Interactive Maps on Variant Phase Spaces– From Measurements - Micro Ensembles to Ensemble Matrices on Statistical Mechanics of Particle Models

1.1.1. White and Black Box Models Input, output and functions are fundamental elements of the wider applications of dynamic systems [3, 5, 21] such applications include: mathematics, probability, physics, statistics, classical logic, and cellular automata. For a pair of N bit vectors X, Y ∈ BN 2 with states, for a given 0-1 function f , the pair of 0-1 vectors are linked by an equation where the function may be expressed by Y = f (X) thus:


White and Black Box Models
Input, output and functions are fundamental elements of the wider applications of dynamic systems [3,5,21] such applications include: mathematics, probability, physics, statistics, classical logic, and cellular automata.
For a pair of N bit vectors X, Y ∈ B N 2 with states, for a given 0-1 function f , the pair of 0-1 vectors are linked by an equation where the function may be expressed by Y = f (X) thus: This is called a white box model [3,15,28]. Using the white box model, a pair (X, Y) can be explicitly calculated by a function f .
If there is no explicit expression for a unknown function U, a pair of vectors (X, Y) could be collected for their correspondences on the pair of input-output relationships. Equation Y = U(X) is still satisfied. This is called a black box model. i.e. A pair of (X, Y) can be measured by a unknown function U, or expressed as In science and engineering [13,28,29], a black box is a device, system, or object which can be viewed solely in terms of its input, output and transfer characteristics without any knowledge of its internal works.
From a cellular automata viewpoint, the black box approach is useful in describing a situation where both input and output are in the form of two bit vectors for an unknown function of a digital system.

Characteristic Point and Phase Space
In mathematics and physics [3,14,18,20,21,29], the concept of a phase space as introduced by W. Gibbs in 1901 is a space in which all possible states of a system are represented. Here, each possible state of the system corresponds to one unique point in the phase space. For cellular automata, the phase space usually consists of all possible values of pairs of input and output vectors in multiple dimensions.
For either a known function f or for an unknown function U, when the states of X, Y reside in the same finite region, it is entirely feasible in principle to undertake an exhaustive procedure to list all pairs of {(X, Y)}. For a given N bit vector X, the vector generates a point with a unique spatial position to indicate the characteristics of the function and by listing all such possible points, a phase space for the function is established.

Historical review on phase spaces of statistical mechanics
Top-down and bottom-up are two distinct strategies of intelligent processing and knowledge ordering used in humanistic and scientific theories [4,13,15,25,28]. In practice, they can be seen as alternative styles of thinking and problem solving. Top-down may be taken to mean an approach based on an analysis or decomposition to identify key components within a global target that has been identified for study and from which there may be constructed a hierarchy of local features. Bottom-up may be taken to describe a process of synthesis via integration working from local features towards a global target. (1804-51) recognized that Liouville's works could be used to describe mechanical systems and so placed Liouville's mathematical theorem into a mechanical context. Pücker working in Germany and Cayley and Sylvester in the UK, extended projective geometry beyond the ordinary three dimensions in the 1840s and Grassmann developed an n dimensional vector space in 1844.
Riemannn's work in 1868 developed the geometric properties of multi-dimensional manifolds. This was followed by further developments in the 1870s by E. Betti, F. Klein, and C. Jordan then more recently by [23,24,31,33].
As it was Lagrange who took the first steps, a bottom-up approach is now often described as a Lagrange expression. Hamiltonian dynamics is a typical representative under this expression as it is founded on a pair of conjugate parameters [31,33].

Top-down Approaches
Robert Rudolf Clausius (1822-88) explored an expression of the second law for which the only function is the transfer of heat to propose the function dQ/T to compare heat flows with heat conversions using Carnot's techniques to derive entropy and show the two laws of thermodynamics were the equivalent of the older caloric theory. Gustav Robert Kirchhoff (1824-1887) derived from the second law of thermodynamics that objects cannot be distinguished by thermal radiation at a uniform temperature to formulate a black body [23,24,31,33].
Leonhard Euler (1707-1783) provided key methodologies in this direction with a top-down approach known as a Euler expression. A Fourier series of a periodic function is a typical representative under this expression founded on a periodic function composed of a set of simple harmonic components [31,33].

Formal Expression of Phase Space in Statistical Mechanics
Following methodologies established by Hamilton, Lagrange, and Euler, [26] and L. Boltzmann (1844-1906) went on to lay the foundations of statistical mechanics from 1871. They introduced the term phase to describe the analogy they saw between the physical trajectories of particles in two dimensional space and Lissajous figures expressed as interactive maps. When two harmonic frequencies exist as rational fractions, period 4 circular patterns occur. However, when the frequency ratio is irrational, the system trajectories visit all points on the plane bounded by the signal amplitude.

Common Interpretations of Quantum Mechanics
Quantum mechanics is a modern legacy with its roots in classical statistical mechanics [11,12,17]. Meanwhile, Bose-Einstein, Fermi-Dirac statistics, and Planck's quantum are deeply connected with the statistical mechanics of Boltzmann and Gibbs [17,19]. In the context of the pursuit of an interpretation of quantum mechanics, the state vector or wave function has been widely discussed as a model for describing the individual components of a system (e.g. an electron).
The most comprehensive descriptions of an individual physical system are to be found in the various versions of the Copenhagen interpretation [8], or in subsequent versions incorporating minor modifications as in the hidden variable interpretations [19,31].
An interpretation according to the state vector based not on an individual system but on an ensemble of identically prepared systems is known as a statistical ensemble interpretation or more briefly just as a statistical interpretation [19,31].
The two alternative strategies of top-down and bottom-up strongly influence the direction of various explorations in the field of quantum mechanism. The Lagrange expression emphasizes single particles in a bottom-up strategy. In contrast, the Euler expression emphasizes complex objects treated as ensembles in a top-down strategy.
From as early as the turn of the 20th century when Plank started his quantum revolution, various interpretations of quantum behaviors have been explored. Following the Heisenberg matrix approach and the Schrödinger wave function equation, the intellectually absorbing anomalies of quantum mechanics have been linked to intrinsic behaviors associated with particle and wave duality.
To address the various paradoxes encountered in the development of quantum mechanics during the course of 20th century, a number of different interpretations may be listed [19,31]. In general, a Lagrange expression is preferred for representing a single quanta while a Euler expression can better describe certain group activities. It is interesting to note that de Broglie's Double-solution with a special interpretation can to be involved in both cases [9,31].

Probabilistic interpretation
According to Einstein's criteria for quantum mechanics [10], an interpretation of quantum mechanics can be characterized by its treatment of: Here, an interpretation is taken to mean a correspondence between the elements of the mathematical formalism M and the elements of an interpreting structure I, where: The Mathematical formalism M consists of the Hilbert space machinery of ket-vectors, self-adjoint operations on the space of ket-vectors, unitary time dependence of the ket-vectors and measurement operations: and ...
The interpreting structure I includes states, transitions between states, measurement operations and possible information about spatial extension of these elements.
Applying Einstein's criteria to this set of interpretations, the ensemble interpretation (statistical interpretation) is a minimalist interpretation. It claims to make the fewest assumptions associated with the standard mathematics. The most notable supporter of such statistical interpretation was Einstein himself [19,22,31].

Statistical Interpretation of Quantum Mechanics
At the 1927 Solvay Congress, Einstein proposed a statistical interpretation in order to avoid conceptual difficulties if the reduction of a wave packet led to the association of wave functions with individual systems. He hoped that someday a complete theory of microphysics would become available to establish a conceptual base as a (preferred) alternative to modern quantum mechanics [9,31].
In 1932, von Neumann established mathematical foundations for quantum mechanics as a standard interpretation on Hilbert space to provide a proof rejecting any hidden variable approach [32].
Influenced by K.V. Nikolskii and V.A. Fock, D. I. Blokhintsev developed a statistical interpretation in the 1940s. He expressed the view that modern quantum mechanics is not a theory of micro-processes but rather a means of studying their properties by the application of statistical ensembles. Menawhile, the approach taken in the publication was borrowed from classical macro physics [6,19,22].
Landé's 1951 book sought to reconcile the contradictions between the two classical concepts of the particle and the wave by providing something equivalent to the descriptions of physical phenomena in either terms. He emphasized that in diffraction experiments, particles exhibit both maximum and minimum intensities of diffraction through a perfectly normal mechanical process that can be described in terms of a wave explanation. Using transition probability, these experimentally-determinable transition probabilities can be shown to map a matrix [19].

Main weaknesses in key interpretations
Compared to continuous approaches, Heisenberg's matrix offers several advances in handling the case of a single particle. In July 1926, the first question Heisenberg asked Schrödinger was, "Can you use your continuous wave equation to explain black body radiation or quantum effects in photoelectric actions?" Due to the inherent differences between the two strategies it is difficult to find a direct answer to the question under the Copenhagen interpretation, "Is the Schrödinger equation a single particle description or an equation for a group of particles?" [16,19,31].
Through statistical interpretation is a minimalist interpretation, it too is not a complete interpretation. During the development of statistical interpretation there were various debates between Blokhintsev and Heisenberg during the 1940s [19,22].
Heisenberg questioned as self contradictory, Blokhintsev's basic contention that quantum mechanics eliminates the observer and becomes objectively significant due to the fact that the wave function does not describe the state of a particle but rather identifies that the particle belongs to a particular ensemble. In this, Heisenberg argued that in order to assign a particle to a particular ensemble, some knowledge of the particle is required on the part of the observer [19], p445.
The main weakness of Blokhintsev's ensemble interpretation is that though its mathematical formula can express wave distributions well it fails to describe particle structure properly. This is a common weakness of similar mathematical constructions based on periodic components of a Fourier series [19,27,31].
Similar to difficulties faced by the Schrödinger equation, ensemble construction is suitable for wave representations but is weak in particle description. On the other hand, the Copenhagen interpretation is preferred for a single particle but comes with inherent limitations of expression with respect to wave behaviours which require further reliance on Born's probabilistic interpretation of the wave-function.

Other Challenges on Statistical Mechanics
Statistical mechanics presents us with several fundemental difficulties [20,23,24,31,33]: Ergodic property: a time sequence average over a large set of local measurements be replaced by space (phase)-average Analytic apparatus the construction of asymptotic formulas.
Computational Efficiency: use of modern computing power in tackling complexity.
Discrete via continuous: relationships between irregular discrete systems and regular continuous systems.
Logic foundation: solid logic foundation for statistical mechanics.

Chapter organization
In this chapter, variant construction comprising variant logic, variant measurement and variant phase space is explored with a view to addressing the main challenges and difficulties associated with statistical interpretations and statistical mechanics. The focus is on a unified model to illustrate a path leading from local measurements to global matrices on phase space via variant construction.
This chapter is organized into 12 sections addressing the following: 1. general introduction (above)

System architecture
In this section, system architecture and its core components are discussed with the use of diagrams.

Architecture
The three components of a Variant Phase Space System are the Creating Micro Ensemble (CME), the Canonical Ensemble (CEIM) together with the Interactive Map and the Global Ensemble Matrix (GEM) as shown in Figure 1. The architecture is shown in Figure 1(a) with the key modules of the three core components being shown in Figures 1 In the first part of the system, a micro ensemble and its eight projections are created for a given vector and function by the CME component. Next, in order to exhaust all possible 2 N vectors, a CE and eight IMs are established by the CEIM component. Then, in order to exhaust all possible 2 2 n functions, a CE matrix and eight IM matrices are generated by the GEM component.
With eight parameters in an input group, there are four parameters in the intermediate group and two parameters in the output group.
The three groups of parameters may be listed as follows.

Input group:
N an integer indicates a 0-1 vector with N elements n an integer indicates n variables for a function

CME creating micro ensemble
The CME component as shown in Figure 1

CEIM canonical ensemble and interactive map
The CEIM component as shown in Figure 1

GIM global ensemble and interactive map matrix
The GIM component as shown in Figure 1 The CIM module further organizes the data to arrange each set as a 2 2 n−1 × 2 2 n−1 matrix with 2 2 n elements and with the specific arrangement determined by FC condition.
After two exhaustive processes through CEIM and GIM activities, a CE matrix and the relevant IM Matrices are generated. Each matrix contains 2 2 n elements as distributions. Further symmetry properties can be identified from each specific configuration.
Since specific components and modules are relevant to the detail of the complex mechanisms, further explanations on each component are presented in Sections 3 through 5.

Creating micro ensemble
The first part of the system is the CME component composed of four modules: VM Variant Measures, PM Probability Measurements, ME Micro Ensemble and IP Interactive Projection respectively.
It is necessary to clearly describe these four modules in order to understand the measurement properties of variant construction [35]- [44]. Relevant information and supporting materials on fundamental levels of variant construction are briefly descried in Section 3.1 and the four modules are investigated in Sections 3.2 through 3.5.

Initial preparation on variant measurements
The variant measurement construction is based on n-variable logic functions and N bit vectors taken as input and output results [40,43,44].

Two sets of states
let a position j be the selected variable 0 ≤ j < n, x j be the selected variable. Let output variable y and n-variable For all states of x, a set S(n) composed of the 2 n states can be divided into two sets: S 0 (n) and S 1 (n).

Four variant functions
For a given logic function f , input and output pair relationships define four variant logic

Two polarized functions
Considering two standard logic canonical expressions: AND-OR form is selected from Any logic function can be expressed as a variant logic form. In f + |x| f − structure, f + selected 1 items in S 0 (n) as same as the AND-OR standard expression, and f − selected 0 items in S 1 (n) as same as OR-AND expression.

n = 2 representation
For a convenient understanding of the variant representation, 2-variable logic structures are illustrated in Table 2 for its 16 functions in four variant functions as follows.
For a pair of { f + , f − } functions selected from the structure, relevant representations are illustrated in Table 3 to show the variant capacity on the full expression of all logic functions.
Checking two functions f = 3 and f = 6 respectively.

Variant measure functions
Let ∆ be the variant measure function [1,35]- [42] For any given n-variable state there is one position in ∆J(x) to be 1 and other 3 positions are 0.

Basic Properties of Variant Logic
For given n 0-1 variables, a given function J and an N bit vector X, the following corollaries can be described [35]- [44].

Corollary 3.1:
For n 0-1 variables, the state set contains a total of 2 n states.

Corollary 3.2:
For n 0-1 variables, the function set contains a total of 2 2 n functions.

Corollary 3.3:
For an N bit vector X, the phase space is composed of a total of 2 N vectors.

VM variant measures
Using defined variant functions, it is possible to describe the VM module in Fig. 1(b) as follows.
Under variant construction, N bits of 0-1 vector X under a function J produce seven Meta measures composed of a measure vector V M(J, X).
From a measuring viewpoint, there are seven measures identified in this set of parameters. They can be expressed in three levels.
In the current system, the output of the VM module is expressed as

PM probability measurements
Measures of V M(J, X) are input as numeric vectors into the PM module. Using variant quaternion and other three core measures, local measurements of probability signals are calculated as eight meta measurements in two groups by following the given equations. For The first group of probability signal vectors ρ and {ρ 0 , ρ 1 } are defined by The second group of probability signal vectorsρ and {ρ 0 ,ρ 1 } is defined by The two groups of probability measurements are key components in variant measurement. The first group corresponds to multiple probability measurements and the second group corresponds to conditional probability measurements. In this Chapter, only two quaternion measurements are used in order to focus attention on the simplest interactive combinations without further measurements of {ρ 0 , ρ 1 } and {ρ 0 ,ρ 1 } involved.
Under such condition, the output signals of the PM module can be expressed as a pair of probability vectors in quaternion forms PM(J, X) = {ρ,ρ}.

ME micro ensemble
The ME module has two inputs. PM(J, X) provides probability measurement vectors to provide the basis of the measurement. The input parameter SM indicates Selected Measurements from PM(J, X).
In this paper, two cases for a pair of measurement selections are restricted to permit an investigation of possible configurations of interactive distributions in their variant phase spaces under simple conditions.
Under these conditions, each (p i (J, X), p j (J, X)) or (p i (J, X),p j (J, X)) determines a fixed position on variant phase space as a Micro Ensemble. The output of the ME module can be expressed as ME(J, X) = (p i (J, X), p j (J, X))|(p i (J, X),p j (J, X)) under a given function J, an N bit vector X and SM conditions.

Variant Phase Space
Since each ME must be located on a certain position in a square area on variant phase space, it is convenient to show the restrictions and specific properties according to the following propositions.

Proposition 3.1:
In the Case A condition, a total of six configurations can be identified in different P selections. For each configuration, its pair of probability measurements can be restricted in a triangle area of a [0, 1] 2 region.
Proof: Any selection of two elements (p i (J, X), p j (J, X)) from P, it satisfies 0 ≤ p i (J, X) + p j (J, X) ≤ 1, there are six distinct selections.

IP interactive projections
Using a micro ensemble ME(J, X), different projections can be identified in an IP module under various interactive conditions. Based on the input micro ensemble for each Case, two groups of eight interactive projections can be distinguished by symmetry/anti-symmetry and synchronous/asynchronous conditions.

Synchronous and Asynchronous Operations
Each ME(J, X) is a pair of probability measurements and it is essential to establish corresponding rules to place their interactive projection in the same probability region i.e.
We can distinguish between Synchronous and Asynchronous time-related operations.
Under a synchronous operation {+, −, ×, /, }, only one merged measurement is located in [0, 1] region to express one activity from a ME.
However, under an asynchronous operation ⊕, two input measurements p + = p − , generate an output result as a vector that has two positions of p + and p − with a weighted value 1 on each position; when p = p + = p − there is a weighted value of 2 on the position p.
Under asynchronous operations, merged results may be distinguished by their position or overlap each other with a cumulative weight value of 2. However, under synchronous operations, two measurements are merged as a unit weight to shift interactive measurements to one position in the [0, 1] region.
From an integrative viewpoint, the two types of operations may be considered capable of either merging two particles (asynchronous) on two positions or integrating two waves (synchronous) on a position.

Case A: Multiple Probability Interactive Projections
For each ME(J, P) = (p i (J, X), p j (J, X)) has a position on a unit square [0, 1] 2 .
Let P = {p + , p − } (or {p x , p y }) locate a pair of measurements, the IP module projects two measurements and its weight into four conditions in different symmetric properties to form two groups of eight weight vectors as interactive projections.

Case B: Conditional Probability Interactive Projections
For each ME(J,P) = (p i (J, X),p j (J, X)) we can note that it has a position on a unit square For the four projections in a square area, the following equations are required.
For the two projections in a diagonal line, the following equations are satisfied.

Key Properties in IP Module
Under Symmetry/Anti-symmetry and Synchronous/Asynchronous conditions, one ME corresponds to eight interactive projections to express their selected characteristics. Under different conditions, one pair of probability measurements can be interactively projected into eight distinct results. However, from a variant viewpoint, it is not sufficient for a serious analysis to use only a single set of measurements from a ME, further extensions are required.

CE canonical ensemble
In the CE module, all the MEs are collected to form a CE in variant phase space according to the following equations.
Using equation CE(J), a canonical ensemble of variant phase space is produced. Each non-zero position has a numeric weight as a value to indicate numbers of MEs collected in a position. Proof: For each probability measurement, N + 1 values may be distinguished; points are located in a triangular area and a total of (N + 1)N/2 points may be distinguished.  Proof: This is the total number of vectors that may be distinguished for ∀X. Proof: For a given SM condition, a CE(J) distribution is independent of special sequences of collection. Its detailed configuration is relevant to {n, N} and SM respectively. All valid positions can be statistically generated.

Key Properties in CE
Under this organization, each CE(J) has a fixed plane lattice with a distinct distribution. This invariant property is useful for our further explorations.

IM interactive map
In an IM module, all possible IP projections of either {u, v} or {ũ,ṽ} are collected. Each projection corresponds to a specific IM distribution.
The IM module provides a statistical means to accumulate all possible vectors of N bits for a selected signal and generate a histogram. Eight signals correspond to eight histograms respectively. Among these, four histograms exhibit properties of symmetry and another four histograms exhibit properties of anti-symmetry.

Statistical distributions
For a function J, all measurement signals are collected from the IP and the relevant histogram represents a complete statistical distribution as an IP map.
Using u and v signals, each u β or v β determines a fixed position in the relevant histogram to make vector X on a position. After completing 2 N data sequences, eight Under the multiple probability condition, β ∈ {+, −, 0, 1} Under the conditional probability condition, β ∈ {+, −, 0, 1}

Normalized probability histograms
Let |H(..)| denote the total number in the histogram H(..), a normalized probability histogram (P H (..)) can be expressed as Here, all interactive maps are also restricted in [0, 1] 2 areas respectively.
Distributions are dependant on the data set as a whole and are not sensitive to varying under special sequences. Under this condition, when the data set has been exhaustively listed, then the same distributions are always linked to the given signal set.

GEIM global ensemble and interactive map matrices
The GEIM component is composed of two modules: SCEIM Sets of CE&IM, and CIM CE&IM Matrices.
{CE(J), I M(J)} and ∀J are put in the SCEIM module to generate sets of CEs and IMs on each given function exhaustively. All generated CEs and IMs are organized by the CIM module under the FC condition in which a special variant coding scheme is applied for a global configuration of output matrices.

SCEIM sets of canonical ensembles and interactive maps
The SCEIM module produces {SCE, SI M} composed of all possible CE and IM sets of ∀J under exhaustive conditions.
Meanwhile, the SCE and the SIM provide output results to the CIM module.

CIM canonical ensemble and interactive map matrices
In addition to using {SCE, SI M} as inputs, the FC also inputs a code scheme to determine a detailed configuration for each matrix.

Global Matrix Representations
In the CIM module, {SCE, SI M} inputs have nine sets of CEs and IMs. Each set is composed of 2 2 n elements and each element is a histogram or a plane lattice. The CIM module arranges all 2 2 n elements generated as a matrix by a given FC code scheme.

The Matrix and Its elements
For a given FC scheme, let FC(J) = J 1 |J 0 , each element

Representative patterns of matrices
Four cases of FC codes are selected for illustrations in this Chapter. Further discussion on the details of variant coding scheme has been previously published in [40,44].
For example, four sample cases are shown in Figure 5 under relevant conditions.  Under each condition, each FC code is a special configuration to make sixteen elements arranged as a 4 × 4 matrix.
For the matrices in this chapter, four configurations are applied to construct sample matrices with elements arranged for illustration purposes.

Symbolic representations on selected cases
Using a representational model, for a given condition, there are two sets of CEM in both MPS and CPS. Each set contains a CEM and eight IMMs. Since each matrix contains 2 2 n elements, the existence of so many possible configurations adds to the difficulties in reaching a satisfactory understanding of the data sets. In this section, symbolic representations are applied to show more clearly the essential symmetric properties of various matrices. Using variant logic, the following equations can be established for an n = 2 condition to apply (a, b, c, d) = (10, 8, 2, 0) and (ã,b,c,d) = (10,14,11,15) for each meta function.
Using this symbolic style, four cases of configurations and their polarized decompositions are represented as follows.

CEM groups
Using n = 2 configurations, relevant CEMs on either MPS or CPS are shown in Figure 8(a-h).

VPS organization
Two groups of matrices are shown in Figure 8. The four matrices shown as 8 (a-d) illustrate MPS conditions and the four matrices shown as 8 (e-h) illustrate CPS conditions. Considering various CEs exhibiting conjugate symmetry properties, such arrangements may be noted to have similar distributions along the diagonal and anti-diagonal directions so that it is possible to find a pair of CEs with each CE pair-matched by a geometric transformation to another CE through either rotation or reflection.

MPS Structures
The four CE matrices in the MPS group as shown in Figure 8(a-d) can be analyzed as follows.

SL in MPS
For the matrix in Figure 8

W in MPS
For the matrix in Figure 8

C in MPS
Also, for the matrix in Figure 8

CPS Structures
The four CE matrices shown in Figure 8(e-h) in the CPS group can be analyzed as follows.  In all, five distinct regions can be identified as significant. It is interesting to note such remarkable symmetry illustrating interactions between and among these meta functions.

SL group in MPS
For the SL group in Figure 9 (M1-M8), the two matrix vectors {u, v} =

M1-M4:
Let us first consider elements M1-M4 where the four matrices of (u + , u − , u 0 , u 1 ) are in a symmetry group, In u + matrix M1, elements in the columns and rows are arranged in what may be described as a periodic crossing structure.
In u − matrix M2, four elements with the same IMs are arranged in a 2 × 2 block with four distinct distributions being observed.
In u 0 matrix M3, each element shows simple additions from elements in u + and u − respectively. It it interesting to note that only two pairs of positions {0:15, 6:9} are similarly distributed in the relevant MPS matrix.

M4-M8:
Let us now consider elements M5-M8 where the four matrices of (v + , v − , v 0 , v 1 ) are in an anti-symmetry group, In v + matrix M5, elements in the columns and rows are arranged as periodic crossing structures.
In v − matrix M6, four elements with the same IMs are arranged in a 2 × 2 block with four distinct distributions observed.
In v 0 matrix M7, each element shows simple additions from elements in v + and v − respectively. Only two pairs of positions {0:15, 6:9} are similarly distributed in the relevant MPS matrix.
In v 1 matrix M8, significant symmetry properties can be observed. Two pairs {0:15, 6:9} have anti-symmetry properties that are the same as the v 0 condition.

W group in MPS
For the W group in Figure 9 (M9-M16), the two matrix vectors {u, v} =

M9-M12:
Let us now consider elements M9-M12 where the four matrices (u + , u − , u 0 , u 1 ) are in a symmetry group, In u + matrix M9, four elements with the same IMs are arranged in a 2 × 2 block with four distinct distributions observed.
In u − matrix M10, elements in the columns and rows are arranged as a periodic crossing structure.
In u 0 matrix M11, each element shows simple additions with elements in u + and u − respectively. It it interesting to note that only two pairs of positions {0:15, 6:9} are similarly distributed in the relevant MPS matrix.

M13-M16:
Let us now consider elements M13-M16 where the four matrices (v + , v , v 0 , v 1 ) are in an anti-symmetry group, In v + matrix M13, four elements with the same IMs are arranged in a 2 × 2 block with four distinct distributions observed.
In v − matrix M14, elements in the columns and rows are arranged as a periodic crossing structure.
In v 0 matrix M15, each element shows simple additions with elements in v + and v − respectively. Only two pairs of positions {0:15, 6:9} are similarly distributed in the relevant MPS matrix.
In v 1 matrix M16, significant symmetry properties can be observed. Two pairs {0:15, 6:9} have anti-symmetry properties the same as under the v 0 condition.

F group in MPS
For the F group in Figure 9 (M17-M24), the two matrix vectors {u, v} =

M17-M20:
Let us now consider elements M17-M20 where the four matrices (u + , u − , u 0 , u 1 ) are in a symmetry group, In u + matrix M17, the horizontal elements are arranged in H-2R patterns and vertical elements are in a periodic crossing structure.
In u − matrix M18, vertical elements are arranged in V-2R patterns and the horizontal elements as a periodic crossing structure.
In In v + matrix M21, the horizontal elements are arranged in H-2R patterns and the vertical elements as a periodic crossing structure.
In v − matrix M22, the vertical elements are arranged in V-2R patterns and the horizontal elements as a periodic crossing structure. In

M25-M28:
Let us now consider elements M25-M28 where the four matrices of (u + , u − , u 0 , u 1 ) are in a symmetry group, In u + matrix M25, the horizontal elements are in a periodic crossing structure and the vertical elements are arranged in V-4R patterns In u − matrix M26, the horizontal elements are arranged in H-4R patterns and the vertical elements as a periodic crossing structure.
In u 0 matrix M27, each element shows simple additions with elements in u + and u − respectively. It it interesting to note that six pairs In v + matrix M29, the horizontal elements are arranged in H-4R patterns and the vertical elements as a periodic crossing structure.
In v − matrix M30, the horizontal elements are arranged in H-4R patterns and the vertical elements as a periodic crossing structure. In

CPS Structures
Four groups of different configurations shown in Figure 10 (C1-C32) are discussed separately as follows.

C1-C4:
Let us now consider elements C1-C4 where the four matrices of (ũ + ,ũ − ,ũ 0 ,ũ 1 ) are in a symmetry group, Inũ + matrix C1, elements in the columns and rows are in a periodic crossing structure.
Inũ − matrix C2, four elements with the same IMs are arranged in a 2 × 2 block with four distinct distributions observed.

C5-C8:
Let us now consider elements C5-C8 where the four matrices of (ṽ + ,ṽ ,ṽ0 ,ṽ 1 ) are in an anti-symmetry group, Inṽ + matrix C5, elements in the columns and rows are arranged in a periodic crossing structure.
Inṽ − matrix C6, four elements with same IMs are arranged in a 2 × 2 block and four distinct distributions are observed.

C9-C12:
Let us now consider elements C9-C12 where the four matrices of (ũ + ,ũ − ,ũ 0 ,ũ 1 ) are in a symmetry group, Inũ + matrix C9, four elements with the same IMs are arranged in a 2 × 2 block and four distinct distributions are observed.
Inũ − matrix C10, elements in the columns and rows are arranged as a periodic crossing structure.

C13-C16:
Let us now consider elements C13-C16 where the four matrices of (ṽ + ,ṽ ,ṽ0 ,ṽ 1 ) are in an anti-symmetry group, Inṽ + matrix C13, four elements with the same IMs are arranged in a 2 × 2 block and four distinct distributions are observed.
Inṽ − matrix C14, elements in the columns and rows are arranged as a periodic crossing structure.

C17-C20:
Let us now consider elements C17-C20 where the four matrices of (ũ + ,ũ − ,ũ 0 ,ũ 1 ) are in a symmetry group, Inũ + matrix C17, horizontal elements are arranged in H-2R patterns and vertical elements are in a periodic crossing structure.
Inũ − matrix C18, vertical elements are arranged in V-2R patterns and horizontal elements as a periodic crossing structure.

C25-C28:
Let us now consider elements C25-C28 where the four matrices of (ũ + ,ũ − ,ũ 0 ,ũ 1 ) are in a symmetry group, Inũ + matrix C25, horizontal elements are arranged as a periodic crossing structure. and vertical elements are arranged in V-4R patterns Inũ − matrix C26, horizontal elements are arranged in H-4R patterns and vertical elements as a periodic crossing structure.

Global symmetric properties
Working from four sets of CEM and IMM results, key global symmetry properties are presented and summarized in Table 4 for CEMs and in Table 5 for IMMs as follows.
Where CP is a conjugate pair, GP is global polarization and a:(0:15,6:9), d: It is interesting to note that significant differences in symmetry properties between MPS and CPS can be observed for CEM conjugate pairs.
In general, we find double the number of incidences of symmetry properties with CPS compared with MPS shown in Table 4.
Where SP is a Symmetric Pair, ASP is an Anti-symmetric Pair, GS is Global It is interesting to note that symmetry properties evident in IMM groups in Table 5

Corresponding structures on variant phase space
Top-down and bottom-up strategies can both be applied to Variant Phase Space. See Table 7.
Top-down and bottom-up strategies can each open a window through which to glimpse the mysteries of Variant Phase Space. Such glimpses do not yet provide a complete picture and further investigation is clearly required.

Main results
It is appropriate to present the results as a series of detailed propositions and predictions as follows.
For an n variable function J ∈ B 2 n 2 and an N bit vector X ∈ B N 2 , following propositions can be established.

Propositions
Proposition 11.1: Two types of probability measurements, Multiple and Conditional probabilities determine two distinct phase spaces, MPS and CPS.
Proof: In a PM module, multiple probabilities generates MPS and conditional probabilities create CPS.

Proposition 11.2:
Two types of operations: symmetry/anti-symmetry and synchronous/asynchronous generate eight interactive projections.
Proof: Results may be generated using a CEIM module and Proposition 11.3 is further supported by Propositions 11.1 to 11.2. Proposition 11.4: Each CE is a statistical distribution and each IM corresponds to one of eight IP modes.
Proof: A pair of probability measurements has one fixed CE combination and each IP mode corresponds to one IM distribution. Proposition 11.5: Both Proposition 11.3 and Proposition 11.4 provide a general Maxwell Demon mechanism.
Proof: For any function, CE and IMs can be fully and exhaustively generated without reference to thermodynamic issues. Proposition 11.6: Exhausting ∀J ∈ B 2 n 2 , two sets of {CE} and 16 sets of {IM} can be generated, each set contains 2 2 n elements and each element is a distribution.
Proof: Using the SCEIM module, they are natural outputs. Proposition 11.7: In a variant logic framework, there are 2 n ! × 2 2 n configurations for arranging a set of {CE} and eight sets of {IM} into a CEM and eight IMMs.

Proof:
Since each IMM has the same organization as the CEM, a total of 2 n ! × 2 2 n configurations can be distinguished and each configuration corresponds to a variant logic matrix. Proposition 11.8: With a top-down approach, either a CEM or an IMM on a proper configuration can be composed of two polarized matrices. Each polarized matrix has periodic structures on its columns and/or rows.
Proof: Since a proper configuration is based on n periodic meta vectors and their combinations, its relative arrangements are invariant under permutation and complementary operations on the vector with 2 2 n bits that determine each polarized structure.

Predictions
Prediction 11.1: Following a bottom-up strategy, it is not possible to determine CE properties using limited numbers of ME.
This prediction points towards a more general intrinsic restriction on uncertainty effects for incomplete procedures applied to random events. Prediction 11.2: For a configuration that is not in a variant logic framework, there may be a square integral configuration capable of providing an approximate solution.
Periodic matrices could play a key role as core components of approximation procedures. Prediction 11.3: A sound statistical interpretation of quantum mechanics can be established using VPS construction.
Since both top-down and bottom-up strategies are included, further exploration is feasible. Prediction 11.4: VPS construction can provide a foundation based on logic and hierarchies of measurement levels for complex dynamic systems, statistical mechanics, and cellular automata.
Through VPS construction clearly offers significant potential, this prediction needs to be tested by solid experimental and theoretical results backed by evidence.

Conclusion
This chapter provides a brief investigation into Variant Phase Space (VPS) construction. Using an n variable 0-1 function and an N bit vector, a VPS hierarchy can be progressively established via variant measures, multiple or conditional probability measurements, and selected pair of measurements to determine a Micro Ensemble (ME) and its eight interactive projections. Collecting all possible 2 N pairs of probability measurements, a Canonical Ensemble (CE) and its eight Interactive Maps (IMs) are generated following a bottom-up approach.
Applying a Maxwell demon mechanism, all possible 2 2 n functions can be calculated to create a result comprising a {CE} and eight sets of {IM}. Using either a CE or an IM as an element, it is possible to use a variant logic configuration to organize each set of distributions to be a 2 2 n−1 × 2 2 n−1 matrix as a CE Matrix (CEM) or IM Matrix (IMM), respectively. Following a top-down approach, a CEM or IMM can be decomposed into two polarized matrices with each matrix having periodic properties that meet the requirements of a Fourier-like transformation.
The main results are presented as ten propositions and four predictions to provide a foundation for further exploration of quantum interpretations, statistical mechanics, complex dynamic systems, and cellular automata.
The chapter does not explore global properties in detail, and further detailed investigations and expansions are necessary.
Anticipating that the principles put forward in this chapter will prove to be well founded, we look forward to exploring advanced scientific and technological applications in the near future.