Abstract

The revised Q4–Q5 plan assumes a native Q3 insertion and a compatible quantum gauge operation. Those application data are not present in the supplied XX-A–XX-D package. This manuscript therefore does not manufacture an actual quantum breaking term or claim cancellation of an undefined coefficient.

We reproduce the frozen original-rod lapse and calculate its coupled classical Ward identity. An exact formula identifies the classical breaking introduced by a compact source cutoff. We correct the output-density mismatch between XX-A and XX-D and distinguish the full compact geometric gauge group from the trivial compact residual subgroup of the gauge-fixed free operator.

Two further domain tests explain why classical symbol invariance is not a quantum Ward theorem. Ordinary compact gauge translations do not descend to the gauge-fixed free equation quotient. Centrally parametrized second jets also do not automatically support the proposed quantum gauge derivation.

We calculate the spatial Newton images of finite contact families, including their derivative coefficients, and the classical Ward images of a specified comparison counterterm family. These are explicit images, not a claim that the family exhausts admissible quantum counterterms. A simultaneous actual-image criterion is proved for insertion and subsidiary repair after their defects and domains have been constructed.

An escaping-source example on the identical classical prescription shows that compact-local convergence alone does not control Newton composition. The native quantum Q3 prerequisites remain unverified, so native Q4 and Q5 are unentered and unclosed. The results are domain and non-determination theorems, not universal impossibility theorems for a future completion.