Which assumption are you referring to? If it's about the states in a finite region, note that I specifically limited this to distinguishable states. Whether or

Which assumption are you referring to? If it's about the states in a finite region, note that I specifically limited this to distinguishable states. Whether or not a finite region can have an infinite number of states (countable or otherwise) is irrelevant; we can only distinguish finitely many in finite time. Two states being indistinguishable means they'll give rise to the same output behaviour (e.g. from a mathematician in a room, carrying out some procedure).