Theorem: VII-H release

For the free massless scalar, vacuum, calibrated Wick square, source sign, test space, and prescription class declared in Sections sec:model–sec:extension:

1. the complete first source-order connected vacuum response exists as a real retarded distribution;

2. every permitted finite normalization difference is a real multiple of $\hslash {\delta }_0$;

3. the test $F_{\ast }$ of (eq:Fstar) annihilates all such differences and has the exact nonzero value $-\hslash t_0/(2{\pi }^2)$;

4. the delayed product experiment (eq:experimental-tests) is prescription independent and strictly nonzero;

5. the finite-bank, calibration, optimization, and error statements apply with the complete ambiguity already classified.

This is a normalization-independent connected quantum response within the declared model and order. It is not a completion of the inherited gravitational quantum-response programme.

Proof. Lemmas lem:wick, lem:kernel, and lem:reference construct the response. Theorem thm:ambiguity gives the complete finite comparison. Theorems thm:main and thm:experiment give the invariant and its delayed realization. Section sec:bank proves the remaining assertions.