Abstract

We construct a local Grade S realization of Cheung's fixed-regulator scalar-only X-B prescription on the supported host presentation already fixed by X-A-ST. The quantum field is the free Klein–Gordon scalar on the declared Minkowski completion; the metric coefficients remain classical and off shell. The host geometry is not reconstructed or evolved.

At every fixed positive ramp width, the bare relational dressing is admitted through second background order, both by a joint polynomial wavefront test and, separately, as a scalar-linear functional with a smooth compact test. For every specified off-shell linear representative $F_c$, the normalized response is

We prove continuity, retarded support, independence from the Hadamard contraction kernel, and the absence of positive-$\hslash$ terms in its output test. The completed first coefficient is invariant in the unchanged free algebra on the gauge algebra frozen before the calculation. An explicit equation-ideal representative separates this identity from the retained edge and switch defects.

The four X-B-ii scope closures are incorporated into the same file. Classical normalization on the entire insertion domain determines the degree-one output class; compatible contact modifications are therefore algebraically zero. Conversely, the one-contraction verification does not construct a time-ordered family: the Feynman kernel's square fails the ordinary wavefront product criterion on the diagonal. This is a limitation of the present construction, not a nonexistence theorem for renormalized products.

The contribution is a supported transfer, normalization audit, and explicit scope separation of the source results, not a new priority claim for them. Its potential value is a reproducible starting domain for later response analysis without conflating free-field admission, classical covariance, singular-product construction, and measurement. A finite destructive-clock interface encodes supplied evaluations; it does not acquire a continuum field.