The HBMP starts from the assumption that physical reality has structure and that this structure has a foundation. Many scientists find it difficult to assume that physical reality applies mathematics because they consider math as a human invention. The fact that foundations tend to have a simple structure that is easily comprehensible can counter this attitude. Thus, a large chance exists that intelligent humans already discovered this structure long ago and archived it in their math library.

A further assumption is that the foundation of the structure of reality must automatically extend into a more complicated structure, which is then also a structure that is featured by reality. The archived mathematical founding structure must show the same quality and will extend in a more complicated structure that will resemble the corresponding structure that is owned by reality. So despite the fact that the human discoverers of the founding structure were not looking for a foundation of physical reality, they accidentally uncovered a structure that automatically emerges into higher levels of a structure that characterizes physical reality.

With other words, it is quite obvious that reality features a structure that also occurs in humanly invented mathematics.
The task is now to uncover the founding structure that has the quality that it emerges into a structure that closely resembles the lower levels of the structure of physical reality. A good candidate is an orthomodular lattice that eighty years ago was discovered by two scientists Garrett Birkhoff and John von Neumann and that they unluckier-wise called "quantum logic." They gave this lattice its name because its lattice structure is very close to the structure of classical logic. The elements of classical logic are logical propositions. In their introductory paper, the scientific duo showed that the set of closed subspaces of a separable Hilbert space has exactly the lattice structure of the orthomodular lattice that they called "quantum logic." The elements of the lattice are closed subspaces and do not represent logical propositions. The orthomodular lattice is atomic, and the atoms correspond to rays (one-dimensional subspaces) A complete set of mutually independent rays correspond to an orthogonal base of the separable Hilbert space.

Via the internal product of pairs of Hilbert vectors the Hilbert space introduces number systems. Operators that map subspaces onto themselves, own eigenspaces that couple eigenvalues to eigenvectors. Hilbert spaces can only cope with number systems that are division rings. Only three suitable division rings exist. Reality uses them all. So we apply a separable Hilbert space that uses the quaternions for the specification of its inner product. Thus, also eigenvalues can be quaternions.

Quaternionic number systems exist in multiple versions. Cartesian and polar coordinate systems can distinguish these versions. The specification of the inner product uses the version that also defines the background parameter space of the separable Hilbert space. The other versions correspond to parameter spaces that cover platforms, which float over the background parameter space.

The background parameter space is eigenspace of a special reference operator. The eigenvectors of this operator that correspond to the same real part of the quaternionic eigenvalue span a subspace that scans over the Hilbert space as a function of the selected real value that we will destine as the progression value. This scanning subspace corresponds to the static status quo of the model and makes from the model a dynamic model.

Each infinite dimensional separable Hilbert space owns a unique companion non-separable Hilbert space that embeds its separable partner.

Quaternionic functions can reuse the eigenvectors of the reference operator and the corresponding eigenvalues as parameter values to specify the eigenvalues of a newly defined operator as its target values. This procedure combines Hilbert space operator technology with quaternionic function theory and indirectly with quaternionic differential and integral calculus.

The combination becomes a very powerful read-only repository in which reality can store all its dynamic geometric data in quaternionic eigenvalues and quaternionic continuums.

Observers that travel with the scanning subspace can read the part of the repository that contains historic time-stamps. The continuum that embeds both the stored event and the observer transfers the information, but the observers perceive this information in space-time format. A Lorentz transform converts the Euclidean (quaternionic) storage format into the perceived space-time format and implements time-dilation and length contraction.

Something very essential is missing from this model. It misses interesting dynamics, and it misses means to keep these dynamics coherent.
Reality appears to apply stochastic processes to achieve this purpose.