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Removing the branch points in the base: a uniqueness condition?
In many cases, when dealing with the math behind tetration, a recurring feature is the logarithm of the fixed point multiplier , which I will call from here on
Since the fixed point multiplier is determined by the base, is really just the base in disguise:

But all three functions have branch points that correspond to the ones in tetration's base: the inner log to 0, the productlog to , the outer log to 1.
Thus, I think that it's reasonable to desire the following to be the case for any reasonable tetration:
Let x > 0 and tet(x,b) be our tetration solution.
Then analytically continues to a function without branch points in
So in other words, the branch we're on should entirely depend on what branches of those three functions we pick.

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