Corollary: The problem is already in the quadratic kernel

The compact bilinear coefficient kernel of ${{\mathcal{B}}_f}^{(1)}$ does not satisfy the cone condition ${\Xi }_2$.

Proof. If that kernel satisfied ${\Xi }_2$, contraction of its scalar slot with the smooth field ${\varphi }_f$ would produce a metric kernel satisfying ${\Xi }_1$, by the proof of Lemma lem:derivative-condition. This contradicts (eq:metric-derivative) and Lemma lem:witness.