Holographic Attractor Dynamics: Banach Fixed-Point Proof and Lyapunov Stability of the Cyclic Cosmological Attractor

We prove rigorously that the coupled entropy-geometry dynamical system governing cyclic bounces possesses a unique global attractor at (g*, γ*) = (0.33 ± 0.02, 0.67 ± 0.03), with relaxation time τ_relax = 300 ± 100 cycles. The proof uses Banach's fixed-point theorem applied to the cycle map on a complete metric space, supplemented by Lyapunov stability analysis. The attractor eliminates fine-tuning: the 3.3% viability window identified in Paper II is not a coincidence but a dynamical inevitability. We further derive the HΣΔ filter from a holographic renormalisation group flow, connecting the phenomenological Sigma-Delta mechanism to UV/IR mixing in the bulk.Part IV of VII.