Abstract

We formulate delayed-query retrieval from Hawking radiation as an operational task on outgoing quantum-field modes in a four-dimensional Schwarzschild exterior. Starting from the affine-vacuum horizon boundary condition and the scalar scattering problem, we derive the normalized greybody-modified emission law and finite-packet covariance. For a trusted four-state qubit preparation, we prove the exact optimal retrieval formula, a message-independent semiclassical null, a sharp measure-before-query bound, and complete-attempt robustness bounds. We construct a finite-energy emission isometry on degenerate scalar occupation sectors whose late-radiation marginal agrees with a truncated Hawking-mode thermal state for every message. With a declared early key resource, its joint radiation permits partial quantum recovery; without that resource, the late radiation is exactly message-independent. A second conserving completion has the same late state and energy distribution but no retrieval advantage even with the early resource. Thus emission spectra, including the complete late marginal in this example, do not determine the assisted recovery channel. We calculate the exact coherent-access, destructive-number, and referenced-recovery optima of the construction. Finite-packet cutoffs, collection channels, certified model errors, and attempted-run sampling are treated explicitly. The results are direct field-mode statements and conditional completion theorems, not a derivation of microscopic black-hole dynamics, an astrophysical decoding experiment, or a resolution of the information paradox.