Declarations

scope. The new results concern the propagated companion's sampling support, two compactly realized diagnostics, the local cancellation subclass, and an explicit nonzero auxiliary pairing on the complete free shell. The inherited admission and source calculations are not repeated.

scope. The counterexample obstructs unqualified classical invariance of the specified coefficient. It does not obstruct every possible additional observable term, consistent co-transformation, or interacting construction.

scope. The auxiliary remainder has a compact microcausal bilinear kernel. Its nonzero value on a complete free configuration proves that its class is not removed by the inherited null ideal. No gauge action on the quotient is inferred.

correction. The switching display's support describes where the propagated companion is sampled. It is not the complete support of the functional's dependence on the original metric field. The copied (CP) is a sufficient condition; the off-condition example does not promote it to an if-and-only-if condition.

scope. XII-E supplies the assembly, sign, variation, and scalar-sourced slot. XII-D supplies the completion roots and causal plateau proposition. XII-A supplies the carrier and null ideal. XI-B supplies the free operator identities. XII-C supplies only the retained source and jet boundaries. The compact realizations use standard smooth Cauchy uniqueness and finite propagation.

. No successful pdflatex execution, proof-assistant check, independently executed symbolic computation, independent referee review, or journal acceptance is asserted. The witnesses and their signs are established by the displayed analytic calculations.