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This paper proposes a rigorous Third-order Enhanced Axiomatic System (TEAS) that resolves the fundamental conflict between quantum mechanics and general rel- ativity within the framework of Woodin cardinals. By establishing an exact corre- spondence between renormalization group flow and categorical duality, we construct a quantum-classical fiber bundle mapping Q : H → Γ(T*M), derive the spacetime emergence mechanism k ~ log(ΛUV /ΛIR), and propose three fundamental axioms: quantum-classical correspondence, noncommutative geometric duality, and topolog- ical order stability. Key innovation: We establish the physical motivation for Woodin cardinals in quantum gravity through entropy scaling and renormalization completeness. The covering property of Woodin cardinals ensures the mathematical consistency of the quantum-to-classical transition, providing a set-theoretic resolution to Haag’s the- orem. This approach differs from ∞-category methods by providing a set-theoretic foundation that resolves Haag’s theorem constraints through determinacy proper- ties. This work provides a mathematically consistent solution to the Haag theorem contradiction, offering a testable framework for quantum gravity theory with exper- imentally verifiable predictions.