Identifiability and the Limits of Empirical Distinction in Physical Theories

This work presents a formal, model-independent analysis of empirical identifiability in physical theories. The central question addressed is whether a physical theory can be uniquely determined from empirical data under physically realistic observational constraints, independent of technological limitations or experimental sophistication. We prove a general classification theorem showing that, under a minimal and unavoidable set of constraints—finite precision, finite measurement time, countability of observables, and finite information content—any infinite-dimensional theory space is fundamentally non-identifiable. In such cases, the forward map from theory space to observable outcome distributions is necessarily non-injective. Consequently, infinitely many distinct theories are observationally indistinguishable in principle. This result constitutes a formal no-go theorem for the unique empirical identification of infinite-dimensional physical theories. The analysis is information-theoretic rather than interpretational. It does not rely on philosophical arguments, subjective priors, or assumptions about experimental inadequacy. Instead, the non-identifiability result follows directly from the mismatch between the infinite information required to specify a point in an infinite-dimensional theory space and the finite information capacity of any physically realizable set of measurements. We further show that finite-dimensional theory spaces may be identifiable in principle, provided a sufficient number of independent observables exist and the corresponding forward map has full rank. This yields a sharp classification: identifiability is determined by the dimensionality of the theory space rather than by experimental ingenuity. The framework is domain-independent and applies uniformly to classical field theories, quantum field theory, general relativity, and other infinite-dimensional theoretical frameworks. Illustrative examples are discussed, including renormalization group universality in quantum field theory and diffeomorphism invariance in general relativity, both of which manifest the structural non-injectivity proven here. This work clarifies the epistemic limits of empirical science by demonstrating that certain forms of underdetermination are not contingent or temporary, but are instead structural features imposed by the nature of observation itself. The results have implications for theory selection, experimental design, model evaluation, and the interpretation of null or convergent experimental outcomes across physics and related disciplines. FUSION OS INNOVATIONS