The Fleming-Viot particle system consists of a finite population of particles evolving on a graph or state space, where particles that are killed are instantaneously reborn at the location of a uniformly chosen surviving particle. The ancestral circulation describes the long-run flow of genealogical lineages through the system. For two particles, this circulation can be computed explicitly, but for three or more particles the system of equations governing the stationary measure becomes entangled in a complex way because the rebirth mechanism creates dependencies across all surviving particles simultaneously, and no closed-form or even tractable recursive description is known for general graphs or killing rate structures.