Abstract

We derive the complete kinematic second jet of the original-rod relational lapse on the power-law Einstein–scalar background reproduced in XX-C. The calculation retains every metric component, both second-jet fields, the clock embedding, the induced spatial metric, and the original slice-elliptic rod equation. The second spatial rod does not enter the quadratic lapse.

The unsmoothed classical coefficient is a continuous polynomial on a spatial Schwartz configuration domain. Its Hessian has noncompact dipole tails. After a compact source cutoff, that Hessian is microcausal: its equal-time off-diagonal covectors occur with opposite signs. Nevertheless its raw internal Hadamard contraction fails the sufficient product criterion at coincidence. We give both a Hadamard-subtracted one-contraction prescription and a transverse Taylor extension of the corresponding punctured numerical kernel.

A conditional theorem constructs the native, cutoff-free Wick symbol through quadratic perturbation order and first order in Planck's constant, together with products against compact regular probes. The theorem states separately the infrared hypotheses for internal and external contractions. Those hypotheses have not been verified for a selected XX-B/XX-C representative. Products requiring two noncompact rod legs need additional control and cannot be assigned a universal inverse-square separation bound.

The formal classical symbol recovers the printed classical coefficients exactly. This is not an expectation-value limit. The requested native quantum Q3 gate remains open wherever its noncompact contraction hypotheses remain unverified; no universal extension no-go is claimed.