Which conservation law determines that a total negative energy results in a closed orbit?

The Conservation of Energy

The shape of a Keplerian orbit is fundamentally governed by the conservation of two key quantities: energy and angular momentum. The Conservation of Energy dictates the overall classification of the orbit. If the total energy, which is the sum of the kinetic energy and the gravitational potential energy of the orbiting body, is calculated to be negative, the resulting orbit must be a closed curve. These closed orbits include both perfect circles ($e=0$) and ellipses ($0 < e < 1$). If the total energy were zero, the orbit would be parabolic ($e=1$), and if the total energy were positive, the orbit would be hyperbolic ($e>1$); both of these latter two cases represent open, unbound paths where the body will never return to the vicinity of the central mass.