Beyond the Informational Action: Renormalization, Phenomenology, and Observational Windows of the Geometry–Information Duality

We extend the Geometry Information Duality by i framing the informational action in a manifestly covariant Renormalization Group RG language ii analyzing the back reaction of quantum informational degrees of freedom on gravitational dynamics beyond leading order iii mapping the resulting corrections onto concrete astrophysical and cosmological observables. We derive scale-dependent coupling functions G of k, Lambda of k, and alpha info of k from a functional RG flow on an informationally extended minisuperspace and compute their imprints on black hole thermodynamics, early universe inflation, and gravitational wave propagation. Finally, we delineate observational windows, including VLBI black hole imaging, space-based gravitational wave detectors, and precision torsion balance experiments capable of constraining the informational parameters of the theory.