Example: A mathematical source rejected by conservation

Let $f\in {C_c}^{\infty }(U)$ satisfy ${\partial }_{0}f\not\equiv 0$. Set

Then ${\partial }^{\mu }{\tau }_{\mu 0}={\partial }_{0}f\ne 0$, so no field solution exists under the comparison of Proposition prop:bianchi.

This is a comparator diagnostic, not a source extracted from an admissible operational datum.