Reconstruction Support and Island Dominance in Black-Hole Information Recovery

The island formula recovers a unitary Page curve in controlled gravitational models, but the dominance of an island saddle, the existence of a radiation recovery channel, and the cost of implementing that channel are logically distinct statements. We formulate this distinction as a two-stage framework. In the first stage, candidate islands and the physical Page transition are determined exclusively by the standard generalized entropy and quantum-extremal-surface (QES) prescription. In the second stage, an explicitly model-dependent reconstruction score ranks the QES candidates by normalized recovery, connectivity, algebraic, and complexity diagnostics. The score is not proposed as a new gravitational entropy formula. For a nondegenerate island/no-island crossing we derive the first- and second-order displacement of the operational reconstruction-support transition relative to the Page transition. We define operator recovery on a code subspace without assuming an interior–exterior tensor factorization, state the Fawzi–Renner recoverability implication with its correct conditional-mutual-information hypothesis, and derive a dimension-normalized mutual-information network diagnostic bounded in <!-- MathType@Translator@5@5@MathML2 (no namespace).tdl@MathML 2.0 (no namespace)@ -->