Incidence curl has zero discrete divergence and gradient curl

Let ${\mathrm{\partial }}_k$ be the absolute oriented cellular boundary from degree $k$ to degree $k-1$. By [F2], ${\mathrm{\partial }}_1{\mathrm{\partial }}_2=0$. Since $\bar{G}={\mathrm{\partial }}_1^T$ and $\bar{C}={\mathrm{\partial }}_2^T$, transposition gives $\bar{C}\bar{G}=({\mathrm{\partial }}_1{\mathrm{\partial }}_2{)}^T=0$. For a polygonal face this is the telescoping sum of endpoint differences around its boundary, including a triangular or rectangular Yee face.

Similarly [F2] gives ${\mathrm{\partial }}_2{\mathrm{\partial }}_3=0$, hence $\bar{D}\bar{C}=({\mathrm{\partial }}_2{\mathrm{\partial }}_3{)}^T=0$. For a tetrahedral or cubical volume, each edge in the boundary of its oriented boundary has two occurrences with opposite induced signs. The cellular identity covers arbitrary cell attachments as well, so this argument does not restrict the claim to tetrahedra.

Periodic identification identifies faces with the prescribed orientation, so the quotient cell structure retains oriented boundary-of-boundary cancellation. For PEC, let $\bar{G},\bar{C},\bar{D}$ be the absolute incidence matrices and let $P_e$ select interior edges and $P_v$ select interior vertices. The matrices are $G=P_e\bar{G}{P}_v$, $C=\bar{C}{P}_e^T$, and $D=\bar{D}$. An interior vertex is incident to no boundary edge, so $P_e^T{P}_e\bar{G}{P}_v=\bar{G}{P}_v$. Hence $CG=\bar{C}\bar{G}{P}_v=0$, while $DC=\bar{D}\bar{C}{P}_e^T=0$. This retains boundary-face magnetic cochains although boundary electric edge values are fixed.