Abstract

We specify a classical Einstein–scalar model compatible with the signature, scalar switching prescription, and plateau gauge algebra of Landmark X-A. The Einstein–Hilbert density is not switched. Its sign is fixed by a positive transverse-traceless kinetic term and an attractive Newtonian comparison. We display the quadratic Fierz–Pauli action, one de Donder gauge-fixing term, and the metric Euler derivative through coupling order ${\kappa }^2$.

Several distinctions are indispensable. Minimal scalar coupling introduces the switch itself, but no switch derivatives, into the metric Euler derivative. Switch derivatives appear in the scalar equation and in the divergence of the metric source. Nonlinear Noether identities require the classical higher gravitational coefficients; the quadratic action alone is insufficient. Gauge fixing supplies a separate breaking term and is not diffeomorphism invariant on the plateau.

We prove the exact fixed-background Noether identity and the tensor-density covariance of the invariant metric Euler operator on the admitted plateau group. X-A's varied-action formula is used as a premise, not presented as a new relational covariance theorem. Finally, for a specified free scalar input, the ungauge-fixed coupled Einstein jet is empty for every nonzero compact switch of the declared class. This is not an emptiness theorem for all inputs, nor for the gauge-fixed wave equations. The manuscript stops at the classical sheet: no propagator, quantum multiplet, tree, or BRST class is constructed.