Geometric Information Dynamics Construction H: Renormalization Group Fixed Points and the Rigidity of Gravity

This paper establishes a rigorous connection between information dynamics andrenormalization group (RG) fixed point theory, proving that the Einstein equationsare the unique RG attractor when information twist exceeds a critical threshold.The core innovation lies in discovering that when the cumulative information twistTcum approaches the critical value Θc ≈ 83ln2 bits, there exists a unique ultravioletfixed point for the RG flow on the theory space, whose classical limit is strictlyequivalent to the vacuum Einstein equations (with cosmological constant). Wederive the RG evolution equation for information twist and prove the rigidity ofits fixed point condition—RG flows satisfying causality, unitarity, and conservationof information distinguishability must converge to Einstein gravity in the infrared.This framework provides an emergence-based answer to “why spacetime is fourdimensional and causally closed” based on information principles and dynamicalphase transitions, and predicts a universal scaling law N−2/3 for quantum gravitycorrections.