Abstract

We complete the fixed-width scalar-only response sector of X-B-ST on the supported host already prescribed by X-A-ST. The new constructions are a second-vertex map and an iterated first-vertex map on specified off-shell linear insertions. Both are defined directly by a finite formal test-level Møller expansion, without a time-ordered product hierarchy. Their smooth compact output tests satisfy retarded support and continuity bounds, and their sum reproduces the inherited second scalar response.

The completed relational field is background invariant through order ${\kappa }^2$ in the unchanged free scalar algebra on the gauge algebra frozen before the calculation. The proof gives an explicit three-term variation: a compact free-equation term, a dressed coordinate defect, and a dressed switch defect. In particular, it retains the propagation of the first-order edge gap into the second coefficient. A complete pure-gauge two-jet closes the remainder left in X-B-ST, while a massless conformal example checks the iteration and its signs.

These coefficients contain no positive power of $\hslash$ in their tests and no Hadamard contraction kernel. This tree-exactness concerns the specified linear rooted prescription, not arbitrary interacting products. The diagonal product boundary of X-B-ST remains unchanged.

The contribution is a supported realization and normalization, sign, and defect audit of Cheung's X-C construction, rather than a priority claim for the underlying quadratic-field method. Its prospective value is an explicit second-order baseline that separates background covariance from singular-product admission and finite readout. A complete destructive-clock instrument encodes supplied evaluations and admits reference-preserving regular transport. It does not acquire a continuum field, evolve the host metric, or establish spacetime emergence.