Fixed-node fourth-term cascade: no tree-safe word on $(0,1,-1),2,3,w_0,\dots,w_m$

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 $w_j=2\cdot3^{j+1}-1$ (so $w_0=5$ and $w_{j+1}=3w_j+2$) and $m\ge0$. No word $Q$ beginning with $0,1,-1$, containing $w_m$, all of whose letters lie in

is $\theta$-safe for any tree witness $\theta$. The pair $(1,-1)$ forces the value $1$ and the pair $(-1,w_m)$ forces the value $0$ at the same node $(1,1)$, and closing $(-1,w_m)$ by its fourth term creates $w_{m+1}$: an infinite fourth-term cascade at a fixed node, so no depth-type potential can make that repair rule terminate (Erdős Problem 195, tree-witness approach).

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.