Appending a far record to an $\varepsilon$-safe word: safe iff the record is $\prec_\varepsilon$-below every earlier letter

Fix $\varepsilon:\mathbb{N}\to\{0,1\}$ and the $2$-adic order $x\prec_\varepsilon y$ iff the $k$-th two's-complement digit of $x$ equals $\varepsilon_k$, $k=\nu_2(y-x)$. A finite word $P$ over $\mathbb{Z}$ is $\varepsilon$-safe if the sequence "$P$, then all remaining integers in $\prec_\varepsilon$-order" has no monotone $4$-term arithmetic progression. Let $t$ be an integer exceeding $2b-a$ and $3b-2a$ for all letters $a,b$ of $P$ (a far new record). Then $P$ followed by $t$ is $\varepsilon$-safe if and only if $P$ is $\varepsilon$-safe and $t\prec_\varepsilon a$ for every letter $a$ of $P$. Since $\varepsilon$ is arbitrary, this also characterises far-record extensions in Ho-style chains where $\varepsilon$ may change between stages. A tool for Erdős Problem 195; it does not decide the problem.