Tetration and p-adic

[Ember Edison]

we have that \( V(0):=0 \), since \(^N 0 = 0 \) iff \( N \) is odd and \(^N 0 = 1 \) otherwise. 

In my personal opinion, this conclusion is sufficient. 

Of course further discussion about the definition of V(0) is also interesting!

So, the best answer to your question above cannot ignore our preliminary assumption based on the fact that \( V(0) \in \mathbb{Z} \) definition, while it would be not even be an element \( \mathbb{R} \) by construction... otherwise the "constant congruence speed analysis" cannot be taken as a way to solve related problems in the p-adics.

And the interesting point is that \( V(0)\not\in\mathbb{Q}_p \not\Leftrightarrow V(0)\not\in\mathbb{R} \not\Leftrightarrow V(0)\not\in\mathbb{C}\not\Leftrightarrow V(0)\not\in\Omega_p \), \( (\mathbb{R},\mathbb{C} ) \) unequal we have seen in the standard Tetration, and the rest of the unequality is a simple known fact.