Since mathematics is the study of patterns in the physical world, pattern’s distinctions are revealed by groups of symmetry. Studying the patterns of elementary particles and their high energy interactions require a handful of distinct symmetry groups, for examples: SU(2), SU(3), as well as Lorentz and Poincaré groups. The first two are classified as groups of internal symmetry. The last two are classified as groups of space-time symmetry. Moreover, the first two are examples of compact symmetry while last two are examples of noncompact symmetry which means the domain is an open set say between zero and infinity. Wherever and whenever these compact and noncompact symmetries are combined together the results are classified under the general name of supersymmetry. Conventionally, a symmetry group must be constraint under one operation (say multiplication) by the following properties: (1) closure, (2) associativity, (3) identity, and (4) reciprocity.

Unfortunately, multiplication of symmetric singular Hadamard matrices can only satisfies associativity and therefore cannot be classified as groups of symmetry by conventional mathematics. Nonetheless, Hadamard matrices can describe a quantum theory of space-time charges as vacuum fluctuations of zero-point energies. Furthermore, the intrinsic symmetries of singular Hadamard matrices are equivalent to the Hopf-Mobius topology of any physical dimension of space-time as squares of energy.