Closed electric and magnetic two-forms transfer their periods to the horizon

For any surface $A$ in the Statement, the fundamental class and smooth-chain comparison supply a cycle $z_A$ as specified. The restricted form $i_A^{\ast }\alpha$ is a smooth top form on the compact oriented boundaryless surface, so the top-integral bridge gives ${\int }_A\alpha ={\int }_A{i}_A^{\ast }\alpha ={\int }_{z_A}{i}_A^{\ast }\alpha$. To compare with a chain in $U$, write $z_A={\sum }_j{a}_j{\sigma }_j$. Functoriality gives ${\sigma }_j^{\ast }{i}_A^{\ast }\alpha =(i_A\circ {\sigma }_j{)}^{\ast }\alpha$ for each simplex; summing the defining simplex integrals gives ${\int }_{z_A}{i}_A^{\ast }\alpha ={\int }_{(i_A{)}_{\mathrm{\#}}{z}_A}\alpha$. Every sum is finite. For each affine face map ${\delta }_k$, $(i_A\circ {\sigma }_j)\circ {\delta }_k=i_A\circ ({\sigma }_j\circ {\delta }_k)$, so the signed face sums also give $\mathrm{\partial }(i_A{)}_{\mathrm{\#}}=(i_A{)}_{\mathrm{\#}}\mathrm{\partial }$. The identical reasoning applies to $\beta$. Choice is used only as required by the cited top-integral and smooth-chain suppliers.

In the cobordism case let $j:B_R\hookrightarrow U$ be its inclusion. The smooth two-form $j^{\ast }\alpha$ has compact support because $B_R$ is compact. The containment hypothesis gives $j^{\ast }(d\alpha )=j^{\ast }(d\beta )=0$ everywhere on $B_R$. Manifold Stokes and commutation of exterior derivative with pullback give ${\int }_{S_R}\alpha -{\int }_S\alpha ={\int }_{\mathrm{\partial }{B}_R}(j{\mathrm{\mid }}_{\mathrm{\partial }{B}_R}{)}^{\ast }\alpha ={\int }_{B_R}d(j^{\ast }\alpha )={\int }_{B_R}{j}^{\ast }(d\alpha )=0$. The boundary components have precisely the stated orientations. The same computation applies to $\beta$; thus the usual compact annulus is covered directly, without identifying a manifold with a singular chain.

In the singular-chain case apply Stokes in $U$ to the actual finite three-chain $C_R$. Each of its nonzero simplices has image in the region of closedness, so the pullbacks of $d\alpha$ and $d\beta$ to every such simplex vanish. Using step 1.1 to identify the boundary evaluations gives ${\int }_{S_R}\alpha -{\int }_S\alpha ={\int }_{\mathrm{\partial }{C}_R}\alpha ={\int }_{C_R}d\alpha =0$. The same argument gives the equality for $\beta$. If closedness is initially known only in the open exterior and the forms extend smoothly to the collar boundary, the coefficients of their exterior derivatives extend continuously there and are limits of zero; hence the same calculation remains valid at that boundary.

For a homologous closed surface $A$, replace the boundary of $C_R$ in step 2.1 by $(i_A{)}_{\mathrm{\#}}{z}_A-(i_S{)}_{\mathrm{\#}}{z}_S=\mathrm{\partial }C$ and use the same bridge to obtain ${\int }_A\alpha ={\int }_S\alpha$ and ${\int }_A\beta ={\int }_S\beta$. If $z_A$ is replaced by another smooth representative of $[A]$, their difference is a smooth boundary by the smooth-chain comparison; its pushforward is a boundary by postcomposition, so the hypothesis and the computed integrals do not change. The same applies to $z_S$. The orientation on $S$ is the one in $S_R-S$, opposite to the inner-boundary orientation of the truncated exterior. In a source-free geometric electrovacuum datum, $\alpha =F_q$ and $\beta ={\star }_{g_q}{F}_q$ satisfy exactly the closedness assumptions; dividing the two period equalities by $4\pi$ therefore preserves both magnetic and electric charges. This proves the Statement.