Theorem: Qualified X-G release

For the frozen coefficients and background class:

1. The free BRST multiplet and differential of Section sec:free are constructed and normalized to X-E/X-F's graviton operator.

2. The unrestricted scalar gauge variation is (eq:B); its leading coefficient has the nonzero free-shell witness of Theorem thm:matter.

3. No exact source-preserving variational upgrade satisfying Definition def:upgrade exists. The explicit obstruction is

   $$\frac{5M_{\chi }}{8}{{\int }_0}^{\epsilon }\zeta ({\sigma }_{\epsilon }^{\prime \prime }{)}^2\mathrm{d}t>0.$$

4. Under Assumption ass:graphs, the two mixed contraction types exhaust the non-tadpole ${\kappa }^2\hslash$ inventory. The local fish admits the extensions of Proposition prop:fish.

5. The scalar nonlocal product criterion has the exceptional configuration of Proposition prop:nullWF. Its failure is not a nonexistence theorem.

6. Theorem thm:conditional is a conditional background-identity result. The existence assumptions needed for an unconditional one-loop completion have not been replaced by an asserted covariance principle.

Proof. Item (i) is Proposition prop:gfnorm, Proposition prop:Ghb, and Theorem thm:freeBRST. Item (ii) is Proposition prop:breaking and Theorem thm:matter. Item (iii) is Theorem thm:obstruction. Item (iv) is Proposition prop:inventory and Proposition prop:fish. Item (v) is Proposition prop:nullWF. Item (vi) is the coefficientwise proof of Theorem thm:conditional and its stated hypotheses.