QM and GR as Two Regimes of One Dissipative Dynamics. Quantum Mechanics and General Relativity As two Regimes of One Dissipative Dynamics

We propose a three-term Fokker–Planck equation on a smooth, infinite-dimensional Hilbert manifold (M, g₀) carrying a positive-definite background metric and a non-negative tension functional Φ: M → ℝ. The equation has three structurally distinct terms — drift, diffusion, and divergence-free circulation — with a universal binding Dₖ = (α/2) ωₖ between the diffusion tensor and the eigenfrequencies of the Witten Laplacian associated with Φ. We establish two structural limits and one dynamical bridge between them. (i) Schrödinger limit. In the circulation-dominated regime ε = ‖∇Φ‖/‖Ω‖ ≪ 1, with the Weyl algebra forced as the unique stable irreducible deformation of the Poisson structure per Darboux chart (Stone–von Neumann uniqueness combined with Fedosov classification under projection stability), the equation is mathematically equivalent to the Schrödinger equation via the Madelung transformation; the parameter α projects to Planck’s constant ℏ, whose numerical value remains empirical. (ii) Einstein limit. On a four-dimensional Lorentzian submanifold MΘ ↪ (M, g₀) whose tangent/normal decomposition is set by the smallest absolute eigenvalues of the Hessian of Φ at pointer states, the Gauss–Codazzi equations — with the Riemann tensor of (M, g₀) vanishing since g₀ is flat — yield the Einstein field equation Gμν + Λgμν = κTμν with derived coupling κ = 2/H̄²ᵥac and Newton constant formula G = c⁴/ (4π H̄²ᵥac). Lorentzian signature follows algebraically from i² = −1 (Weyl algebra) combined with g₀ > 0; dₛpace = 3 is forced by a two-sided closure (Ehrenfest–Tangherlini orbital stability gives d ≤ 3; spin-statistics topology π₁ (SO (d) ) = ℤ₂ gives d ≥ 3). (iii) Dynamical bridge. The two limits are not independent. The cosmological term is dynamical, Λ (λ) = (8πG/c⁴) ⟨Φ⟩ᵥac, carrying the time-dependence of the projected dynamics into the gravitational sector. The contracted Bianchi identity then yields the modified conservation law κ∇ˡTμν = ∇νΛ, i. e. an energy exchange between matter and vacuum that is derived rather than postulated and is consistent with interacting dark energy phenomenology accessible to DESI-class surveys. We propose, but do not strictly prove, that the matter source Tμν feeding the Einstein equation is the energy-momentum of the Schrödinger-limit fields of the same equation. We state honestly what is not derived: the numerical values of ℏ, G, and the magnitude of Λ (specific values depend on detail properties of Φ) ; the existence of a Lagrangian formulation on (M, g₀) ; and the mode-by-mode identification of Tμνψ with the geometric matter contribution. The claim is structural, not numerical: two pillars of fundamental physics share a common mathematical origin with a derived coupling between them.