What question did this study set out to answer?

The aim is to present a coherent framework combining quantum mechanics and general relativity using tensor geometry without conventional assumptions.

March 27, 2026Open Access

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

Key Points

The aim is to present a coherent framework combining quantum mechanics and general relativity using tensor geometry without conventional assumptions.
Developed coherency geometry as a framework deriving both quantum theory and general relativity.
Utilized 3×3 positive-definite symmetric tensors to define geometrical primitives.
Applied the concept of stationarity of a coherency functional to establish dynamical laws.
Demonstrated that the theory produces exact geometric results independent of the coherency functional.
Showed compatibility-induced entanglement results in the Born rule via Gleason's theorem.
Equated cosmological constants and gravitational constants to structural integers, implying deep connections between quantum mechanics and general relativity.

Abstract

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

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

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