Theorem: En07-A local mathematical release

Under the channel and finite-interface definitions of this paper:

1. complete isometric evolution admits the restoration map (eq:restore);

2. exact whole-code recovery from $R$ is equivalent to a constant complete complementary channel;

3. destructive finite readout followed by discard equals partial trace, including references;

4. the specified complete readout dilation is isometric;

5. record-only reconstruction has error at least $1-d^{-1}$;

6. finite semantic realizations return a quantum code without measuring that designated output, and finite acyclic evaluation terminates;

7. replacement-channel proximity gives (eq:featureless);

8. the exact flagged-transfer identity (eq:flagged) and the two-state control formulas (eq:controlR)–(eq:controlleak) hold;

9. observable channel equality does not fix a unique implementation energy, heat, work, or reset ledger;

10. the selected-entropy counterexample and the conditional outgoing-recovery implication hold.

These are local restatements, examples, and separation arguments. They are not a new dynamical recovery theorem.

Proof. Theorems thm:global and thm:exact prove (i)–(ii). Lemma lem:erasure-interface proves (iii)–(iv). Theorem thm:classical proves (v). Proposition prop:finite proves (vi). Proposition prop:featureless proves (vii). Proposition prop:flagged and Example ex:control prove (viii). Proposition prop:no-landauer-from-channel proves (ix). Proposition prop:entropy and Corollary cor:gravity prove (x).