whatThe circular restricted three-body problem, drawn in the frame that rotates with the planet. Massless particles feel the sun, the planet, and the frame's centrifugal and Coriolis terms; the faint contours behind them are the zero-velocity curves of the effective potential -- the geometry of where a slow particle may and may not go.

whyThe claim: trapped by geometry alone. Sixty degrees ahead of and behind the planet lie L4 and L5 -- empty points -- yet particles released nearby cannot leave, tracing slow tadpole loops. Released along the orbit but farther out, they walk the great horseshoe, turning back whenever they near the planet. Scattered anywhere else they fall or are thrown clear. The HUD counts every fate; Jupiter really does keep two camps of asteroids exactly here.

howRK4 at fixed step in the rotating frame; each particle's Jacobi constant is tracked and its worst relative drift reported in the HUD -- the integrator's honesty ledger, a few parts in ten thousand at worst. Fates are read from a 36-bin longitude visit mask: tadpoles stay in a window about their point, horseshoes cover nearly the whole ring yet never the planet's neighborhood. Click to release a handful anywhere. One integer seeds every orbit.