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 τᵣelax = 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.
Jean Beauve (Mon,) studied this question.