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 Eₙ = 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 coshβ (φ−φ★) /MP 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 ρb (φ) = 2MP⁴/coshβ (φ−φ★) /MP structurally provides the upper bound ρₜot ≤ 2MP⁴; 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.