Proposition: Causal loopholes are changes of model

Each of the following permits perfect retrieval but violates the declared comparison:

1. advance access to the actual $J$;

2. a predictable query as in Example ex:predictablequery;

3. a second accessible register retaining the other bit;

4. correlated ancillary assistance as in Example ex:dense;

5. an unrestricted exact-valued classical state storing the input word.

Proof. Store or recover the requested bit using the additional access indicated in each case. The added access either invalidates freshness, the cut, or its capacity accounting.