 Corner-force obstruction for a cubic polytrope

The sine-product cubic polytropic interior has zero mass density gradient at every corner. Its hydrostatic potential $C-2K\rho$ therefore has zero gradient there. In contrast, at a corner of a cube containing a positive mass density, every interior mass element contributes a gravitational-potential gradient pointing into the same coordinate octant. Each component of the gradient is strictly nonzero. This directly proves failure of global isolated gravitational matching, even though the interior hydrostatic and Poisson equations hold.