Proposition: A nonzero retained component

Before summation over tensor components and branches, the temporal incidence (eq:differentiated) has a nonzero coefficient in the frozen pair. Its line differential symbols are nonzero at the selections (eq:nus).

Proof. For a tensor whose only component is ${\gamma }_{00}=h$, the displayed density formula gives

Thus $B_6$ contains the nonzero monomial

Transferring the divergence produces a differentiated retarded line with a nonzero coefficient.

The interaction contains the temporal kinetic monomial $-\frac{1}{4}g{\gamma }_{00}({\partial }_0\varphi {)}^2$. One metric cross-contraction is nonzero because ${\mathcal{P}}_{00,00}=1/2$, and one scalar cross-contraction joins the two displayed temporal scalar derivatives. The fixed species factors are nonzero.

At (eq:nus), $p_0=1$. The temporal retarded derivative has nonzero symbol, and the two scalar derivatives have nonzero product symbol. Their principal conormal singularities are therefore not removed by these differential operators. Finally, the original external weight is nonzero at the retained triple fixed in J0.