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 asks whether the information in a collapsing quantum state survives evaporation and can be read from the Hawking radiation. This paper shows that, for a named set of collapsing states, the question reduces to an exact decision theorem. Model the global final evolution as an isometry from an initial code subspace into the accessible radiation together with a complete hidden final sector; the two are complementary quantum channels. Then the initial code is exactly reconstructable from the radiation if and only if the hidden sector carries no input-dependent information — a direct instance of the exact quantum-error-correction and complementary-channel theorems, which are imported unchanged. Three exhaustive cases follow, with loss defined by the absence of a global recovery rather than by the non-isometry of some channel representation: a globally irreversible evolution admits fundamental loss; a globally reversible evolution with an input-dependent hidden sector stores or exports the information rather than destroying it; a globally reversible evolution with an input-independent hidden sector permits complete exterior recovery — and strong complete evaporation, meaning no remnant, baby universe, or other input-dependent hidden sector, makes the last case automatic. La Profilée’s contribution is the level- and sector-correct placement of the channel theorem: the end of the black hole is not the end of the global quantum state, remnants and baby universes are storage or export models under global isometry, and a reproduced Page curve is entropic consistency in a model, not by itself a proof that a full code is recoverable. What the theorem decides is the in-principle question; whether real four-dimensional black holes satisfy its premises, and whether any recovery is executable with realistic resources, are separate questions, marked open throughout. No new evaporation mechanism is proposed.