Boundary-flux commutation obstructs charged regulator insertions

Consider the idealized attempt to insert a nonzero charge in a region $R$ of the finite-graph model of Exact Ward/Gauss predictions in adopted regulator. Let $\mathrm{\partial }R=\{e\in E:e\text{ has exactly one endpoint in }R\}$, with $ϵ(e,R)=1$ for links directed out of $R$ and $-1$ for links directed into $R$, and put $Q_R={\sum }_{x\in R}(n_x-b_{x}I)\mathrm{.}$ On ${\mathcal{D}}_R={\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\cap {\bigcap }_{e\in \mathrm{\partial }R}D(E_e),$ suppose a bounded operator $O$ preserves ${\mathcal{H}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, maps each boundary domain $D(E_e)$ into itself, and satisfies $E_{e}O\varphi =O{E}_e\varphi$ for all $\varphi \in D(E_e)$ and all $e\in \mathrm{\partial }R$. Then $[Q_R,O]\psi =0(\psi \in {\mathcal{D}}_R)\mathrm{.}$ If, in addition, $[Q_R,O]\psi =qO\psi$ on ${\mathcal{D}}_R$ for a fixed $q\mathrm{\ne }0$, then $O\psi =0$ for every $\psi \in {\mathcal{D}}_R$. Thus a charge-changing insertion cannot act nontrivially on this domain while preserving the physical sector and commuting with every boundary electric operator. This is a fixed-regulator statement; it does not claim a continuum charged-sector or general locality theorem.