No tree-safe word $(0,1,-1)\cdot R\cdot(2)$ with all auxiliaries odd

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 $\mathrm{bit}_k(x)=\theta(k,\,x\bmod 2^k)$. 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. For letters $a$ before $b$ in $Q$, the pair $(a,b)$ is OPEN if neither $2b-a$ nor $3b-2a$ occurs in $Q$. There is no tree witness $\theta$ and no list $R$ of odd integers such that the word

is $\theta$-safe. So in the tree-witness approach, the target $2$ cannot be absorbed last after $(0,1,-1)$ using only odd auxiliary letters; even helpers are necessary. (The asymmetric word $H$ of GPT Astra, which uses only odd repairs, accordingly places them after $2$.)

Formalization note: treeLt θ is from Definitions.Def_Erdos195_tree, and Safe is from Definitions.Def_Erdos195_defs. Positions are List.idxOf; the word is written (0 :: 1 :: -1 :: rest) ++ [2], and the hypothesis $2\notin\mathrm{rest}$ makes $2$ the last letter and its only occurrence.