ΨD =xn+x(D−n): A Single-Parameter Cosmology Unifying Dark Energy, Dark Matter, and the Cyclic Univers

A single-parameter cyclic cosmology governed by ΨD = xn + x(D−n) is presented,where x represents total physical energy and n the continuous geometric dimensionstate (0 < n < 3 in a D = 3 universe). Because x > 0 forces ΨD > 0 always, singularity is mathematically excluded. Two asymptotic geometric barriers at n = 0 andn =3—bothenforced by the residual x0 = 1 ̸ = 0 — cyclically reverse cosmic evolution without requiring additional free parameters. The model explains dark energyas energy released during the n : 2 → 3 dimensional transition; dark matter emergesas the gravitational imprint of incomplete dimensional change. Time is redefinedas displacement-measurement information derived from n(t), not as an independentcoordinate. Entropy grows monotonically throughout cosmic contraction via blackhole mergers, Hawking radiation, and form reduction processes. The contractiondoes not produce a singular Big Crunch; rather, the system reaches ρ = ρc bouncepoint and reverses into a new expansion cycle. The model yields n ≈ 0.014 at 600Myr post-Big Bang, explaining JWST’s observation of anomalously massive earlygalaxies through local crystal bounce seed mechanisms consistent with Loop Quantum Cosmology. Singularity is proven impossible via three independent arguments:AM-GM inequality, potential energy lower bounds, and LQC bounce conditions.Energy conservation and entropy monotonicity are mathematically demonstratedacross expansion and contraction phases. The universe forms a closed, cyclic systemwhere energy is conserved and entropy accumulates monotonically across cycles,with each successive universe inheriting higher entropy from its predecessor.