Theorem: Exact benchmark release

For the frozen model of Section sec:model, this manuscript establishes:

1. the evaluated positive spin-two and spin-zero vacuum spectral densities (eq:r2)–(eq:Wspec);

2. the explicit once-subtracted reduced form factors and real, causal, tempered retarded extension (eq:finalPi), with scaling degree at most eight;

3. the complete three-parameter transverse contact freedom and unique zero-momentum calibration;

4. both distributional linear Ward identities on all compact tests in the declared class;

5. the nonzero contact-independent spectral discontinuity and a positive compact vacuum detector;

6. consistent four-kernel CTP bookkeeping at quadratic source order;

7. the continuous prepared response ${\mathcal{R}}_{\epsilon }(j,h)$;

8. the exact preparation residual, plateau-supported intertwining, and a late contact-independent nonzero residual detector.

These statements do not establish the companion's relational scalar two-point coefficient, its characteristic graph, a graviton–ghost inventory, nonlinear metric seagulls, an interacting BRST/BV identity, a match to Fröb, bulk restoration of O.1, a measurement instrument, or ultraviolet completeness. No inherited companion quantum coefficient is admitted.

Proof. Item (i) is Theorem thm:spectral. Item (ii) is Lemma lem:dispersion, Proposition prop:checks, and Theorem thm:retarded. Item (iii) is Theorem thm:contacts and Corollary cor:unique. Item (iv) is Theorem thm:ward. Item (v) is Theorem thm:nonlocal and Proposition prop:detector. Item (vi) is Proposition prop:ctp. Item (vii) is Proposition prop:prepared. Item (viii) is Theorem thm:residual.

The excluded objects are not among the frozen fields, vertices, insertions, or interfaces. No proof above constructs or identifies them.