A Universal Framework for Compositional Obstruction Theory: The ∞-Operadic Structure of Chaos

We prove that all singularities arising in weakly associative compositional theories are governed by a single obstruction class ω ∈ H²_tot(Chaos), the total cohomology of a canonical double complex associated to Chaos. The framework Chaos is a universal ∞-operadic category: every Weakly Admissible Theory (WAT) embeds fully faithfully into it, and a theory develops singularities iff [ω] ≠ 0. This obstruction is canonically dual to non-zero curvature via a natural isomorphism, establishing an obstruction-curvature duality. The Karr decomposition stratifies any admissible flow into complexity layers; global regularity holds precisely when this decomposition has no singular strata. Singularities are classified into clustering, evasion, or mixed types. The Fundamental Tunnel Γ_Chaos, a canonical (∞,1)-site encoding the categorical structure of time, governs the relationship between local and global flow: anomalies in the Karr decomposition arise from a misalignment between an observer's proper decomposition and the global stratification.