Scaling $x\mapsto 2x+c$ transports tree safety to the shifted subtree witness, both directions

A tree witness is a function $\theta:\mathbb N\times\mathbb Z\to\{0,1\}$ on the nodes $(k,r)$ of the binary tree of residues. For distinct integers $x,y$ with $k=\nu_2(y-x)$, put $x\prec_\theta y$ iff $\mathrm{bit}_k(x)=1 \Leftrightarrow \theta(k,\,x\bmod 2^k)=1$ ($\mathrm{bit}_k$ is the $k$-th two's-complement binary digit). A finite word $Q$ of integers is $\theta$-safe if the completion "the letters of $Q$ in order, then all remaining integers in $\prec_\theta$-order" contains no monotone $4$-term arithmetic progression (positions of $a,a+r,a+2r,a+3r$ increasing, $r\neq0$). Positions in $Q$ are first occurrences. Let $c\in\{0,1\}$ and let $P$ be a finite word. Then the word $2P+c=(2p_1+c,2p_2+c,\dots)$ is $\theta$-safe if and only if $P$ is $\theta'$-safe, where $\theta'(k,r)=\theta(k+1,\,2r+c)$:

The forward direction transports a safe word on one parity class down to a safe word for the shifted subtree witness (e.g. $x\mapsto (x-1)/2$ on odd words), and the backward direction is the lifting: any safe word for any witness $\theta'$ lifts to a safe word on the class $c$ for every $\theta$ whose subtree at $c$ is $\theta'$.