[AntonioLao]

A theory of ‘of theory’ (toot) would be no theory at all. It is similar to a guess of a guess, a dream of a dream, a thought of a thought, or a reflection of a reflection. In logic, a no is a no. But a no and a no is a yes while a no, a no, and a no is a no and a no, a no, a no, and a no is again a yes. The recurring patterns are that odd number of no is a no and even number of no is a yes. On the other hand, any number, even or odd, of a yes is always a yes.

However, in mathematical transformations, a parallel reflection of a parallel reflection is the same as the inverse transformation of an inverse or the reciprocal of a reciprocal. All these indicate no transformations at all signifying a perfect symmetry. If no is represented by -1 and yes is unity then a Boolean algebra exists. It is analogous to the binary number system used in computer science represented by the binary digits 0 and 1. Its multi-dimensional representations would be described by certain Hadamard algebra using singular square symmetric Hadamard matrices.
A point transformation of -1 and 1 generates the ring of integers under the operation of addition. Under the operation of multiplication, this ring solely generates the cyclic values of -1 and 1. A linear vector transformation using the row or column vector basis: (1,1), (-1,-1), (1,-1), and (-1,1) generates arbitrary vector (a, b) where both a and b are integers. This would also include the linearly independent vectors: (0,0), (1,0), (-1,0), (0,1), and (0,-1) under scalar multiplication. Nevertheless, only linear vector-matrix multiplication can generate singular square symmetric Hadamard matrices. These serve as a geometric explanation for LASER amplifications and oscillations and also a starting point for a theory of collimated holographic consciousness.