Proposition: Inaccessible references do not change the certificate

Suppose all controllers use only the declared code, fixed setting, geometry, accessible data, and resource permissions, and every operation extends by the identity on an inaccessible reference $E$. Then:

1. adding $E$ does not change $\delta$, the catalogue, or any matrix entry;

2. the instrument, centered-response, and transport identities remain valid on arbitrary reference-entangled inputs;

3. equal joint inputs on the declared dependency interface give equal corresponding joint outputs.

Proof. No control function has an input from $E$, so its selected operation and classical table remain unchanged. All relevant identities are identities of linear maps and hence remain identities after tensoring with ${\mathrm{id}}_E$. The last assertion follows from Theorem thm:locality.