Macrotypes: a structural criterion for physically meaningful macrovariables

Abstract The paper develops a structural criterion for identifying physically meaningful macrovariables, together with an information-theoretic interpretation. We introduce macrotype maps , defined as quotient maps that identify microconfigurations indistinguishable under admissible transformations and invariant under admissible interactions. This yields a notion of macroscopic identity based on two invariance principles: (i) internal invariance, understood as temporal or symmetry-based stability of defining properties, and (ii) external relational invariance, understood as a stable response profile across admissible coupling contexts. Macrotypes retain precisely the invariant interaction-relevant information for macroscopic behaviour while discarding irrelevant microscopic detail, providing a principled selection criterion for physically meaningful coarse-graining. The point of the construction is not the mere observation that macrostates may be represented as equivalence classes. Rather, the paper proposes a criterion for selecting those equivalence classes whose fibres are physically meaningful because they preserve invariant structure and reproducible interaction behaviour under explicitly specified admissible contexts. The framework is illustrated with examples from both physical and functional domains, and it naturally supports an information-compression interpretation and the possibility of hierarchical and compositional descriptions of macroscopic organization. The approach offers a concise structural contribution to the micro–macro problem by characterising macroscopic identity in terms of transformation– and interaction–stable invariant structures.