Sink steps are blocked only by other sinks' trees

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 and $w=3b-2a$ its fourth term; the pair is fully open in $T$ if neither lies in $T$. 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 $2b_i-a_i$ in $T$.

Theorem. Let $T$ be $\theta$-safe, let $w\notin T$, and assume the C1 condition for $w$: whenever $w=2y-x$ for an in-order pair $(x,y)$ of $T$, we have $3y-2x\in T$ (true for the fourth term of a true sink). If $T\cdot w$ is not $\theta'$-safe for any tree witness $\theta'$, then there are $x\in T$ with $2w-x,\,3w-2x\notin T$ and a fully open in-order pair $(x',y')$ of $T$ with $3y'-2x'\neq w$, $\nu_2(y'-x')=\nu_2(w-x)=k$, $x'\equiv x\pmod{2^k}$ and opposite orientation (the $k$-th binary digits of $y'$ and $w$ differ). This is a node conflict. Moreover:

1. no open path starting at $(x',y')$ has last fourth term equal to $w$;
2. $(x',y')$ lies on an open path ending at a true sink (in the sense of the previous milestone) whose fourth term is different from $w$.

So appending the sink fourth term $w$, the forced first step of closing a path (milestone #149), can only be blocked by a fully open pair belonging to the in-tree of a different sink. In particular, if every fully open pair of $T$ not released by $w$ reaches $w$, then $T\cdot w$ is tree-safe. The proof combines milestones #59, #153 and the previous milestone.