The problem is that on the complex plane, the continuum iteration is multivalued. You have to remember that, say, \( \exp^{1/2}(z) \) is a multivalued "function" in the same sense that \( \log(z) = \exp^{-1}(z) \). The equation \( \exp^u(\exp^v(z)) = \exp^{u+v}(z) \) only holds for general real (or complex) \( u \), \( v \), and complex \( z \) if one chooses the correct branches of the functions involved for the given set of parameters. Failure of this identity to hold (which you assume when iterating "further" from a value obtained by one continuum iteration) is no different than the fact the equation \( \log(\exp(z)) = z \) does not hold for all \( z \).
Let's examine this iteration process more closely. Note that \( \exp^u(z) = \mathrm{tet}(u + \mathrm{slog}(z)) \), and both \( \mathrm{tet} \) and \( \mathrm{slog} \) are multivalued functions. Consider taking the iteration of that value at \( 1 + \mu \). We have \( \exp^{1 + \mu}(x_0) = \mathrm{tet}(1 + \mu + \mathrm{slog}(x_0)) \). Now consider iterating that a little more, say, \( \delta \). We would want to do this:
\( \exp^{\delta}(\exp^{1 + \mu}(x_0)) = \exp^{\delta + 1 + \mu}(x_0) \),
which, in detail, means we'd like to try doing this:
\( \begin{align}
\exp^{\delta}(\exp^{1 + \mu}(x_0)) &= \mathrm{tet}(\delta + \mathrm{slog}(\mathrm{tet}(1 + \mu + \mathrm{slog}(x_0)))) \\
&= \mathrm{tet}(\delta + 1 + \mu + \mathrm{slog}(x_0)) \\
&= \exp^{\delta + 1 + \mu}(x_0)\\
\end{align} \)
But if we mull over these steps, we see that the second equality cannot be justified. We cannot necessarily say that \( \mathrm{slog}(\mathrm{tet}(z)) = z \) for a general complex \( z \) any more than we can say \( \log(\exp(z)) = z \) for a general complex \( z \). It's all because of the ambiguity of the multivalued functions involved. Even the single-valued principal branch of \( \mathrm{tet} \) is not an injective function on the complex \( z \)-plane, as can be seen by inspecting its graph.
Indeed, this paradox shows that no matter how we may try to extend \( \mathrm{tet} \) to the \( z \)-plane, we cannot make it injective, at least if our \( \mathrm{tet} \) is continuous (up to a cut, anyway).
Let's examine this iteration process more closely. Note that \( \exp^u(z) = \mathrm{tet}(u + \mathrm{slog}(z)) \), and both \( \mathrm{tet} \) and \( \mathrm{slog} \) are multivalued functions. Consider taking the iteration of that value at \( 1 + \mu \). We have \( \exp^{1 + \mu}(x_0) = \mathrm{tet}(1 + \mu + \mathrm{slog}(x_0)) \). Now consider iterating that a little more, say, \( \delta \). We would want to do this:
\( \exp^{\delta}(\exp^{1 + \mu}(x_0)) = \exp^{\delta + 1 + \mu}(x_0) \),
which, in detail, means we'd like to try doing this:
\( \begin{align}
\exp^{\delta}(\exp^{1 + \mu}(x_0)) &= \mathrm{tet}(\delta + \mathrm{slog}(\mathrm{tet}(1 + \mu + \mathrm{slog}(x_0)))) \\
&= \mathrm{tet}(\delta + 1 + \mu + \mathrm{slog}(x_0)) \\
&= \exp^{\delta + 1 + \mu}(x_0)\\
\end{align} \)
But if we mull over these steps, we see that the second equality cannot be justified. We cannot necessarily say that \( \mathrm{slog}(\mathrm{tet}(z)) = z \) for a general complex \( z \) any more than we can say \( \log(\exp(z)) = z \) for a general complex \( z \). It's all because of the ambiguity of the multivalued functions involved. Even the single-valued principal branch of \( \mathrm{tet} \) is not an injective function on the complex \( z \)-plane, as can be seen by inspecting its graph.
Indeed, this paradox shows that no matter how we may try to extend \( \mathrm{tet} \) to the \( z \)-plane, we cannot make it injective, at least if our \( \mathrm{tet} \) is continuous (up to a cut, anyway).
