Theorem: Complete local free-scalar benchmark

Consider the real massless scalar in its vacuum Fock representation on $M={\mathbb{R}}^{1,3}$, with the Weyl net and domains specified in Section sec:model. Then:

1. $Q=:{\varphi }^2:$ and $L=-Q/2$ belong to the same physical Wick-field extension of the original scalar theory. Their mixed ordered products exist on a common invariant dense domain containing the original vacuum-word domain. Every real compact smearing of $L$ has a self-adjoint realization affiliated with the original local Weyl algebra containing its support.

2. The improved tensor

   $$t_{\mu \nu }=:{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi :-\frac{1}{2}{\eta }_{\mu \nu }:{\partial }_{\rho }\varphi {\partial }^{\rho }\varphi :+\frac{1}{6}({\eta }_{\mu \nu }\square -{\partial }_{\mu }{\partial }_{\nu })Q$$

   is symmetric, conserved, and traceless on that domain.

3. Its compact-slab currents implement all conformal Killing transformations on every compactly smeared Wick polynomial, all descendants, and all finite ordered mixed products. The elementary retarded contact is

   $${{\partial }_{\mu }}^x{C_X}^{\mu }(x,y)={\delta }^{(4)}(x-y)(X^{\nu }{\partial }_{\nu }+{\sigma }_X)\varphi (y),{\sigma }_X=\frac{1}{4}{\partial }_{\nu }{X}^{\nu }.$$

   Descendant contacts and the corresponding complete field actions are given in Sections sec:contacts and sec:charges.

4. The sign-adjusted local derivations ${\delta }_X$ defined in Section sec:brackets satisfy

   $$[{\delta }_X,{\delta }_Y]={\delta }_{[X,Y]}$$

   on the complete ordered Wick-field algebra.

5. Within the stress-tensor class of Definition def:stressclass, the complete residual family has a proved finite-dimensional span. Two explicit probes are injective on that span. Their exact vanishing forces the displayed tensor $t$, and therefore all the local Ward identities just stated.

No assertion is made that the hypotheses of the supplied Family 282 claim have thereby all been verified. In particular, no global conformal action on the whole bounded net is constructed.

Proof. Assertion (i) is proved in Propositions prop:domain and prop:physicalL. Assertion (ii) is Proposition prop:improvement. The contact calculation is Proposition prop:elementarycontact; implementation and mixed products are Theorem thm:localaction. Lie compatibility is Proposition prop:brackets. Classification and complete separation are Theorems thm:classification and thm:coverage. The exclusions are the scope of those statements, not additional enhancement claims.