Proposition: No transfer from concentration to joint admission

Theorem thm:main constructs no representative of $[u{]}_{\mathrm{sol}}$ in ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, and no interacting gravitational product. It therefore provides no repair of (eq:XIG). In particular, any additional claim assigning such a representative under the unchanged XI-G hypotheses would still contradict (eq:XIG), independently of a classical-limit estimate.

Proof. The constructed map is

The metric argument has already been fixed before the first quantum map. Its output is a scalar-linear field in a free scalar representation. No step has an output with the two quantum metric slots and scalar slot of $u$; no map between those kernel classes is defined. Products of the resulting free scalar operators, when defined on their domains, are not the missing gravitational retarded products.

If a further construction produced a representative in the stated intersection, its existence would contradict (eq:XIG). An estimate on expectation values of a different, already-defined operator does not alter that set-theoretic contradiction.