Remark: Subtraction does not modify the old product by fiat

VII-C's counterterm discussion distinguishes a numerical subtraction from an extension preserving the old Wick product. Subtracting from the positive-cutoff coefficients changes products already fixed in ${\mathcal{P}}_{\mathrm{mc}}$. Assigning a subtracted value only at the sharp kernel does not restore sequential continuity of the unsubtracted coefficient map VIIC (thm:coefficient; Section “Counterterms and the scope of the conclusion'). VII-D explicitly retains this obstruction VIID (Section “Scope audit'). Accordingly, the finite part here is not a counterterm inserted inside the unchanged old polynomial product.