The word $W_{16}$ is a tree-safe true root release

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. An in-order pair $(a,b)$ of $Q$ ($a$ occurs before $b$) is open if $2b-a\notin Q$ and $3b-2a\notin Q$; it lies at the node $(k,\,a\bmod 2^k)$, $k=\nu_2(b-a)$, and forces $\theta$ there to equal $\mathrm{bit}_k(b)$. A root pair is one with $b-a$ odd (node $(0,0)$); $Q$ is a true root release if every open root pair has even second entry, i.e. no open pair forces the value $1$ at the root. The $32$-letter word $W_{16}$ is $\theta$-safe for some tree witness and is a true root release. It consists of $0$; the even powers $3^{2t}$, $t=0,\dots,14$, in the order $t=0,8,4,12,2,10,6,14,1,9,5,13,3,11,7$; the odd powers $3^{2t+1}$ in the order $t=13,5,9,1,11,3,7,12,4,8,0,10,2,6$; then $2\cdot3^{29}$ and $3^{29}$. This is the case $j=16$ of the bit-reversal family $W_j$.

Formalization note: treeLt θ is from Definitions.Def_Erdos195_tree; Safe, bitOne are from Definitions.Def_Erdos195_defs. Positions are List.idxOf; the release property is stated as: every open in-order pair $(a,b)$ with $(b-a) \bmod 2 = 1$ has $b \bmod 2 = 0$.