Abstract

We isolate the actual functional-support calculation needed to apply IX-E-2's evaluated commutator to the retained two-way sector. The local quadratic functional $W_g$ has functional support exactly $\mathrm{supp}g$. For the sharp insertion, a nonzero smooth surface coefficient cannot be cancelled by the locally bounded bulk term in its distributional metric derivative.

A correction is necessary to the proposed solid-ball argument. For the explicitly specified flat massless zero-Cauchy realization, the initial-surface support is the union of the time-zero null spheres generated by $\mathrm{supp}f$, not necessarily the union of their solid balls. We prove this support statement, exhibit an open missed region, and show that the retained null spheres nevertheless suffice to exclude the commutator slab. Thus a gap inside a solid ball does not by itself provide an obstruction witness.

The bridge proof is complete under a displayed boundary-trace certificate. The supplied prerequisite statements, however, do not specify that certificate for the inherited operator ${\mathsf{E}}_0$. We do not identify a standard Green operator with an unspecified inherited operator by notation alone. The supplied-data release is therefore Unresolved at that specific identification. If the certificate is supplied, the bridge follows without another support calculation. No time-ordered product is constructed.