A Mathematical Theory of Identity Persistence_ Identity, Invariance, Drift, and Capacity Under Admissible Transformation

Abstract This paper develops a mathematical theory of identity persistence under admissible transformation. It asks what structure is required for a same/not-same judgment across recurrence to be meaningful, non-arbitrary, and non-trivial. The paper defines identity-bearing units, state spaces, admissible transformations, admissible redescriptions, quotient state spaces, continuation relations, invariant vectors, scalar governance functionals, drift bounds, verdict functions, and finite-state identity capacity. The central claim is structural rather than ontological: a persistence judgment is coherent only when the identity-bearing unit, admissible transformation regime, quotient, invariant basis, continuation structure, governance rule, and drift bounds are specified. Where those are absent, the judgment is not merely vague; it is structurally undefined. The paper introduces a finite verdict set for identity-persistence judgments: PERSIST, BREAK, BRANCH, UNDEFINED, and REGIME-CHANGE. It also introduces a finite-state capacity quantity, C_I, defined as the asymptotic logarithmic growth rate of admissible identity-preserving trajectories in a finite admissibility graph. In local finite regimes, C_I is given by the logarithm of the spectral radius of the admissible-transition matrix. For path-budget constraints, the state space is lifted to include accumulated drift, yielding a lifted spectral-radius capacity. The result does not claim that all systems instantiate the formal regime, that identity persistence settles ontology, or that probability and entropy are invalid. It supplies a formal framework for when “same system” claims are well-defined, plus a finite graph-theoretic measure of how many identity-preserving transformations a declared regime allows.