Theorem: Fixed-input X-D release

For the frozen scalar-only X-C-v3 observable and fixed inputs satisfying (eq:separation):

1. The completed tests converge to (eq:limiting-completed) in ${\mathcal{D}}_C(M)$, at rate $O(\epsilon )$ in every seminorm, and hence in $Q_m$.

2. The limits are built from the sharp action-level transmission coordinates with zero positive-order initial positions and velocities (eq:velocities). Their smeared linear background kernel is $\theta C^{\ast }({{\mathsf{E}}_0}^{\mathrm{A}}a)$, with a possible jump and no hypersurface delta.

3. The limiting data and tests are independent of the choice of fixed smooth ramp profile with the same endpoints.

4. The completed limiting observable is background-gauge invariant through ${\kappa }^2$ on the unchanged plateau algebra.

The limit is generally different from the multiplied-equation prescription. VII-C does not apply and is not contradicted. No moving-edge argument, crude inverse-$\epsilon$ estimate, or residue subtraction supplies the decision.

Proof. Lemma lem:profile constructs the layer data. Proposition prop:coordinate-convergence propagates them. Theorem thm:convergence proves the compact-test limit. Theorem thm:identification and (eq:limit-kernel) identify the prescription. Corollary cor:gauge passes the identity after convergence. Section sec:viic audits the separate VII-C construction.