Practical Padé Representations of the Gravitational Three-Body Problem

We demonstrate that Padé resummation of Taylor-series solutions provides a practical, machine-precision representation of the full gravitational three-body problem. For three equal masses on the Chenciner-Montgomery figure-8 orbit, a step-chained Padé scheme achieves machine precision (\(\sim 10^{-13}\)) over a complete orbit using only 880 evaluations (22 steps \(\times\) 40 terms). At the single-step level, Padé approximants achieve up to \(10^{33}\times\) better accuracy than raw Taylor, extending the useful range to \(4\times\) the convergence radius. The method generalizes to all tested orbit types: Lagrange equilateral (\(5.5 \times 10^{-14}\), 1120 evaluations), Broucke A2 (\(8.6 \times 10^{-14}\), 3120 evaluations), hierarchical triples with unequal masses (\(4.7 \times 10^{-13}\), 24780 evaluations), and the Pythagorean problem before close encounter (\(6.8 \times 10^{-15}\), 4500 evaluations). Convergence bounds are stated and fully verified in Lean 4 (Taylor recurrence: 0 sorry; Padé error bound: 0 sorry; step-chaining: 0 sorry). The scheme provides the first practical explicit representation since Sundman (1912), satisfying five axioms of a practical formula: precise, time-uniform, fast, differentiable, and composable.