Compensator and homogeneous-data dependence

Let $C,C^{\prime }$ be right inverses of $K$ on the admitted compact gauge class, and put $A_C=C^{\prime }-C$. Then $A_{C}K=0$, and, on the same underlying coefficient triple,

If $A_{C}e={\eta }_e$ and $A_{C}r={\eta }_r$ are smooth near $\mathrm{supp}f$, integration by parts gives

These compact smooth smearings have one-particle vectors and change the represented polynomial by

They need not vanish. For example, $C^{\prime }(qe+\pi r+K\xi )=\xi +q\eta$ is another right inverse for any fixed smooth $\eta$. Changing zero correction data by a homogeneous solution instead adds $\kappa \int fH{\varphi }_0$, a distinct change. No identification of these prescriptions follows from their right-inverse property.