Fixed-witness one-letter absorption criterion for tree-safe words

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 $\mathrm{bit}_k(x)=\theta(k,\,x\bmod 2^k)$ (two's-complement digits). 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; it is tree-safe if it is $\theta$-safe for some $\theta$. An in-order pair $(a,b)$ of $Q$ ($a$ occurs before $b$) is open if $2b-a\notin Q$ and $3b-2a\notin Q$; an open pair lies at the node $N(a,b)=(k,\,a\bmod 2^k)$, $k=\nu_2(b-a)$, and $\theta$-safety requires $\theta(N(a,b))=\mathrm{bit}_k(b)$ there. Let $R$ be a $\theta$-safe word and $t\notin R$. Then $R\cdot t$ is $\theta$-safe (for the same $\theta$) if and only if

In particular (Astra's conditional absorption) if $t$ is not a third term of any in-order $3$-AP of $R$ and every letter $x\prec_\theta t$ of $R$ already has $2t-x$ or $3t-2x$ in $R$, then $t$ can be appended without changing the witness. This is the fixed-witness one-letter extension criterion; it contains bottom insertion (all letters $\succ_\theta t$) as the special case where (ii) is vacuous (Erdős Problem 195).

Formalization note: treeLt θ is from Definitions.Def_Erdos195_tree; Safe, bitOne are from Definitions.Def_Erdos195_defs. Positions in a word are List.idxOf; P ++ [c] appends c.