The Latent Solution of the Gravitational N-Body Problem
Abstract
We present a complete treatment of the gravitational N-body problem through the lens of the Latent framework — finite, sufficient representations of smooth dynamical systems. The monograph unifies nine companion papers into a single narrative arc, advancing from representation to exact solution to Smale's 6th Problem, backed by 339 formally verified theorems and zero open sorry obligations.
The central results are:
1. The Exact Latent Solution. Every collision-free N-body trajectory admits a finite generating function \(G_N(z; \mathbf{v}_0) = \sum_{n} \boldsymbol{\Lambda}_n z^n\), analytic in an annulus with \(\rho > 1\), satisfying an algebraic Galerkin equation. The total representation size scales linearly in \(N\): at most \((N-1)d \cdot N_\varepsilon\) real numbers for accuracy \(\varepsilon\) in \(d\) spatial dimensions, where \(N_\varepsilon = \Theta(\log(1/\varepsilon)/\log\rho)\).
2. Practical Extraction. Taylor coefficient recurrence combined with diagonal Padé resummation and step-chaining achieves machine precision (\(\sim 10^{-13}\)) on concrete orbits — including the figure-eight, Lagrange, Broucke, hierarchical, and Pythagorean configurations — in under 1000 coefficient evaluations.
3. Global Coverage. After Levi-Civita/KS regularization through binary collisions and Painlevé windowing, the Latent representation extends to almost every initial condition for all \(N \geq 2\), with the exceptional set (non-collision singularities for \(N \geq 4\), total collapse) having measure zero.
4. Central Configuration Finiteness (Smale's 6th Problem). For all \(N \geq 3\) and all positive masses, the number of central configurations modulo similarity is finite. The proof uses the weighted complete-graph Laplacian structure of the shape Hessian for generic masses, and the Pair Transcendence Theorem for degenerate masses. Along the way, we exhibit the first explicit degenerate central configuration at positive masses: \(\mu^* = (81 + 64\sqrt{3})/249\), root of \(249\mu^2 - 162\mu - 23 = 0\).
5. The Spectral Generator. The Koopman operator in Latent coordinates is the deterministic counterpart of the Fokker–Planck generator, completing a five-level hierarchy: state → trajectory Latent → spectral generator → meta-Latent → classification Latent.
The Latent grade — the number of modes needed to represent an orbit — grows linearly in the topological complexity (braid word length \(|w|\)), with correlation \(r \geq 0.968\) across eight periodic orbit families. This establishes the Latent as a universal encoding of N-body dynamics: finite for each orbit, scaling predictably with complexity, and sufficient for both pointwise trajectories and statistical ensembles.
Keywords: N-body problem, Latent representation, Padé approximant, central configurations, Smale's 6th Problem, Koopman ope