Abstract

We give a self-contained, model-specific continuation of the improvement and finite-fit programme described in Oai282-A. The model is the vacuum Fock representation of the real massless scalar on four-dimensional Minkowski space, with its original Weyl local net. We construct the improvement scalar $L=-\frac{1}{2}:{\varphi }^2:$, prove its mixed-product and common-domain properties, and establish affiliated realizations of its compact smearings in that same net. We calculate the elementary retarded Ward contact, its descendant contacts, and the local stress-current action on the complete Wick-polynomial field algebra. Translations, Lorentz transformations, dilations, and special conformal transformations are included. Successive actions obey the conformal Lie algebra with an explicitly fixed sign convention.

A separate classification theorem identifies every conserved stress tensor in a declared homogeneous, Poincare-covariant local Wick-polynomial class. Its complete residual family is two-dimensional, and two explicitly normalized matrix-element probes separate it. Consequently finite vanishing gives the full local Ward conclusion within that class, rather than merely a fit on named tests.

The conclusions concern local current implementation and ordered Wick-field products. They do not assert a general scale-to-conformal theorem, a complete renormalized time-ordered stress-tensor insertion scheme, global integration of the conformal generators, or an interacting improvement-scalar reconstruction. The distinct characteristic-product programme of XIV-I, XIV-J, and XV-B is not completed. No coefficient from that programme is admitted; O.1 remains NOT_CLOSED.