Astra's four-class absorption certificate (27 letters) is SAFE

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 $\theta$. The word

is SAFE. It is the instance $A=(1,17,49,9,25,57,41)$, $a=0$, $b=3$, $y=18$ of the four-class absorption recipe; $A$ occupies all four residue classes $1,9,17,25$ modulo $32$.