A Coding Theorem for Identity Persistence

This paper proves a coding theorem for finite identity-persistence regimes. The target problem is identity persistence under transformation: when can a same/not-same judgment across recurrence be treated as meaningful, non-arbitrary, and non-trivial? Prior work argued that such a judgment requires a declared identity-bearing unit, admissible transformations, admissible redescriptions, an identity-relevant quotient, continuation structure, invariant basis, governance functional, and drift bounds. This paper adds the constructive layer. For any declared finite identity regime, there exists a deterministic identity code whose accepted language is exactly the admissible identity-preserving trajectory language of that regime. The code maps raw histories into quotient trajectories, verifies local admissibility and drift constraints, handles lifted path budgets, and emits one of five verdicts: PERSIST, BREAK, BRANCH, UNDEFINED, or REGIME-CHANGE. The code is sound, complete, deterministic, replayable, and maximal. For local finite regimes, the accepted language grows at rate C_I = log rho(A), where A is the admissible-transition matrix. For path-budget regimes, the corresponding rate is log rho_reach(A_lift). No sound code can exceed these rates without accepting inadmissible continuations. Thus finite identity capacity is both a converse bound and an achievable bound. The result is a bounded structural theorem. It does not claim that all real systems instantiate the formal regime, nor that identity persistence is reducible to computation. It shows that once a finite identity-persistence regime is declared, persistence has an exact admissible language, a capacity, and a maximal deterministic enforcement code.