Theorem: Compact polarized completion and witness recovery

Retain the fixed data, inherited source and completion interfaces, compact root-kernel assumptions, and witness hypotheses stated in this manuscript.

Then:

1. ${\mathcal{O}}_{\mathrm{pol}}$ is the real symmetric polarization of the diagonal completion, bilinear in the tensor slots and linear in the scalar.

2. The explicit diagonal test-function construction (eq:t-test) defines a compact ambient kernel $t_{\mathrm{D}}$ representing that polarization.

3. The functional is jointly continuous on the chosen smooth configuration spaces, with some finite derivative orders.

4. Its complex extension agrees with the revised XVI-E/F polarization convention and with the current XVI-G witness value at each frozen frequency.

5. Partial contraction of ${t_{\mathrm{D}}}^{\mathbb{C}}$ on the specified tensor witness pair equals $j_{\lambda }$ as a scalar distribution.

These conclusions establish neither a full-kernel wavefront exclusion nor admission of the cubic free-shell class. They establish no BRST/BV observability or quantum product.

Proof. Theorem thm:Opol proves items 1 and 4. Theorem thm:trep proves item 2. Proposition prop:estimate proves item 3. Proposition prop:tmatch proves item 5. The unperformed tests and unspecified quantum data are recorded in Sections sec:admission and sec:quantum.