A Recurrence Converse for Identity Persistence

This paper proves a recurrence converse for finite declared identity-persistence regimes. Given an admissible transition matrix A, it shows that the capacity separating persistence-bearing retention from storage-only retention is carried not by acyclic trajectory growth but by the reachable recurrent core A_rec(q). Under the class-indexed verdict observable, the same/not-same channel anchored at q has capacity log ρ(A_rec(q)), with achievability and converse established over the same mathematical object. The argument separates attribution, mathematical, and dictionary layers. Strongly connected components are characterized as the extremal partition for recurrence-rate attribution, a Perron–Frobenius localization identity isolates the recurrent source of capacity, and a uniform counting bound yields the recurrence converse. The result refines the Coding Theorem for Identity Persistence by identifying where finite identity capacity resides. The paper further computes verdict-only capacity through finite determinization, proves a strict sub-one-bit upper bound for bare verdict streams, resolves small-horizon recurrence as a naming rather than capacity phenomenon, and establishes a sublinear-budget recurrence converse using the zero-cost recurrent core. Weighted extensions, binding linear-budget regimes, and countable-state extensions remain open.