The sink's fourth term never conflicts with its own fully open path

A tree witness $\theta:\mathbb N\times\mathbb Z\to\{0,1\}$ defines a strict total order $\prec_\theta$ on $\mathbb Z$ (milestone framework). A finite word $T$ of integers is $\theta$-safe if the sequence "letters of $T$ in order, then all remaining integers in $\prec_\theta$-order" has no monotone 4-term arithmetic progression; positions are first occurrences. For an in-order pair $(a,b)$ of a word ($a$ before $b$), call $t=2b-a$ its third term, $w=3b-2a$ its fourth term, and $\nu_2(b-a)$ its level. An open path in $T$ is a family of in-order pairs $(a_i,b_i)$, $0\le i<m$, of letters of $T$ with $3b_i-2a_i=2b_{i+1}-a_{i+1}$ for $0\le i<m-1$ and no third term $t_i=2b_i-a_i$ in $T$. Write $d_i=b_i-a_i$, $k_i=\nu_2(d_i)$, and $w=3b_{m-1}-2a_{m-1}$ for the fourth term of the last (sink) pair.

Theorem. Suppose $T$ is $\theta$-safe and contains an open path of length $m$ whose sink fourth term $w$ is absent from $T$. Let $j<m$, put $k=k_j=\nu_2(b_j-a_j)$, and let $x\in\mathbb Z$ satisfy $\nu_2(w-x)=k$ and $x\equiv a_j \pmod{2^k}$ (the pair $(x,w)$ has the level of pair $j$ and sits at its node). Then $\theta\bigl(k,\;x \bmod 2^k\bigr)=1\iff \text{the $k$-th binary digit of $w$ is }1 .$

In the tree characterization of safety, this is exactly the orientation condition of the new pair $(x,w)$ created when $w$ is appended. By milestone #149 every closure of the path must place $w$ first, and this result shows that the path's own fully open pairs never cause a node conflict for that step: any obstruction to appending $w$ comes from pairs off the path. (Proof idea: $w\equiv a_j \pmod{2^k}$ forces $\nu_2(\sum_{l>j}d_l)>k$ by the ruler law, so $w\equiv b_j\pmod{2^{k+1}}$.)