**6.4.3 D-P: particle group**

{*PH*(*v*˜0|*J*)} in Figs 4-5 (g) show six pairs of anti-symmetry distributions: {8 14}, {2 11}, {0 15}, {6 9}, {4 13}, {1 7} four elements are distinguished on the anti-diagonals: {10, 12, 3, 5}. Under this condition, ten classes can be identified.

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

{*PH*(*v*˜1|*J*)} in Figs 4-5 (h) show six pairs of anti-symmetry distributions: {8 14}, {2 11}, {0 15}, {6 9}, {4 13}, {1 7} four pairs of symmetry elements: {3:5}, {10 : 12}, {2:4}, {11 : 13} are distinguished. Under this condition, twelve classes can be identified.

#### **6.5 Odd and even numbers**

From a group viewpoint, only D-P and D-W need to be reviewed as different groups in symmetry conditions. Anti-symmetry conditions are unremarkable.

It is reasonable to suggest that anti-symmetry operations will be much easier to distinguish under experimental conditions, since sixteen groups in D-P conditions and twelve groups in D-W conditions will have significant differences. However, under the symmetry conditions (only) minor differences can be identified.

#### **6.5.1 Single and double peaks**

Single and Double peaks can be observed in Fig 4(5) (h): {3, 5} for even and odd numbers respectively.

For two other members {10, 12}, (only) single pulse distributions are observed in Figs 4-5 (h) to show the strongest interference results.

#### **6.6 Class numbers in different conditions**

To summarize over the different classes, 16 matrices are shown in different numbers of identified classes as follows:


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

#### **6.7 Polarized effects and double path results**

18 Measurement Systems

{*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).

{*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

{*PH*(*v*˜0|*J*)} in Figs 4-5 (g) show six pairs of anti-symmetry distributions: {8 14}, {2 11}, {0 15}, {6 9}, {4 13}, {1 7} four elements are distinguished on the anti-diagonals:

{*PH*(*v*˜1|*J*)} in Figs 4-5 (h) show six pairs of anti-symmetry distributions: {8 14}, {2 11}, {0 15}, {6 9}, {4 13}, {1 7} four pairs of symmetry elements: {3:5}, {10 : 12}, {2:4}, {11 :

From a group viewpoint, only D-P and D-W need to be reviewed as different groups in

It is reasonable to suggest that anti-symmetry operations will be much easier to distinguish under experimental conditions, since sixteen groups in D-P conditions and twelve groups in D-W conditions will have significant differences. However, under the symmetry conditions

Single and Double peaks can be observed in Fig 4(5) (h): {3, 5} for even and odd numbers

For two other members {10, 12}, (only) single pulse distributions are observed in Figs 4-5 (h)

To summarize over the different classes, 16 matrices are shown in different numbers of

13} are distinguished. Under this condition, twelve classes can be identified.

symmetry conditions. Anti-symmetry conditions are unremarkable.

**6.4.1 Left: polarized vertical group**

**6.4.2 Right: polarized horizontal group**

distributions fall within the region [0, 0.5].

{10, 12, 3, 5}. Under this condition, ten classes can be identified.

**6.4.3 D-P: particle group**

**6.4.4 D-W: wave group**

**6.5 Odd and even numbers**

**6.5.1 Single and double peaks**

identified classes as follows:

respectively.

(only) minor differences can be identified.

to show the strongest interference results.

**6.6 Class numbers in different conditions**

Their distributions fall within the region [0.5, 1].

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 distributions can be observed under synchronous and asynchronous conditions.

#### **6.7.1 Particle distributions and representations**

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

$$\begin{cases} P\_H(\vec{u}\_0|J) = \frac{P\_H(\vec{u}\_-|J) + P\_H(\vec{u}\_+|J)}{2} \\ P\_H(\vec{v}\_0|J) = \frac{P\_H(\vartheta\_-|J) + P\_H(\vartheta\_+|J)}{2} \end{cases} \tag{16}$$

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

#### **6.7.2 Wave distributions and representations**

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

$$\begin{cases} P\_H(\vec{u}\_1|J) \neq P\_H(\vec{u}\_0|J) \\ P\_H(\vec{v}\_1|J) \neq P\_H(\vec{v}\_0|J) \end{cases} \tag{17}$$

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 slit experiments.

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

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.

**7.3 The BCP for a special subset of QM**

**7.4 The EPR contribution on variant construction**

invariant properties.

in the Copenhagen interpretation.

⎧ ⎪⎪⎨

⎪⎪⎩

by Einstein as far back as 1935.

⎧ ⎪⎪⎨

⎪⎪⎩

**7.5 Afshar's experiments on variant construction**

�*S*1, *S*2, *IM*(*S*1, *S*2), *SM*(*S*1, *S*2)� →

are available for future theoretical and experimental exploration.

We may deduce that there is (only) a special subset of QM for which the BCP is satisfied. Under the variant construction there are six distinct logical configurations that can be used to support 0-1 vectors. Of these six, Bohr's approach is suitable for only the two schemes of pure static YES or NO. Meanwhile, the other four variant, invariant and mixed configurations lie outside the BCP framework. From this viewpoint, Bohr offers insight into important special

<sup>359</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

Bohr's QM construction is complete and useful in many theoretical and practical environments for static and static-like systems. However, the variant construction provides a more powerful and general mechanism to handle different dynamic systems with variant and

From EPR proposed experiments and other theoretical considerations, Einstein demonstrated a depth of understanding of weakness inherent in the foundations of the QM approach. He clearly identified two operators with non-communication properties that failed to support simultaneous operations and recognized that this type of mechanism was still not explained

> *IM*(*S*1, *S*2) → {*M*(*u*1), *M*(*v*1), *M*(*u*˜1), *M*(*v*˜1)...}; *SM*(*S*1, *S*2) → {*M*(*u*0), *M*(*v*0), *M*(*u*˜0), *M*(*v*˜0)...}.

From this correspondence, many possible configurations of combinations and their subsets

Using the variant construction, rich configurations can be expressed. From such mapping, it can be seen to be nothing less than astounding that such meta constructions were identified

Afshar's experiments apply anti-symmetry signals making the following correspondence:

*ψ*<sup>1</sup> → {*v*+}; *ψ*<sup>2</sup> → {*v*1}; *σ*<sup>1</sup> → {*PH*(*v*1|*J*)}; *σ*<sup>2</sup> → {*PH*(*v*0|*J*)}.

⎧ ⎪⎪⎨

⎪⎪⎩

�{*uβ*, *vβ*, *u*˜*β*, *v*˜*β*...}, {*uβ*, *vβ*, *u*˜*β*, *v*˜*β*...}, {*M*(*u*1), *M*(*v*1), *M*(*u*˜1), *M*(*v*˜1)...}, {*M*(*u*0), *M*(*v*0), *M*(*u*˜0), *M*(*v*˜0)...}�

(18)

(19)

(20)

circumstances of QM rather than provides an all embracing general solution.

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

Using the variant construction, EPR devices have the following correspondence:

*S*<sup>1</sup> → {*uβ*, *vβ*, *u*˜*β*, *v*˜*β*...}; *S*<sup>2</sup> → {*uβ*, *vβ*, *u*˜*β*, *v*˜*β*...};

Four anti-diagonal positions are linked to symmetry and anti-symmetry pairs, twelve other pairs of functions belong to non-symmetry and non-anti-symmetry conditions. Their meta elements can be identified by the relevant variant expressions.
