Coherent Commutativity and Commutativity Islands in Noncommutative Systems

Version 3: This version refines the exposition, strengthens the structural definition of commutativity as a restricted case of transformation without relevant contextual accumulation, and clarifies the notions of structural memory, coherent commutativity, and commutativity islands. It also improves the visual presentation of the key algebraic relations, incorporates stronger continuity with earlier MGQC studies, and sharpens the interpretive distinction between literal commutativity, contextual re-expression, and locally commutative domains, while preserving the article’s central thesis and role within the MGQC research program. Abstract Commutativity is one of the most familiar properties of classical arithmetic. In the real and complex numbers, multiplication is commutative, and this historical regularity encouraged the view that commutativity is a natural property of numerical operations. Yet many important mathematical and physical systems are globally noncommutative, including matrix algebras, quaternions, transformation groups, and operators in quantum mechanics. This article proposes a structural reinterpretation of this contrast. It argues that classical commutativity corresponds to structurally restricted systems in which transformations do not accumulate relevant changes of context, whereas noncommutativity appears when transformations modify the operative setting of the system, generating what is here called structural memory. On this basis, the article introduces the concept of coherent commutativity. The point is not to restore literal commutativity where it fails, but to show that, in certain noncommutative settings, a composition AB may be coherently reorganized as B′A, where B′ is a contextually re-expressed version of B adapted to the structural change introduced by A. Conjugation in group theory provides the clearest algebraic model of this re-framing mechanism, though not its exhaustive form. The article also studies commutativity islands, that is, local commutative domains embedded within globally noncommutative systems. Through examples from matrices, rotations, quaternions, semidirect products, holonomy, and quantum mechanics, the article argues that noncommutativity is often best understood not as an isolated algebraic anomaly but as the manifestation of transformations acting across changing structural contexts. In this sense, the article treats classical commutativity, coherent commutativity, and commutativity islands as distinct but related structural regimes within a broader interpretive framework. This preprint forms part of the Model of General Quasi-Coherence (MGQC) research program.The author publishes under the name Antonio Dominguez-Digat. Earlier records may appear under Antonio Domínguez, Antonio Dominguez, or Antonio Dominguez Digat.