Theorem: Classical admission and the reduced extension family

Retain the exact classical matching inputs, the supplied classical and root cone bounds, the ambient propagator prescription, and the declared cubic density representative.

Then:

1. The classical prepared polynomial has direct microcausal membership, and its free-shell class contains its own microcausal kernel.

2. The oriented off-diagonal wavefront relation of the selected propagator defines every differentiated two-edge product occurring in the explicit reduced sector.

3. For nonzero ${\chi }_{\mathrm{I}}$, the open-index vertex block has a nonzero spacelike component for a specified smooth background. No general nonvanishing of the full contracted sum follows.

4. The source-decorated reduced sector has local transverse scaling degree at most eight in four relative dimensions.

5. Its numerical building blocks admit point extensions; their ambiguity orders are at most four. No normalization constants are chosen.

6. Under the root $Z_2$ premise, compressed root attachment and proper pushforward exist for each extension choice. The resulting smeared coefficients form a family.

No member of that family is selected as a normalized retarded interacting coefficient or a BRST/BV observable.

Proof. Theorem thm:admission proves item 1. Proposition prop:product and Theorem thm:reduced prove item 2. Proposition prop:nonzero proves item 3. Proposition prop:sd proves item 4. Theorem thm:extension proves item 5. Theorem thm:attach and Corollary cor:coefficient prove item 6. The unchosen normalizations and unconstructed interacting identities are recorded in Section sec:limits.