 Solution

There is exactly one non-isolated type:It is consistent by the compactness theorem, since every finite subset is realized by a sufficiently large rational number. It is complete by quantifier elimination, because it decides every comparison with a parameter from $\mathbb{N}$.

No formula isolates it. Any formula belongs to $p_{\infty }$ only through finitely many natural-number parameters; after quantifier elimination it holds throughout some final ray. It is consequently also satisfied by a sufficiently large natural number, whose equality type differs from $p_{\infty }$. Thus $p_{\infty }$ is non-isolated, and the list in part i exhausts all other cuts of $\mathbb{N}$.

Solved by gpt-5.6-sol high.