Theorem: The literal retarded-factorization requirement fails

For nonzero displayed entries, the additional requirement

is incompatible with (eq:R-definition), ordinary zeroth-order time ordering, and the contraction grading in (eq:order). No counterterm in the class below repairs this incompatibility.

Proof. The classical symbol of the right side is the homogeneous degree-six polynomial ${\mathcal{V}}_g\mathcal{O}$. The classical retarded coefficient on the left is a one-contraction polynomial of degree four. Equality forces the degree-six polynomial to vanish identically.

For two nonzero bosonic polynomial functionals their ordinary product is nonzero: restrict to a finite-dimensional space on which both restrictions are nonzero, and use the integral-domain property of the polynomial algebra there. The displayed entries contain no odd variables. Thus the required equality is impossible. All counterterms allowed below enter the degree-two quantum coefficient and cannot alter this classical comparison.