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This paper proposes a unified theory of quantum gravity based on the Woodin cardinal κ ≈ 118, achieving the unification of mathematical infinity and physical measurability through Jiuzhang Con structive Mathematics (JCM). The core contributions include: (1) Designing three desktop exper iments (superconducting processors, atom interferometry, and gravitational wave data reanalysis) to constrain κ = 118 ± 25; (2) First prediction of κ-dependent quantum entanglement oscillations (frequency ν ∝ κ 1/6 ) and light speed medium κ modification (∆n ∝ κ −1/2 ); (3) Constructing a ”theory-experiment-technology” closed loop to promote the engineering of quantum gravity. The theory physicalizes κ into a measurable constant through JCM’s three principles (domain restriction, operational finitization, and dual isomorphism), establishing mathematical connections with Woodin cardinal axioms, with rigorous formula derivation, concrete experimental design, and quantified risk control.