Theorem: Hawk01-B release

Within the declared conserving model:

1. arbitrary-input, reference-preserving recovery is exactly equivalent to a constant complete complement;

2. the KSW framework gives ${ϵ}_{\mathrm{rec}}\le \sqrt{2{\delta }_{\mathrm{leak}}}$ and ${\delta }_{\mathrm{leak}}\le \sqrt{2{ϵ}_{\mathrm{rec}}}$ in half-diamond normalization;

3. the maps $V_1,\dots ,V_N$ form a conserving sequential process with internally generated key, no reset, and explicit vacuum and pending branches;

4. every individual pre-endpoint emitted block is message-independent, but the joint radiation supports optimal recovery error $s_t(1-d^{-2})$;

5. the complete residual leakage distance is exactly the same quantity, with reference-information transfer $I(S:R_{\le t})=2p_t\log d$;

6. the declared information-free endpoint releases every residual degree of freedom and permits exact recovery;

7. complete destructive occupation access instead has the obstruction and exact optima of Theorem thm:number;

8. a second conserving process has identical radiation density operators for one chosen preparation, and identical radiation entropy histories for every pure preparation, but retains the code in a disclosed inaccessible endpoint memory.

None of these statements derives the microscopic information-transfer dynamics or terminal evolution of a realistic black hole.

Proof. Items (i)–(ii) are Theorems thm:exact and thm:approximate. Item (iii) is Theorem thm:physicality. Items (iv)–(v) follow from Theorems thm:prefixchannels, thm:optima, and thm:information. Item (vi) is Theorem thm:endpoint. Item (vii) is Theorem thm:number. Item (viii) is Theorem thm:entropyseparation. The coupling, degeneracies, and endpoint remain explicit model inputs, so no microscopic gravitational derivation follows.