Root releases after inserting $x=4\cdot3^{11}+3^{i-2}$ into $W_8$

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. Let $i\in\{2,4,\dots,12\}$ and $x=4\cdot 3^{11}+3^{i-2}=\tfrac49T+3^{i-2}$ with $T=3^{13}$. Insert $x$ into $W_8$ after the last even power and append $6x$, $3x$:

Each $Q_i$ is $\theta$-safe for some tree witness and is a true root release. The new odd letter $x$ is grounded through the bridge $9x-4T=3^i\in W_8$, which closes the pair $(2T,3x)$ via $3\cdot3x-2\cdot2T=3^i$. These are the $j=8$ analogues of $Q_{109},Q_{117}$ for $j=4$.

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 $b \bmod 2 = 0$.