Abstract

We give a self-contained finite-interface formulation of the endpoint obligation left open by Hawk04-B and En07-A. Complete isometric evolution, recovery from a declared outgoing factor, destructive classical readout, operational discard, and thermodynamic implementation are different assertions. We prove the complete-output restoration theorem, the exact constant-complement criterion with an explicit deterministic decoder, reference-preserving readout-and-drop identities, record-only reconstruction bounds, and a featureless-output obstruction. We state explicitly the external Kretschmann–Schlingemann–Werner input used for approximate recovery, derive its half-diamond normalization, and provide dimension-explicit Choi certificates and access-preserving endpoint stability estimates.

Two defects in the proposed programme are corrected. All quoted errors use half-diamond distance; the corresponding unhalved diamond bounds have twice the stated values. Moreover, approximate constancy of the complete complement supplies a recovery upper bound, whereas approximate constancy of the accessible output supplies a recovery lower bound. These hypotheses cannot be interchanged.

The mathematical release is an exact and approximate endpoint certificate, not a constructed gravitational endpoint. No microscopic gravitational preparation, outgoing receiver, bulk–boundary dictionary, or evaporation endpoint is supplied. The analytic admission obligations recorded in the supplied Cheung manuscripts are unchanged. Companion manuscripts are documentary comparisons, not premises of the local proofs.