Open paths close from the sink backwards: closing one pair forces the whole tail, fourth terms first

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 $Q$ of integers is $\theta$-safe if the sequence "letters of $Q$ 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. An open path in a word $T$ is a family of in-order pairs $(a_i,b_i)$, $0\le i<m$, of letters of $T$ with

(the fourth term of each pair is the third term of the next) and no third term $2b_i-a_i$ in $T$. Theorem. Let $T$ contain such an open path and let $T\cdot R$ be $\theta$-safe for some tree witness $\theta$ (any extension $R$). If the third term $t_i=2b_i-a_i$ of some pair $i$ occurs in $T\cdot R$, then for every $j$ with $i\le j<m$ both $t_j$ and $w_j=3b_j-2a_j$ occur in $T\cdot R$, and $w_j$ comes before $t_j$.

Since $w_j=t_{j+1}$, the closing order is forced: $w_{m-1}$ first, then $t_{m-1}, t_{m-2},\dots,t_i$. So the tail of an open path can only be closed from the sink backwards. The reason is the tree criterion: an in-order pair whose third term appears later must already have its fourth term, placed before the third term. This is the acyclic counterpart of milestone 143 (open cycles never close): for a path the closure must start at the sink's fourth term $w_{m-1}$. It is a structural lemma about the tree-witness approach to Erdős #195, not a construction or an impossibility result.

Formalization note: words are List ℤ, positions via List.idxOf, pairs are a b : ℕ → ℤ on indices i < m; safety is Safe (treeLt θ) (T ++ R).