Extending $(0,1)$ by one even target $t\equiv 0 \pmod 4$ forces $\varepsilon_0=1$ and $2t-1$ before $t$

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 $t>0$ with $4\mid t$, and let $Q$ be an $\varepsilon$-safe word that begins with $0,1$, contains $t$, and whose only even letters are $0$ and $t$. Then

This is a rigidity statement for extending the safe word $(0,1)$ by a single new even target in Ho-style constructions of $4$-AP-free permutations of $\mathbb{Z}$ (Erdős Problem 195): the lowest digit of $\varepsilon$ is forced and the continuation $2t-1$ must be placed before $t$ itself.

Formalization note: Safe, epsLt are from Definitions.Def_Erdos195_defs; "begins with $0,1$" is [0, 1] <+: Q; "before" is comparison of List.idxOf; $\varepsilon_0=1$ is ε 0 = true.