Specified-last-entry ordering: every finite set has a SAFE ordering ending at any prescribed element

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. Let $X\subset\mathbb Z$ be a finite set and $z\in X$. Then there is a duplicate-free list $L$ whose set of entries is exactly $X$, whose last entry is $z$, and which is SAFE for some tree witness $\theta$.

This is the ordering lemma used by GPT 6 Astra in the exceptional case of the four-class restricted absorption theorem: a reflection set must be appended in a safe order with one prescribed entry last. The proof uses the tree order with $\theta(k,\cdot)=\neg\mathrm{bit}_k(z)$, in which $z$ is the greatest element. Sort $X\setminus\{z\}$ increasingly in it and append $z$. A list sorted increasingly for $\prec_\theta$ is SAFE for the reversed order, which is the tree order $\prec_{\neg\theta}$. This is the tree version of Ho's reverse-order lemma.