Theorem: Off-shell non-admission, with a universal smearing witness

For every nonzero real $f\in {C_c}^{\infty }(M_{+}^{\circ })$,

No element of ${\mathcal{P}}_{\mathrm{mc},\mathrm{pol}}[[\hslash ]]$ has classical symbol equal to ${\mathcal{B}}_f$ on all smooth configurations.

Proof. The metric derivative is

Choose

For $b={E^{\mathrm{A}}}_0(f\mathrm{d}{\varphi }_f)$, integration of $\square b_0=f{\partial }_t{\varphi }_f$ over space gives

The advanced solution has zero final data, which fixes the first equality. The last expression is nonzero because compact support and ${\partial }_t(tf)=0$ would imply $f=0$.

Proposition prop:boundary gives a timelike wavefront component of the metric derivative. Partial contraction of an admitted microcausal polynomial kernel with smooth configurations cannot produce such a component: adjoining zero covectors in the contracted slots would violate the original cone exclusion. This proves non-admission.

An off-shell-equal classical symbol has the same derivative. Positive powers of $\hslash$ cannot alter that contradiction.