A static point mass in a gravitational field is already at its lowest energy state (a literal ground state if you will). It can only really do one thing, which

A static point mass in a gravitational field is already at its lowest energy state (a literal ground state if you will). It can only really do one thing, which is to stay. The logarithm of the number of states then is zero. So in some sense it cannot encode information at all. In fact I think more generally speaking, things that are at a global minimum of energy level cannot really be used to encode information (unless you have a mechanism to add entropy w/o increasing energy), because they have one state, and the log of the number of states then gives you zero bits. Probably another way to look at the idea that gravitational potential path-independence means memorylessness means devoid of bits of information.