This revised manuscript develops an algebraic formulation of refinement in a finite measurement framework. It treats refinement operations as partially defined, set-valued maps between context-dependent admissible assignments, with emphasis on composition, quotient structure, finite-resolution equivalence, and the conditions under which relational structures remain stable under contextual refinement. The manuscript provides the algebraic machinery supporting the broader refinement-closure program, including admissibility, refinement histories, path dependence, obstruction, and the emergence of effective structure through stable composition. This version clarifies the distinction between admissible generators, uniquely selected generators, and empirically sufficient generators. A local generator constraint is not treated as uniquely forced by finite measurement structure alone. Instead, candidate generators are restricted by admissibility, refinement composition, finite-resolution quotienting, and empirical refinement stress. The revision introduces residual-based language for assessing whether a candidate generator remains sufficient when confronted with additional finite measurement data under refinement. The manuscript therefore frames physical selection not as proof of a uniquely true generator, but as the identification of surviving equivalence classes of generators that remain sufficient within a finite measurement domain. This strengthens the connection between the algebraic refinement structure and later applications to wave-like dynamics, quantum-like contextuality, and empirically constrained model selection.
Charles Durbin (2026) studied this question.