Theorem: X-C-ST local Grade S release

Relative to the explicitly stated inherited host, classical handoff, and scalar free-field presentation:

1. the second-vertex map $R_1(V_{2,g};F_c)$ and the diagonal iterated coefficient $\frac{1}{2}{R}_2(V_{1,g},V_{1,g};F_c)$ are constructed on all specified off-shell compact linear representatives;

2. both have smooth compact retarded output tests, continuous background and test dependence, and no positive-$\hslash$ remainder in the stated linear rooted prescription;

3. their sum represents the inherited second scalar response exactly once;

4. the completed relational field is background invariant through ${\kappa }^2$ on the unchanged ${\mathcal{G}}_p$, with the explicit equation-ideal representative (eq:ideal-representative);

5. outside that algebra the coordinate and switch defects retain the three-term formula, including the propagated first-order coordinate gap;

6. the complete pure-gauge two-jet closes the X-B-ST remainder with the bare second dressing retained; the conformal diagnostic has the stated classical values;

7. the new finite installed encoding has a complete destructive-clock instrument, scheduled locality, reference-preserving regular transport, and a declared resource/error ledger.

This is local Grade S only.

Proof. Definition def:moller and Theorem thm:maps prove (i)–(iii), with Theorem thm:yf and Proposition prop:hbar supplying the restricted recursion and quantum-order characterization. Proposition prop:variation and Theorem thm:ward prove (iv)–(v). Section sec:diagnostics proves (vi). Theorems thm:instrument, thm:locality, and thm:transport, with Proposition prop:table and (eq:error-ledger), prove (vii). These provide Definition def:grade.