Theorem: Constructed spatial gauge refinement net

The assignment

is an isotone locally commuting spatial net. For each fixed $O$, the increasing algebras ${\tilde{\mathfrak{A}}}_n(O)$ eventually stabilize. Admitted bounded observable sequences whose ${J_n}^{-1}$ images converge in norm have well-defined product- and adjoint-preserving limits.

The full graph algebra is $M_4$, and centerful proper regional algebras persist.

Proof. Isotony holds at every scale. Lemma lem:nestedregions gives an increasing family. Its union is a $\ast$-algebra, and norm closure gives a $C^{\ast }$-algebra. Local commutativity passes to the closure.

The dimensions of the increasing subspaces are integers bounded by $\dim M_4=16$. Only finitely many strict increases are possible. Hence each fixed regional sequence eventually stabilizes.

If $a_n,b_n$ have norm-convergent comparison images, those images are uniformly bounded and

Adjoints behave similarly.

The global physical algebra is $M_4$ by Proposition prop:gaugenet. Take $O_{12}$ containing arcs $1,2$ completely but not arc $3$ completely. Its algebra is $M_2\oplus M_2$. Small portions of arc $3$ add only the already present electric observable $F$. An open interval $O_3$ in the interior of arc $3$ has algebra $C^{\ast }(F)$ for sufficiently fine scales. Thus the shared center persists.