Theorem: XX-A mathematical release

For the prescription explicitly frozen in this manuscript:

1. the power-law background is an on-shell Einstein–scalar background with a nondegenerate calibrated matter clock;

2. the original spatial rods exist uniquely as decaying smooth coefficient fields through second order;

3. the complete raw and relational lapse coefficients are (eq:N1)–(eq:N2) and (eq:n1)–(eq:n2);

4. the second spatial rod is constructed, and its absence from the quadratic lapse is proved;

5. the complete linear functional and Hessian obey (eq:linear-ward)–(eq:quadratic-ward);

6. a compact off-shell witness distinguishes the original rods from the specified lapse-weighted rods;

7. instantaneous singularities, spatial tails, regular Wick comparisons, and exact Ward-repair criteria have the scopes stated in the corresponding propositions;

8. no native quantum covariance, renormalized lapse insertion, complete quantum breaking, admissible cancellation, or retained-order quantum Ward identity is constructed.

The classical freeze is constructed. The quantum milestone is unresolved at Q1.

Proof. Lemma lem:on-shell proves (i). Proposition prop:rods proves (ii). Theorem thm:lapse proves (iii) and (iv). Theorem thm:classical-gauge and Corollary cor:coupled prove (v). Proposition prop:rod-detection proves (vi). Propositions prop:tail and prop:instantWF, Lemma lem:regular-wick, and Theorem thm:repair prove the mathematical assertions in (vii).

For (viii), the formulas defining the classical target, unfixed operator, and gauge-fixing candidate do not supply the missing Q1 realization and contraction data. The regular Wick lemma assumes those data, and the Ward-repair theorem assumes a defined quantum insertion and breaking. Neither assumption is discharged for the native target here. The gate ending is therefore unresolved, not constructed and not a universal impossibility theorem.