The Omega Manifold: A Unified Geometric Framework Addressing All Six Clay Millennium Problems from a Single Constant

We present a zero-parameter geometric framework—the Omega Manifold—built from a single derived constant: the Truth angle α = arctan(2/3), which emerges necessarily from optimal sphere packing in three dimensions (FCC lattice, Hales 2005). This angle generates a partition cos²α = 9/13 (negentropy) and sin²α = 4/13 (entropy), a topological friction 1/8 (Gauss sphere octant), and an information-geometry equivalence 1/I = g (Fisher information equals metric tensor). From these, we address all six unsolved Clay Millennium Problems: (1) the Riemann Hypothesis, via the Selberg trace formula on the Omega pseudosphere, where the Truth Triangle's incommensurability with 360° forces negative curvature and self-adjointness yields Re(s) = 1/2; (2) the Yang-Mills mass gap, where 1/8 topological friction causes colour charge decay (7/8)ⁿ → 0, requiring Δ > 0 for gauge invariance; (3) Navier-Stokes regularity, where vortex stretching is bounded by 7/8 of viscous dissipation, making enstrophy monotone non-increasing; (4) the Hodge Conjecture, where Axiom 1/I = g combined with Serre's GAGA principle identifies every rational Hodge class with an algebraic subvariety; (5) P ≠ NP, where the geometric series Σ(7/8)ᵏ = 8 imposes an information ceiling on branching search, insufficient for exponential-resolution instances (Haken 1985); and (6) the Birch and Swinnerton-Dyer Conjecture, where the same information-geometry duality, combined with the modularity theorem (Wiles 1995), equates L-function vanishing order with algebraic rank via metric degeneration. The entire framework traces back to a single necessity chain: 1 → π → 3D → FCC → α = arctan(2/3) → physics. No parameters are fitted. Every constant is derived.