The Black Hole Information Problem as a Channel Theorem - Exterior recovery, remnants, and the Page curve, decided by a complementary channel

The black hole information problem contains distinct questions about the continuation of the hole, the reversibility of the complete physical evolution, recovery from exterior radiation, and the feasibility of decoding. For a finite collapse code and a specified exterior channel, exact recovery is equivalent to input independence of its complete complementary channel. We establish the exact recovery criterion, the exhaustive loss/storage/recovery trichotomy, the strong-complete-evaporation corollary, the remnant and baby-universe classification, the entropy counterexample and the approximate information–disturbance criterion. The current LP architecture makes the addressed bearer and physical cut explicit before these questions are evaluated. Constitution identity is distinct from Q2a and Q2b, and a code-recovery theorem is not automatically a theorem about the complete identity of a physical bearer. We prove this independence with two different hidden coupling profiles that have identical, perfectly recoverable exterior channels. We also strengthen the Page-curve obstruction: two globally isometric evolutions have the same prescribed radiation entropy curve for every pure code input, while one is perfectly recoverable and the other erases the entire code from the exterior. A reference-system decoupling test distinguishes them exactly. The approximate result now gives dimension-independent diamond-norm bounds with explicit normalization, and separates a sufficient identity-margin certificate from a verdict of actual identity loss. Strong complete evaporation forces exterior recovery only when the complete complement is input independent and global reversibility is established. The cessation of the black hole alone establishes neither premise. The results decide these questions for a fully specified finite model; they do not supply an evaporation mechanism or choose the realized four-dimensional channel.