  The Pauli exclusion principle gives an upper bound of 1 on the natural
occupation numbers. Recently there has been an intriguing amount of theoretical
evidence that there is a plethora of additional generalized Pauli restrictions
or (in)equalities, of kinematic nature, satisfied by these numbers. Here for
the first time a numerical analysis of the nature of such constraints is
effected in real atoms. The inequalities are nearly saturated, or quasi-pinned.
For rank-six and rank-seven approximations for lithium, the deviation from
saturation is smaller than the lowest occupancy number. For a rank-eight
approximation we find well-defined families of saturation conditions.
