One-even bridge criterion: odd batches around a single new even letter (root bit 1)

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 (one-even bridge criterion, root bit $1$). Let $\theta(0,0)=1$. Let $S$ be a word that is $\theta_S$-safe for some witness $\theta_S$ with $\theta_S(0,0)=1$. Let $P$ and $T$ be words of odd integers with no letter of $P$ in $S$ and no letter of $T$ in $S\cdot P$, and let $y\notin S$ be even. Then

if and only if

1. the even half of $S\cdot y$ is $\theta_E$-safe and the odd half of $S\cdot P\cdot T$ is $\theta_O$-safe;
2. (closing pairs into $y$) for every odd $a\in S\cdot P$: $2y-a\in S\cdot P$ or $3y-2a\in S$;
3. (progressions with third term $y$) for every even $a\in S$ and odd $b\in S\cdot P$ with $2b-a=y$ and $a$ before $b$: $3b-2a\in S\cdot P$;
4. (new third terms of pairs of $S$) for odd $a$ and even $b$ in $S$, $a$ before $b$, with $2b-a\notin S$ but $2b-a\in P\cdot T$: either $3b-2a\in S$, or $3b-2a=y$ and $2b-a\notin P$.

This is the exact root-level criterion for absorbing a single new even letter into a root-bit-$1$ word using only odd helpers placed before and after it: nothing is required of the odd batches except odd-half safety, while the letters before $y$ must be (almost) closed under reflection $a\mapsto 2y-a$. It complements the lifting criteria Erdos195.lifting_criterion and Erdos195.lifting_criterion_true (odd target after an even batch). Formalization note: $S$ itself only needs to be safe for some root-bit-$1$ witness, not for $\theta$.