Restricted absorption: a mod-32 sparse SAFE block absorbs any admissible target with $2n+4$ letters

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 $A=(x_1,\dots,x_n)$ be a SAFE word all of whose entries are congruent modulo $32$ to an odd integer $r$. Let $a$ be even, $b\equiv r+2\pmod 4$, and let $y\notin Q=(a)\cdot A\cdot(b)$ satisfy $2y-b\equiv r+16\pmod{32}$. Then

is SAFE, where $2y-A$ and $10y-10b+A$ are taken entrywise in the original order. No size condition on $y$ is needed, $A$ may be empty, and SAFE-ness of $Q$ itself is not assumed (it follows). This is the mod-32 form of GPT Astra's uniform finite absorption theorem: an infinite class of arbitrarily long SAFE words absorbs every target of a residue class with exactly $2n+4$ new entries.