= Solution
There is exactly one non-isolated type:
$$
p_\infty(x)=\{n<x:n\in\mathbb N\}.
$$
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.
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