  Experimental studies with ultracold atoms have enabled major breakthroughs in
understanding three-body physics, historically a fundamental yet challenging
problem. This is because the interactions among ultracold atoms can be
precisely varied using magnetically tunable scattering resonances known as
Feshbach resonances. The collisions of ultracold atoms have been discovered to
have many universal aspects near the unitarity limit. Away from this limit,
many quantum states are expected to be active during a three-body collision,
making the collisional observables practically unpredictable. Here we report a
major development in predicting three-body ultracold scattering rates by
properly building in the pairwise van der Waals interactions plus the
multi-spin properties of a tunable Feshbach resonance state characterized by
two known dimensionless two-body parameters. Numerical solution of the
Schr{\"o}dinger equation then predicts the three-atom collisional rates without
adjustable fitting parameters needed to fit data. Our calculations show
quantitative agreement in magnitude and feature position and shape across the
full range of tuning of measured rate coefficients for three-body recombination
and atom-dimer collisions involving ultracold Cs atoms.
