Theorem: X-H-ST mathematical release

At fixed positive width:

1. the field differential of Section sec:stageA is explicitly constructed and nilpotent off shell;

2. the relative classical BV master action is constructed through ${\kappa }^2$, modulo divergences on the plateau, with independent auxiliary field and the signed antifield lift;

3. the unchanged free causal algebra is not identified with this interacting classical complex; the natural first-order scalar rule fails free-equation descent;

4. the unrestricted switched master residual and the positive reciprocity pairing are separately defined and are not identified;

5. the X-G exact source-preserving variational upgrade obstruction remains in force;

6. finite supplied classical residual reports and the separate reciprocity reports have the complete destructive instruments proved above;

7. Stage D remains a conditional consistency and encoding interface, not a constructed quantum lift;

8. scheduled locality, regular reference-preserving transport, and the stated resource/error ledger hold for the complete finite interface.

These statements form local Grade S.

Proof. Theorem thm:kinematic proves (i). Lemmas lem:action and lem:sign, and Theorem thm:BV, prove (ii). Theorem thm:nondescent and Proposition prop:compactghost prove (iii). Equations (eq:swCME) and (eq:W), with Proposition prop:distinct, prove (iv). The reciprocity argument and Proposition prop:recip give (v). Propositions prop:residualvalues and prop:threshold, Theorem thm:instrument, and Corollary cor:obstructioninstrument prove (vi), with the interpretation limited by Proposition prop:noacquisition. Definition def:quantuminterface and Proposition prop:quantumcriterion give exactly the conditional conclusion in (vii). Section sec:locality proves (viii).