Mathematical Operations and Physical Interactions

The Physics–Mathematics Transformation (PMT) framework investigates whether restricted, explicitly defined correspondences between mathematical and physical systems can support valid scientific inferences in either direction. This third paper in the PMT research series examines the relationship between mathematical operations and physical interactions, focusing on the preservation of composition laws, operation ordering, and observable distinctions. Using transformations of physical states and their mathematical representations, the study investigates operation-preserving mappings, faithful representations, noncommuting operations, and the information loss that may arise when distinct physical processes receive identical mathematical descriptions. Two restricted case studies address order-sensitive sequential transformations and the composition of independent or coupled systems. The analysis also examines physical realizability, approximation errors, measurement requirements, and comparison with established mathematical and physical methods. The paper develops a structured procedure for assessing candidate PMT correspondences through admissibility, structural preservation, observable fidelity, and independent validation. It does not claim a universal equivalence between mathematical operations and physical forces or establish new physical laws. Instead, it identifies conditions under which such correspondences may support scientifically justified reasoning and future investigations. Physics–Mathematics Transformation research series — Paper 3.