Abstract

At fixed regulator width, we apply the XI-track inverse $-\mathcal{P}{E_0}^{\mathrm{R}}$ to the actual cubic source frozen in XII-C, including its ramp-derivative terms. Finite propagation gives a cutoff-independent smooth response without assuming source conservation. We then construct the first-order completion from both the divergence-form coordinate leg and the switched scalar-response leg.

The transposed root is smooth but need not be compact before localization. A compact ambient collar root is constructed; multiplication by the actual source coefficients produces compact effective roots supported at nonnegative times. The completed degree-$(2,1)$ kernel has a covector-sum-zero bound. This proves membership in XII-A's polynomial microcausal domain and gives carrier admission. Smooth changes of ordering add only regular linear fields, so the formally completed sector generated by the new insertion is ordering-stable.

The plateau calculation requires a sign correction. With the prescribed forcing response $u_{\mathrm{D}}={G_{\mathrm{gr}}}^{\mathrm{R}}{\mathcal{J}}_{\mathrm{D}}$, the finite two-jet term enters with a minus sign. We prove the corresponding exact variation, display the gauge-fixing companion and Einstein-shell term separately, and give a conditional companion cancellation. This is not an unqualified invariance theorem for the cubic insertion alone.

The new admission neither changes XI-G's original exclusion nor repairs XII-C's empty fixed-input jet. No quantum retarded product, interacting differential, state, loop, or sharp-width limit is constructed.