Open cycles force closures at their nodes, for every tree witness

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; the pair is closed if one of them occurs. An open cycle in a word $T$ is a family of in-order pairs $(a_i,b_i)$, $0\le i<m$, of letters of $T$ with

and no third term $2b_i-a_i$ in $T$. Write $k_i=\nu_2(b_i-a_i)$ for the 2-adic level of pair $i$; its node is the residue $a_i \bmod 2^{k_i}$. Theorem (open cycles force closures at their nodes). Let $(a_i,b_i)_{i<m}$ be an open cycle in $T$, fix $i<m$ and put $k=k_i$. Let $\theta$ be any tree witness and $R$ any word with $T\,R$ $\theta$-safe. If $x$ comes before $y$ in $T\,R$, with

then $2y-x$ or $3y-2x$ occurs in $T\,R$.

Every pair of an open cycle stays open forever (the open-cycles-never-close milestone), so it fixes the witness bit at its node for every safe extension. This theorem turns that into $\theta$-independent forcing: every pair at the same node with the opposite orientation must be closed, whatever witness is used. 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 a b : ℕ → ℤ on indices i < m, the successor index is (i+1) % m, and safety is Safe (treeLt θ); the level is padicValInt 2 (b i - a i) and the congruences are stated as divisibility by powers of 2.