Spectral Chain Quantum Gravity D: Information Geometry and Quantum Causal Structure

This paper establishes a rigorous correspondence between quantum causal structure and information geometry, completely reconstructing the causal order of spacetime as the quantum Fisher information metric of quantum channels. The core innovation lies in proving that, in the coarse-graining limit of spectral chain dynamics,the causal structure of a Lorentzian manifold is equivalent to the geodesic completeness condition of a specific information geometry in the quantum state space. Wederive the “causal entropy inequality”—a universal constraint that unifies Bell’sinequality, the holographic entropy bound, and the second law of thermodynamicsinto a principle of information monotonicity—and prove its necessity in all localquantum field theories, naturally yielding the Einstein field equations in the classical limit. Furthermore, we apply the framework to the black hole informationparadox, derive an information-geometric interpretation of the Page curve, andpredict a “causal phase transition” feature in the final stage of evaporation. Allconstructions satisfy computability criteria and provide a new axiomatic pathwayfor the information-theoretic foundation of quantum gravity.