Proposition: Full B 6 leading fiber numerator

The complete density variation satisfies

After the metric propagator fiber and the principal kinetic stress variation are contracted, the corresponding $B_6$ leading differential numerator is a fixed nonzero conventional factor times

It is not identically zero as a fiber-tensor polynomial.

Proof. The principal kinetic variation of the stress tensor has fiber coefficient

In four dimensions ${\mathrm{tr}}_{\eta }S=-\lambda$, so trace reversal gives

The signs and factors from ${{\mathcal{V}}_g^{\prime \prime }}_{\gamma \varphi }$ and the fixed metric line are nonzero. The mass term has lower scalar-line differential order and is not part of this principal differential numerator.

Because $p^2=0$, the $b_2{\eta }^{\alpha \mu }$ term makes no contribution after contraction with $p_{\alpha }{p}_{\mu }$. Write $\tau ={\mathrm{tr}}_{\eta }G$. The remaining quadratic expression is

For variation $r$,

Its directional derivative is therefore

proving (eq:B6identity).

The retarded divergence and observation scalar derivative supply the two factors of $p$ in (eq:B6identity). The frozen scalar line normalizations and the nonzero kinetic coefficient supply only fixed nonzero factors.

For example, $G_{00}=1$ with all other components zero, $p=(1,-1,0,0)$, and $k=(1,0,0,0)$ give $Q=\lambda =1$. The original weights are nonzero at the retained configuration.