Theorem: III-B local Grade S release

Under Definition def:laboratory and the finite interface specified above:

1. the exhibited unchanged-product symbol fails to descend, with a fixed nonzero representative gap;

2. massive free-shell tangency on ${\mathcal{D}}_0$ is exactly local Killing symmetry;

3. the retained first-order jet admits the stated unique past-trivial harmonic section and quotient functional;

4. the supported ${\mathsf{C}}_1$ Cauchy quotient is Hausdorff and well posed;

5. compact coupled Ward pairs exist with strictly positive on-shell value, while the displayed adjoint-exact family vanishes;

6. the complete destructive clock instrument supplies history labels through explicit dictionaries;

7. the finite three-qubit encoding realizes four descended Ward evaluations with values $j(k+1)$;

8. payload and classical-code transport preserve the complete declared instrument, with all dependencies inside the prescribed metric cone and with resource bound (eq:resource).

These conclusions satisfy Definition def:grade.

Proof. Items (i)–(v) are Theorems thm:product, thm:killing, thm:jet-descent, thm:reduced, and thm:compact-pair, together with Proposition prop:exact-zero. Theorems thm:instrument and Proposition prop:dictionary prove (vi). Theorem thm:pilot proves (vii). Theorems thm:locality and thm:transport, the printed code-forwarding schedule, and the ledger prove (viii).

The geometry, analytical domain, classical quotients, finite encoding, complete instrument, and transport permissions have all been specified. These are precisely the five clauses of the local Grade S contract.