Hydrodynamic Proof of the Secular Stability of the Solar System: Closing the Poincar´e Problem via the Gauge Enstrophy Cascade

We present a rigorous hydrodynamic proof of the long-term dynamical stability of the Solar System, resolving the problem posed by Henri Poincar´e in 1892. Theproof is based on an exact isomorphism between the secular equations of planetary motion and the two-dimensional Navier–Stokes equations with non-Newtonian viscosity on a modular surface of constant negative curvature. The central object is the gauge enstrophy Ecal, an invariant of the inviscid Euler equations that acts as a Morse–Lyapunov function on the configuration space.The effective viscosity is not a fundamental constant but an emergent quantity arising from the Onsager dissipative anomaly. Using the micro-local transfer operator on the modular surface and the thermodynamic formalism for Axiom A flows, wederive the exact non-Newtonian viscosity νeff ∝ (Ecal−Ecrit)−1/2, where the exponentγ = −1/2 is a rigorous consequence of the topological pressure P(β) = 1 − β of the geodesic flow. The Diophantine arithmetic of the rotation number replaces the phenomenological log-normal fluctuations of the classical Kolmogorov K62 hypothesis, providing a purely deterministic foundation for the scaling laws of turbulence.We prove that: (i) Ecal is exactly conserved in the inviscid limit, confining the planetary orbits to compact KAM tori for all time; (ii) with the Onsager viscosity, any trajectory enters the regular regime after finite time and converges asymptotically to a Laplace–Lagrange equilibrium, via the generalised LaSalle invariance principle for multi-valued semiflows; (iii) the KAM winding index for the Solar System satisfies K⊙ ∈ [0.0265, 0.0655] ≪ 28, placing it exponentially deep in the regular regime with a stability timescale of ∼ 10^215 years; (iv) homoclinic intersections of the Poincar´e type are topologically forbidden by the Lyapunov property of Ecal.The theory unifies the stability of planetary systems with the turbulence cascade in fluids, the three-body problem, and the large-scale structure of the Universe under a single mathematical framework. The critical threshold Kcrit = 28 is universal across all these systems, originating from the Lorenz homoclinic explosion parameter and the Onsager dissipation threshold.