  Logical proofs and definitions are developed to establish (i) the
energy-level spacings, \xi for each chemical element (from the periodic table
of chemical elements) can be converted to the ionization energies, (ii) both
\xi and the ionization energies are unique, and (iii) the averaged ionization
energies of any quantum matter is proportional to the averaged ionization
energies of its constituent chemical elements, if and only if \xi not = 0 and
\xi is not an irrelevant constant. Physical sets are then constructed such that
they are members of a specific physical class where each class belongs to a
specific physical category, P. However, there is not a single
structure-preserving functor from one energy-level spacing physical category,
P(\xi) to another P'(\xi). Therefore, the existence of many P(\xi) imply the
existence of different categories of physical systems and quantum matter.
