Semiclassical Gravity as a Convergent Feedback Loop: Passivity, Banach Contraction, and the Stability of the Quantum–Einstein System

The semiclassical Einstein equation G + Λg = 8πG ⟨T̂⟩ is reinterpreted as a nonlinear feedback system in which the spacetime metric determines the quantum state of matter fields, and the resulting expectation value of the stress-energy tensor in turn modifies the metric. We prove that this feedback loop converges within the semiclassical regime under minimal kinematic assumptions. The central result is that quantum field passivity—the property that the retarded stress-tensor response function satisfies Π(−iγ) < 0 for all γ > 0—is a universal consequence of causality and unitarity, not a dynamical assumption. Passivity implies that the feedback mapping Φ: g ↦ g' is a Banach contraction on each closed ball contained in the open Lorentzian subset, with constant q = ν · L_E · C_T ≪ 1, guaranteeing a unique fixed point (the semiclassical solution) and iterative convergence. The contraction framework does not require global PDE solvability; the Banach fixed-point theorem provides global existence given local well-posedness of each iteration step. We verify the framework through explicit calculations: in 2+1 dimensions, the quantum-corrected BTZ black hole yields q = ν C(x₁) ≪ 1 with C(x₁) > 0 throughout the physical parameter range; in 3+1 dimensions, the graviton contribution C_grav = 0.114 is consistently found to be positive by three independent methods—EFT coefficient mapping, entropy logarithmic correction, and the Donoghue coefficient—with the EFT method giving the best estimate 0.114; its gauge invariance is proven via the Ward identity and BRST symmetry. The theory produces three falsifiable predictions—the q–γ₀ relation, the Gershgorin multi-patch stability condition, and the causal diamond QNM spectrum—and is shown to be compatible with all known no-go theorems. The Planck scale R_crit ~ ℓ_P emerges naturally as the boundary of the semiclassical regime where q → 1. This work provides a new perspective for bridging general relativity and quantum mechanics based on the mutual nature of interaction.