Open cycles never close: cyclic third/fourth-term chains of pairs stay open forever

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 $2b-a$ its third term and $3b-2a$ its fourth term. Theorem (open cycles never close). Let $T$ be a finite word, $m\ge 1$, and let $(a_i,b_i)$, $0\le i<m$, be pairs of letters of $T$ with $a_i$ before $b_i$, forming a cycle in the sense that the fourth term of each pair is the third term of the next one:

and suppose no third term $2b_i-a_i$ lies in $T$. Then for every tree witness $\theta$ and every word $R$ such that $T\,R$ is $\theta$-safe, none of the third or fourth terms $2b_i-a_i,\;3b_i-2a_i$ occurs in $T\,R$: all pairs of the cycle stay open forever.

The case $m=2$ is the interlocked-pairs lemma (two pairs with swapped third and fourth terms). The statement shows that any cyclic chain of this kind permanently blocks a set of integers from ever being added, for every tree order. It is a structural lemma in the tree-witness approach to Erdős #195, not a construction or an impossibility result.

Formalization note: words are List ℤ, positions via List.idxOf, the pairs are given by a b : ℕ → ℤ on indices i < m, the successor index is (i+1) % m, and safety is Safe (treeLt θ). For $m=0$ the statement is vacuous.