Mathematical Formulation of the Structural Universe

This paper provides the mathematical formulation for the structural cosmology model introduced in Paper 1 (Alshareif, 2026, DOI: 10.5281/zenodo.23440971) and Paper 2 (Attractive Folds, 16 Oct 2026). The goal is to derive the observed dark energy fraction (68.3% per Planck 2018) from pure geometry, without invoking a cosmological constant Λ. We model the universe as a thin spherical dome shell with radius R_u = 46.5 Billion Light Years (observable radius). Using membrane dome theory, we show that a small peripheral compression shell (12% of radius, 32% of volume) creates distributed tension in the inner 68% of volume, which is observed as dark energy. Result: Ω_DE = (r_n / R)^3 = 0.68 exactly matches Planck within 0.3%. This proves dark energy is not a mysterious energy but structural tension: T_dark = k * (E_center - C_periphery).