Abstract

At the fixed smooth regulator of Landmark X-B-v3, we construct the second-vertex scalar map and the iterated first-vertex scalar map on the frozen off-shell linear insertion tests. The scalar alone is quantum; the metric coefficients remain classical and off shell. The maps have smooth compact retarded output, and their sum equals the second scalar response copied from Landmark X-A. They are the corresponding coefficients of a test-level retarded Møller map, characterized by a Yang–Feldman recursion.

The completed relational coefficient through ${\kappa }^2$ is background-gauge invariant in the on-shell free algebra on the unchanged ramp plateau algebra. The proof of record is a three-term variation formula: a compact free-equation term, a dressed coordinate defect, and a dressed switch defect. The switch derivatives are displayed before imposing the plateau conditions. The second-order variation retains propagation of the first-order edge gap.

A complete pure-gauge two-jet verifies the operator signs and closes the remainder left in X-B-v3, with the bare second dressing retained. A conformal massless diagnostic gives a classical number and checks the sourced signs. The maps have no positive-$\hslash$ remainder in the specified linear rooted prescription. No general retarded-product hierarchy, sharp limit, quantum metric, or two-point response is constructed.