Tree orders $\prec_\theta$ on $\mathbb{Z}$: strict total, no monotone 3-AP, Ho's identity $(\ast)$

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. For every tree witness $\theta$:

These are the three properties of the tail order used in Ho-style constructions of $4$-AP-free permutations; they show that the $\varepsilon$-orders (the case $\theta(k,r)=\varepsilon_k$) can be replaced by the much larger family of tree orders.

Formalization note: treeLt θ is from Definitions.Def_Erdos195_tree; Safe, bitOne, No3AP, Star, HasMonotoneAP are from Definitions.Def_Erdos195_defs. Positions in a word are List.idxOf; $\nu_2$ is padicValInt 2.