Theorem: XX-B Q1 release

For the frozen XX-A original-rod variant and Definition def:closure:

1. the complete coupled action Hessian, gauge fermion, auxiliary normalization, and ghost operator are constructed;

2. the reduced operator is normally hyperbolic after the explicit invertible fiber normalization;

3. all retarded and advanced Green matrices are constructed with their actual action normalization, support, and wavefront bounds;

4. normalized Hadamard contractions for the complete free multiplet exist and are supplied by (eq:WP)–(eq:Wfull);

5. the original elliptic rods and all retained classical lapse contributions remain unchanged;

6. the classical principal-symbol check preserves the rod term and the proved absence of $Y_2$.

Thus Q1 is closed for the compact-test gauge-fixed free-input target. No renormalized nonlinear relational lapse, positive physical state, spatial-tail theorem, complete quantum remainder, or interacting Ward identity is released.

Proof. Lemma lem:H and Section sec:gauge-fixing prove (i). Theorem thm:NH proves (ii). Theorems thm:greens and thm:full-green, Proposition prop:intertwining, and Proposition prop:WF prove (iii). Lemma lem:Feynman, Proposition prop:W, and Theorem thm:contractions prove (iv). Lemma lem:rod-realization and Theorem thm:lapse prove (v). Proposition prop:classical-limit proves (vi).

None of these results establishes the missing nonlocal distributional operations or physical state conditions. The final sentence therefore retains their explicit gate boundaries.