The Three-Body Problem Solved Distributionally: Spectral Fokker-Planck for the Circular Restricted Three-Body Problem

Poincaré (1890) proved that the three-body problem admits no globally convergent power-series trajectory solution. We do not contradict that result. Instead, we change the object: from individual trajectories to the evolving probability density of outcomes. For the stochastic Circular Restricted Three-Body Problem (CR3BP), we show that the law admits a finite spectral-generator representation

\[p(\mathbf{x}, t) \approx \sum_{k=0}^{N-1} c_k e^{\lambda_k t}\varphi_k(\mathbf{x}),\]

where the generator \(M\) can be constructed either by an integration-by-parts weak form in the overdamped setting or by Kronecker-structured phase-space operators with a specular-reflection penalty in the kinetic setting. The representation is reusable: once \(M\) is built, stationary distributions, spectral gaps, first-passage times, and capture-versus-escape probabilities become linear-algebra queries rather than fresh Monte Carlo campaigns. Numerically, the overdamped Earth-Moon model matches Monte Carlo to 0.05\% accuracy, while the phase-space model with the boundary penalty achieves 4.7\% error at roughly \(950\times\) speedup. The stationary law identifies L1 as a probability gateway between the two primaries, and the linearized spectral picture recovers the classical Routh stability threshold for L4/L5. We also isolate a boundary-condition phenomenon specific to kinetic spectral methods: Neumann boundary conditions do not enforce specular reflection, while the boundary-penalty construction does. Theoretical approximation control is inherited from the spectral representation framework through dimension-insensitive mode-count bounds, and the project is accompanied by an ongoing Lean formalization of the structural generator and boundary arguments. The main claim is therefore law-level, not trajectory-level: the stochastic CR3BP is unsolvable pointwise in Poincaré's analytic sense, yet tractable distributionally through a reusable spectral generator.