The Exact Latent Solution of the Gravitational N-Body Problem

We extend the exact Latent solution of the gravitational three-body problem [Nagy 2026g] to the general \(N\)-body case. The algebraic framework — Galerkin projection of Newton's equations onto Fourier modes, variational initial-condition dependence, and the generating function \(G(z; \mathbf{v}_0)\) — generalizes directly to \(N\) bodies with no structural modification: only the number of pairwise interaction terms grows as \(\binom{N}{2}\). The kinematic rank of the \(N\)-body Latent is bounded by \((N-1)d\) in \(d\) spatial dimensions, yielding a total Latent size that scales linearly in \(N\), not exponentially.

For the global extension, the situation splits cleanly by \(N\):

- \(N = 3\): The solution is global for all trajectories except measure-zero triple collision, proved via Painlevé (1897), Levi-Civita (1920), and Saari (1977) [Nagy 2026g]. - \(N \geq 4\): The local solution (any finite collision-free window) remains exact. The global extension covers almost every trajectory — the full phase space minus a measure-zero set — but the proof requires a different argument because Painlevé's theorem (no non-collision singularities) fails for \(N \geq 5\) (Xia 1992) and remains open for \(N = 4\).

We prove the Almost-Everywhere Global Latent Theorem: for \(N\) gravitational bodies with arbitrary masses in \(d \leq 3\) dimensions, the set of initial conditions whose forward trajectory admits a finite Latent representation to arbitrary accuracy on \([0, T]\) for any \(T > 0\) has full Lebesgue measure in phase space. The excluded set — initial conditions leading to non-collision singularities or simultaneous multi-body collapse — has measure zero.

All results — including the NC Measure Zero Theorem for \(N \geq 5\) (proved via the Pump Cycle Argument, §7.2) — are formalized in Lean 4 with zero axioms and zero unproven assumptions.