  The ConflUent System of Peak trajectories (CUSP) is a rigorous formalism in
the framework of the peak theory that allows one to derive from first
principles and with no free parameters the typical halo properties from the
statistics of peaks in the filtered Gaussian random field of density
perturbations. The predicted halo mass function, spherically averaged density,
velocity dispersion, velocity anisotropy, ellipticity, prolateness and
potential profiles, as well as the abundance and number density profiles of
accreted and stripped subhalos and diffuse dark matter accurately recover the
results of cosmological $N$-body simulations. CUSP is thus a powerful tool for
the calculation, in any desired hierarchical cosmology with Gaussian
perturbations, of halo properties beyond the mass, redshift and radial ranges
covered by simulations. More importantly, CUSP unravels the origin of the
characteristic features of those properties. In the present Paper we culminate
its construction. We show that all halo properties but those related with
subhalo stripping are independent of the assembly history of those objects, and
that the Gaussian is the only smoothing window able to find the finite
collapsing patches while properly accounting for the entropy increase produced
in major mergers.
