Abstract

We give a supported fixed-width realization of Cheung's Landmark X-E on the conical host and Minkowski completion already retained by X-A-ST, X-C-ST, and X-D-ST. The host is not reconstructed or evolved.

Starting from the Palatini identity, we derive the Einstein–Hilbert quadratic form, its positive transverse-traceless kinetic sign, the Feynman-weight de Donder operator ${\mathsf{P}}_{\mathrm{gr}}=-\frac{1}{4}{\mathsf{P}}_{\mathrm{tr}}\square$, and its retarded inverse. The switched total action has an explicit Noether defect. Its invariant two-jet equations require $({\partial }_{\mu }g)T^{\mu }{}_{\rho }[{\varphi }_0]=0$. For every nonconstant fixed switch, an open dense set of smooth free-scalar inputs violates a pointwise necessary condition; their coupled invariant-jet fibres are empty, although the gauge-fixed retarded response exists.

We construct the isolated scalar-sourced tree by inserting $-2{\mathsf{P}}_{\mathrm{tr}}{{\mathsf{E}}_0}^{\mathrm{R}}(g:T[\mathbf{\varphi }]:)$ into the affine second-metric slot of the completed scalar observable. An explicit three-slot distribution proves microcausal admission of the resulting Wick cubic. We distinguish its distributional kernel from its smooth contracted tests, correct the support bound to $C^3$, and prove fixed-support continuity. Its coordinate response is

With the free graviton held fixed, the complete one-tree expression is invariant through order $\kappa$, but has the order-${\kappa }^2$ defect $-{\hat{A}}_1[{\mathcal{L}}_{\xi }\mathbf{r}]$. A supported free-solution witness proves that this defect is not an equation-zero representative.

The principal action and tree constructions are credited to X-E. The contribution of this transfer is an explicit host-compatible proof and a corrected support, topology, jet-order, and truncation audit. These distinctions provide a usable baseline for subsequent tree completions without mistaking one admitted scalar cubic for an interacting quantum metric or a loop construction. A finite supplied evaluation dictionary has a complete destructive-clock instrument and reference-preserving scheduled transport. The release is local Grade S only.