Four-class restricted absorption: a SAFE block in one odd class mod 8 absorbs a 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 $A$ be a SAFE word all of whose entries are congruent modulo $8$ to an odd integer $r$. Let $a$ be even and $b\equiv r+2\pmod 4$, and put $Q=(a)\cdot A\cdot(b)$. Let $y\notin Q$ be such that $c=2y-b\notin Q$, $c\equiv r\pmod 8$, both $A\cdot(c)$ and $(c)\cdot A$ are SAFE, and $7b-6y\notin A$. Then

is SAFE, where $2y-A$ and $10y-10b+A$ are entrywise images in the original order. This is GPT Astra's mod-8 generalization of the mod-32 absorption theorem: the block $A$ may occupy all four residue classes modulo $32$ inside one odd class modulo $8$. No size condition is needed and $A$ may be empty.