Odd affine maps $x\mapsto mx+s$ transport tree orders and tree safety exactly

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 $m$ be an odd integer (of either sign) and $s$ any integer, and let $f(x)=mx+s$. Call $\theta'$ an $f$-transport of $\theta$ if for every $k$ and every $0\le r<2^k$

Then (1) an $f$-transport exists (it is unique on the residues $0\le\rho<2^k$, since $r\mapsto (mr+s)\bmod 2^k$ is a bijection of $\{0,\dots,2^k-1\}$); and for every $f$-transport $\theta'$: (2) $f$ is an order isomorphism, $x\prec_\theta y\iff f(x)\prec_{\theta'}f(y)$; (3) for every finite word $Q$, $Q$ is $\theta$-safe iff $f(Q)=(f(q_1),f(q_2),\dots)$ is $\theta'$-safe.

So tree safety is exactly invariant under the odd affine group $x\mapsto mx+s$; with $m=-1$ this is the mirror $x\mapsto s-x$, under which an open pair of $Q$ at node $(k,a\bmod 2^k)$ becomes an open pair of $s-Q$ at node $(k,(s-a)\bmod 2^k)$. Together with the scaling lemma for $x\mapsto 2x+c$ this covers all maps $x\mapsto 2^j m x+s$.

Formalization note: the xor condition is written as $\theta'(\dots)=1\iff(\theta(k,r)=1\iff\neg\,\mathrm{bit}_k(mr+s)=1)$; % is Int.emod; bitOne x k is $\lfloor x/2^k\rfloor\bmod 2=1$.