  A finite physical system has a finite entropy -- a finite number of distinct
possible states. Here we show that finite maximum counts of distinct
(orthogonal) states also define other basic quantities of physics, including
energy, momentum and Lagrangian action. A finite number of distinct states also
limits the resolution of measurements and makes classical spacetime effectively
discrete. Our analysis generalizes speed limits on time evolution: we count the
distinct states possible in a finite-length of unitary transformation. As in
Nyquist's bound on distinct signal values in classical waves, widths of
superpositions bound the distinct states per unit length. Maximally distinct
transformations are effectively discrete, allowing us to simplify analysis and
simulation of maximally distinct dynamics.
