Coherence as the Governing Invariant of Persistence_ Minimal Structure, Scalar Measures, and Cross-Domain Realizations

Abstract Persistence is assumed across physics, biology, cognition, and engineered systems, yet it is never formally accounted for. Identity is treated as primitive, and collapse is explained post hoc in domain-specific terms such as error, noise, maladaptation, or randomness. This paper identifies coherence as the scalar governing invariant that orders persistence under perturbation. It proves the minimal structural requirements necessary to support persistence, establishes SO(2) as the minimal sufficient symmetry enabling recurrence, interference, and scalar misalignment detection, and formalizes coherence using a triplet state space. Diverse domains are shown to be substrate-specific realizations of the same underlying coherence structure. Failure, across all domains, is demonstrated to be drift beyond recoverable coherence bounds. The result is a unifying, falsifiable ordering principle that operates prior to physics, evolution, cognition, and optimization, without replacing them.