Structural Coarsening and Compatible Representations: A Formal Framework for Transformation Classes, Invariants, and Quotient Partitions

Abstract

Many systems studied in science are accessible only indirectly, through partial measurements, reconstruction procedures, and model-dependent representations. This preprint develops a formal framework for determining which structural distinctions are supported by a specified operational language and under what conditions external representations preserve the resulting quotient structure coherently. Beginning with a primitive set-theoretic system comprising a configuration set, an ambient transformation family, a value set, and an admissible property family, the framework establishes an antitone Galois connection between transformation classes and property families. Enlarging a transformation class restricts the corresponding invariant family and thereby induces a kernel equivalence that is no finer than the equivalence associated with the smaller class. The resulting internal quotient structure is related to external target sets through a factorization theorem: every structurally compatible representation factors uniquely through the corresponding quotient set. For nested transformation classes, the internal theory determines a coherent system of canonical surjections between quotient sets. The corresponding morphisms between external target sets, however, are not supplied automatically; their existence is formulated as a separate external compatibility criterion. This demarcation distinguishes internally necessary mathematical structure from model-dependent representational adequacy. The framework is further extended to ordered regimes. A regular stratification requires both strictly nested transformation classes and strictly nested kernel equivalences. When such a stratification is equipped with compatible external representations and coherent external morphisms throughout the hierarchy, the Representation Spectrum arises as a derived idealized realization: a linear organization in which internal quotient coarsening is mirrored by externally compatible transitions. The framework is then used to formulate testable structural hypotheses concerning grokking in deep learning and the Hubble tension. These applications are treated not as confirmations of a universal law, but as falsifiable hypotheses whose evaluation depends on explicit operationalization and on the existence of the required permitted external morphisms. The framework therefore provides a rigorous formal mechanism for structural coarsening and representation factorization, together with a precise diagnostic language for representational compatibility and incompatibility.