Lemma: The completed Einstein response on the witness

On the observation region,

and

Proof. The trace of (eq:main-w) is $8a^{\prime }$. Consequently $\mathcal{P}{w}_{\xi }$ has only its $22$ and $33$ components nonzero, both equal to $4a^{\prime }$. The spatially homogeneous zero-data inverse satisfies ${E_0}^{\mathrm{R}}{a}^{\prime }=B$, giving (eq:main-e).

This local calculation equals the compact-source calculation: all sources capable of influencing the observation have spatial coordinates in $B_R$, where $\zeta =1$. The transitions of $\zeta$ are outside that causal past. Nested retarded operations do not enlarge the overall causal dependence region. Finite propagation therefore justifies the homogeneous formulas BGP.

For (eq:main-e), the relevant density coefficients are

The entire response is supported after $t_a>\epsilon$. Thus the coordinate ramp is one and

On the causal region where $e_{\xi }$ contributes, $g=1$ by (eq:witness-diamond). Since ${\partial }_t\varphi =2t$ and ${\partial }_1\varphi =2x^1$,

Hence

Subtracting proves (eq:main-integrand).