Six corrections to the production sheet

The sheet anticipated an admission proof by composition of retarded kernels, a gauge-fixing companion equal to $-I$, and a three-term sum ${\delta }_{\mathrm{split}}+{\delta }_0{\mathcal{C}}_3+{\delta }_{\mathrm{gf}}$. Each of these must be corrected before the gates can be decided.

1. The source $J_3=2{\mathcal{H}}_2[\gamma ]$ is local, unswitched and unramped, and the second coefficient has zero data on $\Sigma$. Its retarded response is ${{\mathsf{E}}_0}^{\mathrm{R}}{J}_3$ with the zero-data inverse, which truncates the source at $t=0$. The resulting kernel has a sharp onset at $\Sigma$ in a metric slot. That onset, not the finite-rank mismatch, is the obstruction (Section sec:F1).

2. The gauge-fixing companion is the retarded response to the order-$\kappa$ de Donder variation $2D^{\ast }D{s}_{\xi }$. It equals $K{\zeta }_{\mathrm{gf}}[s_{\xi }]$ with ${\zeta }_{\mathrm{gf}}=-{{\mathsf{E}}_0}^{\mathrm{R}}D{s}_{\xi }$: pure gauge, zero data, supported in $J^{+}(\mathrm{supp}\xi )$. Its ${\mathcal{A}}_1$-pairing on the plateau is zero, not $-I$ (Section sec:F2).

3. XI-E's cancellation theorem restored $\gamma \mapsto \gamma +\kappa {\mathcal{L}}_{\xi }\gamma$ by hand in a parametrization with an external second coefficient $k$. Once $k=u_3[\gamma ]$ is generated by the response, the covariant part of $u_3[\gamma +K\xi ]-u_3[\gamma ]$ is that shift. Adding ${\delta }_{\mathrm{split}}=0$, which already contains the by-hand $+I$, to ${\delta }_0{\mathcal{C}}_3=I$ counts the same term twice (Section sec:F3).

4. The quadratic flow term $\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$ of the single-field law is supplied by the response through the flatness identity $J_3[K\xi ]=-A(\frac{1}{2}{\mathcal{L}}_{\xi }K\xi )$. It is not added separately.

5. The second-order parameter selected by gauge fixing is not in ${\mathfrak{g}}_U$. Its support is $J^{+}(\mathrm{supp}\xi )\cap M_T$. The resulting remainder is a switching term of X-A's and VI's type, with no ramp layer, and it vanishes under an explicit causal plateau condition.

6. The identity is stated with the Einstein-shell term $\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}{{\mathcal{L}}^{\mathrm{dens}}}_{\xi }(A\gamma )$ retained. On the gauge-fixed free shell $A\gamma =-2D^{\ast }b$; the term vanishes when $D^{\ast }b=0$ on $U$, which holds for the witness and for every linearized Einstein solution. XI-A's distinction between the gauge-fixed and the Einstein shell is not erased.