Abstract

We freeze one density-before-variation prescription at fixed regulator width. A necessary correction is made explicit: weighting the entire Einstein–Hilbert density would change the lower-order sheet and would not retain the XI-B free operator. To preserve that handoff, only the specified cubic density coefficient is multiplied by the existing ramp. All other gravitational coefficients, the switched scalar action, and the gauge-fixing term remain those of XI-A and XI-B.

A finite, unintegrated prescription fixes the cubic density representative. Its actual Euler source is computed as ${\sigma }_{\epsilon }{J}_3+{\mathcal{D}}_{{\sigma }_{\epsilon }}$. The complete tensor formula displays all first and second derivatives of the ramp and reproduces XII-B's pure-trace layer term. The changed fixed-background Noether identity contains both the scalar-switch defect and the additional breaking caused by weighting only one gravitational coefficient. Its apparent third ramp derivatives cancel algebraically.

For the named free inputs ${\varphi }_m$, the coupled invariant metric–scalar jet is empty for every switch of the declared plateau class. This is proved anew: the compatibility condition has a layer correction, but a nonzero late switching defect survives where the ramp is already one. Arbitrary leading vacuum metric data are allowed. The zero-field vacuum remains a counterexample to universal jet emptiness.

No Green operator is applied. No completed insertion is admitted, no wavefront theorem or response plateau law is proved, and no interacting differential or sharp-width limit is introduced.