The copied switching display

For free $\varphi$, put $q_{\xi }={{\rho }_{\xi }}^{\mu }{\partial }_{\mu }\varphi$ and $v_f={E_m}^{\mathrm{A}}f$. The display from XII-E is

where

It follows from the free-scalar identity $V_{1,1}[K\rho ]\varphi =-P_m(\rho \cdot \partial \varphi )$, the zero-data relation ${E_m}^{\mathrm{R}}{P}_m(g{q}_{\xi })=g{q}_{\xi }$, and Lemma lem:no-ramp. We use it as the handed switching formula, not as a new cubic response.