For the physical rational equation ����=(��-��)��, if both �� and �� are physical dimensions of length then �� equals one unit of length. If �� is equal to the Planck length of 10¯³⁵ meter then �� is 1/(10³⁵-1), which is a very tiny bit larger than 10¯³⁵ meter. Firstly, if �� is exactly zero then �� is also exactly zero. On the other hand, if �� is 10³⁵ then �� is 10³⁵/(10³⁵+1), almost unity but always remains a proper fraction. Secondly, as �� approaches infinity then �� approaches unity but never becomes an improper fraction. Thirdly, if �� is exactly unity then �� is exactly ½. Fourthly, if �� =-2 then ��=+2. These four pairs of unique solutions suggest the validity of a few physic-mathematical theorems.

The first singular solution where ��=0 and ��=0 suggests Nernst heat theorem, a statement of the third law of thermodynamics in a restricted form: If a chemical change takes place between pure crystalline solids at absolute zero there is no change of entropy. The second suggests the existence of a singular universe as its size indicated by �� approaches infinity, its existential probability indicated by �� approaches from zero to exactly unity but never becomes an improper fraction. The third suggests the physical validity of the virial theorem of classical mechanics, which can be generalized by the special theory of relativity such that the total energy minus the rest energy is equal to the negative of the average value of the kinetic energy. The fourth suggests the quantized spin of the hypothetical graviton such that the quantization of space-time charges by second order symmetric singular Hadamard matrices gives the geometrical equivalence for all three generations of neutrinos and gravitons. In a sense, a neutrino is simply a disguised graviton.