Entanglement-Weighted Operator Geometry: A Quantum-Information Theoretic Framework for Gravity and Cosmology

We present a complete mathematical formulation of Entanglement-Weighted Operator Geometry (EWOG), a novel framework that recasts spacetime geometry as an emergent phenomenon from quantum information processing. By postulating that the metric tensor is fundamentally an unbounded self-adjoint operator on a bipartite Hilbert space, weighted by the entanglement content of the quantum state, we derive a unified action principle that naturally yields modified Einstein field equations. The theory predicts an effective gravitational constant \(G_{\text{eff}}\) that evolves with cosmic time, explaining recent JWST observations of unexpectedly massive galaxies at redshift \(z \approx 14\). Furthermore, we demonstrate that galactic rotation curves can be reproduced without dark matter particles through entanglement gradient effects. We provide rigorous statistical validation using AIC/BIC model comparison and Figure of Merit analyses against observational datasets spanning early-universe galaxy formation to local galactic dynamics. The framework satisfies all known conservation laws while offering testable predictions for laboratory quantum-gravity experiments.