What can legitimately be substituted

Let $S_{ϵ}$ be a smooth compact-kernel regularization of the field arguments. For finite $ϵ,R$, the Gaussian identity for a quadratic polynomial is

It is an algebraic identity once its displayed pairing exists.

This reproduces the quadratic Gaussian content needed for a regularized Wick lemma. The exact unprovided wording of an XX-A lemma is not invented.

Taking $ϵ\to 0$ requires either the subtraction proof of Section sec:Wick or a specified extension theorem. Taking $R\to \infty$ additionally requires the estimates of Section sec:IR. The two limits are not identified merely because every finite expression is defined.

A joint limit needs joint uniform bounds. None are supplied by the free commutator, by positivity of a comparison sector, or by a finite approximation.