Tetration and p-adic

[marcokrt]

I've definitely lost the plot so far. I'm pretty shit at p-adic stuff. But \(V(0) = -\infty\) seems like a non issue. This is absolutely standard. The same way every valuation satisfies this--this is how we make p-adic norms. Even adding in the fact that \(^N 0 = (N \pmod{2})\).  The reason this happens is because you have assumed that \(0^1 = 0\) and \(0^0 = 1\). And we have a strange oscillation. This is still a singular behaviour though, which is better written with \(V(0) = - \infty\)...

I tend to lean towards Ember's analysis here. As there's room for error about \(V(0)\)... and \(V(0) = -\infty\) seems perfectly natural. Especially because it appears to be acting as a valuation. The same way the degree of a constant polynomial \(\deg( C) = 0\) and the degree of the zero polynomial is \(\deg(0) = -\infty\)... It may seem wrong at first glance, but it describes the algebra perfectly...

I'm out of my depth here though, so correct me if I'm being an idiot! But I see it as being much more natural to just set \(V(0) = -\infty\)....

Regards, James

This is not a real issue, indeed. Let me go back to the origin of the problem for a while and you will clearly see the big picture.
My original goal (and the goal of my trilogy of papers on NNTDM) was to find, in radix-10, the number of new stable digits (i.e., "frozen digits") of any integer tetration \( ^N a \), such that \( a \not\equiv 0 \pmod{10} \), for a unitary increment of \( N \).
So, for this purpose, I simply disregarded the solutions \( \alpha_{00} = 0 = \dots 00000 \) and \( \alpha_{01} = 1 = \dots 000001 \), since we usually cannot see any arbitrarily large number of zeros before \( N \), but we generally agree that \( \dots 000000N = N \), otherwise we would have overcomplicated a solved problem for the two mentioned trivial cases, while the congruence speed is not clearly constant for any \( N \equiv 0 \pmod{10} : N \neq 0 \). It is just a matter of how we decide to define V(N) at the beginning, and this follows from what we are going to study... but after we have finally reached the end of the third paper, we can agree that the next goal may be to focus ourselves on the p-adics as an ending point, rather than as a tool to solve the original problem of the number of stable digits in the integers \( \pmod {10^N} \), so we have to just remove Def. 1.3 from https://arxiv.org/pdf/2210.07956.pdf and set all the \( 15 \) solutions of \( y^5=y \) as a gold standard... it follows that \( V(0) \) is the p-adic order of \( 0 \), which is equal to \( \infty \) (while \( V(1)=0 \) still holds as it has previously been defined).
Now, working backwards from this point, we can also agree that the number of digits that \( ^N 1 \) freezes going from \( N \) to \( N+1 \) is as big as we wish, since \( 1=01=001=0001=\dots \) and also \( 1^1=01=001=0001=\dots \) as \( 1^{(1^1)}=01=001=0001=\dots \), and so forth.