Abstract

We identify one relational lapse in an explicitly specified Einstein–scalar theory on a power-law inflationary background. The clock is the scalar field. Spatial labels are harmonic coordinates on its level sets, with the asymptotic Cartesian boundary condition. We use the original inflaton-slice harmonic rods, rather than the modified rods employed in some simplified cosmological formulas.

The complete classical expansion through second order is derived in tensor components. It retains arbitrary metric perturbations, second-order field data, the clock displacement, and the spatial coordinate correction. At this order the lapse depends on the first spatial rod correction but not on the second. The resulting quadratic functional has an explicit local part and an instantaneous spatial-Green part.

We prove that compact external smearing does not generally make the functional derivatives spatially compact. We also calculate the off-diagonal wavefront set of the instantaneous Green kernel and show, by an exact distributional example, that an internal diagonal correction can have a non-diagonal observable-level image. Quantitative tail and regulator-comparison criteria separate spatial integration from ultraviolet removal.

The quantum input gate is not passed: a complete gauge-fixed covariance, a state or parametrix with the required spatial control, an admissible counterterm classification, and the retained-order quantum gauge identity are not constructed. Accordingly, the release is an incomplete quantum audit, not a regulator-removal theorem or a universal obstruction. The classical formulas and auxiliary distributional results are proved independently of the declared models in Em02-C and Em02-D.