Open cycles are never tree-safe: every open cycle has a node conflict at its minimal level

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. An open cycle in a word $T$ is a family of in-order pairs $(a_i,b_i)$, $0\le i<m$, $m\ge 1$, of letters of $T$ with

and no third term $2b_i-a_i$ in $T$. Theorem (open cycles are never tree-safe). If $T$ contains an open cycle, then no word $T\,R$ is $\theta$-safe, for any tree witness $\theta$ and any word $R$; in particular $T$ itself is not safe for any tree witness.

Equivalently, the points $t_i=2b_i-a_i$ satisfy $t_{i+1}-t_i=d_i:=b_i-a_i$, so $\sum_i d_i=0$ and the minimal 2-adic valuation $k$ of the $d_i$ is attained by an even number of pairs, all at the same node modulo $2^k$; consecutive such pairs around the cycle point in opposite directions. Hence the node-conflict hypothesis of milestone 146 always holds. This removes that hypothesis: open cycles of every length (including the interlocked configuration of milestone 141) are forbidden configurations, and milestone 144 becomes vacuous for safe stages. 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, the pairs are a b : ℕ → ℤ on indices i < m, the successor index is (i+1) % m; the conclusion is ¬ ∃ θ, Safe (treeLt θ) (T ++ R).