Abstract

At fixed $\epsilon >0$, with $\varphi$ the free quantum Klein–Gordon field and $h,k$ classical backgrounds, the bare ramp dressing is admitted through ${\kappa }^2$. We distinguish joint microcausal admission of its polynomial kernels from the scalar-only linear test domain actually used for the quantum construction.

The single map $R_1(V_{1,g};\cdot )$ is constructed on specified off-shell insertion tests. Its output is a free field with a smooth compact test, it has the stated retarded support, and it is tree-exact: all positive-$\hslash$ corrections vanish. Its action on the first dressing coefficient equals the mixed term of the classical ramp expansion.

The completed coefficient ${\hat{A}}_1$ inherits the ramp background-gauge identity on the algebra frozen in Landmark X-A. The bare ${\kappa }^2$ dressing is admitted and its classical transformation is retained, but no full quantum identity at that order is claimed: ${\rho }_{2,g}$ is displayed and is not supplied by a constructed map. Edge and switch defects remain outside the claimed algebra. The dressing survives at zero switch. No sharp limit, graviton tree contribution, loop triangle, or two-point response is constructed.