Theorem: Classical and distributional release

For the theory, background, clock, and harmonic boundary prescription fixed in this paper:

1. the displayed background solves the Einstein–scalar equations and has a nondegenerate scalar clock;

2. the clock displacement, induced spatial metric, harmonic-coordinate coefficients, and relational map are constructed through second order;

3. the full-tensor relational lapse coefficients are (eq:n1)–(eq:n2);

4. the complete classical quadratic functional has the local-plus-spatial-Green representation (eq:kernel-bridge);

5. its classical covariance and coefficient identities are those of Proposition prop:covariance and (eq:classical-Ward);

6. compact external smearing does not generally give spatially compact Hessian support;

7. the instantaneous kernel has the off-diagonal wavefront set (eq:WF), and an internal diagonal correction can have the non-diagonal image (eq:image);

8. the tail bounds and conditional joint-limit, Dini, and gauge-repair criteria hold on their stated domains.

These statements do not establish quantum regulator removal, quantum gauge closure, gauge-fixing independence, or a universal obstruction to the relational lapse.

Proof. Proposition prop:background proves (i). Lemma lem:clock, Proposition prop:induced, Lemma lem:laplacian, Proposition prop:rods, and the series inversions prove (ii). Theorem thm:lapse proves (iii). Theorem thm:kernel proves (iv). Proposition prop:covariance and differentiation of its identity prove (v). Propositions prop:noncompact, prop:instantaneous, and prop:image prove (vi)–(vii). Lemma lem:tail, Theorem thm:joint, and Propositions prop:Dini and prop:gauge prove (viii).