Output reservation and the admission criterion

Every compact-distribution polynomial output has a canonical place in ${\mathcal{P}}_{\mathrm{dist}}$. To place it in ${\mathcal{P}}_{\Sigma }$, one must verify (eq:admission), including its directional descendants.

Thus the carrier construction gives a concrete output test rather than a promise that every future graph passes it. If a later graph has a first metric derivative outside ${\mathcal{S}}_1$, then that graph requires a further domain enlargement or supplies an admission obstruction for the current domain.

No unconditional ${\hslash }^3$ output-closure theorem follows from the module (eq:module).