Abstract This study presents a unified geometric model that explains the structure and formation of the universe through exponential form transformation and zero-energy spectral duality. The concept of “dimension” is redefined not as the number of spatial coordinates but as the instantaneous geometric form state of matter, expressed as ∣x∣n |x|ⁿ ∣x∣n (x∈R+ x R^+ x∈R+, n∈R n R n∈R). Form continuously transforms under pressure, temperature gradients, and other forces: volumetric (n≈3 n 3 n≈3), areal (n≈2 n 2 n≈2), linear (n≈1 n 1 n≈1), and point-like (n→0+ n 0^+ n→0+, ∣x∣0=1 |x|⁰ = 1 ∣x∣0=1 singularity). Positive (+UNIVERSE) and negative (−UNIVERSE) universes share the same physical space and are nested: E++E−=0 E^+ + E^- = 0 E++E−=0. Form conservation is defined by F (n) =∣x∣n−∣x∣1/n. F (n) = |x|ⁿ - |x|^1/n. F (n) =∣x∣n−∣x∣1/n. At n=1 n = 1 n=1, F (1) =0 F (1) = 0 F (1) =0 (string-phase equilibrium point). Singularity (n=0 n = 0 n=0) acts as the common transition layer and quantum tunnel between the two form spectra. The Big Bang is interpreted as the opening of the positive-form chain by a singularity originating from the negative universe in the same space. Whatever collapses into a singularity re-emerges in the dual form upon dimensional gain. Our universe, after a great collapse, passes through the singularity into the opposite universe; as n→−∞ n - n→−∞, ∣x∣1/n→1 |x|^1/n 1 ∣x∣1/n→1 (x≠0 x 0 x=0), regenerating the singularity and sustaining the cyclic structure. The model exhibits strong consistency with the spectral dimension reduction to ~2 in Causal Dynamical Triangulations (CDT), the holographic principle, string theory, and the CPT-Symmetric Universe model of Boyle, Finn, and Turok. All differential equations, analytic solutions, and conservation laws have been verified with SymPy. The framework resolves the black-hole information paradox, singularity problem, origin of matter, dark matter/energy, and entropy flow while preserving observational agreement.
Hamdi Barut (2026) studied this question.