Single-inversion criterion: U·(b,c)·V (classes mod 4) has a SAFE extension iff 4c−3b ∉ b·V or 5c−4b ∉ U·c

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 $W$ of integers is SAFE for $\theta$ if the sequence "letters of $W$ in order, then all remaining integers in $\prec_\theta$-order" has no monotone 4-term arithmetic progression. Theorem (single-inversion criterion). Let $b$ be odd and $c$ even, and let $T=U\cdot(b,c)\cdot V$ be a duplicate-free word in which every entry of $E=U\cdot(c)$ is $\equiv c\pmod 4$ and every entry of $O=(b)\cdot V$ is $\equiv b\pmod 4$. Assume $E$ and $O$ are each SAFE (for some tree witnesses). Put $p=4c-3b$ and $q=5c-4b$. Then $T$ has a SAFE end-extension $T\cdot R$ (for some tree witness) if and only if

Moreover, when this holds, $R$ can be chosen with exactly $|V|+2=|O|+1$ entries, all new and distinct.

This is GPT 6 Astra's single-inversion repair lemma (astra_fourclass) as a standalone exact criterion. With $w=3c-2b$ and $t=2c-b$: if $p\notin O$, $R=(2w-O)\cdot(w)$ (reflections in inherited order); if $p\in O$ and $q\notin E$, $R=L\cdot(w,t)$ where $L\cdot t$ is a SAFE ordering of the reflection set $2w-O$ ending at $t$ (milestone #133). The necessity half is the interlocked root obstruction (milestone #135): if $p\in O$ and $q\in E$, the pairs $(b,c)$ and $(q,p)$ can never close and force opposite root bits. Milestones #122 and #136 are the two positive cases, in the coordinates of the four-class absorption theorem.