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 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) = H₀²Ωm (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.