A prefix of a $4$-AP-free enumeration of $\mathbb{Z}$ is safe for some $\varepsilon$-order iff its open pairs are digit-consistent

Let $f:\mathbb{N}\to\mathbb{Z}$ be a bijection containing no monotone $4$-term arithmetic progression, and fix a cut $N\in\mathbb{N}$. Call a pair of positions $i<j<N$ open at $N$ if both values $2f(j)-f(i)$ and $3f(j)-2f(i)$ occur in $f$ at positions $\ge N$. For $\varepsilon:\mathbb{N}\to\{0,1\}$ let $\prec_\varepsilon$ be the $2$-adic order on $\mathbb{Z}$ (for $x\neq y$ with $k=\nu_2(y-x)$: $x\prec_\varepsilon y$ iff the $k$-th two's-complement digit of $x$ equals $\varepsilon_k$), and call the prefix $f(0),\dots,f(N-1)$ $\varepsilon$-safe if the completion "prefix, then all remaining integers in $\prec_\varepsilon$-order" has no monotone $4$-AP. Then

This eliminates the order from Ho-style constructions with stage-dependent $\varepsilon$: a chain of prefixes, each safe for its own $\varepsilon$-order, exhausting $\mathbb{Z}$ is exactly a $4$-AP-free enumeration of $\mathbb{Z}$ with infinitely many cuts at which the open pairs impose consistent binary-digit constraints (each open pair $(i,j)$ forces $\varepsilon_k$ to be the $k$-th digit of the later value $f(j)$). It reduces the stage-dependent-$\varepsilon$ approach to Erdős Problem 195 to a purely combinatorial condition on the permutation; it does not by itself decide whether such permutations exist.