First off, the graph I give is not regular, it is the Cauchy-integral tetrational (which *may* be equivalent to Kneser's tetrational and the intuitive Abel-matrix one.). This tetrational is real at the real axis and decays to the conjugate fixed points toward \( \pm i \infty \). It is, however, asymptotic to regular tetrationals in the imaginary direction (but two different ones, for the two smallest-magnitude conjugate fixed points). Though your graph from the regular looks sort of like mine. (I presume you used the positive-imag-part conjugate fixed point, which would be roughly like the upper half of the Cauchy-integral tetrational.)
The "change of branch" I had been talking about at \( h \approx 1.492 \) is not of \( \log \) but \( \mathrm{slog} \). It refers to the point at which in order to advance along the path by \( \delta \), i.e. to compute \( \exp^{\delta}(\exp^{\alpha}(z)) \) and get \( \exp^{\delta + \alpha}(z) \), you need to use a different branch of \( \mathrm{slog} \) in \( \exp^u(z) = \mathrm{tet}(u + \mathrm{slog}(z)) \), which corresponds to using a different branch of \( \exp^\delta \).
That there exist paths which self-cross is unavoidable, since the function cannot be injective and I believe must take every complex value (except, perhaps, 0) infinitely many times at infinitely many places. This non-injectivity is the source of the "paradox" you originally mentioned. For the regular this is easy to prove since it is entire and transcendental, then just use Picard's great theorem. Not sure how to prove it for the non-regular Cauchy-tetrational, but the functional equation \( F(z+1) = \exp(F(z)) \) may be useful (\( \exp \) is not injective).
EDIT: just saw your second post -- seems you already realized the bit about the functional eq. and how the non-injectivity of \( \exp \) contributes.
The "change of branch" I had been talking about at \( h \approx 1.492 \) is not of \( \log \) but \( \mathrm{slog} \). It refers to the point at which in order to advance along the path by \( \delta \), i.e. to compute \( \exp^{\delta}(\exp^{\alpha}(z)) \) and get \( \exp^{\delta + \alpha}(z) \), you need to use a different branch of \( \mathrm{slog} \) in \( \exp^u(z) = \mathrm{tet}(u + \mathrm{slog}(z)) \), which corresponds to using a different branch of \( \exp^\delta \).
That there exist paths which self-cross is unavoidable, since the function cannot be injective and I believe must take every complex value (except, perhaps, 0) infinitely many times at infinitely many places. This non-injectivity is the source of the "paradox" you originally mentioned. For the regular this is easy to prove since it is entire and transcendental, then just use Picard's great theorem. Not sure how to prove it for the non-regular Cauchy-tetrational, but the functional equation \( F(z+1) = \exp(F(z)) \) may be useful (\( \exp \) is not injective).
EDIT: just saw your second post -- seems you already realized the bit about the functional eq. and how the non-injectivity of \( \exp \) contributes.
