Coherency Geometry: Quantum Mechanics and General Relativity as Orthogonal Sectors of Tensor Geometry

Coherency Geometry (CG) derives quantum mechanics and general relativity as orthogonal sectors of the Riemannian geometry of positive-definite symmetric 3×3 tensors. CG's three primitives are: a coherency functional C = α∑ᵢρ² (qᵢ, q₀) forced at quadratic order by S₃ invariance, positive-definite symmetric rank-2 tensors encoding local coherency flux, and a compatibility constraint ∑Tᵢ = T_Σ coupling them. The sole dynamical law is the functional's stationarity. No quantum axioms or spacetime manifold are assumed. Zero free parameters remain after gauge-fixing. The affine-invariant metric decomposes orthogonally into 1 scale, 2 eigenvalue-ratio, and 3 eigenframe-orientation degrees of freedom. The 2D shape sector is Kähler, and compatibility-induced entanglement forces the Born rule through Gleason's theorem. The quantum Hamiltonian HN = -Δ₆㶁₍₃ + ∑f (ρᵢ) generates unitary evolution for any smooth radial f on the finite-dimensional Dirichlet spectral tower. The 3D orientation sector carries a spatial metric that vanishes at isotropy, with Lorentzian signature forced by the transport channel's existence at isotropy. Gₒbs Λₒbs ℏ/c³ = 3/N² determines both Λgrav ≈ 10⁻¹²² and G ≈ 10⁻⁶¹ from a single structural integer N ≈ 10⁶¹. Thirteen of fifteen quantum recovery results are exact geometric facts independent of the coherency functional. The frame-sector equations satisfy all four Lovelock hypotheses at every finite N, forcing the Einstein form. The Fisher metric curvature produces a red tilt nₛ = 0. 965 and a gauge connection with bare coupling ratio gV/gT = 1/2. Chevalley's classification forces SU (3) × SU (2) × U (1) from the octant root datum. Three S₃ irrep sectors produce distinct generation hierarchies, with the trivial sector steeper than the standard sector, consistent with observed lepton-quark ordering. Keywords: Tensor geometry, Fisher information metric, Kähler manifolds, ADM formalism, Born rule, spectral theory on manifolds with boundary, gauge group