Proposition: Early-query and side-channel counterexamples

The bound $3/4$ need not hold if the storage or access boundary is changed. In particular:

1. if $J$ is known before encoding, one classical bit can store $X_J$ and succeed with probability one;

2. if a one-bit carrier stores $X_1$ and an uncounted accessible register stores $X_2$, success is one;

3. an unrestricted exact-valued classical state can store the entire input and succeed with probability one.

Proof. Use the indicated stored value at decoding.