Two different specifications

Let ${\ell }_3$ be a cubic density normalized so that $\mathsf{EL}({\ell }_3)=J_3$. There are then two distinct prescriptions:

1. Source weighting: prescribe the local expression ${\sigma }_{\epsilon }{J}_3$ as the new forcing term.

2. Density weighting: prescribe ${\sigma }_{\epsilon }{\ell }_3$ before variation, whose Euler expression is ${\sigma }_{\epsilon }{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon }[{\ell }_3]$.

The production sheet asks for a density-ramped vertex but also writes a response driven only by ${\sigma }_{\epsilon }{J}_3$. That combination needs justification; it is not a consequence of multiplication by a smooth ramp.

The next section shows that the omitted term is genuinely nonzero for the displayed Einstein–Hilbert convention. It is not merely a possible correction for unrelated models.