Topological Dynamics of Cosmological Evolution: From Commensurable Quantum Cycles to Pseudo-Anosov Spacetime Chaos

This paper proposes a novel topological framework for classifying cosmological modelsusing the Nielsen-Thurston classification of mapping class groups (Mod(Σg)). By mappingspacetime evolution functions to surface homeomorphisms, we demonstrate that a quantumcyclic universe with commensurable energy eigenvalues (as recently proposed by Carroll et al.,2026) perfectly structurally corresponds to a periodic mapping class, fundamentally suppressing Boltzmann Brain fluctuations through topological constraints. Furthermore, we establisha direct phenomenological link between these topological mapping classes and the observable universe. We propose that the asteroid-mass Primordial Black Holes (PBHs) generatedduring a non-singular matter bounce act as Reducible invariant submanifolds, accounting forthe dark matter abundance. Concurrently, the scalar-induced stochastic gravitational-wavebackground (SIGW), scaling as ΩGW ∝ f2 in the infrared, serves as the physical manifestation of the Pseudo-Anosov stretch-squash dynamics, rendering our topological frameworkdirectly testable by upcoming interferometers such as LISA, SKA, and ET.To rigorously support this phenomenological link, we resolve the theoretical gap between microscopic quantum fluctuations and macroscopic topological friction by introducing a synergistic Wheeler-DeWitt Hamiltonian. By mapping the extreme amplification of scalar perturbations to quantum squeezed states and topological string winding modes, we provide a definitive quantum-gravitational pathway for the emergence of the aforementioned Reducible PBH knots and the Pseudo-Anosov SIGW background."