Every open path in a stage of an exhaustive tree-safe chain is eventually closed backwards

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 is $\theta$-safe if it followed by all remaining integers in $\prec_\theta$-order has no monotone 4-term AP (positions = first occurrences). Consider a chain of words $P_0\sqsubseteq P_1\sqsubseteq\cdots$, each a prefix of the next and each $\theta_n$-safe for some tree witness $\theta_n$ (which may change with $n$), such that every integer occurs in some $P_n$. An open path in $P_N$ is a family of in-order pairs $(a_i,b_i)$, $0\le i<m$, of letters of $P_N$ with $3b_i-2a_i=2b_{i+1}-a_{i+1}$ for $i<m-1$ and no third term $t_i=2b_i-a_i$ in $P_N$. Theorem. For every such open path there is a stage $n\ge N$ such that $P_n$ contains all third terms $t_j$ and all fourth terms $w_j=3b_j-2a_j$ ($j<m$), and in $P_n$ each $w_j$ comes before $t_j$.

So in any exhaustive chain of tree-safe words, every open path is eventually closed completely, in reverse order starting from the sink's fourth term $w_{m-1}$. Together with milestone 144 (open cycles block exhaustive chains), this describes what must happen to every chain of linked open pairs. It is a structural lemma about the tree-witness approach to Erdős #195, not a construction or an impossibility result.

Formalization note: stages are P : ℕ → List ℤ with P n <+: P (n+1); coverage is ∀ x, ∃ n, x ∈ P n.