The words $B_t=(0,1,6,3,-8-32t,-2-32t,-1-16t,-70-160t,-78-192t,-36-96t)$ are tree-safe true root releases for all $t\ge0$

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. For every natural number $t$ the word

is $\theta$-safe for some tree witness $\theta$ and is a true root release. One witness works for every $t$: $\theta(1,\cdot)=1$, $\theta(2,0)=\theta(2,3)=1$, $\theta(3,0)=\theta(3,6)=1$, and $\theta=0$ everywhere else. All $t$-coefficients are multiples of $16$ and the letters are distinct mod $16$, so every pair meets at level $\le3$, and the nodes and forced bits do not depend on $t$. So the batch can be translated $2$-adically to give an infinite family.

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 $b \bmod 2 = 0$.