Theorem: What a complete evaporation identification would imply

Suppose a gravitational model supplies:

1. a justified finite code and complete microscopic dilation;

2. unitary complete evolution with the preparation ledger retained;

3. a derived outgoing access and endpoint construction;

4. a proof that the endpoint complete residual channel is constant.

Then the outgoing radiation exactly recovers the code, including arbitrary reference entanglement.

If instead all nonradiative output degrees of freedom are proved to be a single fixed pure state, the complete radiation channel is an isometric encoding. A constant final radiation channel on a nontrivial code is incompatible with these hypotheses.

Proof. The complete microscopic evolution supplies an isometry by Theorem thm:global. Apply Theorem thm:exact to the derived endpoint partition. The rank-one case follows from Proposition prop:pure-endpoint. A constant radiation channel cannot have a CPTP left inverse, by Lemma lem:replacement.