Theorem: En07-B mathematical release

For the finite interface fixed in Definition def:interface, this manuscript establishes:

1. complete-output restoration and trace-norm preservation, including references;

2. exact outgoing recovery if and only if the complete complement is constant, with an explicit deterministic decoder and residual-rank dimension bound;

3. destructive readout followed by forgetting equals partial trace, while its coherent complete realization is isometric;

4. record-only and featureless-accessible-output recovery obstructions in the stated half-diamond normalization;

5. channel equality does not determine thermodynamic implementation cost;

6. approximate recovery/leakage conversion conditional on the explicitly imported KSW theorem, together with finite Choi and distinguishability certificates;

7. access-preserving endpoint and finite-time recovery implications under the four declared same-model obligations;

8. finite counterexamples showing that global preservation, conserved complete energy, selected zero entropy, and programme termination do not supply outgoing recovery.

The gravitational endpoint package is not constructed. No thermal implementation, unrestricted resource optimum, bulk–boundary dictionary, or interacting gravitational quantum coefficient is released.

Proof. Assertion (i) is Theorem thm:global. Assertion (ii) is Theorem thm:exact. Assertion (iii) is Lemma lem:erasure-interface. Assertion (iv) is Theorem thm:classical and Proposition prop:featureless. Assertion (v) is Proposition prop:no-landauer-from-channel. Assertion (vi) is Theorem thm:approx, Lemma lem:choi, Proposition prop:certificates, and Lemma lem:witness, with the external input specified in Convention conv:ksw. Assertion (vii) is Theorem thm:endpoint and its corollaries. Assertion (viii) is Proposition prop:entropy and Proposition prop:insufficiency.

The model-specific physical and analytic objects listed after (viii) are not among the constructions in these proofs. The documentary sections do not introduce them as premises or prove them as conclusions.