Proposition: The dropped admission-continuity hypothesis

The mixed kernel satisfies $\mathrm{WF}(K_f)\subseteq {\Gamma }_B$. On a sufficiently large common compact support stage,

The identity from these admitted mixed kernels to their actual functionals in $({\mathcal{P}}_{\Sigma },{\tau }_{\Sigma })$ is not sequentially continuous at $K_f$ for nonzero real $f$.

Proof. The retarded kernel has diagonal conormals and null-propagation pairs $(\xi ,-\xi )$. Multiplication by the step factor in the $z$-slot adds

At this boundary branch $x^0>0=z^0$, so a diagonal contribution cannot meet the boundary there. Finite differentiation and smooth coefficients preserve the upper cone (eq:GammaB).

Strong convergence was proved above. For a localized Fourier seminorm on a closed cone separated from ${\Gamma }_B$, use a slightly larger separated angular cone. The localized kernel has arbitrary polynomial decay in that larger cone and polynomial growth globally. Convolution regularization introduces bounded smooth multipliers converging to one on fixed frequency balls. Split the Fourier integral into a fixed ball, a near-frequency region in the larger separated cone, and its angularly separated complement. The first part tends uniformly to zero. Arbitrary rapid decay controls the second, and rapid decay of the localization transform controls the complement. Choosing the decay order larger than the requested seminorm order and the distribution growth order gives a tail bound uniform in $\epsilon$. This proves normal convergence, the standard compact regularization argument for the stated constant angular cone BDH.

Theorem thm:separator gives nonconvergence of the image functionals and hence the last assertion.