Odd $u$, target $t\equiv0\pmod 4$: $(0,u,2t-u,t)$ is $\varepsilon$-safe when $\varepsilon_0=1$, $\varepsilon_1\neq u_1$, $\varepsilon_{\nu_2 t}=1$

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. Write $u_k$ for the $k$-th binary digit of $u$. Let $u$ be odd and $t\neq0$ with $t\equiv0\pmod 4$. If $\varepsilon_0=1$, $\varepsilon_1\neq u_1$ and $\varepsilon_{\nu_2(t)}=1$, then

In particular, the single order with $\varepsilon_0=1$, $\varepsilon_1=1-u_1$ and $\varepsilon_k=1$ for $k\ge2$ (which depends only on $u$) lets every target $t\equiv0\pmod4$ be absorbed into $(0,u)$ with one auxiliary letter, the reflection $2t-u$ of $u$ through $t$.

Formalization note: Safe, epsLt, bitOne are from Definitions.Def_Erdos195_defs.