  The density profiles of dark matter haloes can potentially probe dynamics,
fundamental physics, and cosmology, but some of the most promising signals
reside near or beyond the virial radius. While these scales have recently
become observable, the profiles at large radii are still poorly understood
theoretically, chiefly because the distribution of orbiting matter (the
one-halo term) is partially concealed by particles falling into halos for the
first time. We present an algorithm to dynamically disentangle the orbiting and
infalling contributions by counting the pericentric passages of billions of
simulation particles. We analyse dynamically split profiles out to 10 R200m
across a wide range of halo mass, redshift, and cosmology. We show that the
orbiting term experiences a sharp truncation at the edge of the orbit
distribution. Its sharpness and position are mostly determined by the mass
accretion rate, confirming that the entire profile shape primarily depends on
halo dynamics and secondarily on mass, redshift, and cosmology. The infalling
term also depends on the accretion rate for fast-accreting haloes but is mostly
set by the environment for slowly accreting haloes, leading to a diverse array
of shapes that does not conform to simple theoretical models. While the
resulting scatter in the infalling term reaches 1 dex, the scatter in the
orbiting term is only between 0.1 and 0.4 dex and almost independent of radius.
We demonstrate a tight correspondence between the redshift evolution in LCDM
and the slope of the matter power spectrum. Our code and data are publicly
available.
