Theorem: Compact one-contraction realization

For every finite $R$, (eq:IR) is a well-defined free Wick symbol through the retained orders. Its quadratic field kernel is $K_{f,R}$, and its scalar correction is exactly (eq:Gamma). All terms in the clock-and-rod jet (eq:ncomplete) are represented.

Proof. Proposition prop:microcausal gives compact microcausal kernels. The finite-component algebraic Wick construction is therefore available, using the contractions of (eq:W); its distributional multiplication uses the positive-frequency relation and does not require a positive state representation.

The change from the declared symmetric subtraction to the $W$-symbol coordinates contracts the symmetric Hessian once with $D$. For the local terms this is (eq:gammaloc); for the rod term it is (eq:Gamma). The latter is finite by compact source support and local integrability.

Since ${F_{f,R}}^{(2)}=s^2{K}_{f,R}$, the normal-ordering change is $(\hslash /2)⟨D,{F_{f,R}}^{(2)}⟩$, giving the coefficient in (eq:IR). The central linear second-jet term generates no internal free contraction.