Theorem: Operational-island existence and its precise scope

Under the finite preparation, nested access, known-unitary, and positive-cutoff declarations of Section sec:model:

1. Haar scrambling has the exact complementary moment (eq:gamma) and leakage-square allowance (eq:K).

2. A retained cut satisfying (eq:thresholdexact) certifies, with probability at least $1-p$, a nontrivial operational island for every later retained cut, with ideal half-diamond recovery error $\min \{a_d,\sqrt{2\zeta }\}$.

3. The decoder is the complete deterministic finite program (eq:SDP), with a stated batch realization and workspace allowance.

4. The early-entangled coherent entropy obeys (eq:Sbounds)–(eq:ranktent); its destructive record obeys (eq:Hbounds) and append-only monotonicity. The pure Page–Sen specialization is restricted to $h=1$.

5. The explicit family of Theorem thm:joint proves positive-measure coexistence of accurate recovery and strict coherent turnover before a cutoff retaining one physical qubit.

6. The controlled-Pauli construction of Theorem thm:exactexample proves exact operational-island existence at a positive-dimensional residual, with an exhibited decoder and saturating coherent entropy.

7. No second commuting copy of the same logical algebra is produced. Residual readout and discard leave the accessible channel unchanged.

8. A purely classical returned interface has permanent whole-diary error at least $1-1/d$. A record plus quantum early access is a different interface and can recover through the teleportation example.

These conclusions prove existence of the operational objects defined in Definition def:island. They do not prove existence of a gravitational island, a quantum extremal surface, a replica wormhole, or spacetime emergence.

Proof. Items (i)–(iii) are Theorems thm:moment, thm:threshold, and thm:decoder, with Lemmas lem:Choi and lem:approx and Proposition prop:workspace. Item (iv) is Theorem thm:entropy and Proposition prop:record. Items (v)–(vi) are the two independent constructive existence results. Item (vii) is Propositions prop:residual, prop:noduplicate, and prop:discard. Item (viii) is Proposition prop:EB and Example ex:teleport. The excluded gravitational data are not among the model declarations or constructions.