Abstract

Landmark XI-E named, and excluded from its release, the classical cubic Einstein response $u_3=\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}(2{\mathcal{H}}_2[\gamma ])$ and its scalar-completed insertion ${\mathcal{C}}_3={\mathcal{A}}_1[u_3;\varphi ]$. We decide two gates for that object at fixed regulator width, in the normalization ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$.

Admission. The second metric coefficient carries the frozen zero Cauchy data on $\Sigma$, and the Einstein–Hilbert density carries neither the switch $g$ nor the ramp ${\sigma }_{\epsilon }$. The slab kernel of ${\mathcal{C}}_3$ therefore has the form $\theta (y^0)\beta (y)j_3(y;\cdot ,\cdot )$: a sharp onset at the initial surface in a metric slot. For every frozen switch and every mass we exhibit admissible observation data for which this kernel has a timelike conormal covector $\pm \mathrm{d}t$ in its wavefront set, so it leaves the polynomial microcausal class of XI-E. The admission map of XI-E does not apply to ${\mathcal{C}}_3$. The failure is not the finite-rank mismatch and not a loop; it is the sharp-Cauchy singularity of VII-B, now in the gravitational slot, which the ramp regulates only in the coordinate operator. Whether the free-shell restriction of ${\mathcal{C}}_3$ has some other microcausal representative is not decided.

Plateau sum. From XI-B's gauge fermion, the order-$\kappa$ variation of the de Donder term under the single-field law is $2D^{\ast }Ds$, with divergence $-\square Ds$; its retarded response through the frozen inverse is the pure gauge $-2\mathcal{R}{{\mathsf{E}}_0}^{\mathrm{R}}Ds=K{\zeta }_{\mathrm{gf}}$, with ${\zeta }_{\mathrm{gf}}=-{{\mathsf{E}}_0}^{\mathrm{R}}Ds$ the de Donder second-order parameter. Hence $u_3[\gamma +K\xi ]-u_3[\gamma ]=s_{\xi }+K{\zeta }_{\mathrm{gf}}[s_{\xi }]+\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}{{\mathcal{L}}^{\mathrm{dens}}}_{\xi }(A\gamma )$, $s_{\xi }={\mathcal{L}}_{\xi }\gamma +\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$: the complete second-order metric law of X-A, with $\zeta$ selected by gauge fixing. The cubic-completed second coefficient then shifts, for a free scalar and $A\gamma =0$ on $U$, by ${\mathcal{A}}_1[K{\zeta }_{\mathrm{gf}};{\varphi }_0]$: a pure switching term carrying $1-g$ or $\partial g$, with no ramp term, vanishing whenever $J^{-}(\mathrm{supp}f)\cap J^{+}(\mathrm{supp}\xi )$ lies in the plateau. On the XI-E witness the ledger reads $-I$ (held fixed) $+I$ (covariant cubic) $+0$ (gauge-fixing response) $+0+0$ (quadratic terms) $=0$. The production sheet's literal sum ${\delta }_{\mathrm{split}}+{\delta }_0{\mathcal{C}}_3+{\delta }_{\mathrm{gf}}=0+I+0$ double counts: XI-E's by-hand co-transformation and the covariant part of the cubic response are one mechanism. The corrected plateau sum vanishes; ${\delta }_{\mathrm{gf}}$ is not $-I$, it is $0$.

Neither result constructs an interacting differential, a quantum retarded product of the cubic source, a loop value, a sharp limit, or an Einstein-jet repair.