A Symmetric Nested Two-Universe Model: Dynamic Dimensionality, Form Conservation, and Zero-Energy Spectral Duality

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.