The Beginning, End and Infinite Cycle of the Universe: A Singularity-Free Evolution via the Dimension–Form Equation xⁿ and a Z₂-Symmetric cosh Potential Framework

This paper presents a cosmological model based on the trinity of being–dimension–time, which explains the entire process from the formation of the universe to its contraction within a single minimal covariant framework. The framework is built upon three fundamental principles: (i) x is the unitless total quantity of being possessed by the universe and is conserved in a closed system; (ii) n is a single parameter that simultaneously represents both the effective geometric dimension and the form taken by being, and varies linearly; (iii) time is not an independent quantity but the measure of change in form (motion). Built upon these three principles, the main formula E_n = xⁿ and its symmetric complement x^(D−n) are derived directly into the cosh identity under Z₂ symmetry; thus, within a canonical scalar–tensor field theory framework, a physically motivated potential V(φ) = Λ⁴ cosh[β(φ−φ★)/M_P] is obtained. The framework reads — under a single consistent roof — the four fundamental problems that the ΛCDM concordance model of modern cosmology cannot solve: singularity, dark energy, dark matter, and the cycle of the universe — from the same family of equations. The model physically rejects zero, infinity and singularity; it accepts that all physical quantities remain within Planck-scale finite positive bounds. The covariant bounce density ρ_b(φ) = 2M_P⁴/cosh[β(φ−φ★)/M_P] structurally provides the upper bound ρ_tot ≤ 2M_P⁴; curvature singularities are structurally absent. The main results are as follows: (a) the structural absence of curvature singularities; (b) the structural bound w_φ ≥ −1 (exclusion of phantom crossing); (c) the central prediction w(z) = −1 + ε₀(1+z)^(2β²), controlled by only two additional parameters (ε₀, β); (d) the reading of the transition between expansion and contraction via the transformation energy Q = xⁿ⁻¹(x−1) of the form transition n: 2 ↔ 3 (Q > 0 in expansion, Q < 0 in contraction); (e) the reinterpretation of black holes as an accelerated local version (n: 3 → 0) of the global collapse of the universe; (f) the inaccessibility of the n = 0 and n = 3 endpoints as a geometric necessity in a D = 3 universe. The framework produces concrete predictions that are testable with Pantheon+, DESI and Planck data.