Theorem: VI-B local Grade S release

For the supported presentation and finite interface specified above:

1. the local Einstein–scalar jet has the displayed second-order equations and the compact initial-preserving formal gauge action;

2. the finite-time harmonic relational scalar descends through second order;

3. the compactly switched candidate has the exact classical defect (eq:defect), the negative plateau gap (eq:gap), and the coefficientwise order obstruction;

4. the explicit constraint-compatible family is nonempty and has a positive second-order descended evaluation;

5. the finite membership–separation alternative holds with its stated coverage and sampling limits;

6. the finite evaluation encoding has the complete destructive-clock instrument, explicit dictionaries, fixed-table approximation limit, and reference-preserving causal endpoint transport;

7. the quantum certificate remains RG-open.

Items (i)–(vi) satisfy Definition def:grade. No cited companion theorem is a premise of their proofs.

Proof. The coefficient equations and action are Definition def:jet and Proposition prop:gauge-action. The coordinate construction and Theorem thm:descent prove (ii). Theorems thm:switch-defect and thm:plateau-obstruction, with Corollary cor:order-obstruction, prove (iii).

Theorems thm:sourced and thm:sourced-nonzero prove (iv). Theorem thm:certificate and the subsequent quantifier audit prove (v). Sections sec:encoding–sec:transport prove (vi).

Section sec:quantum constructs neither the required quantum products nor a complete evaluated response catalogue, so the proposed quantum certificate has not acquired a closed status. This is (vii), not a quantum nonexistence theorem.