Combinatorial Barriers to a Unique Theory of Everything

Abstract

We identify a purely combinatorial limitation on the unique empirical identification of a fundamental physical theory, particularly in the context of a Theory of Everything. Under minimal, physically unavoidable assumptions (specifically, a finite set of finite-valued properties) we show that any infinite space of candidate theories must contain infinite subsets that are empirically indistinguishable. This result is independent of dynamics, spacetime structure, or microscopic laws: unique empirical identification based solely on finite observational access is prevented at a combinatorial level. While recent undecidability results suggest the algorithmic impossibility of identifying such a theory, our result shows that even with an idealized observational record, finite empirical access prevents unique determination from observation alone. We conclude that the meaningful content of fundamental physics is naturally organized into equivalence classes defined by finite observational invariants.