Geometrical approach to the argumentum of bijection of one coordinate-threshold reflection

The application of threshold operations is the perspective direction of the construction of discrete information processing nodes considering the potential possibility of realization of calculating the scalar product directly in the carrier signal, for instance, perspective optical computing medium.
The article analyzes the reflection of bijective binary vectors with simple implementation of both the initial and inverse transformation by means of so-called quasi-hadamard matrices $A_n$ in threshold basis. Currently bijection of such reflection is empirically shown for $n = 4, 6, 8$, however there was no relevant strict proof. The first relevant proof based on the study of the geometrical properties of the reflection generated by quasi-hadamard matrice $A_4$ is provided in this work.
During the proof it was found that it is unique and possible as proposed only for $n = 4$. The article highlights the interesting features of its geometrical interpretation together with the proof of important applied statement about bijection of reflection generated by quasi-hadamard matrice $A_4$.