Abstract

We examine the proposed one-loop continuation of the fixed-regulator relational scalar of Landmarks X-E and X-F. The graviton normalization is retained: $P_{\mathrm{gr}}=-\frac{1}{4}\mathcal{P}\square$ and ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}=-4\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$. The fermion $\int \bar{c}(C-b)$ constructs the corresponding free auxiliary–ghost multiplet. It gives $b=C[h]/2$, the required Feynman-weight gauge fixing, and the ghost term $-\bar{c}\square c$. The independent auxiliary field makes the free differential nilpotent off shell. Its Green matrix and compatibility with the causal field algebra are proved.

Two obstacles must be distinguished. The switched scalar action has an explicitly evaluated gauge-breaking coefficient. Separately, X-F multiplies gravitational sources by the ramp after variation. An exact variational upgrade retaining that field response on all allowed backgrounds is impossible: a compact pure-gauge background in the ramp yields a nonzero reciprocity pairing on genuine free metric solutions. This obstruction is classical and cannot be cancelled by a positive-quantum-order counterterm.

Under a stated Wick graph convention, two non-tadpole contraction graphs exhaust the ${\kappa }^2\hslash$ sector. Their coefficients contain no classical background. This proves a conditional background-identity statement, not existence of the complete observable. The local scalar–graviton fish admits finite-scaling-degree extensions. A scalar model of the nonlocal coordinate contraction has a null-collinear exceptional set for the sufficient wavefront product criterion. Failure of that criterion is not called nonexistence. The required full retarded product is not supplied by local covariance alone.

The free BRST construction and the exact-upgrade obstruction are released. An unconditional renormalized one-loop background identity for the unchanged X-F prescription is not claimed.