CYCLIC QUANTUM COSMOLOGY: TOPOLOGICAL PHASE-TRANSITION MODEL WITH Ψ=xn ORDER PARAMETER AND DIRECTIONAL TUNNELING

This study introduces a new state function Ψ = xn that describes the thermodynamicexistence and expansion dynamics of the universe. In the model, x > 0 represents the universal conformal potential (x ≡ Ω2 = R/R0), while n denotes the quantum-geometric formstate of matter, independent of spatial dimensionality. Since zero and infinity are treatedas undefined, all mathematics—including the Lagrangian, equations of motion, Friedmannequation, and physical interpretations—is expressed exclusively in limit form. The modelbreaks the classical coincidence of +∞ and −∞ on the Riemann sphere, establishing anasymmetric Directed Topological Circle. This structure ensures that when the universe approaches maximum tension it tunnels directly into the negative sub-form; in the negativeregion it contracts to maximum quantum-crystalline compression before an instantaneousjump to n = 1 triggers a new cosmic inflation.Black holes are defined as local phase gates: at limn→0+ Ψ(xBH,n) = 1 matter transitionsdirectly into the global negative phase. Dark energy arises naturally as the negative pressurereflected in the metric from the internal potential energy lost as n increases. WKB tunnelinganalysis demonstrates a classical barrier (S ≈ 8198) that renders topological jumps mathematically inevitable. The model is fully consistent with Loop Quantum Cosmology, f(R)quantum cosmology, and Penrose’s Conformal Cyclic Cosmology. It offers testable predictions such as low-ℓ CMB anomalies, deviations from Bekenstein-Hawking entropy, and mildn-dependent deviations in the H(z) curve. By uniting dark energy, the black-hole information paradox, and the ultimate fate of the universe within a single singularity-free topologicalframework, this approach provides a novel perspective for cosmology.