TPID 4

[sheldonison]

alternate fixed point post I made, #19.
\( \theta(z)=\text{slog}(\text{tet}_{\text{alt}}(z))-z \) is a 1-cyclic analytic function at the real axis, even though such an alternate fixed point tet(z) doesn't have an analytic inverse at integers.... And, as you might expect, theta(z) is not entire since the fixed points are completely different in the complex plane. \( \theta(z) \) is analytic if \( |\Im(z)|\approx<0.38008 \), since tet_alt(0.38008-0.51390)~=0.3181+1.337, which is the fixed point for the Kneser primary tetration function....

Yes, it would be interesting to see a graph.

\( \theta(z) \) is real valued at the real axis, where the slope=-1 at the integers. I generated the initial seed for tet_alt using \( \theta(z)=\frac{-sin(2\pi z)}{2\pi} \), which is approximately what this graph looks like.
   

Contour plot of \( z+\theta(z) \) at \( \Im(z)=0.35i \), \( -1<\Re(z)<1 \). Notice how it conveniently loops around the origin. This is really important because when z+theta(z) loops around -2, you go around a logarithmic branch of \( +2\pi i \), which is necessary to get the secondary fixed point as real(z) goes to minus infinity.
   

Here we have a complex plot of \( \theta(z) \) at \( \Im(z)=0.375i \), \( -1<\Re(z)<1 \), where you can clearly see the influence of the singularity is at z=-0.51389+0.38009i. Red is real, and green is imaginary.
   

I dont think so ; if " tommy's theorem " is correct
...
then analytic tet(z) IMPLIES tet ' (z) > 0.

Does your theorem imply that the secondary fixed point tet_alt(z) function is a numeric fiction? I think it can be made as rigorous as Tetration from the primary fixed point, and like the primary fixed point tet(z) solution, tet_alt(z) is analytic in the upper and lower halves of the complex plane, except for the integers -2,-3,-4.... And at every integer>-2, the 1st and 2nd derivatives are zero, tet_alt'(n)=0 and tet_alt''(n)=0, which contradicts your theorem. At non-integer point for Re(x)>-2, tet_alt'(x)>0.