Coefficient definition

The domain is not all of $({\mathcal{P}}_{\Sigma }\otimes R_3){\otimes }_{R_3}({\mathcal{P}}_{\Sigma }\otimes R_3)$. It is not claimed closed under arbitrary intermediate multiplication. The definition is a free Wick-style formal partial product whose coefficients are actual classical polynomial functionals. It is not an interacting renormalized product, nor is it itself a finite physical quantum channel.