Lemma: Nested regional comparison algebras

For every open $O$,

For disjoint open $O_1,O_2$, the two algebras commute at every scale.

Proof. By Lemma lem:paulibasis, a supported invariant algebra is spanned by electric monomials and cycle-type $Z$ monomials. Under one subdivision, represent an old electric operator on one of its two child links and replace every old cycle $Z$ factor by the product on both children. The cycle condition remains valid at the new degree-two vertices. On the physical basis this realizes exactly the same operator.

If an old closed link lies in $O$, both children lie in $O$, so support is preserved. This proves the inclusion. Disjoint $O_1,O_2$ have disjoint retained link sets, and Proposition prop:gaugenet proves commutation.