Theorem: X-D-ST local Grade S release

For the inherited supported host and the fixed-input family satisfying (eq:separation):

1. the shrinking coordinate layer has the endpoint data (eq:velocities); the finite quadratic normal-flux contribution is retained;

2. the limiting coordinates solve the uniquely identified continuous-coordinate, continuous-normal-flux transmission problem;

3. the completed scalar tests converge in one fixed ${\mathcal{D}}_C$ at rate $O(\epsilon )$ in every seminorm, and hence in $Q_m$;

4. the limiting tests are independent of the fixed profile interior, although finite-width position moments and error constants can depend on it;

5. the background identity passes through order ${\kappa }^2$ on the unchanged frozen algebra, with the explicit compact equation representatives (eq:limit-ideal);

6. the edge and switch distinctions retain the three-term variation and its propagated first gap, with the fixed-versus-moving parameter quantifier stated explicitly;

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

These statements form local Grade S only.

Proof. Lemma lem:flux, Proposition prop:layer, and Lemma lem:profile prove (i). Theorem thm:transmission and Proposition prop:kernel prove (ii). Proposition prop:propagation, Theorem thm:convergence, and Lemma lem:closed prove (iii)–(iv). Theorem thm:ward proves (v). Lemma lem:variation, Proposition prop:defect-limit, and (eq:moving-edge) prove (vi). Sections sec:instrument–sec:transport prove (vii), supplying the remaining clauses of Definition def:grade.