Theorem: X-G-ST local Grade S release

On the retained supported host, with fixed positive width:

1. the independent free auxiliary–ghost multiplet, normalized fermion, and causal Green matrix are constructed on $\hat{M}$;

2. $s_0$ is off-shell nilpotent and descends to a nilpotent derivation of the free causal algebra;

3. the natural switched-scalar breaking is evaluated, with its full off-shell support and a nonzero completion-level leading coefficient;

4. the source-switching prescription is distinguished from density switching;

5. the exact variational upgrade class of Definition def:upgrade is obstructed by the supported positive free-solution witness (eq:positive);

6. the inherited direct tree identity survives on the unchanged plateau algebra, without being promoted to a variational or interacting-BRST identity;

7. the quantum inventory has the explicitly restricted and conditional status stated in Section sec:loops;

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

For the arbitrary-switch matter witness, laboratory detection additionally requires the support certificate stated after Theorem thm:matter. This is local Grade S only.

Proof. Lemmas lem:s0nil and Proposition prop:gfnorm, together with Proposition prop:Ghb and (eq:ghostGreen), prove (i). Theorem thm:freeBRST proves (ii). Proposition prop:breaking and Theorem thm:matter prove (iii). Proposition prop:symmetric proves (iv). Lemma lem:reciprocity, Propositions prop:Dsigned and prop:localwitness, and Theorem thm:obstruction prove (v). Proposition prop:plateau and Corollary cor:survival give (vi). The stated hypotheses of Propositions prop:inventory, prop:fish, prop:nullWF, and Theorem thm:conditional give exactly (vii). Sections sec:instrument–sec:transport prove (viii). The support qualification was stated explicitly and is not removed by this summary.