The base model of the Hilbert Book Model consists of an infinite dimensional separable Hilbert space and it's unique non-separable companion Hilbert space that embeds its separable companion. The version of the quaternionic number system that specifies the values of the inner product of these Hilbert spaces also defines the background parameter space of the base model. The rational values of this background parameter space form the eigenspace of a special reference operator that resides in the separable Hilbert space, and the full background parameter space is the continuum eigenspace of the companion reference operator in the non-separable Hilbert space. The real values of the quaternionic eigenvalues in these eigenspaces that correspond to a selected progression value define a subspace that scans as a function of progression over the base model. This subspace specifies the current static status quo of the model.
Mutually independent Cartesian and polar coordinate systems define other independent versions of the quaternionic number system. These versions correspond with other parameter spaces and corresponding reference operators. These parameter spaces float over the background parameter space. They represent floating platforms. Apart from their varying center location these platforms own characteristics that are defined by their ordering symmetry. The Cartesian coordinate system discerns sixteen different ordering types. For the symmetry-related properties of the platform, only the differences with the symmetry of the background parameter space count. For the Cartesian ordering, this results in a short list of charges and anisotropies.
If we enumerate the possible orderings with binary numbers, then can every bit correspond with a dimension. Thus, binary number 1010 corresponds to ordering ⇧⇩⇧⇩ and the background parameter space corresponds to 0000, which corresponds to ordering ⇩⇩⇩⇩. Now we can fill a table with all possible orderings. Handedness switches with every direction change. The third column lists the handedness. Next, we count the number of bit differences between 0000 and the binary numbers in the first column. The fourth column lists these differences. The differences in the lower half of the table have switched their sign. The short list of charges that occur in the Standard Model is -1, -2/3, -1/3, 0, 1/3, 2/3, and 1. When we multiply these fractals with 3, then these numbers also occur in column four. 0 is the electric charge of the neutrinos. Charge size 1/3 belongs to the down quarks. Charge size 2/3 belongs to the up quarks. Charge size 1 belongs to the electron and the positron. The lower half of the table represents the antiparticles. In that part, also the RGB color is changed into the RGB anti-color. N represents neutral colors.
|1011||⇧⇩⇧⇧||L||2||B||A_up quark |
|1101||⇧⇧⇩⇧||L||2||G||A_up quark |
|1110||⇧⇧⇧⇩||R||2||R||A_up quark |
The table shows that a discrepancy exists between what the symmetry flavor sees as an up-quark or anti-up-quark and what the Standard Model interprets as an up-quark or anti-up-quark. Observers cannot detect who is right.
The table also shows that color charge is related to the direction of anisotropy. Quarks are anisotropic. The other particles are isotropic.
The Hilbert Book Model suggests that the named particles reside on the platforms of which the ordering symmetry corresponds to the second column. These particles inherit the symmetry-related charges and the color charges from the platforms on which they reside. The charges locate on the geometric centers of the platforms.
These charges form sources for corresponding symmetry related fields.
Polar ordering brings extra properties. Polar ordering starts with a Cartesian coordinate system. It may start with running over the 2π radians of the azimuth, or polar ordering may start with rotating over the π radians of the polar angle. Rotation can occur upward or downward. This indicates that polar ordering may explain the origin of spin.
The explanation why the differences in ordering lead to properties of the platforms is quite tricky.
The ordering differences in the symmetry of the versions of the number systems affect the handedness and it affects integration behavior. Integral balance equations explain this last effect. The balance equations convert volume integrals into surface integrals. These equations are known as Gauss theorem and Stokes theorem. The background parameter space together with the floating platforms constitutes a mixed domain. Integration over this mixed domain involves the separation of the sub-domains that are covered by the platforms. The separation is best done by a cube shaped encapsulation that is aligned along the Cartesian coordinate axes. The encapsulation surface must not cross artifacts. Now the surface integrals can be divided into three parts that may react differently on the ordering. Instead of summation, it applies subtraction in the discrepant dimensions. The resulting contribution may be 0, 1/3, 2/3, or 3/3 of the full contribution. The background parameter space envelops the discrepant sub-domains.