Pure gauge and limits of the emptiness statement

For every smooth vector field,

Substitution into (eq:Glin) verifies the first identity. For $\xi \in {\mathfrak{g}}_U$ this is the required pure-gauge diagnostic. It checks the linearization, not the existence of a sourced jet.

The unrestricted family of Einstein jets is not empty: ${\varphi }_0=0$ and all metric and scalar coefficients zero solve the equations. Other inputs satisfying (eq:compatibility) are not classified here. Necessity is not being promoted to a general sufficiency theorem.

No second-order correction can cancel the nonzero order-one compatibility coefficient. The stress tensor is not modified, and the Einstein–Hilbert density is not switched. A possible removal or controlled comparison of compact cutoffs belongs to a successor local-coupling or adiabatic problem; no such limit is taken.