The Physics–Mathematics Transformation: A Conceptual Framework for Bidirectional Scientific Discovery

Physics and mathematics are closely connected, but a shared formal description does not by itself establish an equivalence between physical and mathematical systems. This paper introduces the Physics–Mathematics Transformation (PMT) as a conceptual framework for investigating whether explicitly defined, restricted structural correspondences can facilitate discovery in either direction. In the mathematics-to-physics direction, a result obtained within a mathematical representation may yield a physical inference only when the relevant properties and transfer conditions are justified and the resulting claim is assessed against physical evidence. In the physics-to-mathematics direction, an admissible physical system may suggest conjectures, constructions, or counterexamples, but exact mathematical claims require rigorous justification beyond experimental observation. The framework distinguishes analogy from formal correspondence and discovery, and emphasizes admissible domains, preserved structure, information loss, realizability constraints, and validation. Illustrative thought experiments concern composition, interactions, and possible generalization; a companion study of passive analog networks provides a restricted setting in which physical realizability constrains mathematical transformations. PMT does not establish a universal physics–mathematics duality or a new physical law. Its contribution is a set of methodological criteria for evaluating whether particular cross-domain transformations yield insights beyond existing representations and techniques.