Abstract

We give a supported realization of Cheung's Landmark X-F on the conical host and independently declared Minkowski completion retained by X-A-ST, X-D-ST, and X-E-ST. At fixed positive ramp width, background de Donder gauge is applied to the quantum fluctuation, while the classical metric coefficients remain unconstrained.

The mechanism is a typed commutator identity:

It yields a retarded second-order response whose background variation is ${\mathcal{L}}_{\xi }\mathbf{r}$ modulo the free graviton equation ideal. The mixed tree consequently cancels the one-tree defect of X-E-ST. Together with the matter and self-interaction trees, the completed observable is background invariant through order ${\kappa }^2{\hslash }^0$, with both free fields held fixed. The exact Einstein-density covariance identity retains its off-shell Lie term; no linearized Einstein constraint is imposed on the gauge-fixed free graviton.

We construct all three kernels, prove compact support and microcausal admission by a three-slot sum-zero argument, and exhibit the graviton-slot equation representative needed for the cancellation. The matter kernel retains support $C^3$; the mixed and self kernels can reach the initial-layer causal reserve. Smooth scalar-slot contractions lie in ${\mathcal{D}}_C$, whereas the complete kernels are distributions in a polynomial carrier.

A necessary correction concerns interpretation: source switching defines an equation-level retarded response, not automatically a variational Møller map. A local formal-adjoint test shows why that stronger identification cannot be asserted for the unchanged off-shell source law. Thus the direct tree identity survives without claiming a switched-action or Bogoliubov construction.

The principal tree mechanism is credited to X-F. The contribution here is its supported proof and explicit type, support, topology, and variational audit. This provides a reproducible tree-level baseline for successors without identifying it with loop admission or spacetime emergence. A finite supplied-evaluation dictionary has a complete destructive-clock instrument and regular reference-preserving transport. The release is local Grade S only.