Theorem: Non-entailment of quantum environmental memory

A violation of (eq:classicalbound) in the admitted one-receiver task does not imply quantum non-Markovianity or a quantum-memory environment.

There is a process with a single qubit, no persistent environmental register, and memoryless storage that attains (eq:quantumvalue).

Proof. Use Theorem thm:quantumattainment and identity storage. The storage intervals are fixed channels depending only on the current carrier state. Fresh independent ancillas, if used to realize those channels, need not persist across intervals. The winning resource is the retained qubit, not a history-bearing environment.