Lemma: Minimal extension and form domain

Every numerical cross-contraction in this insertion pair has relative scaling degree at most three in four dimensions. Its extension preserving that coefficientwise degree is unique. The compact quadratic vertex preserves ${\mathcal{D}}_{\varphi }$, and the required two-insertion expressions define continuous forms in the finite-sector Schwartz topology.

Proof. There is only one scalar field at the second insertion. The cross-contractions are therefore $W_m$, ${W_m}^F$, or one first derivative of either distribution. No double cross-line coefficient occurs.

Rescaling momentum gives $W_m(ϵz)={ϵ}^{-2}{W}_{ϵm}(z)$. After spacetime testing, rapid momentum decay and the locally integrable bound $(|p{|}^2+{\nu }^2{)}^{-1/2}\le |p{|}^{-1}$ prove convergence to the massless distribution as $\nu \downarrow 0$. The same argument on the two time half-spaces applies to ${W_m}^F$. Thus their scaling degrees are two, and a first derivative has degree at most three.

A difference of extensions is supported at the relative origin. Every nonzero such distribution has scaling degree at least four, so it is excluded. This is the subcritical extension mechanism of BF.

For the domain statement, expand the compact quadratic vertex in creation and annihilation operators. Pair kernels contain the Fourier transform of a compact smooth coefficient at a sum of positive energies and decrease rapidly in both momenta. Exchange kernels have polynomial growth along the momentum diagonal and rapid decay in differences; they map Schwartz wavefunctions to Schwartz wavefunctions. Finite particle shifts preserve the algebraic domain.

Completely normal-ordered finite-sector matrix elements are smooth in the insertion coordinates. In the contracted terms, smearing the numerical line against $f$ first produces a smooth function; compact coefficient multiplication and integration by parts leave a compactly smeared linear field. These operations give the stated finite-sector form bounds.