Introduction and governing stance

The terms “recycling,” “erasure,” and “holography” can refer to different mathematical and physical tasks. This note fixes their interfaces before comparing them. Its immediate source is the supplied Hawk04-B version 2 manuscript CheungHawk04B, with the predecessor Hawk04-A providing the complete-complement formulation and featureless-radiation comparison CheungHawk04A.

The present task is not to derive a new radiation decoder. It is to prevent three changes of meaning:

1. replacing a quantum-return decoder by a classical readout;

2. replacing a declared marginal by a claim of physical deletion or thermodynamic reset;

3. replacing a channel-correction criterion by an area law or a gravitational bulk–boundary identification.

Throughout, $\mathsf{Enter}$ means destructive computational-basis readout. Its measured quantum names are consumed and its observable output is classical. A decoder instead has a quantum-return interface

A prescribed observation may subsequently consume $C_{\mathrm{out}}$, but that subsequent operation has a measurement channel, not ${\mathrm{id}}_C$, as its target.

Unitary uncomputation, destructive readout-and-drop, and residual leakage are not three versions of the same operation. The first can coherently reverse a specified computation when the required registers are available. The second is generally many-to-one at its declared observable interface. The third is a numerical channel diagnostic, not an operation at all.