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Can you show why it should follow from that???This stems from the exact same arguments as in quantum field theory: there has to be some high-energy cutoff.

If you look at the books of statistical mechanics, you'll find that microstates means only ''energy eigenstate'', and not ''arbitrary state''.Now you're mixing different descriptions of the same system. But it isn't true in any event. The computation of entropy has to count the full set of microstates, which for real particles also includes things like spin and angular momentum, as well as energy.