The General Theory of Irreversible Distinguishability I: Formal Foundations

This paper establishes a minimal formal foundation for structure under irreversibledistinguishability. Starting from a single primitive—distinction—we derive equivalence,refinement, irreversible loss, persistence, probability, Hilbert space structure, unitaryadmissible transformations, counting, and intrinsic temporal ordering. No physical,probabilistic, or dynamical axioms are assumed. Probability, time, and reversible evolution arise necessarily as constrained structures induced by unresolved refinement andirreversible elimination. The formal core is shown to be minimal, irreducible, andcomplete with respect to admissible structure under loss. This paper constitutes thefoundational core of a multi-paper series deriving physics, chemistry, biology, cognition,and society as successive constrained extensions of irreversible distinguishability.