Admissibility Constraint and Finite Distinguishability in Equivalence Classes and Bounded Transformation Structure

admissibility constraint finite distinguishability admissibility-constrained equivalence bounded transformation structure equivalence classes under constraint admissible transformations preserve identity finite representation constraint finite resolution constraint finite capacity constraint equivalence classes mathematics structure-preserving transformations bounded mapping spaces transformation composition constraints identity-preserving mappings local representation without global labeling gauge-equivalent configurations symmetry-equivalent configurations encoding-equivalent descriptions constrained physical evolution representation bounds information theory constrained dynamical systems bounded state space transformations representation-invariant outputs input-output invariance breakdown finite capacity models transformation invariance AI equivalence mapping machine learning unbounded transformations violate finite representation equivalence breakdown finite resolution input-output invariance failure finite capacity expansion of equivalence classes without bound elimination of equivalence under constraint violation