Corollary: Metric variation of the scalar-sourced slot

The metric variation of (eq:full-coefficient), at fixed $\varphi$, equals ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}$. In particular the witness (eq:main-value) also detects that variation of the full displayed coefficient.

Proof. XII-E's scalar-sourced-slot corollary states its admission and its absence of a metric argument XIIE. Its difference under $\gamma \mapsto \gamma +K\xi$ is therefore zero.