Proposition: Monomial audit

Every nonzero displayed monomial of ${\mathcal{D}}_{\sigma }$ has exactly two metric-jet factors. There is no constant or linear term. Consequently, for every real $s$,

Using the separately imported bulk premise gives

The real source polarization

is symmetric and bilinear.

Proof. A derivative of a metric factor still has field degree one. The coefficients ${\mathsf{u}}_j$ and ${\mathsf{b}}_j$ have field degree $j$. Each summand of $C_r$ has degree $r$, and ${\Gamma }_n$ has degree $n$. Thus $C_2$, $C_0{\Gamma }_2$, and $C_1{\Gamma }_1$ all have degree two. Multiplication by fixed derivatives of $\sigma$ does not change field degree. This proves (eq:Dscale).

The first statement in (eq:Jscale) is the imported bulk premise, not a curvature expansion performed here. Adding the two source contributions proves the second statement.

More than homogeneity is used for bilinearity. Each component of the source is a finite sum of products $L(\gamma )M(\gamma )$ of linear differential monomials. For one such product,

This is symmetric and bilinear. Summing proves the assertion. Degree-two homogeneity of an arbitrary nonlinear map would not justify this step.