contour integral. If we use the same contour as in the paper http://www.ams.org/mcom/2009-78-267/S002.../home.html
then we need the values at \( 1+i y \) and \( -1+iy \) for real \( y \).
It is easy to estimate values at \( 1+i y \); use estimate \( \exp(E(y)) \).
As for \( -1+i y \); we use estimate \( \log(E(y))+2\pi i m \) for positive \( y \) and \( \log(E(y))-2\pi i m \) for negative \( y \). Either we have singularity (jump) at \( -1 \), or \( m=0 \).

This explains, why I did not construct such "another tetration", but this is not a proof that this is impossible. Suggest the holomorphic probe function to begin with. Such a function should have some smooth kink of the phase, in order to avoid the jump;
but allow such a jump for the principal branch of its logarithm.
Then we can run the same algorithm, with additional control of the branch of the logarithm. Such a control should recover \( F(-1+iy)=\log(F(iy))+i\pi m \),
adding unity or minus unity to \( m \) each time when \( E(y) \) passes through negative real values. We need the holomorphic kinky probe function, then we can run the algorithm to recover the kinky tetration.
I think about something like \( E(y)=
\Re(L_1) +(1-\Re(L_1))/ \cosh(y) + \Im(L_1) \tanh(y) \)