A certificate is more than a formula

The order in which these slots are solved may vary in a particular construction. The distinction among them does not. A state is not necessary for every algebraic statement, but a state-dependent expectation cannot be evaluated by naming a parametrix alone.

For a finite register, Q1–Q5 are not imposed by pretending that a register already has a graviton quadratic operator. Its native quantum input problem is a complete finite channel or instrument. A claimed field-theoretic or continuum interpretation adds the corresponding comparison and limit obligations.

Closure in this paper means that all operations and conclusions required by one declared target have been supplied on matching domains, with its allowed changes and verification scope stated. It does not mean all-orders renormalizability, an RG fixed point, ultraviolet completeness, or a quantum theory of spacetime.