Theorem: Above-ramp coefficient identity

For every odd $c\in {C_c}^{\infty }(\{t>\epsilon \};TM)$,

off shell in $\gamma$ and $\varphi$.

Proof. The tensor $a_1[\mathcal{R}c]$ is supported where ${\sigma }_{\epsilon }=1$. Consequently

Both fields have zero initial data because $c$ vanishes near $\Sigma$. Cauchy uniqueness gives the first identity. The second follows from $s_0{\mathcal{B}}_f=-\int f(L_{\epsilon }\mathcal{R}c)\cdot \partial \varphi$.