Pentation's definitional ambiguity

[andydude]

what are the

branches above and below everywhere else

?

For example, suppose we know that \( {}^{b}a = c \), then we also know that \( {}^{b+1}a = a^c = a^{c + 2 \pi i k/\ln(a)} \), so we can work backwards and say that for every k, there exists a branch such that \( {}^{b}a = c + \frac{2 \pi i k}{\ln(a)} \) is this the same as the indexing scheme above?