Abstract

We investigate the first geometric coefficient of a relational scalar defined using harmonic coordinates with fixed zero Cauchy data. The quantum fields are a scalar and a gauge-fixed spin-two field, together with the ghost and auxiliary sectors of perturbative gravity. The proposed insertion space is the algebra of compactly supported microcausal polynomial kernels, with a fixed graded Hadamard contraction.

For the coordinate-response operator $L={\mathsf{E}}_{0}C$, we compute its distributional adjoint exactly. It contains a hypersurface delta distribution whose nonzero conormal covectors are timelike. We prove an exact criterion for a linearly smeared coordinate response to belong to the declared domain. More importantly, for every nonzero real test function supported strictly to the future of the initial surface, the first geometric insertion fails the microcausal condition. The proof exhibits a smooth scalar configuration at which the metric derivative has a nonvanishing hypersurface delta term. This excludes a coefficientwise, classically normalized quantization of that insertion in the declared off-shell algebra, and hence excludes the proposed termwise second-order construction in that algebra.

The obstruction is not a claim that the relevant distributions do not exist. Smooth-edge regularization is constructed and its sharp limit is analyzed. A smooth spacetime representative is also constructed for each linear coordinate observable on the free mass shell. Neither result supplies the missing nonlinear interacting products. The paper therefore closes the boundary-domain alternative, not the second-order existence alternative.