What a continuity claim would require

A possible source topology can be specified on kernel stages. For compact $K_n$ and a closed cone ${\Gamma }_n$ satisfying (eq:mc), use compactly supported distributions with $\mathrm{WF}(u_n)\subset {\Gamma }_n$ and the normal distribution topology. Besides strong distribution seminorms, its microlocal seminorms have the form

with $(\mathrm{supp}\psi \times V)\cap {\Gamma }_n=\varnothing$. For the insertion, the flat ramp permits the kernel to be viewed as a compact distribution on the surrounding Minkowski space, including its possible closed support at $t=0$.

This names a source-side continuity question. It does not name a quantum target topology or prove continuity of $Q$. The distributional operation results of BDH concern specified wavefront spaces and specified operations; they are not an automatic completion theorem for a quantum algebra.

Likewise, the causal commutator on linear fields does not uniquely define multiplication of all distributional polynomial kernels. The formal causal-Weyl expansion would require repeated contractions. Even for local quadratic symbols, its second contraction involves products whose usual wavefront multiplication criterion is not satisfied. That observation calls for additional product data; it is not a universal nonexistence theorem.

No state is required merely to ask this admission question. Conversely, the absence of a selected state supplies neither the missing product nor an obstruction to every possible state-independent extension.