A sign-reversal diagnostic

Reversing only the second-vertex sign would add

to the correctly completed pure-gauge coefficient. It is not generally legitimate to replace this by a fixed multiple of $\mathbf{\varphi }(D^{2}f)$, because the term $D{P}_m(X{\mathsf{E}}_{\mathrm{A}}f)$ in (eq:pg-operators) is not generally an equation zero.

A concrete classical check is available on admissible instances with a sufficiently large plateau cylinder and causal reserve. Let $m=0$, and choose $\xi =a(t){\partial }_t$ near the relevant causal compact, with spatial cutoff outside it and $a\in {C_c}^{\infty }((\epsilon ,t_0))$, $a\ne 0$. Take the observation times after $t_0$. The time reparametrization gives locally

and zero-data retarded integration gives

After the support of $a$, sign reversal therefore produces the strictly positive evaluation

against a nonnegative observation density of integral one. All required reserve and plateau conditions are conditions on this diagnostic subfamily, not changes made to a frozen instance after discovering a defect.