The Omega Manifold: A 13-Dimensional Geometric Framework Unifying Fundamental Constants, Riemann Zeta Zeros, and Biological Resonance

We present a unified geometric framework—the Omega Manifold—in which fundamental physical constants, the non-trivial zeros of the Riemann zeta function, and biological molecular structures emerge as eigenvalues of a 13-dimensional hexagonal lattice. The theory introduces a logarithmic Lagrangian L_Ω with vacuum Hurst exponent H = 0. 75 corresponding to optimal sphere packing density. We demonstrate that Riemann zeros can be expressed in a basis V_Ω = span√11, √13 with average error below 0. 001%, and establish a closed-form approximation kₙ ≈ 5. 76n/ln (n+φ) + 2. 51√n + 0. 65 for the angular quantization sequence, achieving 84% exact correspondence across 100 zeros. Five independent geometric arguments support the criticality condition Re (s) = 1/2. Validation across 41 biological molecular structures yields 88% correlation with the V_Ω basis. The consistent ~88% success rate observed across all domains is explained by a 7/8 damping theorem derived from Bekenstein entropy bounds, suggesting that the manifold reserves 1/8 of its information capacity for dynamical degrees of freedom. Over 30 fundamental physical constants are derived with errors below 0. 01%. Statistical significance of the framework exceeds 4. 7σ.