Proposition: Operational discard is not annihilation

Equation (eq:enter-drop) does not assert physical annihilation of a state or its complete information. The complete dilation (eq:enter-dilation) preserves the unknown state and its references, whereas the restricted output after forgetting need not recover it. It also does not determine a heat, work, or memory-reset cost.

Proof. Theorem thm:global applies to $W$ and gives a complete-output left inverse. Restriction by partial trace changes the available output but not the isometry identity. No Hamiltonian, bath temperature, work reservoir, or resource-return condition occurs in (eq:enter-drop). The thermodynamic non-identification is made explicit in Proposition prop:no-landauer-from-channel.