Lemma: Real polarized response

For real smooth symmetric tensor fields $a,c$, define

Then $B$ is symmetric and real bilinear,

Proof. The imported response-existence assertion applies to $a+c,a,c$. Linearity of $L$ and (eq:sourcepolarization) give the displayed equation. The initial data pass through the same linear combination.

The polarization of a homogeneous quadratic differential polynomial is symmetric and bilinear. Therefore each additivity or real-homogeneity defect of $B$ satisfies the homogeneous equation $LZ=0$ with zero initial data. Lemma lem:uniqueness makes every such defect zero. Symmetry also follows directly from (eq:B).