Theorem: Native one-rod Green-response tail

Let $f\in {C_c}^{\infty }(\mathcal{M};\mathcal{V})$, and put

For every $J⋐I$, $z^{\pm }$ has uniformly compact spatial support on $J$. Its original first rod

satisfies (eq:dipolebound), uniformly on $J$.

Thus the distributional kernels ${\mathcal{R}}_x{G}_{\mathcal{B}}^{\mathrm{ret},\mathrm{adv}}(x,y)$, interpreted by testing the input slot with compact $f$, have a proved inverse-square output tail.

Proof. Causal support gives

If the spatial projection of $\mathrm{supp}f$ is contained in a ball of radius $R_f$, then on $J$ the support radius is at most

The supremum is finite on a compact interval inside $I$. The solution is smooth and its required derivatives are bounded on that compact spatial-time region.

The matrix $\mathcal{C}$ is local with smooth time coefficients. Therefore ${\text{calCz}}^{\pm }$ has the same support property. Apply Theorem thm:dipole.