Proposition: The defect is not generally an equation zero

There are admissible plateau data and smooth free multiplet configurations for which ${\mathcal{A}}_1[{\mathcal{L}}_{\xi }\gamma ;\varphi ]\ne 0$. Therefore this classical defect does not belong to the ideal of functionals vanishing on all free configurations.

Proof. Take $m=0$ and the free tensor and scalar

They satisfy $-\square \mathcal{P}\gamma =0$ and $\square \varphi =0$. Set $b=D\gamma$ and the ghosts to zero; these choices satisfy the auxiliary free equations.

Choose a nonnegative nonzero $a\in {C_c}^{\infty }((\epsilon ,T))$. On a sufficiently large causal cylinder let $\xi =a(t){\partial }_t$, with a compact spatial cutoff outside the observation dependence region. Choose $U$ and $g=1$ near its support, and observe after the support of $a$. All calculations in that region are unaffected by the cutoff, by finite propagation.

Put $r={\mathcal{L}}_{\xi }\gamma$. Its components there are

Consequently

There is no layer contribution, and

Direct differentiation gives

and its scalar retarded response is

Thus the completed first coefficient has density

A nonnegative nonzero $f$ in the described observation region gives a strictly positive pairing. An equation-ideal functional would vanish on these free configurations, proving the assertion.