Logical Construction of the Ionization Energy Theory and the Origin of   Physical Categories
  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. 
Andrew Das Arulsamy
physics.gen-ph