Free algebra and the representative warning

Use X-B-v3's free polynomial Wick algebra ${\mathcal{A}}_{\mathrm{off}}$ and its equation ideal ${\mathfrak{J}}_0$, generated by $F_{P_{m}v}$ for $v\in \mathcal{D}(M)={C_c}^{\infty }(M;\mathbb{C})$. Set

The relevant relations are

where ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ are the Green operators of $P_m$. The linear output topology is the quotient test topology. The insertion domain is instead

identified with actual off-shell tests.

This distinction cannot be suppressed. The copied first map satisfies

and the second expression need not vanish. Consequently no iteration in this paper is performed through an on-shell input class.