Proposition: The free-shell algebra has no exact off-shell shadow

There is no classical-symbol map on an extension of ${\mathcal{A}}_0$ which both preserves its zero elements and sends every old scalar linear generator $\varphi (u)$ to the off-shell functional $\int u\varphi$ on all smooth scalar configurations.

Proof. Take real $w\in {C_c}^{\infty }({M_T}^{\circ })$ with $P_{m}w\ne 0$. Such tests exist. The equation relation in ${\mathcal{A}}_0$ is

Every algebra extension preserves this equality, and every symbol map preserving zero must assign it the zero functional.

The required off-shell interpretation would instead assign

Evaluate at the smooth configuration $\varphi =P_{m}w$. Its value is

This contradicts the zero relation.