A One-Parameter Cosmology

This study presents a one-parameter cyclic cosmological model governed by the equation ΨD = xⁿ + xᴰ⁻ⁿ. Here x is a dimensionless positive quantity defined as x = a(t)/aP, the ratio of the cosmological scale factor of the system to the Planck scale factor; n is a continuous real parameter that defines the geometric form state of the system and is connected to observability through the relation n = Dφ/MP (in a D = 3 universe, 0 < n < 3). ΨD is derived as the minimal analytic realization of the n ↔ D − n duality symmetry; the identity ΨD = 2xᴰ/² cosh[(n − D/2) ln x] connects the model to standard scalar-field cosmology. Since the condition x > 0 forces ΨD > 0, the singularity is mathematically excluded. The critical density ρc = 2MP⁴/ΨD is intrinsically derived; the model becomes independent of Loop Quantum Cosmology (LQC). The singularity is excluded by three independent arguments: the AM–GM inequality, the lower bound on the potential energy, and the bounce condition consistent with LQC. The model explains dark energy as energy released during the n : 2 → 3 dimensional transition; dark matter is interpreted as the gravitational imprint of an incomplete dimensional change. In the slow-roll approximation, the equation of state w(z) = −1 + ε(z), ε > 0, and the modified Hubble parameter H²(z) = H₀²[Ωm(1+z)³ + ΩΛ(1+z)^(3ε)] are derived analytically; these predictions are testable at DESI, Euclid, and LiteBIRD precision. Time is not an independent coordinate but the displacement–measurement information derived from n(t). Entropy monotonically increases throughout the cosmic contraction via black hole mergers, Hawking radiation, and form-reduction processes. Contraction does not produce a singular Big Crunch; the system reaches the bounce point ρ = ρc and returns to a new expansion cycle. The model gives n ≈ 0.014 about 600 Myr after the Big Bang; the James Webb Space Telescope (JWST) observations of large mature galaxies in the early universe are qualitatively consistent with the local crystal-bounce seed mechanism. Energy conservation is consistently expressed in four contexts as the Noether current arising from the time-translation symmetry of the action. In the ontological framework of the model, 0 and ∞ are mathematical limits; they are physically inaccessible. The universe operates within a finite, nonzero positive numerical range defined by Planck values.