Theorem: X-F-ST local Grade S release

On the fixed supported host and at fixed positive width:

1. the inherited action, normalization, scalar completion, and invariant-jet switch obstruction remain those explicitly stated above;

2. background de Donder gauge on the fluctuation gives the typed F1 covariance and F2 commutator identities;

3. the prescribed second-order response is uniquely retarded, satisfies the dynamical jet law, and introduces no independent field;

4. the matter, mixed, and self trees are constructed with the corrected supports, normalization, and two- or three-slot microcausal bounds;

5. their generated kernels and typed smooth contractions have the continuity properties of Theorem thm:continuity;

6. the all-tree observable is background invariant through ${\kappa }^2{\hslash }^0$ modulo $\mathcal{J}$, with the explicit graviton-slot preimage (eq:preimage);

7. the flat-gauge residual is admitted and displayed, with its class left open;

8. the finite supplied-evaluation interface has a complete destructive clock, scheduled locality, regular reference-preserving transport, and the declared resource/error ledger.

The retarded source law is not promoted to a variational Møller or Bogoliubov construction. These statements form local Grade S only.

Proof. Lemmas lem:affine and lem:action, and Proposition prop:Noether, prove (i). Lemma lem:cov2, Proposition prop:Q, and Theorem thm:key prove (ii). Past-compact Green theory and Theorem thm:jet prove (iii). Theorem thm:admission and Proposition prop:classical prove (iv). Theorem thm:continuity proves (v). Lemma lem:gslot, Proposition prop:remainderideal, and Theorem thm:main prove (vi). Proposition prop:flat proves (vii). Sections sec:instrument–sec:transport prove (viii). Proposition prop:nonvariational supplies the interpretive restriction.