The Beginning, End, and Infinite Cycle of the Universe: Singularity-Free Evolution with Form-Dimension State ΨD and Covariant Bounce Density in a Z₂-Symmetric cosh Potential Framework

This article presents a Z₂-symmetric cosh potential single-field scalar-tensor framework for late-time cosmology and dark-energy phenomenology. The theory's fundamental structure rests on the cosmological evolution of a canonical scalar field φ, with potential V (φ) = Λ⁴ coshβ (φ − φ★) /MP, where Λ and β are dimensionless parameters. At low energies and low matter-radiation density, the effective-field-theory (EFT) approach is fully valid. In this regime, the theory predicts a redshift-dependent deviation in the dark1 energy equation of state: w (z) = −1 + ε₀ (1 + z) ^ (2β²), where ε₀ and β² parameters are constrained by observational data. At high energy densities, inspired by the structure of loop quantum cosmology, a leadingorder effective correction mechanism ρb (φ) = 2MP⁴Λ⁴/V (φ) is introduced. This mechanism provides automatic avoidance of cosmological singularities. A singularity-free bounce occurs, and after the bounce, the system can evolve into cyclic or attractordominated evolution. The model is structurally minimal and fully covariant. It automatically satisfies the equation-of-state boundary w ≥ −1. The Bianchi identity and energy-momentum conservation are explicitly verified. Linear perturbation analysis demonstrates the absence of ghost and gradient instabilities. The model is testable against Pantheon+ Type Ia supernovae, DESI BAO data, and Planck CMB observations through MCMC analysis. The central innovation is a minimal framework unifying late-time dark-energy dynamics with bounded high-energy behavior.