Definition: Operational island certificate

For an isometry $V:L\to A\otimes B$, an $e$-accurate operational island certificate for the initially interior logical factor $L$ consists of:

1. the specified noncommutative target algebra $\mathcal{B}(L)$, with $d\ge 2$;

2. the accessible interface $A$, excluding physical $B$;

3. one deterministic CPTP quantum-return map $\mathcal{D}:A\to L_{\mathrm{out}}$;

4. the reference-preserving bound

   $$d_{\diamond }(\mathcal{DN},{\mathrm{id}}_L)\le e.$$

Exact certificates have $e=0$. The term “island” refers to this logical reconstruction assignment. It does not designate an independently variable physical residual factor.