H-CLASS: A Representation Theoretic Criterion for Hierarchy Formation via Dynamical Instability

H-CLASS: A Representation-Theoretic Criterion for Hierarchy Formation via Dynamical Instability We introduce H-CLASS, a representation-theoretic criterion that characterizes hierarchy formation through dynamical instability in systems with permutation symmetry. The key insight is that hierarchy emerges when the dominant Lyapunov direction lies in the Standard (Std) representation rather than the Trivial (Triv) representation. We prove this condition is equivalent to positivity of a dominance functional Φ, and provide numerical evidence across three benchmark models. Main results:- Definition of dominance functional Φ ∈ [-1, 1] (Lemma 1)- H-CLASS criterion: Φ(v_Λ) > 0 (Definition 2)- Numerical verification with deliberate counterexample Hierarchy formation in H-CLASS systems requires no catalyst, no fine-tuning, and no specific mechanism—it emerges as a structural consequence of representation dominance. Keywords: hierarchy formation, representation theory, dynamical systems, Lyapunov exponent, permutation symmetry, symmetry breaking