Theorem: Scope of the finite Hawking analogies

Under Definition def:interface and the explicitly specified finite Born readout convention:

1. Complete isometric evolution preserves the code on $RB$. Residual-only relocation holds exactly when the accessible channel is a replacement; it is an additional property.

2. The access declaration fixes the decoder domain. Ignored local operations on the residual do not change the accessible channel.

3. Destructive Enter returns a classical record and the explicitly retained quantum remainder. It does not return the consumed quantum input.

4. Finite scheduled emissions and adaptive readout trees define reference-preserving CPTP channels. Their storage and termination properties do not supply a mass, energy, or geometric law.

5. Added residual readout followed by classical discard is exactly partial trace. It leaves the pre-existing accessible recovery error unchanged.

6. Exact full-code recovery from an accessible quantum factor is equivalent to constancy of its complete complement. A purely classical returned record has recovery error at least $1-1/d$, while selected thermal or entropy data do not determine quantum recovery.

These are statements about the declared finite interfaces. They do not identify those interfaces with a gravitational process.

Proof. Item 1 is Theorem thm:global, Corollary cor:nohiding, and Example ex:relocation. Item 2 is Lemma lem:local. Item 3 is Definition def:enter and Proposition prop:instrument. Item 4 is Proposition prop:sequential, Theorem thm:soundness, and the separate defining data of the three ledgers. Item 5 is Theorem thm:drop. Item 6 is Theorems thm:exact and thm:classical, together with Propositions prop:thermal and prop:entropy.

The geometric and dynamical objects needed for a gravitational identification are not constructed by any of these definitions or proofs. The last sentence is a scope statement, not a universal nonexistence theorem for possible future correspondence maps.