Hilbert Book Model Project/Modules and Modular Systems

<Hilbert Book Model Project

= Modules and Modular Systems =

Existence
Careful inspection of the environment shows that all observable objects in the universe behave as modules or as modular systems.

A hierarchy of modules exists, with on the bottom a set of elementary modules that together constitute all other modules. Modular systems locate at the top of this hierarchy.

Intelligent species represent a particular category of modular systems. At the lower end, a category of super-tiny objects exists that withdraw from observation and are not considered to be modules. These objects are shock fronts. Warps are one-dimensional shock fronts. Clamps are spherical shock fronts.

Huge numbers of clamps constitute, combined in coherent swarms the elementary particles. Sophisticated measuring equipment can detect the elementary particles. Warps constitute, combined equidistant in huge strings, the photons. A human eye can detect the photons.

The Hilbert Book Model reserves the name "module" for observable objects that coherent swarms of clamps can constitute. Thus, the Hilbert Book Model does not classify warps, photons, and single clamps as modules. Elementary particles are point-like objects and they are elementary modules. These objects are represented by their hopping path and by their hop landing location swarm. Each elementary module is controlled by a private stochastic process and resides on a private platform that floats on top of the background parameter space of the base model.

Modular design and construction
The Hilbert Book Model impersonates a creator. At the instant of the creation, the creator stored the dynamic geometric data of his creatures in a read-only repository.

This act makes the creator a modular designer and a modular creator.

Diving deep into the fundamental structure of physical reality requires a deep dive into advanced mathematics.

Usually, this goes together with formulas or other descriptions that are incomprehensible to most people.

The beautiful thing about the deepest levels of the structure of physical reality is that the foundation of reality must be rather simple and therefore it can be described in a simple way and without any formulas.

For example, if the observed indication characterizes physical reality, then the most fundamental and most influential law of physical reality can be formulated in the form of a commandment

“THOU SHALT CONSTRUCT IN A MODULAR WAY.”

This law is the direct or nearly direct consequence of the structure of the deepest levels of physical reality. That foundation restricts the types of relations that may play a role in physical reality.

The structure of the foundation does not yet contain numbers. Therefore, it does not yet contain notions such as location and time.

The Hilbert Book Model expresses the above law intentionally in the form of a commandment. It is not possible to express this law in the form of a formula, such as $$K=m \,a$$ or $$E = m c^2$$.

At the lowest level, numbers that can serve as variables in formulas do not yet exist. The impact of the commandment is far more influential than the impact of these famous formulas.

Modular construction acts very economical with its resources and the law thus includes an important lesson

"DO NOT SPOIL RESOURCES!"

Modular design
Understanding that the above statements indeed concern the deepest foundation of physics requires deep mathematical insight. Alternatively, it requests belief from those that cannot (yet) understand this methodology. On the other hand, intuition quickly leads to trust and acceptance that the above major law must rule our existence! Modular design has the intention to keep the relational structure of the target system as simple as is possible. This does not take away that modular design is a complicated concept.

Successful modular construction involves the standardization of module types, and it involves the encapsulation of modules such that the method hides internal relations from the outside.

Systems become complicated when many relations and many types of relations exist inside that system, which the method must reckon when the system is analyzed, configured, operated, or changed. The reduction of relational complexity plays a significant role during system configuration.

The ability to configure modular systems relies heavily on the ability to couple modules and on the capability to let these modules operate in concordance.

The modular design method becomes very powerful when modules configure from lower level modules.

The standardization of modules enables reuse and may generate type communities. The success of a type community may depend on other type communities.

Modularity in the base model
The base model consists of the combination of a separable Hilbert space and its unique non-separable companion that embeds the separable Hilbert space.

This base model does not yet contain modules. Stochastic mechanisms that access the base model from its outside provide the insertion of modules.

Separable Hilbert space
A complete set of atoms of the orthomodular lattice corresponds to an orthogonal base of the separable Hilbert space that realizes the orthomodular lattice.

The rational values of a number system can enumerate the members of an orthonormal base of an infinite dimensional separable Hilbert space.

If the Hilbert space applies a version of the quaternionic number system to generate the values of the inner products of pairs of Hilbert vectors, then the rational values of this version can enumerate a selected orthonormal base of the complete separable Hilbert space.

A dedicated operator can apply the base vectors as eigenvectors and the enumerators as eigenvalues. The eigenspace of this operator becomes the background parameter space of the Hilbert space. A suitable Cartesian coordinate system orders this parameter space.

The real parts of the eigenvalues define a subspace that scans the Hilbert space as a function of the value of this real part. This value plays the role of the progression of the scan.

Elementary modules
An important category of modules are the elementary modules. These are modules, which are themselves not constructed from other modules.

Elementary modules reside on a private platform. This platform is a separable Hilbert space that shares its vectors with the infinite dimensional separable Hilbert space that embeds in its non-separable companion and acts as a background platform. The parameter space that sequences the ordering of the private platform is eigenspace of a private operator that resides in this floating separable Hilbert space. This eigenspace corresponds to a selected version of the quaternionic number system. The floating Hilbert space applies an inner product that uses values which are taken from this version of the quaternionic number system. Consequently the normal operators that reside in this floating Hilbert space have eigenvalues that belong to this version of the number system.

A mechanism that constructs the elementary module must generate the subsequent locations of the module. Each elementary module type owns a private generation mechanism. The mechanism produces locations that are also taken from the private version of the number system. Together with the time-stamp, the generated locations are immediately stored into the eigenspace of a dedicated operator.

In fact, the mechanism makes the object modular. Without that mechanism, the object is just an arbitrary element of the orthomodular lattice.

Elementary modules are atoms of the orthomodular lattice that own a private mechanism, which provides their dynamic geometric data. These elements form a sub-lattice of which all elements are modules.

At any progression instant, a complete set of atoms that are elementary modules is forming an orthonormal base of a subspace of the background separable Hilbert space.

That progression instant forms a timestamp, and together with a unique spatial location, it represents the property set of a selected elementary module at that progression instant.

At every instant, a quaternion contains the property set of this elementary module. That quaternion is an eigenvalue of the private operator whose corresponding eigenvector is the base vector that spans the ray, which represents the elementary module at that instant.

Within the scanning subspace, a smaller subset of the Hilbert vectors represents all elementary modules that exist at that progression instant. The members of that subset are mutually orthogonal, and each member spans a ray.

Over time, the locations of an elementary module form a hopping path and a coherent location swarm. This period defines the regeneration cycle period of the elementary module.

Each elementary module resides on its private floating platform that carries a private parameter space.

The platform, the hopping path and the swarm characterize the elementary module and define the type diversity of the elementary modules.

All modules and all modular systems that feature the same timestamp, move with the scanning subspace.

At the selected instant, the elementary modules configure all other modules and all modular systems that own the same timestamp. Together the modules form a subspace of the scanning subspace.

Modular systems and modular subsystems are conglomerates of bounded modules. Often the modules are coupled via interfaces that channel the information paths that are used by the relations.

At all levels, the mechanisms that provide the locations of the modules ensure coherence and install part of the binding. The binding effect will be explained below.

Modular subsystems can act as modules and often they can also act as independent modular systems.

The hiding of internal relations inside a module eases the configuration of modular (sub)systems.

If a monolith consists of N objects, then the designer and the constructor must in principle resolve $$N(N-1)$$ relations. Also, these relations can exist in different types.

Modular design and construction encapsulate internal relations inside modules, such that the capsule hides them from the outside. Also, the modular design combines external relations in interfaces. This fact eases system configuration. Modular configuration depends on the availability of resources, such as type communities of suitable building blocks.

In complicated systems, modular system generation can be several orders of magnitude more efficient than the generation of equivalent monoliths.

This efficiency means that stochastic modular system generation gets a winning chance against monolithic system construction.

The generation of modules and the configuration of modular (sub)systems can accomplish in a stochastic or in an intelligent way.

Stochastic (sub)system generation takes more resources and requires more trials than intelligent (sub)system generation.

An inexperienced modular designer must first learn to discern which relations are relevant and which relations are negligible.

Predesigned interfaces that combine provide-relations and require-relations can save many resources.

If all discrete objects are either modules or modular systems, then intelligent (sub)system generation must wait for the arrival of intelligent modular systems.

Intelligent species can take care of the success of their private type. This attention includes the care about the welfare of the types on which its type depends.

Thus, for intelligent modular systems, modularization also includes the lesson

“TAKE CARE OF THE TYPES ON WHICH YOU DEPEND.”

In physical reality, mechanisms that apply stochastic processes appear to generate the elementary modules.

In most cases, system configuration occurs in a trial and error fashion.

Only when intelligent species are present that can control system configuration will intelligent design occasionally manage the system configuration and binding process.

Thus, in the first phase, stochastic evolution will represent the modular system configuration drive.

Due to the restricted speed of information transfer, intelligent design will only occur at isolated locations.

On those locations, intelligent species must be present.

Coherence and binding
The locations that elementary modules take, form coherent swarms.

The cause is the fact that the mechanism that provides these locations applies a stochastic process that owns a characteristic function.

That function equals the Fourier transform of the location density distribution of the swarm.

A gauge factor in the characteristic function acts as a displacement generator.

Consequently, at first approximation, the swarm moves as one unit.

The elementary modules configure higher level modules.

These higher-level modules occupy locations that form combined swarms.

Coordinated stochastic processes generate these combined swarms.

These processes join into an overall stochastic process which again owns a characteristic function that is the Fourier transform of the combined location density distribution of the joined swarms.

This overall characteristic function equals the superposition of the characteristic functions of the stochastic processes of the participating elementary modules.

The superposition coefficients determine the internal movement of the components.

This characteristic function again acts as a displacement generator, and the complete module moves at first approximation as one unit.

The binding via spectral coupling acts attractive.

However, via the Pauli exclusion principle, it can also act in a blocking way. Thus, the Pauli exclusion principle must find its explanation in the spectral binding of modules.

The binding via spectral coupling is not the only type of binding.

Elementary modules reside on platforms that carry parameter spaces, which serve to define the locations of the elements and the location density distribution of the hop landing location swarm of the elementary modules.

The ordering of the parameter space defines the symmetry flavor of the platform, and on its turn, this symmetry flavor determines the symmetry-related charge of the platform.

This charge locates at the geometric center of the platform.

This charge is the source of the contribution of the platform to the symmetry-related field.

The symmetry-related field and the symmetry related charges interact. This interaction represents another form of binding.

The symmetry-related interaction can attract or repel.

Via the platform, the swarm and the symmetry related charge interact.

The symmetry-related charge locates at the geometric center of the platform, and that locates about the geometric center of the swarm.

Further, clamps that deform the embedding continuum represent reactions of the embedding field on the hop landings.

This deformation also represents an attracting binding effect.

Fermions
Fermions are modules that feature half-integer spin. The platform of elementary fermions applies Cartesian ordering and subsequently applies polar ordering in which the polar angle plays a major role and can order upwards or downwards over π radians. All fermions obey the Pauli exclusion principle, which forbids binding of fermions that share all properties.

Bosons
The Hilbert Book Model sees photons as strings of equidistant warps, and it does not see separate warps and clamps as modules.

Clamps appear to play the role that the Standard Model reserves for the Higgs boson.

So only mesons and other conglomerates that show integer valued spin, appear to play the role of bosons.

Atomic modular subsystems
Atomic modular systems are conglomerates of elementary modules. The scope of the Pauli principle is the considered atomic modular system.

Here we postulate that free atomic modular systems consist of fermions.

The stochastic mechanisms that determine the locations of the constituting elementary modules cooperate in an overall stochastic process.

An overall characteristic function characterizes that process.

This characteristic function is the superposition of a set of different characteristic functions that each form the characteristic function of a constituting component elementary module.

Constituents that are equal apart from a set of clamps must differ according to the pattern of the solutions of the Helmholtz equation.

This pattern corresponds to energy levels of photons that the atomic modular system can emit or absorb.

In atomic modular systems, the electric charges and color charges superpose at the geometric center of the mutual platform.

Evolution
The ongoing stochastic design and construction of modules result in increasingly complicated modules that show more functionality, and that functionality may help them and their type community to survives against attacks from competition or environment.

Finally, evolution resulted in the generation of intelligent species. This enabled these species to actively take part in the design and construction of modules.

Intelligent design and construction
Reality offers huge resources in available time and in numbers of building components. In this way, even stochastic design as is applied by nature can reach high levels of complexity.

In the beginning, the model will apply a stochastic design as its generation strategy. This approach will change when the model has achieved a level in which intelligent species appear.

From that instant on, the efficiency of the modular construction strategy, will on some locations increase significantly. Intelligent design and construction will use far less design and generation time and other required resources.

This change will clearly affect the evolution of the model at those locations. Due to the limited speed of information spread, these effects will appear at isolated locations.

As indicated earlier the selection of modular configuration by the creator includes important lessons for intelligent designers.

Where Darwin predicted the survival of the fittest, does modular design and construction predict the survival of the most successful type community.

Actuality shows that taking care of your living environment involves taking care of how the climate inside the planet evolves.

How it should
The hardware industry shows how products can successfully be configured from modules.

Especially the electronics industry has developed ways to construct very complicated systems and subsystems that apply components, which are generated by a lively components market.

This organization provides a large resource and a diversity of components that compete at quality/price ratio. Interface standardization ensures the coupling capability of the offerings.

Especially the easy configuration of these components into systems generate many useful products with great functionality.

The semiconductor industry optimizes these ideas by integrating the components in integrated circuits.

How not
The software industry never found a proper way to apply modularization. Instead, it applies object orientation. However, object orientation does not hide relations inside capsules. Instead, it creates very complicated extra relations that support the inheritance mechanism.