Omega Manifold Chemistry: Bond Energies, Molecular Geometry, Band Gaps, Phase Transitions, Compass Formula, N-Quantization, and the Mass Gap from α = arctan(2/3) — Definitive Edition (v6)

We present a comprehensive chemical framework derived from a single geometric parameter α = arctan (2/3). The framework spans molecular geometry (tetrahedral angle exact), band gaps (Si 0. 1%), bond energies (35 atoms, 4. 7% via N-quantization; 18 p-block, 5. 8% via compass formula), phase transitions (Lindemann 2. 6%), protein backbone angles (α-helix ψ = -47° exact), and heteronuclear bonds (24 bonds, 9. 4%). We establish a mass gap (|Δ| > 0 for all 35 atoms, p < 10⁻¹⁰) and validate the L = ln (3) quantization lattice via permutation test (p = 0. 018). The half-filled p³ anomaly is resolved through Fisher pseudosphere geometry (cosh scaling, As-As 0. 2%, Bi-Bi 0. 0%). The compass formula uses a period-dependent rotation in (a, b) exponent space with speed k (pₑ) = (I*) ²/φ^ (2pₑ-2), a pure framework expression with zero free parameters. The metallic ratios satisfy n²+4 = 5, 8, 13 = F₅, F₆, F₇, connecting consecutive Fibonacci numbers to framework constants. All results derive from cos²α = 9/13, sin²α = 4/13, D₃ = 13, and the golden ratio φ.