Open cycles of pairs block exhaustive chains of tree-safe words

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 block exhaustive chains). Let $P_0, P_1, P_2,\dots$ be finite words with each $P_n$ a prefix of $P_{n+1}$, each $P_n$ $\theta_n$-safe for some tree witness $\theta_n$ (which may depend on $n$), and such that every integer occurs in some $P_n$. Then no stage $P_N$ contains a cycle of in-order pairs $(a_i,b_i)$, $0\le i<m$, $m\ge1$, with

and all third terms $2b_i-a_i$ absent from $P_N$.

So an exhaustive chain of tree-safe words (the kind of chain that would give a permutation of $\mathbb Z$ with no monotone 4-term AP) must never create such a cycle, of any length, at any stage. This is a forbidden-configuration result for the tree-witness method for Erdős #195, not a resolution of the problem.

Formalization note: the chain is P : ℕ → List ℤ with P n <+: P (n+1), safety is ∃ θ, Safe (treeLt θ) (P n), coverage is ∀ x, ∃ n, x ∈ P n; positions via List.idxOf, successor index (i+1) % m.