Proposition: Raw causal-Weyl multiplication does not extend by the stated continuity

Consider an extension containing compact local quadratic polynomials and requiring their products to be obtained by continuous distributional limits of the raw causal-Weyl contractions. Such an extension does not exist without changing that prescription.

Proof. For two local squares the second contraction requires, up to a nonzero constant,

Choose two small disjoint neighbourhoods of distinct null-related points. In suitable coordinates transverse to the light cone, the leading singularity of ${\delta }_m$ is a nonzero smooth multiple of $\delta (u)$. The massive tail does not remove it.

Let $\rho \ge 0$ be a smooth mollifier of integral one. The regularizations ${\rho }_a(u)=a^{-1}\rho (u/a)$ converge to $\delta (u)$, but

for $\psi (0)>0$. The second contractions therefore have no distributional limit. In wavefront language, both signs of the conormal occur and the diagonal pullback of the tensor square is not defined by the multiplication criterion.

A prescription requiring this limit cannot assign the proposed product. Renormalized or reordered products are different prescriptions and are not excluded by the argument.