Theorem: IX-A-ST local Grade S release

For the prescribed cone and the supported data of Definition def:support, the following are established:

1. a local zero-Cauchy response independent of auxiliary extensions on the complete truncated causal support, without a global conical inverse;

2. the exact off-shell insertion $B_f$, coordinate descendants $J(a)$, and an actual compact-distribution polynomial admission domain;

3. a canonical boundary trace modulo microcausal linear kernels, imposed on all directional descendants;

4. faithful bulk inclusion, classical presentation independence, and closure under arbitrary finite pointwise words, smooth directional derivatives, and conjugation;

5. a Hausdorff locally convex graph topology with continuous differentiation, conjugation, test admission, and separate classical multiplication;

6. for every nonzero real $f$, a normalized continuous separator proving that unchanged regular approximants do not converge to $B_f$, despite strong and specified normal kernel convergence;

7. membership of the selected already-defined coefficient outputs in Proposition prop:Wickoutputs, without a quantum closure claim;

8. a complete finite destructive clock, fixed evaluation-table channels and their fixed-interface approximation bound, residual compatibility criteria, and recovery safeguards;

9. scheduled locality, exact no-signalling criteria, and an explicit reference-preserving regular transport route with independently certified physical containment and resource accounting.

These statements constitute local Grade S only. The free quantum-product gate is unresolved in this release. The quantum certificate remains -open.

Proof. Lemma lem:saturation proves (i). Sections sec:carrier–sec:admission prove (ii)–(iv). Section sec:topology proves (v). Theorem thm:separator and Proposition prop:normal prove (vi). Section sec:coefficients proves (vii) and the supported transfer. Sections sec:finite–sec:residual prove (viii). Section sec:transport proves (ix). None of those results defines a renormalized multiplication on the enlarged domain, so the final gate statement is unchanged.