The nonzero pairing is retained

Take $m=0$ and the free configurations

They satisfy the reduced metric and scalar wave equations. Set $b=D\gamma$, with ghosts and antighost zero.

Choose nonnegative nonzero $a\in {C_c}^{\infty }((\epsilon ,T))$ and, on a sufficiently large causal cylinder, $\xi =a(t){\partial }_t$. Multiply it by a spatial cutoff whose derivatives lie outside the observation dependence region. Choose the plateau to contain its compact support, and observe after the support of $a$. Finite propagation permits the following calculation in that cylinder.

For $r={\mathcal{L}}_{\xi }\gamma$,

There is no ramp-layer contribution. Direct zero-data integration gives

Also,

Consequently

for nonnegative nonzero observation tests in the stated region.

This is an evaluation on a free multiplet configuration. The defect is therefore not generally an equation zero. A cancellation of (eq:forward) must supply the omitted co-transformation, not replace ${\mathcal{L}}_{\xi }\gamma$ by $\mathcal{R}c$. Those transformations have different dependence on the metric.

Equation (eq:backgroundwitness) concerns the completed second-order classical candidate. Equation (eq:witness) concerns the free variation of the bare insertion. Neither is a quantum Ward identity, and neither can replace the other.