Abstract

We freeze a compactly switched Einstein–scalar density, a plateau-supported gauge algebra, a zero-Cauchy prescription, and one explicit fixed regulator. The regulator convolves the switched metric arguments before they enter an auxiliary harmonic-coordinate problem. The resulting relational scalar is a well-defined off-shell formal functional through order ${\kappa }^2$. Its coefficient kernels are smooth and compactly supported, and its metric dependence obeys an explicit regulator-thickened support bound.

The regulated coordinate problem does not intertwine the frozen gauge action. On a pure-gauge first coefficient, its coordinate response is $S_{\epsilon }\xi$, rather than $\xi$. We prove an exact first-order observable defect and exhibit one fixed observation test, one plateau-supported gauge vector, and one free scalar for which its smearing is strictly positive at every allowed fixed regulator. The same gap occurs when the observable is evaluated on the scalar solution selected by the switched action. The action itself satisfies the retained classical plateau Noether identity; consequently the failure is attributable to the coordinate regulator, not to omitted switch derivatives or changed Cauchy data.

We also derive the second-order coordinate and relational coefficients, the retained infinitesimal defects, and the complete second-order pure-gauge test. A separate nonzero forced scalar example checks the action-derived response. The release is a proved obstruction for this prescription on its originally declared gauge algebra. No quantum product, quantum Ward identity, sharp-regulator limit, or general no-go theorem for regulated relational constructions is claimed.