Abstract

We prove existence of finite operational islands, defined here as reference-preserving reconstruction certificates for a specified logical factor initially located in an inaccessible vault. The definition is deliberately not a gravitational definition of an island. The final physical residual remains a tensor factor and is not silently granted to the receiver.

The complete model contains an unknown $d$-dimensional diary, an $h$-dimensional old-vault register maximally entangled with accessible early radiation, and a pure blank register of dimension $f$. One known, input-independent unitary scrambles the vault; a nested schedule releases its cells. For Haar scrambling we derive the exact complementary Choi Hilbert–Schmidt moment and convert it to a worst-case diamond certificate. A single-cut probability estimate, together with nested access, certifies all subsequent retained cuts. A deterministic decoder is specified by a finite semidefinite program, and its quantum workspace and implementation allowance are stated.

The entangled preparation changes the entropy calculation: the late radiation is a reduction of a random rank-$h$ projector, not generally a Haar pure-state reduction. We derive its exact purity moments and rigorous entropy bounds. A concrete parameter family has positive Haar measure on which accurate whole-diary recovery and strict coherent entropy turnover both occur before a cutoff leaving one physical residual qubit. We also construct an exact entanglement-assisted example with a positive-dimensional residual and an explicit controlled-unitary decoder.

Append-only Enter records cannot turn over. Their record-only quantum recovery obstruction does not apply when a genuine quantum early register remains accessible; teleportation supplies a precise counterexample to that overextension. No quantum extremal surface, area term, replica wormhole, gravitational Hamiltonian, or spacetime emergence is constructed.