The copied ledger and the evaluation identity

For $\xi \in {\mathfrak{g}}_W$ and a free scalar, (eq:ledger-input) is the finite variation of (eq:O) under $\gamma \mapsto \gamma +K\xi$ at fixed $\varphi$; the scalar-sourced slot has no metric argument and contributes zero to ${\Delta }_{\xi }$. The two rows are the propagated companion and the auxiliary pairing; nothing else.