Proposition: The admission distinction is substantive

For suitable nonzero $f$ supported above the ramp, the actual off-shell mixed kernel $b_f$ is not a kernel of the form (eq:finite). The local vertex (eq:vertex) likewise does not have a smooth finite-rank off-shell kernel. Consequently the joint bound does not place either input in XI-B's declared off-shell generator algebra.

Proof. Choose $f$ nonzero near a point above $\epsilon$, where ${\sigma }_{\epsilon }=1$. Microlocally at a noncharacteristic diagonal covector, the wave inverse has principal symbol proportional to $({\eta }^{\alpha \beta }{p}_{\alpha }{p}_{\beta }{)}^{-1}$. The two endpoint derivatives in $b_f$ give principal symbol proportional to

For $p=\mathrm{d}t$ and $ab=00$, the numerator is nonzero: ${\mathfrak{a}}^{00,00}=-1/2$. The kernel therefore has a nonzero diagonal singularity and is not smooth. Every kernel in (eq:finite) is smooth.

The vertex kernel contains differentiated diagonal delta distributions linking its three field slots. Its nonzero principal derivative coefficients cannot be represented by a smooth kernel. This gives the same distinction for the vertex.