Abstract

XII-D admitted the completed density-weighted cubic insertion ${\mathcal{C}}_{3,\mathrm{D}}$ to XII-A's carrier and proved its signed plateau variation. We assemble the second-order relational coefficient that XII-D named and withheld, on the scalar-linear sector: X-A's scalar completion with its second metric slot filled by the gauge-fixed gravitational coefficient generated by the actual source.

Two corrections to the production brief are proved rather than assumed. With XII-D's forcing convention $u_{\mathrm{D}}={G^{\mathrm{R}}}_{\mathrm{gr}}{\mathcal{J}}_{\mathrm{D}}$, the gauge-fixed second-order metric coefficient is $-u_{\mathrm{D}}$, so the admitted insertion enters the relational coefficient with coefficient $-1$: ${{\Phi }^{(2)}}_{\mathrm{D}}={{\mathcal{A}}_2}^{\mathrm{sc}}[\gamma ,0;\varphi ]-{\mathcal{C}}_{3,\mathrm{D}}$. The literal sum with $+{\mathcal{C}}_{3,\mathrm{D}}$ carries a doubled two-jet defect $-2{\mathcal{A}}_1(s_{\xi };\varphi )$ that survives every plateau hypothesis; XI-E's evaluated witness detects it. Moreover the slot is determined by the metric, so its two-jet row is the co-transformation; a separate co-transformation row would be the double count.

The exact plateau variation is the sum of a gauge-fixing companion and an Einstein row. The companion is displayed with its factors $1-g$ and $\mathrm{d}g$, lives above the ramp, and vanishes under the causal plateau condition. The Einstein row is a compact plateau test of $A\gamma$; on the gauge-fixed shell it is a compact test of the auxiliary field, $-2\int (D{\mathcal{L}}_{\xi }\beta )\cdot b$, and the carrier does not delete it. A tree-composition lemma proves that the scalar-only completion kernel is compact with a covector-sum-zero bound, so $Q_H{{\Phi }^{(2)}}_{\mathrm{D}}\in {\mathcal{E}}_H$ and the enlarged named sector is ordering-stable.

No quantum retarded product of ${\mathcal{J}}_{\mathrm{D}}$, interacting differential, state, loop, sharp-width limit, or Einstein-jet repair is constructed. XI-G's exclusion and XII-C's empty jet are retained.