Functional Sparsity in State Spaces: A Pre-Dynamic Structural Principle

This work formalizes functional sparsity as a minimal structural principle of constraint-defined state spaces. Across a broad class of systems, configurations satisfying layered functional constraints occupy a vanishingly small subset of the ambient state space prior to any dynamics, optimization, or learning process. The paper develops a representation-agnostic framework unifying discrete and continuous settings under normalized measure, bounded dependence, and multiplicative contraction. Key contributions: Functional sparsity is identified as a pre-dynamic geometric property of constrained systems. A general quantitative framework is provided for analyzing sparsity under layered constraints. Bounded dependence is introduced as a structural stability condition preserving multiplicative sparsity. Emergent directional effects, negative long-run drift, and entropic asymmetry are derived as downstream consequences rather than primitive assumptions. This foundation paper positions functional sparsity as a domain-agnostic structural perspective relevant to combinatorial systems, biological viability, optimization landscapes, and learning environments.