ΨD Cosmology: A Phenomenological Z₂-Symmetric Scalar Field Framework for Nonsingular Cyclic Evolution

We present a single-field scalar-tensor framework for late-time cosmology in which a canonical scalar field φ evolves under a Z₂-symmetric cosh potential V (φ) = Λ⁴ coshβ (φ − φ★) /MP, built from a dimensionless order parameter ΨD = xⁿ + x^ (D−n) through the identity ΨD = 2 x^ (D/2) cosh (n − D/2) ln x. In the covariant formulation adopted here, x and n are auxiliary, and only V (φ) enters the action, ensuring full general covariance and a standard EFT structure. The framework yields a late-time evolution dynamically close to Λ-CDM, with small redshiftdependent deviations in the dark-energy equation of state of the form w (z) = −1 + 𝒪 (ε₀ (1 + z) ^3β²), where ε₀ ≪ 1, leading to potentially observable departures from a pure cosmological constant at intermediate redshifts. At high energy densities the dynamics is supplemented by an effective nonsingular bounce inspired by loop-quantum-cosmology-type corrections: the bounce arises when ρ approaches a Planck-scale critical density ρc, modifying the Friedmann equation while preserving the classical limit. The novelty lies in providing a unified Z₂-symmetric parametrisation that connects late-time darkenergy dynamics, bounded high-energy behaviour, and nonsingular cosmological evolution within a single minimal framework. The predictions are testable against precision cosmology surveys (Pantheon+, DESI, Euclid, LSST). A complementary phenomenological reading of (x, n) organises the geometric-form ladder, the cyclic-bounce structure, and the entropy account, with comparative analysis against Λ-CDM, LQC, and quintessence collected in Tables 1–8 and J. 1.