Two no-escape centres force a translate of the earlier odd letters (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); its root bit $\theta(0,0)$ decides the order between integers of different parity. 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. We say an odd letter $o$ before an even letter $v$ in $Q$ escapes at $v$ if $3v-2o\in Q$.

Theorem (two no-escape centres). Let $\theta(0,0)=1$ and let $Q$ be $\theta$-safe. Let $u,u'$ be even letters of $Q$ with $u$ before $u'$, such that no odd letter before $u$ escapes at $u$ and no odd letter before $u'$ escapes at $u'$. Put $\delta=2(u'-u)$. Then

and if at least one odd letter precedes $u$, then some odd letter of $Q$ lies strictly between $u$ and $u'$.

The proof rests on the reflect-or-escape principle: with root bit 1, every odd $o$ before an even $v$ satisfies either $2v-o\in Q$ before $v$, or $3v-2o\in Q$. Without escapes, the odd letters before $u$ form a set symmetric about $u$, and those before $u'$ form a set symmetric about $u'$. Composing the two reflections gives the translation by $\delta$. A finite nonempty set cannot be invariant under a nonzero translation, so new odd letters must be placed between the two centres. This is a structural constraint on repairs that use no escape letters. It is neither a construction nor an impossibility result for Erdős #195.

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