The Beginning, the End, and the Eternal Cycle of the Universe

This paper develops a single minimal covariant framework that addresses the entire cosmological process — from the formation of the universe to its contraction — based on a triadof existence (x), the dimension–form parameter (n) and time interpreted as the operationalmeasure of motion. The proposed framework offers a coherent reading, drawn from the sameequation family, of four foundational problems that the ΛCDM concordance model does notresolve under a single roof: the singularity problem, dark energy, dark matter, and the cyclicnature of the universe. The framework rests on five principles: (i) x is the dimensionlesstotal amount of existence, conserved under the closed-system condition of the universe; (ii)n is a single dimensionless parameter that simultaneously denotes the effective geometricdimension and the form taken by existence at that dimension, and varies linearly — thisdual meaning is a geometric necessity stemming from the spatial structure of the D = 3universe; (iii) time is not an independent quantity but the operational measure of the changeof form (dn/dt); (iv) zero, infinity, and singularity are physically rejected at the Planckscale; (v) the endpoints n = 0 and n = 3 are geometrically forbidden in the D = 3 universe,since they would respectively require a negative dimension and a fourth spatial dimension.The main formula En = xn and its symmetric counterpart xD−n are derived directly to thecosh(·) identity under Z2 symmetry; in the canonical scalar–tensor framework the potentialV(φ) = Λ4coshβ(φ−φ⋆)/MP is obtained on physical grounds. The covariant bounce density ρb(φ) = 2M4/coshβ(φ − φ⋆)/MP structurally enforces the upper bound ρtot ≤ 2M4,curvature singularities are eliminated as a structural consequence, and a phantom crossing(w < −1) is excluded. The central prediction, defined by only two additional parameters(ε0, β), is w(z) = −1 + ε0(1 + z)2β2, testable against Pantheon+, DESI, and Planck data.The expansion–contraction transition is read through the conversion energy Q = xn−1(x−1);black holes are reinterpreted as the local version of the universe’s global collapse (n : 3 → 0).