Proposition: Weighting need not preserve unweighted equivalence

For a smooth weight $\chi$ and compactly supported vector density $B^{\mu }$,

Consequently, the map “multiply a density by $\chi$” need not descend to the unweighted integration-by-parts quotient.

Proof. Integrate the derivative of $\chi B^{\mu }$. For an explicit quartic example, take a compactly supported scalar $\phi$ and $B^{\mu }={\phi }^3{\partial }^{\mu }\phi$. Then

The unweighted divergence has zero Euler derivative. The weighted expression has scalar Euler derivative $(\square \chi ){\phi }^3$, which need not vanish.