Two bodies orbiting under gravity is a solved problem. Kepler and Newton handed us closed-form ellipses, parabolas, and hyperbolas — exact, eternal, predictable. Add a third mass and that clean structure collapses into one of the deepest problems in physics. The three-body problem asks something deceptively simple: given three masses with known positions and velocities, where are they at any future time? Newton’s law of gravity hands you the equations of motion instantly. Solving them is another matter entirely. In 1890 Poincaré proved why it resists us. The system carries only 10 classical integrals of motion — energy, linear momentum, angular momentum, center of mass — but full integration demands more. Lacking enough conserved quantities, no general closed-form solution in elementary functions exists. Worse, he showed that nearly identical starting states diverge exponentially. This was the birth of chaos theory, decades before the term existed. What we do have are exact special-...