Every fully open pair reaches a true sink

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 for some tree witness and let $(a_0,b_0)$ be a fully open in-order pair of $T$. Then there is an open path $(a_i,b_i)_{0\le i<m}$ with $m\ge1$ starting at $(a_0,b_0)$ whose sink fourth term $w=3b_{m-1}-2a_{m-1}$ is not in $T$ (so every pair of the path is fully open), such that $w$ is a true sink: whenever $w=2y-x$ for an in-order pair $(x,y)$ of $T$, the fourth term $3y-2x$ lies in $T$. Moreover $m+1\le 2^r$, where $r$ is the number of distinct levels $\nu_2(b_i-a_i)$ on the path.

Following "fourth term = third term of the next fully open pair" from any fully open pair always stops at a true sink. Its fourth term then satisfies the first (C1) condition of the tree append criterion (milestone #59). By milestone #153 it never conflicts with its own path. Proof: chains that never stop would repeat a pair (pigeonhole over $T\times T$), which gives an open cycle, impossible by milestone #147. The length bound is milestone #151.