Two-entry target-last absorption: every two-letter word absorbs any new target with at most one helper

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 $a\ne b$ be integers and let $y\notin\{a,b\}$. Put $k=\nu_2(b-a)$ and $t=\nu_2(y-a)$, and define

Then $W$ has distinct entries and is SAFE. So any two-letter word can absorb an arbitrary new target placed last, with at most one auxiliary letter $2y-b$. Note that the hypotheses do not require $(a,b)$ itself to have any property, since every two-letter word is SAFE. This is the base case of the local absorption recipes (Astra) in the tree-witness approach to Erdős Problem 195.

Formalization note: treeLt θ is from Definitions.Def_Erdos195_tree, and Safe is from Definitions.Def_Erdos195_defs. Positions are List.idxOf, and $\nu_2$ is padicValInt 2.