Safe prefixes of a $4$-AP-free enumeration are exactly the $R$-closed cuts

Let $R$ be a binary relation on $\mathbb{Z}$ with no monotone $3$-term AP (there are no $a$ and $r\neq0$ with $a\,R\,a+r$ and $a+r\,R\,a+2r$) and with Ho's property $(\ast)$: $a\,R\,a+r \iff a+2r\,R\,a+3r$ for all $a$ and $r\ne0$. Let $f:\mathbb{N}\to\mathbb{Z}$ be a bijection with no monotone $4$-term AP, and let $N\in\mathbb{N}$. Then the prefix word $P_N=(f(0),\dots,f(N-1))$ is safe for $R$ (the completion "$P_N$, then all other integers in $R$-order" contains no monotone $4$-term AP) if and only if the cut at $N$ is $R$-closed:

In words: for a $4$-AP-free enumeration, safety of a prefix reduces to a single closure condition on the cut. Consequently a Ho-style construction over $\mathbb{Z}$ with a fixed completion order $R$ (a prefix-increasing chain of $R$-safe words exhausting $\mathbb{Z}$) is the same thing as a $4$-AP-free enumeration of $\mathbb{Z}$ having infinitely many $R$-closed cuts, which isolates exactly what the fixed-order method demands beyond $4$-AP-freeness.