Logical construction of the ionization energy theory and the origin of physical categories

Logical proofs and definitions are developed to establish (1) that the energy-level spacings,for each chemical element (from the periodic table of chemical elements) can be converted to the ionization energies, (2) both and the ionization energies are unique, and (3) the averaged ionization energy of any quantum matter is proportional to the averaged ionization energy of its constituent chemical elements, if and only if 6= 0 and is not an irrelevant constant. Physical atoms are then constructed to define the physical sets such that these sets 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 to another P′. Therefore, the existence of many P implies the existence of different categories of physical systems and quantum matter.