A compact joint test with the same exact value

Choose any $v\in \mathcal{D}(M;\mathbb{R})$ with $\int v=1$, and set

This is compact and smooth; (eq:joint-relative) gives $F_{H_{\ast }}=F_{\ast }$. Hence

This is an exact joint-test certificate. A general joint weight is not automatically one product $f\otimes h$. The next subsection provides a single product experiment instead.