Proposition: Sampling support of the companion

After transposition onto $v_f$, the terms in (eq:companion-display) sample ${\rho }_{\xi }$ and its derivatives only in

This set is above the ramp. Under (CP), ${\mathfrak{C}}_{\xi }=0$.

Proof. The local term is supported in $\mathrm{supp}((1-g)f)\cap \mathrm{supp}{\rho }_{\xi }$. The advanced test $v_f$ is supported in $J^{-}(K_f)$. Every coefficient of the second term contains a derivative of $g$, whose support lies in $\mathrm{supp}\mathrm{d}g$; derivatives of ${\rho }_{\xi }$ have support contained in $\mathrm{supp}{\rho }_{\xi }$. Apply (eq:rho-support). Cancellation under (CP) is Input statement input:CP.