EFT: Entropic Field Theory — Quantum Gravity from Mutual Information Consistency

We derive the Einstein field equations of general relativity from a single scalar function on the space of quantum states. Starting from the von Neumann entropy S defined on the Bures geometry of density matrices, we construct a Lorentzian manifold whose causal structure, signature, and curvature emerge from the gradient of S. We prove that the Einstein tensor of this reconstructed spacetime equals the trace-reversed Hessian of the entropy field. The mutual information consistency condition — that local time arrows defined as entropy gradient directions must agree across coupled quantum subsystems — is shown to be equivalent to energy-momentum conservation. Newton's constant G emerges as the renormalized coefficient of the stress-energy tensor in the mutual information kernel of quantum fields. The central equation, the Entropic Self-Consistency Equation E[S] = 0, is a single nonlinear PDE for S from which Lorentzian signature, causal consistency, the Einstein equations, and the cosmological constant all follow as consequences. Three parameter-free predictions are derived: the measured value of Newton's constant G, the dark energy equation of state w(a) confirmed against DESI DR2 with 0.03% error, and a non-monotonic gravitational anomaly at the BCS superconducting transition with a parameter-free sign change at 0.88 Tc. Companion paper: EQ(vT) — Entropic Quantum (vectorized Time). DOI: 10.5281/zenodo.18917317