After a sink step, the sink passes back along the path: the previous third term becomes a true sink

A tree witness $\theta:\mathbb N\times\mathbb Z\to\{0,1\}$ defines a strict total order $\prec_\theta$ on $\mathbb Z$: for $x\ne y$ with $k=\nu_2(y-x)$, $x\prec_\theta y$ iff the $k$-th binary digit of $x$ is $1$ exactly when $\theta(k,x \bmod 2^k)=1$. 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); $T$ is tree-safe if it is $\theta$-safe for some $\theta$. For an in-order pair $(a,b)$ of $T$ ($a$ before $b$) call $2b-a$ its third term and $3b-2a$ its fourth term. An open path in $T$ is a family of in-order pairs $(a_i,b_i)$, $0\le i<m$, $m\ge1$, with $3b_i-2a_i=2b_{i+1}-a_{i+1}$ for $i<m-1$ and no third term $2b_i-a_i$ in $T$. Its sink fourth term $w'=3b_{m-1}-2a_{m-1}$ is a true-sink fourth term of $T$ if $w'\notin T$ and every in-order pair $(x,y)$ of $T$ with $2y-x=w'$ has $3y-2x\in T$ (condition C1 for appending $w'$).

Theorem. Let $T$ be tree-safe and let $(a_i,b_i)_{i<m}$, $m\ge1$, be an open path in $T$ whose sink fourth term $w=3b_{m-1}-2a_{m-1}$ is not in $T$. Put $t=2b_{m-1}-a_{m-1}$. Assume uniqueness: every in-order pair $(x,y)$ of $T$ with $2y-x=t$ and $3y-2x\notin T$ has $3y-2x=w$. Then in the word $T\,w$:

1. every path pair $(a_i,b_i)$, $i<m$, is still an in-order pair, and no third term $2b_i-a_i$ lies in $T\,w$ (in particular $t\notin T\,w$), so $(a_i,b_i)_{i<m-1}$ is an open path of $T\,w$ whose sink fourth term is $3b_{m-2}-2a_{m-2}=t$ (for $m\ge2$);
2. $t$ satisfies C1 in $T\,w$: every in-order pair $(x,y)$ of $T\,w$ with $2y-x=t$ has $3y-2x\in T\,w$.

So $t$ is a true-sink fourth term of $T\,w$. Together with milestone #149 (an open path can only be closed from the sink backwards) this shows that the backward closure of a path is a sequence of true-sink steps. The fact $2b_i-a_i\ne w$ uses the ruler law (#148) and the exact valuation of the telescoping sum.