**6. Analysis of results**

In the previous section, results of different statistical distributions and their global matrix representations were presented. In this section, plain language is used to explain what various visual effects might be illustrated and to discuss local and global arrangements.

#### **6.1 Statistical distributions for a given function**

It is necessary to analyze the differences among the various statistical distributions for a given function.

#### **6.1.1 Symmetry groups for a function**

For the selected function *J* = 3, four distributions in symmetry groups are shown in Fig 3 (a-d). (a) *PH*(*u*˜+|*J*) for Left; (b) *PH*(*u*˜−|*J*) for Right; (c) *PH*(*u*˜0|*J*) for D-P; and (d) *PH*(*u*˜1|*J*) for D-W respectively.

Under Symmetry conditions, *PH*(*u*˜+|*J*) = *PH*(*u*˜−|*J*), both Left and Right distributions are the same. *PH*(*u*˜0|*J*) generated with both paths open under asynchronous conditions simulates a D-P. Compared with distributions in (a-b) , it is possible to identify the components from original inputs.

However, for *PH*(*u*˜1|*J*) under synchronous conditions and with the same Left and Right input signals, the simulation shows a D-W exhibiting interferences among the output distributions that are significantly different from the original components.

Matrices for D-P in Fig 4-5 (g) show additional effects for each distribution according to the relevant position with components that can be identified as corresponding to identifiable

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Matrices for D-W in Fig 4-5 (h) show different properties. In general, only one peak can be observed for each element especially for the *J* ∈ {10, 12, 3, 5} condition. Spectra appear to be much simpler than the original distributions in Fig 4-5 (e-f), and significant interference

Pairs of relationships can be checked on symmetry matrices in Figs 4-5 (a-d), four groups are

{*PH*(*u*˜+|*J*)} elements in Figs 4-5 (a) show (only) four distinct distributions. Each column contains only one distribution. Sixteen elements in the matrix can be classified into four vertical classes: {0, 2, 8, 10}, {4, 6, 12, 14}, {1, 3, 9, 11}, {5, 7, 13, 15} respectively. Four meta

{*PH*(*u*˜−|*J*)} elements in Figs 4-5 (b) show (a further) four distinct distributions. Each row contains only one distribution. Sixteen elements in the matrix can be classified into four horizontal classes: {0, 4, 1, 5}, {2, 6, 3, 7}, {8, 12, 9, 13}, {10, 14, 11, 15} respectively. Four meta

{*PH*(*u*˜0|*J*)} elements in Figs 4-5 (c) illustrate symmetry properties. There are six pairs of symmetry elements: {8 : 14}, {2 : 11}, {0 : 15}, {6:9}, {4 : 13}, {1:7}. In addition, four elements on anti-diagonals provide different distributions: {10, 12, 3, 5}. Under this condition,

{*PH*(*u*˜1|*J*)} elements in Figs 4-5 (d) illustrate symmetry properties. There are six pairs of symmetry elements: {8 : 14}, {2 : 11}, {0 : 15}, {6:9}, {4 : 13}, {1:7}. In addition, four elements on diagonal positions provide the same distribution: {0, 6, 9, 15}. Two elements on anti-diagonals: {12, 3} have the same distribution in Fig 4 (d). Under this condition, nine or ten classes of different distributions can be identified for Fig 4 (d) and Fig 5 (d) respectively.

Figures 4-5 (e-h) represent anti-symmetry properties, four groups can be identified.

inputs in many cases. Anti-symmetry signals are generated in merging conditions.

Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

properties are observed.

identified.

**6.3 Four symmetry groups**

**6.3.1 Left: polarized vertical group**

distributions are given as {10, 14, 11, 15}.

**6.3.2 Right: polarized horizontal group**

distributions are given as {0, 2, 8, 10}.

ten classes of distributions are distinguished.

**6.3.3 D-P: particle group**

**6.3.4 D-W: wave group**

**6.4 Four anti-symmetry groups**

#### **6.1.2 Anti-symmetry groups for a function**

Four distributions are shown in Fig 3 (e-h) as asymmetry groups. A pair of equations *PH*(*v*˜+|*J*) = *PH*(1 − *v*˜−|*J*) shows that one distribution is a mirror image of the other. *PH*(*v*˜+|*J*) distribution is shown in Fig 3 (e) for Left signals and *PH*(*v*˜−|*J*) distribution is shown Fig 3 (f) for Right signals.

*PH*(*v*˜0|*J*) is shown in Fig 3 (g) for both paths open under asynchronous conditions to simulate a D-P. Compared with (e-f) distributions, it is feasible to identify the same components from the original inputs.

However *PH*(*v*˜1|*J*) is shown in Fig 3 (h) under synchronous condition with both path signals as inputs to simulate a D-W exhibiting interferences among the output distributions that are significantly different from the original components.

To differentiate between even and odd numbers, *N* = 12 cases are shown in Fig 3 (II, a-h) and *N* = 13 cases are shown in Fig 3 (III, a-h) respectively.

#### **6.2 Global matrix representations**

Sixteen matrices are represented in Fig 4-5 (a-h) with eight signals generating two sets of 16 groups for *N* = {12, 13} respectively.

#### **6.2.1 Symmetry cases**

Matrices for the Left in Fig 4-5 (a) show elements in a column with the corresponding histogram showing polarized effects on the vertical.

Matrices for the Right in Fig 4-5 (b) show elements in a row with the corresponding histogram showing polarized effects on the horizontal.

Matrices for D-P in Fig 4-5 (c) provide asynchronous operations combined with both distributions from Fig 4-5 (a-b) to form a unified distribution. From each corresponding position, it is possible to identify each left and right component and the resulting shapes of the histogram.

Matrices for D-W in Fig 4-5 (d) provide synchronous operations combined with both distributions from Fig 4 -5 (a-b) for each element.

Compared with Fig 4-5 (c) and Fig 4-5 (d) respectively, distributions in Fig 4-5 (d) are much simpler with two original distributions especially on the anti-diagonal positions: *J* ∈ {10, 12, 3, 5}. Only less than half the number of spectrum lines are identified.

#### **6.2.2 Anti-symmetry cases**

In a similar manner to the symmetry conditions, four anti-symmetry effects can be identified in Fig 4-5 (e-h). Matrices in Fig 4-5 (e) are Left operations for different functions; elements are polarized on the vertical and matrices in Fig 4-5 (f) are Right Operations; elements are polarized on the horizontal. Spectrum lines in Fig 4-5 (e) appear in the right half and spectrum lines in Fig 4-5 (f) appear in the Left half respectively.

Matrices for D-P in Fig 4-5 (g) show additional effects for each distribution according to the relevant position with components that can be identified as corresponding to identifiable inputs in many cases. Anti-symmetry signals are generated in merging conditions.

Matrices for D-W in Fig 4-5 (h) show different properties. In general, only one peak can be observed for each element especially for the *J* ∈ {10, 12, 3, 5} condition. Spectra appear to be much simpler than the original distributions in Fig 4-5 (e-f), and significant interference properties are observed.

#### **6.3 Four symmetry groups**

16 Measurement Systems

Four distributions are shown in Fig 3 (e-h) as asymmetry groups. A pair of equations *PH*(*v*˜+|*J*) = *PH*(1 − *v*˜−|*J*) shows that one distribution is a mirror image of the other. *PH*(*v*˜+|*J*) distribution is shown in Fig 3 (e) for Left signals and *PH*(*v*˜−|*J*) distribution is shown Fig 3 (f)

*PH*(*v*˜0|*J*) is shown in Fig 3 (g) for both paths open under asynchronous conditions to simulate a D-P. Compared with (e-f) distributions, it is feasible to identify the same components from

However *PH*(*v*˜1|*J*) is shown in Fig 3 (h) under synchronous condition with both path signals as inputs to simulate a D-W exhibiting interferences among the output distributions that are

To differentiate between even and odd numbers, *N* = 12 cases are shown in Fig 3 (II, a-h) and

Sixteen matrices are represented in Fig 4-5 (a-h) with eight signals generating two sets of 16

Matrices for the Left in Fig 4-5 (a) show elements in a column with the corresponding

Matrices for the Right in Fig 4-5 (b) show elements in a row with the corresponding histogram

Matrices for D-P in Fig 4-5 (c) provide asynchronous operations combined with both distributions from Fig 4-5 (a-b) to form a unified distribution. From each corresponding position, it is possible to identify each left and right component and the resulting shapes of

Matrices for D-W in Fig 4-5 (d) provide synchronous operations combined with both

Compared with Fig 4-5 (c) and Fig 4-5 (d) respectively, distributions in Fig 4-5 (d) are much simpler with two original distributions especially on the anti-diagonal positions: *J* ∈

In a similar manner to the symmetry conditions, four anti-symmetry effects can be identified in Fig 4-5 (e-h). Matrices in Fig 4-5 (e) are Left operations for different functions; elements are polarized on the vertical and matrices in Fig 4-5 (f) are Right Operations; elements are polarized on the horizontal. Spectrum lines in Fig 4-5 (e) appear in the right half and spectrum

{10, 12, 3, 5}. Only less than half the number of spectrum lines are identified.

**6.1.2 Anti-symmetry groups for a function**

significantly different from the original components.

*N* = 13 cases are shown in Fig 3 (III, a-h) respectively.

histogram showing polarized effects on the vertical.

showing polarized effects on the horizontal.

distributions from Fig 4 -5 (a-b) for each element.

lines in Fig 4-5 (f) appear in the Left half respectively.

**6.2 Global matrix representations**

**6.2.1 Symmetry cases**

the histogram.

**6.2.2 Anti-symmetry cases**

groups for *N* = {12, 13} respectively.

for Right signals.

the original inputs.

Pairs of relationships can be checked on symmetry matrices in Figs 4-5 (a-d), four groups are identified.

### **6.3.1 Left: polarized vertical group**

{*PH*(*u*˜+|*J*)} elements in Figs 4-5 (a) show (only) four distinct distributions. Each column contains only one distribution. Sixteen elements in the matrix can be classified into four vertical classes: {0, 2, 8, 10}, {4, 6, 12, 14}, {1, 3, 9, 11}, {5, 7, 13, 15} respectively. Four meta distributions are given as {10, 14, 11, 15}.

#### **6.3.2 Right: polarized horizontal group**

{*PH*(*u*˜−|*J*)} elements in Figs 4-5 (b) show (a further) four distinct distributions. Each row contains only one distribution. Sixteen elements in the matrix can be classified into four horizontal classes: {0, 4, 1, 5}, {2, 6, 3, 7}, {8, 12, 9, 13}, {10, 14, 11, 15} respectively. Four meta distributions are given as {0, 2, 8, 10}.

#### **6.3.3 D-P: particle group**

{*PH*(*u*˜0|*J*)} elements in Figs 4-5 (c) illustrate symmetry properties. There are six pairs of symmetry elements: {8 : 14}, {2 : 11}, {0 : 15}, {6:9}, {4 : 13}, {1:7}. In addition, four elements on anti-diagonals provide different distributions: {10, 12, 3, 5}. Under this condition, ten classes of distributions are distinguished.

#### **6.3.4 D-W: wave group**

{*PH*(*u*˜1|*J*)} elements in Figs 4-5 (d) illustrate symmetry properties. There are six pairs of symmetry elements: {8 : 14}, {2 : 11}, {0 : 15}, {6:9}, {4 : 13}, {1:7}. In addition, four elements on diagonal positions provide the same distribution: {0, 6, 9, 15}. Two elements on anti-diagonals: {12, 3} have the same distribution in Fig 4 (d). Under this condition, nine or ten classes of different distributions can be identified for Fig 4 (d) and Fig 5 (d) respectively.

#### **6.4 Four anti-symmetry groups**

Figures 4-5 (e-h) represent anti-symmetry properties, four groups can be identified.

Class No. Left Right D-P D-W SE 4 4 10 10 SO 4 4 10 10 AE 4 4 16 14 AO 4 4 16 14

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Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

where Left:Left Path, Right: Right Path, D-P: Double Path for Particles, D-W: Double Path for Waves; SE: Symmetry for Even number, SO: Symmetry for Odd number, AE: Anti-symmetry

In order to contrast the different polarized conditions, it is convenient to compare distributions {*PH*(*u*˜+|*J*), *PH*(*u*˜−|*J*} and {*PH*(*v*˜+|*J*), *PH*(*v*˜−|*J*} arranged according to the corresponding polarized vertical and horizontal effects. This visual effect is similar to what might be found when using polarized filters in order to separate complex signals into two channels. Different

For all symmetry or non-symmetry cases under ⊕ asynchronous addition operations, relevant values meet 0 ≤ *u*˜0, *v*˜0, *u*˜−, *v*˜−, *u*˜+, *v*˜<sup>+</sup> ≤ 1. Checking {*PH*(*u*˜0|*J*), *PH*(*v*˜0|*J*)} series,

*PH*(*u*˜0|*J*) = *PH* (*u*˜−|*J*)+*PH* (*u*˜+|*J*)

*PH*(*v*˜0|*J*) = *PH* (*v*˜−|*J*)+*PH* (*v*˜+|*J*)

Interference properties are observed in {*PH*(*u*˜+|*J*) = *PH*(*u*˜−|*J*)} conditions. Under + synchronous addition operations, relevant values meet 0 ≤ *u*˜1, *v*˜1, *u*˜−, *v*˜−, *u*˜+, *v*˜<sup>+</sup> ≤ 1. Checking {*PH*(*u*˜1|*J*), *PH*(*v*˜1|*J*)} distributions and compared with {*PH*(*u*˜+|*J*), *PH*(*u*˜−|*J*)} and

*PH*(*u*˜1|*J*) �<sup>=</sup> *PH*(*u*˜0|*J*)

Spectra in different cases illustrate wave interference properties. Single and double peaks are shown in interference patterns and these are similar to interference effects in classical double

However, for the {*PH*(*u*˜+|*J*) �= *PH*(*u*˜−|*J*)} non-symmetry cases, there are significant differences between {*PH*(*u*˜0|*J*), *PH*(*v*˜0|*J*)} and {*PH*(*u*˜1|*J*), *PH*(*v*˜1|*J*)}. Such cases have interference patterns that exhibit greater symmetry than single path and particle distributions.

2

(16)

2

*PH*(*v*˜1|*J*) �<sup>=</sup> *PH*(*v*˜0|*J*) (17)

distributions can be observed under synchronous and asynchronous conditions.

{*PH*(*u*˜+|*J*), *PH*(*u*˜−|*J*)} and {*PH*(*v*˜+|*J*), *PH*(*v*˜−|*J*)} satisfy following equation.

{*PH*(*v*˜+|*J*), *PH*(*v*˜−|*J*)}, non-equations and equations are formulated as follows:

for Even number, AO: Anti-symmetry for Odd number.

**6.7 Polarized effects and double path results**

**6.7.1 Particle distributions and representations**

The equation is true for different values of *N* and *n*.

**6.7.2 Wave distributions and representations**

**6.7.3 Non-symmetry and non-anti-symmetry**

slit experiments.

#### **6.4.1 Left: polarized vertical group**

{*PH*(*v*˜+|*J*)} elements in Figs 4-5 (e) show that (only) four classes can be distinguished. Elements within these groups members are the same as for symmetry groups in Figs 4-5(a). Their distributions fall within the region [0.5, 1].

#### **6.4.2 Right: polarized horizontal group**

{*PH*(*v*˜−|*J*)} elements in Figs 4-5 (f) show that (only) four classes can be distinguished. Elements within these groups are the same as for symmetry groups in Figs 4-5 (b). Their distributions fall within the region [0, 0.5].
