Theorem: Persistence of the free-equation non-descent

The natural positive-order scalar field rule $s\varphi =\kappa c^{\mu }{\partial }_{\mu }\varphi$ does not preserve the unchanged free scalar equation ideal. The additional vertex $\lambda {\kappa }^3{V}_a$ does not repair this failure.

Proof. For arbitrary smooth coefficient fields,

Choose $p>0$, $\omega =\sqrt{p^2+m^2}$, and

Both $P_m{\varphi }_0=0$ and $\square c=0$, but

which has a nonzero compact pairing on a sufficiently small test region.

Thus the variation of a free equation is nonzero on a free configuration and cannot belong to that unchanged equation ideal. The new vertex changes no field rule and begins only at order three, whereas this failure occurs at order one.