A Recurrence Converse for Identity Persistence

Zenodo abstract This paper develops a recurrence converse for finite identity-persistence regimes. Building on the Coding Theorem for Identity Persistence, it asks what separates persistence-bearing retention from storage-only retention when the same/not-same verdict is treated as an information channel. The answer is the reachable recurrent-core capacity: for a named anchor class q, the class-indexed verdict channel has capacity log ρ(A_rec(q)). The paper proves that strongly connected components form the extremal partition for recurrence-rate attribution, establishes a Perron–Frobenius localization identity for successor-closed subregimes, and derives a uniform counting bound that recovers the local converse of the coding theorem while yielding a reachability-qualified recurrence converse. It distinguishes three structurally distinct regimes: no reachable recurrence, bare recurrence with zero asymptotic rate, and branching recurrence with positive verdict capacity. The paper also closes several frontier variants. It computes verdict-only capacity through finite determinization, proves that bare verdict streams carry strictly less than one bit per step in finite regimes, characterizes small-horizon behavior as a dictionary-layer rather than capacity phenomenon, and establishes a sublinear-budget recurrence converse through the zero-cost recurrent core. Binding linear-budget regimes, weighted and stochastic extensions, and countable-state regimes remain explicitly identified as open.