Escape chains terminate: o + 3^n(w−o) ends in a reflection or precedes o (root bit 1)

A tree witness $\theta:\mathbb N\times\mathbb Z\to\{0,1\}$ defines a strict total order $\prec_\theta$ on $\mathbb Z$ (milestone framework); it satisfies Ho's identity $(*)$: $x\prec x+r \iff x+2r\prec x+3r$. 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. With root bit $\theta(0,0)=1$, every odd $o$ before an even $v$ is either reflected ($2v-o$ occurs before $v$) or escapes ($3v-2o$ occurs). The escape $3v-2o$ is again even.

Theorem (escape chains terminate). Let $\theta(0,0)=1$, let $Q$ be $\theta$-safe, and let $o$ odd come before $w$ even in $Q$. Put $e_m=o+3^m(w-o)$, so $e_0=w$ and $e_{m+1}=3e_m-2o$ is the escape of $o$ at $e_m$. Then there is $n\ge 0$ such that $e_0,\dots,e_n\in Q$, $o$ comes before each of $e_0,\dots,e_{n-1}$, and either (i) $o$ comes before $e_n$ and $2e_n-o\in Q$ comes before $e_n$ (a genuine reflection), or (ii) $n>0$ and $e_n$ comes before $o$.

So escapes cannot be used indefinitely. Each inverted (odd, even) pair starts a geometric chain of escapes with ratio 3 about $o$, and the chain must end in a reflection or in a letter placed before $o$. In particular, if no even letter precedes $o$, some $e_n$ is genuinely reflected. This is a structural lemma for Erdős #195, not a construction or an impossibility result.

Formalization note: words are List ℤ, positions via List.idxOf, safety is Safe (treeLt θ).