An evaluated temporal check

Take $m=0$, ${\varphi }_0=t$, and a nonnegative temporal bump $g_0$ above the ramp. Use a compact spatial cutoff equal to one on a sufficiently large causal cylinder containing the observation dependence region. In that cylinder,

Indeed, ${{\mathsf{E}}_0}^{\mathrm{R}}F(t)={{\int }_0}^t(t-s)F(s)\mathrm{d}s$ for a spatially constant source, and integration by parts in ${{\mathsf{E}}_0}^{\mathrm{R}}{{\mathsf{E}}_0}^{\mathrm{R}}({\partial }_t{g}_0)$ gives (eq:temporal).

A nonnegative observation bump after the source detects a strictly negative value of $\mathcal{T}$. The bare response $-\mathcal{T}$ has the opposite sign. Replacing the inverse coefficient $-1$ by $-1/2$ would halve this number and fail the copied normalization.

This is a classical evaluation on a smooth free solution, not a quantum expectation value or an admitted cubic element of ${\mathcal{A}}_0$.