If the observer has an infinite capacity we can reduce the problem to the single measurement or a pure two-state system.  The single measurement capacity implies relativistic limits i.e. zero entropy ⇒ t ≈ 0*.  In that case matrix X reduces to zero-entropy space-time matrix X_o, Writing X_o into its components, we get the following matrices: X₁, X₂, and X₃ are known as Pauli Spin Matrices.  The matrix X₃ corresponds to the actual physical measurement of a two state system with eigenvalues (EVs) [1, -1] or the measurements along z-axis.  The off-diagonal elements equal to '0' represent zero probability of tunneling between the states represented by EVs 1 and -1.  Pauli matrices are conventionally represented as σ_x ,  σ_y ,  and σ_z.

   
    (We can also say that Weyl's picture SL(2,$ℂ$) - entropy  $≡$  Pauli's picture SU(2).  In j-space language, Weyl's picture represents the description as measured by the macroscopic observer AKU or Obs_M, and Pauli's picture represents the description as measured by the relativistic observer Obs_c.) 

    An important property to note is that Pauli Spin Matrices are their own inverse(s), which means that the definition of the origin is mandatory within the structure described by these matrices.  (It is not necessarily true for a simple reflection of a structure in which case the reference origin is in the plane of the mirror, not in that of the structure itself).  As we recall that the VT symmetry is required to determine the origin, hence the quanta for the measurement space is pre-defined for the structure described by these matrices.  Please also note that Pauli spin matrices do not represent Majorana states.

    When we see a structure like the measurement triangle ΔOXY in anharmonic coordinates, Pauli's spin matrices come to mind immediately.  Let us try to evaluate the measurement triangle ΔOXY by placing σ_x , σ_y , and σ_z at X, Y, and O respectively and measure the unit point U.  Effectively we have replaced the α, β, and γ by spin matrices, and moved the system into a two-component measurement space.  We should be able to find scalars l, m and n such that,
 

We plug in the values of matrices and solve to find,

                                                              l² + m² + n² = 0_j .                      -3

Above equation represents a sphere of finite radius √0_j, a radius whose measurement is not possible for Obs_M.  We should be using the quantity i√0_j or an imaginary value for radius, but we are not.  Why?  Because in measurement space only PE1 measurable quantities are reported, hence only 0_j which actually is the radius squared, is of value.   In other words, the information on the internal structure of this sphere is not available to the macroscopic observer Obs_M.  In a later blog, this argument will result in shell-structure we are so familiar in classical picture. 

    Interestingly enough, we can replace Pauli's spin matrices with the components of Quaternion matrix in equation-1.  In this case also the equation-2 and the equation-3 remain unchanged.  Therefore in both cases, in which the component matrices represent convex surfaces, the geometric net results in spheres of finite radii.  Both these spheres represent stable structures.  One just wonders, what type of current will result from such symmetry?  It should be noted that the value of 0_j will be different in each case.

    The unit-point sphere S_U has a symmetry of 4π or 720^o, centered around the unit point U.  The 4π-symmetry is an essential feature of the systems with interacting components, or the systems whose components are not completely independent of each other.  We can later build a coordinate system with independent axes with S_U with 4π- symmetry, as the origin.  We can visualize the symmetry of 4π as a fundamental internal symmetry of a two-state system, measured as S_U by Obs_M at Λ_∞.

    One may want to think of the symmetry with respect to the integral multiple or half- integral multiple rotation of 360^o, but then not all the information contained within S_U would be measured.  Had we considered only a classical single-state system, the 360^o rotation would have been sufficient to describe the corresponding symmetry.  Obviously, two 360^o rotations do not represent 4π-symmetry which is the symmetry of the origin itself.  In fact the information measured by ∞_j × 360^o rotations, will always be less than the information contained within the 4π-symmetry.  In other words the resources available to a classical observer are not enough to re-define the origin and change the information structure being measured.