A Recurrence Converse for Identity Persistence: The Verdict Channel and the Storage–Persistence Capacity Gap

Abstract This paper characterizes recurrence and distinguishable-history capacity in finite declared identity regimes. For the class-indexed observable, the central quantity is the logarithm of the spectral radius of the reachable recurrent subgraph. A recurrence converse bounds sound retention codes by this capacity, with matching attainability under the declared conventions, and separates extinction, bare recurrence and branching recurrence. For the continuing verdict-only channel, finite determinization gives an exact computable capacity. Every finite regime lies strictly below one bit per step, while an explicit family approaches that ceiling. A separate stopped-graph construction characterizes the terminating channel: operational words are label-images of walks, at most two words occur at each positive length, and the asymptotic rate is zero or negative infinity according to a reachable-cycle test. For transitions with nonnegative costs and budgets enforced at every prefix, the paper establishes a sublinear-budget characterization, a fixed-tilt spectral converse, and a restricted attainability result for a single recurrent block in the large-buffer limit. Countermodels show why feasible entry, expenditure order and the counted language cannot be omitted. The results are scoped to declared finite models; they do not establish physical resource costs or general autonomous self-improvement. Broader composition and capacity questions remain explicitly open.