Proposition: The compact-interior source theorem is insufficient

Both $\mathrm{Re}{F_{\mathrm{co}}}^0$ and $\mathrm{Im}{F_{\mathrm{co}}}^0$ are spatially noncompact on the slab. In particular, $F_{\mathrm{co}}\notin {C_c}^{\infty }({M_T}^{\circ };{\mathbb{C}}^4)$.

Proof. On the plateau,

Neither the real nor the imaginary part of the last expression vanishes identically on $(\epsilon ,T)$. Hence neither corresponding part of ${F_{\mathrm{co}}}^0$ can vanish identically there. Each is spatially constant, and is nonzero at every spatial point at some plateau time. It cannot have compact spatial support.