When the gauge-fixing companion vanishes

Without the stated causal condition the gauge-fixing row is not deleted. Without the Einstein condition the last row is not deleted. In particular, no universal zero variation of ${\mathcal{C}}_{3,\mathrm{D}}$ alone is asserted.

These are classical identities for named polynomial lifts. The compact shift $K\xi$ need not preserve the gauge-fixed free shell, so no action of ${\mathfrak{g}}_U$ by automorphisms of ${\mathcal{E}}_H$ is inferred. There is no interacting differential or quantum Ward identity hidden in the ledger.