Declarations

. This single source contains the model, proofs, and internal bibliography. No external data are required for typesetting.

. No compilation, numerical calculation, independent symbolic execution, or formal proof checking is claimed.

interpretation. The work is a finite-channel benchmark motivated by information-flow questions. It does not identify the residual subsystem with a black-hole interior, the accessible bath with Hawking radiation, or either parameter with gravitational evolution.

. Quantum recovery, complementary channels, entropy identities, concurrence, negativity, and dephasing are established frameworks. The model-specific calculations are proved here without claiming a new general foundation or exhaustive novelty verification.