Bounded localized repair: even-only extensions against an old root force stay within half the diameter

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. $Q$ is SAFE if it is $\theta$-safe for some tree witness $\theta$. Let $Q$ be a finite word containing an even $p$ before an odd $s$ such that $2s-p\notin Q$ and $3s-2p\notin Q$; thus $(p,s)$ is an open pair forcing the odd child at the root. Let $D$ be a word of even integers such that $Q\cdot D$ is SAFE. Then for every odd $x\in Q$ and every $M$ with $|q-x|\le M$ for all $q\in Q$, every entry $z$ of $D$ not in $Q$ satisfies

This is GPT Astra's 3-adic confinement bound (B), at the root node: repairs on the wrong side of an immutable old force cannot use large helpers.