Introduction

The black-hole information problem asks whether a proposed evaporation description is compatible with complete quantum evolution. Hawking's radiation calculation and information-loss proposal provide its historical setting Hawking1975,Hawking1976. A finite circuit semantics does not derive that physical evolution.

This paper supplies an operational formulation in which the following questions have distinct mathematical objects:

1. Is the finite quantum code preserved by the complete evolution?

2. Does the declared accessible subsystem admit a quantum decoder?

3. What instrument produces a specified classical record?

4. Which outputs are omitted from the observer's description?

5. Does the same physical model have an access-preserving endpoint?

Every code comparison retains arbitrary external references.

The exact correction and no-hiding principles are established results KnillLaflamme,BraunsteinPati. The approximate channel conversion used here is due to Kretschmann, Schlingemann, and Werner KSW. We prove the exact statements locally, identify the imported approximate theorem explicitly, and make no priority claim for these principles. Superoperator semantics for quantum languages also has established precedent Selinger.

The contribution is the joint accounting of typed destructive readout, complete complementary outputs, access-preserving regulator certificates, and an exact dynamical negative control. In particular, a nonconstant residual is not promoted to a recoverable residual-only copy of the entire unknown code.