A hierarchy of mathematical structures that are discovered by humans exists that starts from a founding structure.
it evolves into more complicated structures that together constitute a model,
which shows the structure and behavior of the physical reality that we can observe.

This indicates that physical reality applies similar mathematical structures that implement its structure and behavior.

The base model represents a powerful and very flexible platform for modeling quantum physical and cosmological theories. 

It derives from a simple and easily comprehensible foundation that automatically extends in a repository for dynamic geometric data of point-like objects. 

The discoverers called the foundation quantum logic.

Mathematicians called it an orthomodular lattice.

Later the mathematicians called it a Hilbert lattice because it equals the set of closed subspaces of a separable Hilbertspace.

This set constitutes the Hilbert space.

A Hilbertspace features an inner product that closes the underlying vector space.

A huge number of quaternionic separable Hilbertspaces constitute the final repository. 

These separable Hilbertspaces share the same underlying vectorspace. 

They distinguish in the version of the quaternionic number system that they apply to specify their inner product. 

This version determines the private parameterspace of the separable Hilbertspace. 

The parameterspace determines the symmetry of the separable Hilbertspace. 

A dedicated normal operator manages the parameter space in its eigenspace.

A category of normal operators share the eigenvectors of the reference operator. The corresponding eigenvalues are parameter values.

The new operators replace the eigenvalues of the reference operators by the corresponding target values of a selected quaternionic function.

The eigenspaces of the newly defined operators represent sampled dynamic fields.

One of the separable Hilbertspaces acts as a background platform and provides the background parameterspace. 

All other separable Hilbertspaces float with the geometrical center of their parameterspace over the background parameterspace. 

The difference in the symmetry between the floating parameterspace and the background parameterspace generates a symmetry-related charge.

This charge locates at the geometric center of the floating parameterspace. 

This charge results in a source or a sink that produces a corresponding quaternionic symmetry-related field.


In physical reality, elementary particles reside on the floating platforms and are represented by the eigenspace of a dedicated footprint operator.

The hop landing locations of the elementary particles are archived in quaternionic eigenvalues that act as storage bins.

These storage bins combine a timestamp and a spatial location.

The hopping path recurrently regenerates a coherent hop landing location swarm.

A location density distribution describes this swarm and equals the square of the modulus of the wavefunction of the particle.

A stochastic process that owns a characteristic function generates the hop landing locations.

The characteristic function equals the Fourier transform of the location density distribution.

Consequently, the location density distribution is a wave package and the hop landing location swarm can produce interference patterns.

The eigenspace of the footprint operator contains the complete life-story of the elementary particle.


The dimension of the background separable Hilbertspace is infinite. 

Therefore, this Hilbertspace owns a unique quaternionic non-separable Hilbertspace that embeds its separable companion.

Non-separable Hilbertspaces support normal operators that possess continuum eigenspaces.

Quaternionic functions describe these continuums.

The continuums represent physical fields.


The embedding of the floating platforms into the background platform tells the life-story of the elementary particle in a different way.

Every hop landing of an elementary particle may cause a spherical pulse response.

The spherical pulse response only occurs when the hop landing location is isotropic discrepant with the background continuum.

This restriction causes color confinement.

The spherical pulse response acts as a spherical shock front.

Over time the spherical pulse response integrates in the Greenís function of the embedding continuum.

Thus, the pulse injects the volume of the Greenís function into the embedding field.

Initially this locally deforms the field.

The front spreads the injected volume over the field.

Consequently, the deformation quickly fades away.

The volume stays inside the field and persistently expands the field.

The recurrently regenerated hop location swarm causes a persistent deformation that is described by the gravitational potential of the particle.

The gravitational potential of the particle approaches the convolution of the Greenís function of the embedding field and the location density distribution of the swarm.

The gravitational potential of the particle is everywhere a smooth function.

Far from the geometrical center the gravitational potential gets the shape of the Greenís function.


Shock fronts only appear in odd numbers of participating dimensions.

All shock fronts travel with light speed.

During travel the front keeps its shape.

In isolation, field excitations that act as shock fronts cannot be detected.

These field excitations are candidates for black objects.

Spherical shock fronts act as dark matter objects. They deform and expand their carrier field.

During travel, the amplitude of the spherical front diminishes with increasing distance from the pulse.

Only isotropic pulses can generate spherical shock fronts.

Dark matter objects constitute the footprint of elementary particles.

The first kind of stochastic processes generate the pulses that cause dark matter objects.


One-dimensional shock fronts are generated by one-dimensional pulses.

During travel the one-dimensional shock front keeps the shape and the amplitude of the front.

One-dimensional shock fronts represent a package of a standard amount of energy.

All energy packages possess a normalized polarization vector.

One-dimensional shock fronts carry a standard amount of energy.

A string of equidistant one-dimensional shock fronts constitutes a photon.

The emission duration of all photons is the same.

The frequency of the string determines the energy that is contained in the photon.


Elementary particles behave as elementary modules.

Together, they constitute all modules and some modules constitute modular systems.


A second type of stochastic process controls the composition of composite modules.

This process also owns a characteristic function.

The characteristic function equals a dynamic superposition of the characteristic functions of the components.

The superposition coefficients act as displacement generators.

The displacement generators determine the internal locations of the components.

All modules, including the elementary modules, feature a displacement generator, which determines the position of the module.


The fact, that the composition of modules is defined in Fourier space, where location does not play the role that it plays in configuration space, causes the entanglement of components of composite modules.

The gravitational potential of a unit mass at a large distance r from a point-mass of mass M can be defined as the work W, that needs to be done by an external agent to bring the unit mass in from infinity to that point equals MG divided by r.


G is the gravitational constant.

A black hole is an encapsulated region that deforms the surrounding continuum.

The encapsulation has a minimal surface.

Inside the region field excitations do not exist.

At the boundary of the region the gravitational energy of energy packages equals their kinetic energy.

This defines the radius of the region as its mass multiplied by the gravitational constant and divided by the square of the speed of light.

mMG/rbh=mc2 thus rbh=MG/c2