  It has been known for nearly 20 years that the pseudo phase-space density
profile of equilibrium simulated dark matter halos, $\rho(r)/\sigma^3(r)$, is
well described by a power law over 3 decades in radius, even though both the
density $\rho(r)$, and the velocity dispersion $\sigma(r)$ deviate
significantly from power laws. The origin of this scale-free behavior is not
understood. It could be an inherent property of self-gravitating collisionless
systems, or it could be a mere coincidence. To address the question we work
with equilibrium halos, and more specifically, the second derivative of the
Jeans equation, which, under the assumptions of (i) Einasto density profile,
(ii) linear velocity anisotropy - density slope relation, and (iii)
$\rho/\sigma^3\propto r^{-\alpha}$, can be transformed from a differential
equation to a cubic algebraic equation. Relations (i)-(iii) are all observed in
numerical simulations, and are well parametrized by a total of 4 or 6 model
parameters. We do not consider dynamical evolution of halos; instead, taking
advantage of the fact that the algebraic Jeans equation for equilibrium halos
puts relations (i)-(iii) on the same footing, we study the (approximate)
solutions of this equation in the 4 and 6 dimensional spaces. We argue that the
distribution of best solutions in these parameter spaces is inconsistent with
$\rho/\sigma^3\propto r^{-\alpha}$ being an fundamental property of
gravitational evolution, and conclude that the scale-free nature of this
quantity is likely to be a fluke.
