Tree-safe words starting $(0,1,-1)$ containing $t\equiv1 \pmod 4$ must meet $\{-3,-5,2t+1,3t+2\}$

A tree witness is a function $\theta(k,r)\in\{0,1\}$ on the nodes $(k,r)$, $0\le r<2^k$, of the binary tree of residues. For distinct integers $x,y$ with $k=\nu_2(y-x)$, put $x\prec_\theta y$ iff $\mathrm{bit}_k(x)=\theta(k,\,x\bmod 2^k)$ (two's-complement digits). A finite word $Q$ of integers is $\theta$-safe if the sequence "the letters of $Q$ in order, then all remaining integers in $\prec_\theta$-order" contains no monotone $4$-term arithmetic progression. Let $Q$ be a $\theta$-safe word (for any tree witness $\theta$) beginning with $0,1,-1$, and let $t\in Q$ with $t\equiv1\pmod 4$, $t\neq1$. Then

Indeed the in-order pairs $(1,-1)$ and $(-1,t)$ both sit at the tree node $(1,1)$ and, if both were open (neither continuation $2b-a$, $3b-2a$ present), would force opposite values there. This is the fixed-node cascade behind the failure of the rule "close the newest conflicting pair by its fourth term" when extending $(0,1,-1)$ in Ho-style constructions of $4$-AP-free permutations of $\mathbb{Z}$ (Erdős Problem 195); the case $t=5$ gives the necessary condition $Q\cap\{-3,-5,11,17\}\neq\varnothing$.

Formalization note: treeLt θ is from Definitions.Def_Erdos195_tree; Safe, bitOne are from Definitions.Def_Erdos195_defs. "Begins with" is the list prefix relation <+:; positions in a word are List.idxOf. The statement quantifies over every witness θ, so it holds in particular for all ε-orders.