Abstract

XII-E assembled the scalar-linear second-order relational coefficient and left its gauge-fixed remainder explicit. We decide that remainder without changing the source, inverse, insertion sign, or parameter space.

The propagated companion has no ramp-layer contribution. Its switching display is retained, its cancellation under the copied causal plateau condition is stated with that hypothesis, and an explicit off-condition diagnostic shows why universal cancellation cannot omit a support assumption.

The Einstein remainder equals a compact auxiliary pairing on the gauge-fixed free shell. It vanishes on configurations satisfying $D^{\ast }b=0$ near the parameter support, but it does not vanish on the whole shell. We exhibit a massless free scalar, a free pure-trace metric with affine auxiliary field, a compact plateau parameter, and an observation satisfying the causal plateau condition for which the assembled variation is strictly negative. The calculation also fixes the value of the auxiliary pairing. Its compact microcausal kernel defines a nonzero class in the inherited carrier.

Unqualified classical invariance of the named coefficient is therefore Proved obstructed. A cancelling interacting co-transformation is not supplied by the free differential; its interface is recorded, not constructed. The scalar-sourced slot is included once and has zero metric variation. No quantum retarded product of the cubic source, state, loop, interacting differential, sharp-width limit, old-class repair, or Einstein-jet repair is introduced.