Compact IIA D4 Flux Charge Balance

Let $X_6$ be a compact oriented boundaryless six-manifold, $\gamma$ a compact smooth closed oriented one-cycle, and ${\omega }_1\in {\mathrm{\Omega }}^1(X_6)$ a smooth closed one-form. Assume $G_2\in {\mathrm{\Omega }}^2(X_6)$ and $H_3\in {\mathrm{\Omega }}^3(X_6)$ are smooth, and let ${\tilde{G}}_4$ be a degree-four de Rham current, meaning a continuous linear functional on smooth compactly supported two-forms with the test-form topology. For a degree-$k$ current $T$, use the distributional exterior derivative defined by $⟨dT,{\eta }_{5-k}⟩:=(-1{)}^{k+1}⟨T,d{\eta }_{5-k}⟩\mathrm{.}$ Interpret the adopted D4 Bianchi law as the weak equality of degree-five currents $d{\tilde{G}}_4+[H_3\wedge G_2]=2{\kappa }_0^2{\mu }_4{\delta }_5(\gamma ),$ where $⟨[H_3\wedge G_2],{\eta }_1⟩:={\int }_{X_6}{H}_3\wedge G_2\wedge {\eta }_1$ and $⟨{\delta }_5(\gamma ),{\eta }_1⟩:={\int }_{\gamma }{\eta }_1$ for smooth one-form test forms ${\eta }_1$. Then ${\int }_{X_6}{H}_3\wedge G_2\wedge {\omega }_1=2{\kappa }_0^2{\mu }_4{\int }_{\gamma }{\omega }_1\mathrm{.}$ This is a conditional weak flux-charge balance; it is not a proof of global source cancellation or a claim about the net charge in the main ${\varphi }^{2\mathrm{/}3}$ orientation.