A Coding Theorem for Identity Persistence

Abstract This paper proves a coding theorem for finite identity-persistence regimes. Prior work established that coherent identity persistence under transformation requires explicit specification of an identity-bearing unit, state space, admissible transformations, admissible redescriptions, an identity-relevant quotient, continuation structure, invariant basis, governance functional, and drift bounds. A finite-state identity capacity was then defined as the asymptotic logarithmic growth rate of admissible identity-preserving trajectories. The present paper proves the constructive achievability theorem: every declared finite identity regime admits an exact deterministic identity code whose accepted language is precisely the set of admissible identity-preserving trajectories. The encoder maps state histories into canonical quotient trajectories; the verifier checks local admissibility, redescription invariance, continuation legality, invariant comparability, scalar drift bounds, and lifted path budgets; the decoder emits one of five verdicts: PERSIST, BREAK, BRANCH, UNDEFINED, or REGIME-CHANGE. For local finite regimes, the achieved growth rate equals CI = log rho (A), where A is the admissible-transition matrix. For path-budget regimes, the lifted code achieves log rhoᵣeach (Aₗift). No sound identity code can exceed these rates without accepting inadmissible trajectories and thereby violating the declared identity regime. Finite identity capacity is therefore both an upper bound and an achievable bound. The result completes the finite-regime identity-persistence stack: structural necessity, capacity definition, deterministic verdict procedure, exact enforcement, converse bound, achievability, maximality, and replay determinism. The theorem is bounded to finite declared regimes; continuous, stochastic, weighted, nonstationary, and infinite-state generalizations remain open.