Theorem: En07-C finite mathematical release

For the finite interface of Definition def:interface, this manuscript establishes:

1. complete restoration and reference-stable trace-norm preservation;

2. exact outgoing recovery if and only if the complete complement is constant, with the explicit decoder (eq:exact-decoder) and dimension bound (eq:dimension-bound);

3. destructive basis Enter followed by forgetting equals partial trace, with a complete isometric dilation and no annihilation or thermodynamic-cost conclusion;

4. record-only and featureless-accessible-channel lower bounds;

5. approximate recovery/leakage conversion conditional on the explicitly imported KSW theorem, with the local half-diamond normalization and finite Choi and distinguishability certificates;

6. explicit finite separations of selected thermal statistics, spectra, entropy, purifier marginals, energy drain, and termination from outgoing recovery;

7. separation of complete energy, implementation-dependent erasure cost, and residual leakage;

8. access-preserving endpoint, finite-time, and receiver error budgets conditional on the four same-model obligations;

9. compatibility inequalities for simultaneous accessible replacement and complement-decoupling allowances.

No gravitational preparation, complete gravitational evolution, outgoing gravitational receiver, bulk–boundary dictionary, Planck-scale endpoint, or inherited interacting gravitational quantum coefficient is constructed.

Proof. Theorem thm:global proves (i). Theorem thm:exact and Corollary cor:dimension prove (ii). Lemma lem:erasure-interface and Proposition prop:not-annihilation prove (iii). Theorem thm:classical and Proposition prop:featureless prove (iv). Convention conv:ksw, Theorem thm:approx, Lemma lem:choi, Proposition prop:certificates, and Lemma lem:witness give (v). Examples ex:one-state and ex:purifier, Proposition prop:entropy, and Proposition prop:insufficiency prove (vi). Definition def:energy, Proposition prop:energy, and Propositions prop:landauer-package and prop:no-landauer-from-channel establish (vii). Theorem thm:endpoint and Corollary cor:receiver prove (viii). Corollaries cor:compatibility and cor:endpoint-compatibility prove (ix).

The excluded gravitational and analytic objects are not among these constructions. The documentary edges do not supply them.