On-shell closure and finite-region symplectic flux

Let $\mathcal{L}$ have a chosen first-variation decomposition with Euler functional $E_{\varphi }$. Take a smooth two-parameter field family and its commuting coordinate variations ${\delta }_1,{\delta }_2$. If $E_{\varphi }=0$ and each variation satisfies the linearized equation, meaning $({\delta }_i{E}_{\varphi })[\eta ]=0$ for every smooth test variation $\eta$ with the input held fixed as in the definition, then $d_M\omega =0$. For a finite oriented smooth singular four-chain $C$ with $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+B$, where the ${\mathrm{\Sigma }}_i$ are oriented three-chains and $B$ is the remaining oriented boundary chain,

${\mathrm{\Omega }}_{{\mathrm{\Sigma }}_2}-{\mathrm{\Omega }}_{{\mathrm{\Sigma }}_1}=-{\int }_B\omega \mathrm{.}$

Thus on-shell closure alone gives independence between cuts only when the side flux vanishes.

Some Properties of Noether Charge, §3

Advanced Lectures on General Relativity, §1.3.1

Covariant phase space with boundaries, §2.2

The chain is finite and compact in the sense of being a finite sum of compact simplex domains. No limit at spatial infinity, falloff assumption, or convergence of an exhaustion is included.