Exact lifting criterion with root bit 1: absorbing an odd target after an even batch

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. The halves of a word are the subsequences of its even letters and of its odd letters, each mapped by $x\mapsto\lfloor x/2\rfloor$. The half witnesses are $\theta_E(k,r)=\theta(k+1,2r)$ and $\theta_O(k,r)=\theta(k+1,2r+1)$.

Theorem (exact lifting criterion, root bit $1$). Let $\theta(0,0)=1$. Let $S$ be a $\theta$-safe word, $R$ a word of even integers, and $y\notin S$ an odd integer. Write $P=S\cdot R$. Then $S\cdot R\cdot y$ is $\theta$-safe if and only if

1. the even half of $P$ is $\theta_E$-safe;
2. the odd half of $S\cdot y$ is $\theta_O$-safe;
3. (closing new even letters) for every $b\in R\setminus S$ and every odd $a\in S$: $2b-a\in S$, or $2b-a=y$, or $3b-2a\in P$;
4. (no new third terms of open pairs) for all $a,b\in S$ with $b-a$ odd and $a$ before $b$ in $S$: if $2b-a\in R\setminus S$ then $3b-2a\in S$;
5. (in-order progressions ending at $y$) for every even $b\in P$ with $2b-y\in S$ preceding $b$ in $P$: $2y-b\in P$.

This is the companion of the root-bit-$0$ lifting criterion (Erdos195.lifting_criterion). When the root bit is $1$, open odd-difference pairs must end at an odd letter. So every new even letter must be closed against every odd letter of $S$ (condition 3), and a new even letter must not become the third term of a pair of $S$ that is open at the root (condition 4). Odd affine maps $x\mapsto mx+s$ with $s$ odd flip the root bit (Erdos195.safe_odd_affine_iff). The two criteria together therefore cover absorbing a target of either parity after a batch of the opposite parity.