Theorem: X-B-ST local Grade S release

For the inherited X-A-ST host presentation and the frozen scalar-only sheet:

1. the bare ramp dressing is admitted through ${\kappa }^2$, jointly on the completion and separately on the scalar-linear domain actually used;

2. the single normalized map $R_1(V_{1,g}[h];F_c)$ is constructed on all specified off-shell compact linear representatives; its output test is smooth, compact, continuous, retarded, and $\hslash$-independent;

3. the completed coefficient ${\hat{A}}_1$ obeys the background identity on the originally frozen ${\mathfrak{g}}_p$;

4. the mixed term and its transformation are constructed; the classical ${\rho }_{2,g}$ and its role are displayed, without a constructed second-order quantum completion;

5. the one-contraction pair passes its restricted symmetry and causal-factorization checks, but the present tools do not define the required second quadratic pair or a time-ordered family;

6. the whole-domain normalization uniquely determines the degree-one response class; every sheet-compatible contact modification vanishes in the free algebra;

7. Branch B is recorded and not opened;

8. the finite installed encoding has a complete destructive-clock instrument, scheduled locality, reference-preserving regular transport, and the stated resource and error ledger.

These statements constitute local Grade S only.

Proof. Propositions prop:joint and prop:linear prove (i). Theorems thm:retarded and thm:hbar, with Proposition prop:patch, prove (ii). Proposition prop:variation and Theorem thm:ward prove (iii). Section sec:second proves (iv). Lemma lem:DF, Propositions prop:check and prop:causal, and Theorem thm:nonextension prove (v). Theorem thm:unique and Corollaries cor:contact–cor:ward prove (vi). Section sec:branchB records exactly (vii). Theorems thm:instrument, thm:locality, and thm:transport, together with the finite-table and resource statements, prove (viii). These provide the clauses of Definition def:grade.