Vertex closure of the retained hash family

The family retained at one affine hash state is vertex-closed.

Fix an integral ten-class profile $\beta$, a scale $m$, an odd modulus $p$, a weight vector $w$ and an affine offset $b_0$, and a progression-free set $S\subseteq\{0,\dots,\lfloor p/2\rfloor-1\}$. Retain those marginal-supported addresses whose three mode hashes $X,Y,Z$ all take a common value in $S$. Then the retained family is vertex-closed: whenever three retained addresses $x,y,z$ have a coordinatewise-supported mixed address $(x_1,y_2,z_3)$, that mixed address is itself retained.

The mechanism is the one Davie--Stothers use in Lemma 3.3. On any supported mixed edge the three $d=8$ hashes satisfy $X + Y = 2Z$ identically, because the grades at each position sum to $8$; so the three retained values $s_x, s_y, s_z \in S$ form a three-term arithmetic progression modulo $p$, and since $S$ lies below $p/2$ and is progression-free, $s_x = s_y = s_z$. The mixed address is marginally regular because each of its three mode words is inherited from a marginally regular address, so it is a legitimate member of the ambient family, and it carries the common hash value.

Vertex closure is exactly the hypothesis the deterministic pruning step needs: it guarantees that a supported mixed address of retained vertices is again a retained edge, so inducedness can be certified inside the retained family.