The Universal Scaling Law of Topological Stiffness: A UV-Complete Proof of the Complexity Ratchet

We present the definitive demonstration of Universal Fractality in the Coherent Field Model (CFM). By deriving the scale-invariant Stiffness Functional, denoted Xi, we show that subatomic stability and cosmic morphology are governed by the same topological recovery constant, approximately 0.9994. Using a dual-tier simulation suite—Sim-Alpha (3-shell baryon collisions) and Sim-Omega (cosmic-scale filaments)—we establish that structural persistence is a geometric property of the space-lattice, not the result of ad hoc assumptions. Key results include: Scale-Invariant Recovery: The 0.9994 mass-energy recovery fraction holds from femtometre to megaparsec scales, confirming a universal geometric memory. Cosmic Web Reproduction: Sim-Omega produces filamentary structures matching observations from JWST and Hubble without invoking Dark Matter particles; the extra rotational support emerges from topological hysteresis of the lattice. Complexity Ratchet Mechanism: Topological knots resist entropic decay once the stiffness functional Xi exceeds the stochastic vacuum coupling (eta), explaining the natural accumulation of structure in the universe. This work unifies subatomic confinement (the Strong Force) with cosmic binding (Dark Matter) as manifestations of the same Topological Stiffness Functional, providing a UV-complete, scale-invariant framework for understanding the emergence and persistence of complexity across the cosmos.