Abstract

We establish the prepared response on the counter-propagating pure-trace witness family. The source is smooth and past compact relative to compact observation sets, but is not a compact-interior source. Real polarization and complex bilinear extension produce the response $U_D=-\mathcal{P}{V}_{\lambda }$, where $V_{\lambda }$ is an explicit zero-data time integral. A local energy proof supplies uniqueness among all smooth zero-data solutions on the slab, without a spatial decay assumption.

The spatial traceless response contraction is computed exactly. For the completion, the switched scalar forcing is a compact-interior source, whereas the coordinate forcing is spatially noncompact. An explicit coordinate candidate is nevertheless a unique smooth solution of the required Cauchy problem. We also construct a restricted auxiliary extension of the compact-source wave operator, while keeping its identification with the inherited coordinate prescription conditional.

The single-metric root identity is not directly applied to a polarized source. Its legitimate polarization, and the later root–auxiliary compatibility result of XVI-F-v3, are recorded separately. No compact class kernel, quantum retarded coefficient, or new admission theorem is constructed.