Every $t\equiv 3 \pmod 4$ can be appended to $(0,1,-1)$

Fix $\varepsilon:\mathbb{N}\to\{0,1\}$ and the $2$-adic total order on $\mathbb{Z}$ given by $x\prec_\varepsilon y$ iff $x\neq y$ and the $k$-th two's-complement digit of $x$ equals $\varepsilon_k$, where $k=\nu_2(y-x)$. A finite word $Q$ of integers is $\varepsilon$-safe if the sequence "the letters of $Q$ in order, then all remaining integers in $\prec_\varepsilon$-order" contains no monotone $4$-term arithmetic progression. Let $t\equiv 3\pmod 4$ with $t\neq-1$, put $h=\nu_2(t+1)\ (\ge 2)$, and define $\varepsilon_h=0$ and $\varepsilon_k=1$ for $k\neq h$. Then

The three-letter word $(0,1,-1)$ is a natural 'trap' prefix in Ho-style constructions of $4$-AP-free permutations of $\mathbb{Z}$ (Erdős Problem 195); this shows every target $t\equiv 3\pmod 4$ can be appended directly, with no auxiliary letters, by choosing a single digit of $\varepsilon$ depending on $t$.

Formalization note: $\varepsilon$ is fun k => decide (k ≠ padicValInt 2 (t + 1)); Safe, epsLt come from Definitions.Def_Erdos195_defs.