Theorem: X-A-conv-ST release

For the fully specified completion data (eq:switch)–(eq:witness), at fixed $0<\epsilon \le 1/8$:

1. the coordinate and scalar Cauchy coefficients exist uniquely through order two;

2. the convolution response is $S_{\epsilon }\xi$, and the fixed witness has the strictly positive coefficient (eq:energy);

3. the coefficient survives scalar-solution evaluation and has the real-surjective first-order orbit (eq:orbit);

4. the retained second-order defects are the displayed ones, and a higher-order addition preserving the first coefficient cannot repair its nonzero defect;

5. under the appropriate source-specific chart certificate, density transport preserves that obstruction on the unchanged prescribed host;

6. any scalar-only complete readout invariant on the full witness orbit is uninformative about that scalar;

7. a complete destructive instrument reports finite supplied evaluations and statuses, with all unmeasured finite residuals and references retained;

8. the two-coefficient circuit is constructed only conditionally on the independently supplied charge and matching hypotheses;

9. scheduled locality and reference-preserving regular transport hold under the declared finite permissions.

With (SC-2) and the retained operational reserves, this is local Grade S.

No convolution repair, field-to-qubit acquisition, interacting differential, inherited characteristic quantum coefficient, or spacetime emergence is constructed.

Proof. Lemma lem:Cauchy, Proposition prop:coordinates, and (eq:rho1)–(eq:rho2) prove (i). Lemma lem:preflight and Theorem thm:positive prove (ii). Theorem thm:evaluated proves (iii). Propositions prop:seconddefect and prop:pure, with Corollary cor:nohigher, prove (iv). Theorem thm:hosttransfer proves (v). Theorem thm:orbitclosure proves (vi). Theorem thm:instrument and Proposition prop:noacquisition give the construction and its precise limitation in (vii). Proposition prop:matching and Theorem thm:relational prove (viii). Theorems thm:locality and thm:route prove (ix).