Proposition: Rebasing, enlargement, and weighting

The following distinctions hold.

1. Under the same response prescription, replacing $m$ by $m+{\sum }_{\ell }{\rho }_{\ell }{n}_{\ell }$, with constant allowed coefficients, preserves global membership and finite augmented rank.

2. A compatible enlargement of the contact class preserves the existing finite fit. Its complete rank and left kernel must be assessed anew.

3. Multiplication of a density by a nonconstant weight need not descend to the unweighted integration-by-parts quotient.

Proof. For (i), the change lies in $\mathcal{C}$; finitely it is an elementary augmented-column operation. Fitting vectors change by $\rho$, and every old separator annihilates the change.

For (ii), the same allowed combination still fits the target. Any annihilator of the enlarged image also annihilates that combination, but the image may be larger.

For (iii), for compactly supported $B^{\mu }$,

For $B^{\mu }={\phi }^3{\partial }^{\mu }\phi$,

which is not generally a divergence. One may take $\phi$ to be the trace of a compactly supported tensor field.