Theorem: Finite-cutoff resolution certificate

Suppose the retained schedule satisfies the threshold hypothesis of Theorem thm:threshold, and choose a unitary in its success set. At the positive-residual endpoint $K$, the model supplies:

1. complete preparation: the variable diary, fixed $HE$ purification, pure $F$, and every release placeholder;

2. complete input-independent evolution: the declared $U$ and release SWAPs;

3. declared outgoing access: $A_K=R_{K}E$, excluding $B_K$, with the explicit endpoint isometry ${V_K}^U$;

4. a complete-complement certificate:

   $$d_{\diamond }({{\mathcal{C}}_{\mathcal{K}}}^U,{\mathcal{T}}_{{{\sigma }_{B_K}}^U})\le \zeta .$$

The endpoint admits the deterministic quantum-return decoder of Theorem thm:decoder. Its ideal half-diamond error is at most $\min \{a_d,\sqrt{2\zeta }\}$; an implementation allowance $\iota$ adds by Proposition prop:transfer. The returned classical record, if subsequently produced, has its separately stated entropy and access contract.

No conclusion about later physical evaporation follows.

Proof. The four items are explicit objects of the fixed model, not unconstructed physical identifications. The endpoint leakage is the threshold theorem, and the decoder and implementation statements have already been proved. The schedule stops with $b_K\ge 2$; no subsequent dynamics or gravitational endpoint has been specified.