NOTE:The first part of this was posted on 2/1/16, and that part is reposted here again so that the whole derivation is presented together. The new material starts where noted by "NEW MATERIAL FROM HERE ON"
THE SIMPLE MATH OF TRUE UNITS
THE ELEMENTARY MATH OF TRUE UNITS
ELEMENTARY PARTICLES AND UNITS OF MEASUREMENT
In order to see how the minimal quantum extent and content of our smallest possible elementary distinction relates to known elementary particles, we develop equations that can be used to describe the combination of up and downquarks to form the proton and neutron and the Hydrogen atom. We choose the Hydrogen atom to start with because it is the simplest, most stable, and most abundant known element in the universe. If all forms of substance are quantized, then in order for quarks to combine in stable structures, they must satisfy the Diophantine (integer) forms of the equations of Dimensional Extrapolation conveying the logic of the transfinite substrate into the spacetime domain of our experience. This family of Diophantine equations is represented mathematically by the expression
Σ^{n}_{i=1} (X_{n})^{m} = Z^{m}.
The Pythagorean Theorem equation, the Fermat’s Last Theorem equation and other important equations are contained within this general expression. We mention this fact here because these theorems play key roles in the geometry and mathematics of Dimensional Extrapolation and the combination of elementary particles to form stable physical structures. Because the various forms of this expression as m varies from 3 to 9 conveys the geometry of 9dimensional reality to our observational domain of 3S1t, we call this expression the “Conveyance Expression”, and individual equations of the expression ”Conveyance Equations”.
When n = m = 2, the expression yields the equation
(X_{1})^{2} + (X_{2})^{2 }= Z^{2}
which, when related to areas, describes the addition of two square areas, A_{1 }and A_{2} with sides equal to X_{1 }and X_{2 }respectively, to form a third area, A_{3}, with sides equal to Z. When these squares are arranged in a plane with two corners of each square coinciding with corners of the other squares to form a right triangle, we have a geometric representation of the familiar Pythagorean Theorem demonstrating that the sum of the squares of the sides of any right triangle is equal to the square of the third side (the hypotenuse) of that triangle.
We use this simple equation in Dimensional Extrapolation to define the rotation and orthogonal projection from one dimensional domain into another, in the plane of the projection. There are an infinite number of solutions for this equation, one for every conceivable right triangle, but in a quantized reality, we are only concerned with the integer solutions. Considering the Pythagorean equation as a Diophantine equation, we find that there exists an infinite subset of solutions with AB = X_{1},_{ }BC = X_{2 }and AC = Z equal to integers. Members of this subset, e.g. (3,4,5), (5,12,13), (8,15,17), etc. i.e., (3^{2} + 4^{2} = 5^{2}, . 5^{2} + 12^{2} = 13^{2}, 8^{2} + 15^{2} = 17^{2}, … ) are called “Pythagorean triplets”.
To describe the combination of two threedimensional particles, we have the Conveyance equation when n = 2 and m = 3:
(X_{1})^{3} + (X_{2})^{3 }= Z^{3}.
When we define X_{1}, X_{2} and Z as measures of volumes, just as we defined them as measures of areas when n = m = 2, we can apply this equation to quantal volumes in a threedimensional domain. Using the minimal quantal volume of the electron as the unit of measurement, and setting it equal to one, we have a Diophantine equation related to our hypothetical elementary particle with minimal spinning volume containing uniform substance: if it is spherical, we can set its radius equal to r_{1}, and if there is a second uniform spinning particle rotating at maximum velocity, with radius r_{2}, we can describe the combination of the two particles by the expression 4/3π(r_{1})^{3} + 4/3π(r_{2})^{3}. If this combination produces a third spinning spherical object we have:
4/3π(r_{1})^{3} + 4/3π(r_{2})^{3} = 4/3π(r_{3})^{3},
where r_{3} is the radius of the new particle. Dividing through by 4/3π, we have:
(r_{1})^{3} + (r_{2})^{3} = (r_{3})^{3}, which is a Diophantine equation of the form of the Fermat equation,
X^{m} + Y^{m} = Z^{m} when m =3.
Notice that the factor, 4/3π cancels out, indicating that this equation is obtained regardless of the shape of the particles, as long as the shape and substance is the same for all three particles. (This is an important fact because we found in investigating the Cabibbo angle that the electron, while symmetrical, is not necessarily spherical.) Note also, that the maximum rotational velocity and angular momentum will be different for particles with different radii, because the inertial mass of each particle will depend upon its total volume. In a quantized reality, the radii must be integer multiples of the minimum quantum length. Since this equation is of the same form as Fermat’s equation, Fermat’s Last Theorem tells us that if r_{1} and r_{2} are integers, r_{3} cannot be an integer. This means that the righthand side of this equation, representing the combination of two quantum particles, cannot be a symmetric quantum particle. But, because Planck’s principle of quantized energy and mass tells us that no particle can contain fractions of mass and/or energy units, the righthand side of the equation represents an unstable asymmetric spinning particle. The combined highvelocity angular momentum of the new particle will cause it to spiral wildly and fly apart. This may lead us to wonder how it is that there are stable particles in the universe, and why there is any physical universe at all. Again, we are faced with Leibniz’s most important question: why is there something instead of nothing?
The answer turns out to be relatively simple, but is hidden from us by the limitations of our methods of thinking and observation if we allow them to be wholly dependent upon our physical sense organs. For example, we think of a sphere as the most perfect symmetrical object; but this is an illusion. Spherical objects can exist in a NewtonLeibniz world, but we actually exist in a PlanckEinstein world. In the real world, revealed by Planck and Einstein, the most perfectly spherical object in three dimensions is a regular polyhedron. (polyhedron = multisided threedimensional form; regular; all sides are of equal length.) The most easily visualized is the sixsided regular polyhedron, the cube. In the NewtonLeibniz world, the number of sides of a regular polynomial could increase indefinitely. If we imagine the number of sides increasing without limit while the total volume approaches a finite limit, the object appears to become a sphere. But in the quantized world of Planck and Einstein, the number of sides possible is limited, because of the finite size of the smallest possible unit of measurement (which we are defining here) is relative to the size of the object. And because the “shape” factor cancels in the Conveyance Equation for n = 3, Fermat’s Last Theorem tells us that, regardless of the number of sides, no two regular polyhedrons composed of unitary quantum volumes can combine to form a third regular polyhedron composed of unitary quantum volumes.
To help understand the physical implications of this, suppose our true quantum unit exists in the shape of a cube. Using it as a literal building block, we can maintain particle symmetry by constructing larger cubes, combining our basic building blocks as follows: a cube with two blocks on each side contains 8 blocks; a cube with three blocks on each side contains 27 blocks; a cube with four blocks on each side contains 64 blocks; etc. Fermat’s Last Theorem tells us that if we stack the blocks of any two such symmetric forms together, attempting to keep the number of blocks on all sides the same, the resulting stack of blocks will always be at least one block short, or one or more blocks over the number needed to form a perfect cube. Recall that if these blocks are elementary particles, they are spinning with very high rates of angular velocity, and the spinning object resulting from combining two symmetric objects composed of unitary quantum volumes will be asymmetric, causing its increasing angular momentum to throw off any extra blocks until it reaches a stable, symmetrically spinning form.
This requirement of symmetry for physical stability creates the intrinsic dimensionometric structure of reality that is reflected in the Conveyance Expression. It turns out that there can be stable structures, because when n = m =3, the Conveyance Expression yields the equation:
(X_{1})^{3} + (X_{2})^{3 }+ (X_{3})^{3}= Z^{3},
which does have integer solutions. The first one (with the smallest integer values) is:
3^{3} + 4^{3 }+ 5^{3}= 6^{3}
^{}
^{}
It is important to recognize the implications of Σ^{n}_{i=1} (X_{n})^{m} = Z^{m}. When n, m, the X_{i} and Z are integers, is an exact Diophantine expression of the formof the logical structure of the transfinite substrate as it is communicated to the 3S1t domain. For this reason, we call it the Conveyance Expression. It should be clear that the Diophantine equations yielded by this expression are appropriate for the mathematical analysis of the combination of unitary quantum particles. When the Diophantine expressions it yields are equations with integer solutions, they represent stable combinations of quantum equivalence units, and when they do not have integer solutions, the expressions are inequalities representing asymmetric, and therefore, unstablestructures.
In the quantized ninedimensional domains of TDVP, the variables of the Conveyance Equations are necessarily integers, making them Diophantine equations, because only the integer solutions represent quantized combinations. When n = m = 2, we have the Pythagorean Theorem equation for which the integer solutions are the Pythagorean Triples. When n = 3 and m = 2, the Conveyance Equation yields the inequality of Fermat’s Last Theorem, excluding binomial combinations from the stable structures that elementary particles may form. On the other hand, the Diophantine Conveyance Expression when n = m = 3, integer solutions produce trinomial combinations of elementary particles that will form stable structures. This explains why there is something rather than nothing, and why quarks are only found in combinations of three.
Embedded within the transfinite substrate are three dimensions of space and three dimensions of time that are temporarily contracted during observations, and condensed into the distinctions of spinning energy (energy vortices) that form the structure of what we perceive as the physical universe. In the humanly observable domain of 3S1t, this spectrum ranges from the photon, which is perceived as pure energy, to the electron, with a tiny amount of inertial mass (0.51 MeV/c^{2} ≈ 1 x10^{47} kg.) to quarks ranging from the “up” quark at about 2.4 MeV/c^{2}, to the “top” quark at about 1.7 x10^{5 }MeV/c^{2}, to the Hydrogen atom at about 1x10^{9} MeV/c^{2} (1.67 x10^{27}kg.), to the heaviest known element, Copernicum (named after Nicolaus Copernicus) at 1.86 x10^{24}kg . So the heaviest atom has about 10^{23} times, that is, about 100,000,000,000,000,000,000,000 times heavier than the inertial mass of the lightest particle, the electron. All of the Elements of the Periodic Table are made up of stable vortical distinctions that are known as fermions, “particles” with an intrinsic angular spin of 1/2, or they are made up of combinations of fermions. Table One, above, lists the fermions that make up the Hydrogen atom and their parameters of spin, charge and mass based on experimental data.
Bohr’s solution of the EPR paradox, validated by the Aspect experiment and many subsequent experiments refined to rule out other possible explanations, tells us that newly formed fermions do not exist as localized particles until they impact irreversibly on a receiver constituting an observation or measurement. In the TDVP unified view of reality, every elementary particle, every distinct entity in the whole range of particles apparently composed of fermions, is drawn from the continuous transfinite substrate of reality when it is registered as a finite distinction in an observation or measurement. Our limitations of observation and measurement and the dimensional structure of reality result in our perception of fermions as separate objects with different combinations of inertial mass and energy. What determines the unique mix that makes up each type of observed particle? To answer this question, we must continue our investigation of the rotation of the minimum quantal units across the four dimensions of space, time and the additional dimensions revealed by the mathematics of TDVP.
One of the most important invariant relationships between dimensional domains is the fact that each ndimensional domain is embedded in an n+1 dimensional domain. This means that all distinctions of extent, from the ninthdimensional domain down, and the distinctions of content within them, are inextricably linked by virtue of being sequentially embedded. Because of this intrinsic linkage, the structure of any distinction with finite extent and content, from the smallest particle to the largest object in the universe, reflects patterns existing in the logical structure of the transfinite substrate. Such a distinct object will always have in its content, combinations of the forms reflecting those patterns. In a quantized reality, the dimensionometric forms of such objects will be symmetric and a multiple of the smallest unit of measurement,
STABLE VORTICAL FORMS AND TRUE QUANTAL UNITS
Chemists trained in the current paradigm think of the combination of elementary particles and elements as forming atoms and molecules by the physical bonding of their structures, and model these combinations in tinkertoy fashion with plastic or wooden spherical objects connected by single or double cylindrical spokes. This is helpful for visualizing molecular compounds in terms of their constituents prior to combining, but that is not necessarily what actually happens. Inside a stable organic molecule, volumetrically symmetric atoms are not simply attached; their subatomic spinning vortical “particles” combine, forming a new vortical object. Elementary particles are rapidly spinning symmetric vortical objects and when three of them combine in proportions that satisfy the threedimensional Conveyance Equation, they do not simply stick together  they combine to form a new, dimensionally stable, symmetricallyspinning object. Because they are spinning in more than one plane, these objects are best conceived of as closed vortical solitions.
The triadic combinations of elementary vortical objects, like up and downquarks, form new vortical objects called protons and neutrons; the combinations of electrons, protons and neutrons form new vortical objects called elements; and the triadic combinations of volumetrically symmetric elements form new vortical objects called organic molecules. Thus, the dimensional forms of symmetricallyspinning objects formed by the combining of smaller vortical objects form closed vortices in 3S1t with new physical and chemical characteristics, depending upon both their internal and external structure. We will take the volume of the smallest possible quantized vortical object as the basic unit of measurement as the true quantal unit.
THE TRUE UNIT, THE CONVEYANCE EQUATION AND THE THIRD FORM OF REALITY
Conceptually, the true quantum unit in TDVP is therefore a subquark unitary extent/content entity spinning in the mathematically required nine dimensions of quantized reality. When we choose to measure the substance of a quantum distinction, the effects of its spinning in the three planes of space register as inertia or mass, spin in the timelike dimensional planes manifests as energy, and spinning in the additional planes of reality containing the space and time domains, requires a third form of the stuff of reality, in addition to, but not registering as either mass or energy, to complete the minimum quantum volume required for the stability of that distinct object. Because this third form of the stuff of reality is unknown in current science, we need an appropriate symbol to represent it. Every letter in the English and Greek alphabets has been used as a symbol for something in math and science, so we have gone to the historically earlier PhoenicianAramaicHebrew alphabet. We will represent that potential third form of reality here with the third letter of the Aramaic alphabet, ג (Gimel), and we will call the subquark unitary measure of the three forms of reality the Triadic RotationalUnit of Equivalence, or TRUE Unit.
The mix of the three forms, m, E and ג, needed to maintain symmetric stability, present in any given 3S1t measurement, will be determined by the appropriate Conveyance Equation, as demonstrated below. When n = m = 3, Σ^{n}_{i=1} (X_{n})^{m} = Z^{m} yields:
(X_{1})^{3} + (X_{2})^{3 }+ (X_{3})^{3}= Z^{3}
^{}
^{}
The integer solutions of this Diophantine equation in TRUE units represent the possible combinations of three symmetric vortical distinctions forming a fourth threedimensional symmetric vortical distinction.
THE PRIMARY LEVEL OF SYMMETRIC STABILITY – QUARKS
With the appropriate integer values of X_{1},^{ }X_{2}, X_{3}, and Z, in TRUE units, this equation represents the stable combination of three quarks to form a Proton or Neutron. There are many integer solutions for this equation and historically, methods for solving it were first developed by Leonhard Euler ^{ref}. The smallest integer solution of this Conveyance Equation is 3^{3} + 4^{3 }+ 5^{3}= 6^{3}.
Trial Combination of Two UpQuarks and One DownQuark, i.e.
The Proton, With Minimal TRUE Units
Particle

Charge^{*}^{}

Mass/Energy

ג

Total TRUE Units

MREV^{**}

u_{1}

+ 2

4

1

3

27

u_{2}

+ 2

4

0

4

64

d

 1

9

4

5

125

Total

+ 3

17

5

12

216=6^{3}

* For consistency in a quantized reality, charge has also been normalized in these tables.
^{** }Minimum Rotational Equivalent Volume (MREV)
If we attempt to use the smallest integer solution, 3^{3} + 4^{3 }+ 5^{3}= 6^{3}, to find the appropriate values of ג for the Proton, we obtain negative values for גfor the first upquark and the downquark and zero for the second upquark. It is conceivable that some quarks may contain no ג units, but negative values are a problem, because a negative number of total ג units would produce an entity with fewer total TRUE units than the sum of mass/energy units of that entity, violating the conservation of mass and energy, destroying the particle’s equilibrium and identity. When we try to use the smallest integer solution of the conveyance equation to describe the combination of one upquark and two downquarks in a neutron, all of the quarks have negative ג units. See the table below:
Trial Combination of One UpQuark and Two DownQuarks in TRUE Units
Particle

Charge

Mass/Energy

ג

Total TRUE Units

MREV

u

+ 2

4

1

3

27

d_{1}

 1

9

5

4

64

d_{2}

 1

9

4

5

125

Totals

0

22

10

12

216=6^{3}

NEW MATERIAL FROM HERE ON
The redistribution of TRUE units cannot result in
the appearance of negative ג units
in the internal structure of an entity. A triadic entity with negative total ג units is not possible because a negative
number of total ג units would
violate the conservation of mass and energy, destroying the particle’s
equilibrium and identity. Analogous to the axiom ‘nature abhors a vacuum’, a
result of the second law of thermodynamics, just as the electrons of an
incomplete shell rush around the entire volume of the shell trying to fill it, negative
ג units would pull a TRUE units out
of the mass/energy of the particle to fill the void and the measurable mass/energy
of the particle would no longer be that of a proton or neutron and conservation
of mass/energy in 3S1t would be violated.
Attempting to use the smallest integer solution, (3,
4, 5, 6) of the Conveyance Equation to find the appropriate values of ג for the proton and neutron, we obtain
negative total ג unit values. This
solution would change the particle’s measurable mass/energy identity and
violate conservation of mass and energy, so we continue to look for an
appropriate solution. The next numerically smallest integer solution for the
Conveyance Equation is 1^{3} + 6^{3
}+ 8^{3}= 9^{3}, but, using it also results in negative
values. The smallest
integer solution of the Conveyance Equation that produces no negative values of
ג for the Proton is 6^{3} + 8^{3 }+ 10^{3}=
12^{3}, using this solution we have the electrically and
symmetrically stable Proton:
The Proton (P^{+})
Particle*

Charge

Mass/Energy

ג

Total TRUE
Units

MREV

u_{1}

+ 2

4

2

6

216

u_{2}

+ 2

4

4

8

512

d_{1}

 1

9

1

10

1,000

Total

+ 3

17

7

24

1728=12^{3}

* u_{1 }and u_{2} have the same number of TRUE units of mass and energy, and
therefore will register as upquarks in the collider data, but have different
numbers of TRUE units of equivalent volume participating as ג
to produce the volumetrically
symmetric, and therefore stable, Proton.
Nature, reflecting the patterns
of the dimensional substrate, does not have to rely upon random particle
encounters to build complex structural forms. Compound structures are formed
within the mathematical organization of the Conveyance Equation, and useful
building blocks have a significant level of stability in 3S1t for protons to combine
with other compound particles and create structures sufficiently complex to
support life. To see how other structures arise from quarks, protons and
electrons, we need to know how protons, neutrons and electrons relate to the
Conveyance Equation: (X_{1})^{3} + (X_{2})^{3 }+
(X_{3})^{3}= Z^{3}. If the number of TRUE
units in the proton is equal to the integer X_{1}, the number of TRUE units in the neutron = X_{2}, the number of TRUE units
in the electron = X_{3}, then
the resulting compound entity, will be stable in the 3S1T domain of physical
observations.
We know that the 24 TRUEunit Proton
must combine with an electron to form a Hydrogen atom, and with other protons, electrons
and neutrons to form the other elements. In order to find the smallest solution
of the conveyance equation that can include the 24 TRUE units of the proton, we
may start by trying the solutions we’ve used so far. 24 is a multiple of 2, 3,
4, 6, and 8, any one of which can be a factor of X_{1} in the conveyance equation solutions we’ve used so
far. Up to this point we’ve only used the first two of the smallest primitive
integer solutions of the equation: 3^{3}
+ 4^{3 }+ 5^{3 }= 6^{3} and 1^{3} + 6^{3 }+ 8^{3 }= 9^{3}. (A
primitive Diophantine solution is defined as one without a common factor in all
terms.) We have also used 6^{3}
+ 8^{3 }+ 10^{3}= 12^{3}, an integer solution
obtained by multiplying all of the terms of the smallest primitive solution by
2. The first 36 integer solutions of the conveyance equation (X_{1})^{3}
+ (X_{2})^{3 }+ (X_{3})^{3 }= Z^{3} are listed below in ascending order. Primitive
solutions are in bold.
3^{3}
+ 4^{3} + 5^{3} = 6^{3}
1^{3}
+ 6^{3} + 8^{3 }= 9^{3}
6^{3} + 8^{3}
+ 10^{3} = 12^{3}
2^{3}+ 12^{3}
+ 16^{3} = 18^{3}
3^{3}
+ 10^{3} + 18^{3 }= 19^{3}
7^{3}
+ 14^{3} + 17^{3 }= 20^{3}
12^{3} + 16^{3}
+ 20^{3 }= 24^{3}
4^{3}
+ 17^{3} + 22^{3} = 25^{3}
3^{3}
+ 18^{3} + 24^{3 }= 27^{3}
18^{3}
+ 19^{3} + 21^{3 }= 28^{3}
11^{3}
+ 15^{3} + 27^{3} = 29^{3}
15^{3} + 20^{3}
+ 25^{3} = 30^{3}
4^{3} + 24^{3}
+ 32^{3} = 36^{3}
18^{3} +
24^{3} + 30^{3} = 36^{3}
2^{3}
+ 17^{3} + 40^{3} = 41^{3}
6^{3}
+ 32^{3} + 33^{3} = 41^{3}
16^{3}
+ 23^{3} + 41^{3} = 44^{3}
5^{3} + 30^{3}
+ 40^{3} = 45^{3}
3^{3} + 36^{3}
+ 37^{3} = 46^{3}
27^{3} + 30^{3}
+ 37^{3} = 46^{3}
24^{3} +
32^{3} + 40^{3} = 48^{3}
8^{3} + 34^{3}
+ 44^{3} = 50^{3}
29^{3}
+ 34^{3} + 44^{3} = 53^{3}
12^{3}
+ 19^{3} + 53^{3} = 54^{3}
36^{3}
+ 38^{3} + 42^{3} = 56^{3}
15^{3} + 42^{3}
+ 49^{3} = 58^{3}
21^{3} + 42^{3}
+ 51^{3 }= 60^{3}
30^{3} + 40^{3}
+ 50^{3} = 60^{3}
7^{3} + 42^{3}
+ 56^{3} = 63^{3}
22^{3}
+ 51^{3} + 54^{3} = 67^{3}
36^{3}
+ 38^{3} + 61^{3} = 69^{3}
7^{3}
+ 54^{3} + 57^{3} = 70^{3}
14^{3}
+ 23^{3} + 70^{3} = 71^{3}
34^{3} + 39^{3}
+ 65^{3} = 72^{3}
38^{3} + 43^{3}
+ 66^{3} = 75^{3}
31^{3} + 33^{3}
+ 72^{3} = 76^{}
^{
}
The
numbers appearing in the totals in the tables describing quarks, protons,
neutrons and atoms are the smallest possible nonnegative integers consistent
with the empirical data and the requirement for symmetry that the sum of the
three totals cubed must equal an integer cubed. Thus, we can calculate the
number of ג units involved, and the totals of TRUE
units required by the conveyance equation to yield results consistent with
empirical particle collider data. Note that the TRUE units in these tables are
measurements of threedimensional objects in multiples of the unitary linear
measure of their volumes, and their minimal rotational equivalence volumes
(MREV), listed in the last column, are equal to the TRUE unit values cubed.
Negative values for ג cannot occur
because of the conservation of mass and energy. Negatives would destroy the
mass/energy/ ג balance and turn the quarks into unstable
combinations which would decay quickly. So we must find the smallest unique conveyance equation solution for
each combination of subatomic particles. The correct unique solution can be
found for each triadic subatomic particle by starting with the smallest
integer solution of the conveyance equation and moving up the scale until no
negative values are obtained. Using the solution 6^{3}
+ 8^{3 }+ 10^{3}= 12^{3}, the first
attempt to find the TRUE unit configuration of the neutron is shown below:
Trial Combination of One UpQuark and Two DownQuarks in
TRUE Units
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

MREV

u

+ 2

4

2

6

216

d_{1}

 1

9

1

8

512

d_{2}

 1

9

1

10

1000

Totals

0

22

2

24

1728=12^{3}

Since this solution still produces a
negative value of ג for d_{1}, we must move to the
next larger solution to represent the Neutron. The smallest unique Conveyance
Equation solution with no negative or zero values of ג for the stable Neutron is 9^{3}
+ 12^{3 }+ 15^{3}= 18^{3 }
Second Trial of
Quark Combinations for the Neutron
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

MREV

u_{3}

+ 2

4

5

9

729

d_{2}

 1

9

3

12

1,728

d_{3}

 1

9

6

15

3,375

Totals

0

22

14

36

5,832=18^{3}

These TRUE unit numbers give us a
stable neutron; but we have another problem: None of the solutions with a term
equal to 24 have a second term equal to 36. Nor do any of the solutions listed
have two terms with the ratio 24/36 =2/3. This is a problem because it means
that combinations with equal numbers of protons and neutrons could not be
stable, and we know that Hydrogen, the element Helium, and other elements are
stable combinations with equal numbers of protons and neutrons. Looking at the
TRUEunits analysis of Helium as an example, we have:
Attempt to Construct a Helium Atom with P^{+ }= 24
and N^{0} = 36
Particle

Charge

Mass/Energy

ג

Total TRUE
Units

MREV

2e

 6

2

78

80^{*}^{}

512,000

2P^{+ }

+ 6

34

14

48

110,592

2N^{0}

0

44

28

72

373,248

Totals

0

80

120

200

995,840=(99.861…)^{3}

*Note: The number of TRUE units
making up the electron is unknown at this point. This value was chosen because
it is the integer value that produced a total MREV nearest to a cube, as it must
be for a stable Helium atom. The smallest integer value in TRUE units for the
proton is 24.
Since a neutron of 36 TRUE units
produces an unstable Helium atom, contradicting the empirical fact that stable
Helium atoms exist, we have to seek another integer solution of the conveyance equation
for the neutron.
Going back to the list of conveyance
equation solutions, we see that the next smallest solution that does not
generate negatives for the neutron is the primitive solution 7^{3}
+ 14^{3} + 17^{3 }= 20^{3}.
Third Trial of
Quark Combinations for the Neutron
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

MREV

u_{3}

+ 2

4

3

7

343

d_{2}

 1

9

5

14

2,744

d_{3}

 1

9

8

17

4,913

Totals

0

22

16

38

8,000=20^{3}

Next, we need to see if this quark
combination for the neutron combined with protons and electrons will yield
stable atomic structures. Using the values we derived for P^{+} and N^{0},
the first integer solution of the conveyance equation containing the values X_{1 }= 24 and X_{2 }= 38 is obtained by
multiplying both sides of the primitive solution 12^{3} + 19^{3}
+ 53^{3} = 54^{3}
by 2, yielding the integer solution 24^{3}
+ 38^{3} + 106^{3} = 108^{3}.
Helium Atom with P^{+ }= 24 and N^{0} = 38
Particle

Charge

Mass/Energy

ג

Total TRUE
Units

MREV

2e

 6

2

210

212^{*}^{}

9,528,128

2P^{+ }

+ 6

34

14

48

110,592

2N^{0}

0

44

32

76

438,976

Totals

0

80

256

336

10,077,696=216^{3}
^{}

^{*}With the TRUE units determined for
protons and neutrons, the Helium atom is stable only if the total number of TRUE
units for the electron is 106.
Besides the TRUE units that appear
as mass/energy in given elementary particles, because of the embedded nature
(dimensional tethering) of dimensional domains in TDVP, there must be a minimum
number of ג units associated with each particle for stability. Consistent with up
and downquark decay from the strange quark, the stabilization requirement of
an integer solution for the conveyance equation, and the additional TRUE units of
ג
needed for particle stability, the
following table describes the electron, proton and neutron in TRUE units, with up
quarks composed of a total of 24 TRUE units, down quarks composed of a total of
38 TRUE units and electrons composed of a total of 106 TRUE units. It therefore represents the normalized
mass/energy, minimum ג
and total volumes for stable electrons, protons and neutrons, the building
blocks of the physical universe.
The Building Blocks of the Elements in TRUE Units
Particle

Charge

Mass/Energy

ג

Total TRUE
Units*

Volume

e

 3

1

105**

106

1,191,016

P^{+}

+ 3

17

7

24

13,824

N^{0}

0

22

16

38

54,872

* Whether mass, energy or gimmel (ג), upon measurement, each TRUE unit occupies
the same volume, i.e. the minimal volume for an elementary particle as a spinning
object, as required by relativity and defined in TDVP as the basic unit of
volume. Each TRUE unit is capable of contributing to the structure of physical
reality as m, E or ג to form a particle, according
to the logical pattern in the substrate reflected in the Conveyance Equation, and
the relative volume of each particle (in the three dimensions of space) is
equal to the total number of TRUE units cubed times the shape factor. As noted
before, the shape factor cancels out in the Conveyance Equation. For this
reason, the righthand column in these tables contains cubed integer amounts representing
the Minimum Relative Equivalence Volume
(MREV) for each particle making up the combination of subatomic particles.
**The TRUE unit values for the elementary
particles are uniquely determined by conditions necessary for a stable universe.
The values for up and downquarks are the necessary values for the proton and
neutron, as determined above, and the number of ג
units and the total TRUE units for the electron are determined by calculating
the ג
units necessary to form a stable Helium atom. They also determine the smallest
possible stable atoms, Hydrogen, Deuterium and Tritium, as shown below.
THE
SECONDARY LEVEL OF SYMMETRIC
STABILITY – ATOMS
Atoms
are semistable structures composed of electrons, protons and neutrons. They
are not as stable as protons and neutrons, but they are generally more stable
than molecules.
The Elements of the Periodic Table
The Hydrogen atom is unique among
the natural elements in that it has only two mass/energy components, the electron
and proton. Thus, because Fermat’s Last Theorem prohibits the symmetrical
combination of two symmetrical particles; they cannot combine to form stable
structures like the combination of quarks to form the proton and neutron. The
electron, with a small fraction of the mass of the proton, is drawn by electric
charge to whirl around the proton, seeking stability. This
means that the Hydrogen atom, the elemental building block of the universe,
composed only of the mass and energy of an electron and a proton, is inherently
unstable. So why is it that we have any stable structures at all; why is
there a universe? As Leibniz queried: “why
is there something rather than nothing”?
One of the X_{n} integers must be 24 to represent the TRUE unit value
of the proton, and among the integer solutions of the m = n = 3 conveyance equation listed above there are four solutions
with 24 as one of the X_{n} solution
integers. Nature is parsimonious, and we must never make a mathematical
description or demonstration any more complicated than it has to be. Therefore,
we start with the smallest solution with 24 as one of the X_{n} integers. It is 3^{3}
+ 18^{3} + 24^{3 }= 27^{3}. But it does not
contain an X_{n} equal to
38, so we must continue, searching for an integer solution that contains both
24 and 38 on the left side of the equation. Since there are no smaller integer solutions
with comultiples of 24 and 38 as terms in the left side of the equation, we
can use the solution that provided a stable Helium atom: 24^{3}
+ 38^{3} + 106^{3} = 108^{3}. Using it to represent the Hydrogen
atom, we have:
TRUEUnit Analysis
for Hydrogen 1 (Protium), Valence =  2 + 1 = 1
Particle

Charge

Mass/Energy

ג

Total TRUE
Units

Volume

e

 3

1

105

106

1,191,016

P^{+}

+ 3

17

7

24

13,824

C_{ג}*

0

0

38

38

54,872

Totals

0

18

150

168^{}

1,259,712=108^{3}

* Since the Proton
required 17 mass/energy units and 7 ג units, adding up to
24 Total TRUE units, to achieve triadic stability (see the Tables describing
the Proton), to achieve the same level of stability as the proton and neutron,
the Hydrogen atom must have a third additive component, C_{ג},
consisting of 38 ג units, the third form
of the ‘stuff’ of reality, not measureable as mass or energy in 3S1t. This
satisfies the conveyance equation and produces a stable Hydrogen atom with a total
volume of 108^{3}.
Without the ג units needed by Hydrogen to achieve stability, we would have no
universe. The TRUE units of two symmetrically stable entities, the electron and
proton, could not combine to form a third symmetrically stable entity (Fermat’s
Last Theorem). Because of the asymmetry of their form as two symmetric entities
of different sizes in TRUE units, they could not combine; they would spiral and
be easily separated by any external force. Even if they could adhere to other
particles, the resulting universe would be very boring. All multiples of such a
building block would have the same chemical characteristics. With the input of
the appropriate number of ג units, Hydrogen
is a basic building block of symmetrically stable forms in the 3S–1t observable
domain of the physical universe.
In 3S1t, TRUE units can manifest as
mass, energy or ג, in order to form symmetrically stable
particles and the 168 total TRUE units of the Hydrogen atom may be arranged in another
stable structural form, observed as the simple combination of one electron, one
proton and one neutron, known as Deuterium, an isotope of Hydrogen (an atom with the same chemical properties).
Hydrogen 2 (Deuterium), Valence = 2 + 1 = 1
Particle

Charge

Mass/Energy

ג

Total TRUE Units

Volume

e


3

1

105

106

1,191,016

P^{+}

+
3

17

7

24

13,824

N^{0}

0

22

16

38

54,872

Totals

0

40

128

168

(108)^{3}

Hydrogen 2 (H2) is held together by
electrical charge and 128 ג units, 22
less than the H1 atom. This means that H2 is not as stable as H1. What about
other isotopes of H1? Is it possible that the TRUE units of a Hydrogen atom or
a Deuterium atom can combine with one or more additional neutrons to form stable
isotopes? Hydrogen 3 (H3), known as Tritium, is a second isotope of Hydrogen.
Its form in TRUE units is represented below.
Hydrogen 3 (Tritium), Valence =  2 + 1 = 1
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

Volume

e

 3

1

105

106

1,191,016

P^{+ }

+ 3

17

7

24

13,824

2N^{0}

0

44

32

76

438,976

Totals

0

62

144

206

(118.018…)^{3 }^{*}^{}

^{*}We see that H3
is an asymmetric structure. One electron, one proton and two neutrons, brought
together by attractive forces, cannot combine volumetrically to form a
symmetrically stable structure, and as a result, it is unstable and there are
very few H3 atoms. Looking at the TRUE unit structure for H1, H2 and H3, we see
that all three are bonded by electrical charge, but H1 has volumetric stability
and 150 ג units holding it together;
H2 has volumetric stability, more mass/energy units and fewer ג units than H1; and H3 has more
mass/energy units and ג units, but no volumetric stability. This explains why
H1 is the most abundant, H2 less abundant, and H3 correspondingly less stable. The
atomic weights of the elements of the periodic table, in amu (atomic mass units), are actually the mean values of atomic
masses calculated from a great number of samples. The accepted mean atomic
weight for Hydrogen to four significant figures is 1.008. This includes H1 and
all isotopes of Hydrogen. If all hydrogen atoms were H1 atoms, this number
would be exactly 1. H1 is by far the most stable, and therefore, most abundant,
of the Hydrogen family, making up more than 99.99% of all Hydrogen in the
universe. Other H isotopes make up the remaining 0.01%, mostly H2, with H3 and
other isotopes heavier than H2 occurring only rarely in trace amounts.
Using TRUEunit analysis, we can
investigate every possible combination of H1 atoms and neutrons and determine
which combinations are the most stable. After Tritium, the next stable combination
of TRUE units, Helium, involves 336 TRUE units, as shown below.
HELIUM Valence =  2 + 2 = 0 (Inert)
Particle

Charge

Mass/Energy

ג

Total TRUE
Units

Volume

2e

 6

2

210

212^{*}^{}

9,528,128

2P^{+ }

+ 6

34

14

48

110,592

2N^{0}

0

44

32

76

438,976

Totals

0

80

256

336

(2x108)^{3}
^{}

Why is this not called “quadrium”, a third isotope of
Hydrogen? It is a new element because it has two electrons filling its outer
(and only) shell, so that it is not attracted to other atoms.
New elements
arise when a unique new combination of TRUE units, constructed using multiples
of the basic building blocks of electrons, protons and neutrons is formed. The
next element is the combination of the inert atom, Helium, with the asymmetric
atom, H3 with to form Lithium.
LITHIUM, Valence = – 2 + 3 = +1
Particle

Charge

Mass/Energy

ג

Total TRUE
Units

Volume

3e

 9

3

315

318

32,157,432

3P^{+ }

+ 9

51

21

72

373,248

4N^{0}

0

88

64

152

3,511,808

Totals

0

142

400

542

(330.32…)^{3
}^{*}^{}

^{*} Since the total volume is not an
integer cubed, Lithium, like Tritium, is volumetrically asymmetric. It has a
stronger electrical bond than H3 and more ג
units connecting it with the multidimensional substrate for added stability, but
it is less stable because it is asymmetric.
THE TERTIARY LEVEL OF SYMMETRIC STABILITY – MOLECULAR
BONDING
We’ve
seen how quarks combine in very stable symmetric triads of TRUE units and how
atoms form stable or semistable vortices, spinning structures consisting of
stable triads of protons, neutrons and electrons. A third level of stable and
semistable structures occurs as molecules are formed from more complex combinations
of elemental atoms.
The Role of Valence
The number of electrons in the outer shell of an atom
determines the observable identifying chemical characteristics of an element
and with which other elements it can combine. Due to the quantized attractive
force of electrical charges, arising from quantized angular momentum and spin,
electrons are attracted to the oppositely charged protons in the nucleus of an
atom. Electrons, having a fraction (1/17) of the mass of photons, are pulled
into orbit around the protons of an atom, forming specific finite, graduated
concentric dimensional domains called “shells” enclosing the atom.
Using TRUE
unit analysis, we find that, as a consequence of the size of the atom and the electron
in TRUE units, the first shell has a volume of 212 TRUE units, the exact volume
of two electrons. The second shell, with a larger diameter, has a volume of 848
TRUE units, and thus can contain 848/106 = 8 electrons. The maximum number of
electrons that each shell can accommodate can be found by determining the volumetric
equivalence of each shell in TRUE units. The maximum number of electrons in
shells 1 through 6, respectively, is 2, 8, 18, 32, 50, and 72. As more complex
atomic structures are formed by the addition of more of the building blocks,
the finite volumes of the electron shells are filled with electrons, one after
the other.
Atoms
combine to form stable or semistable molecules in mathematically predictable
ways, depending on the number of electrons in their outermost shells. If an
atom, even though electrically neutral and symmetrically stable, has room for
one or more electrons in its outer shell, it can combine with another atom with
that number of electrons in its outer shell to form a new structure. For
example, an H1 Hydrogen atom, which has one electron in its
twoelectroncapacity shell, can combine with Lithium, which has its first
shell filled, and one electron in its second shell. In another example of electron
bonding, two Hydrogen atoms, with a combined two electron deficiency in the
outer shells, can bond with one Oxygen atom which has two electrons in its
outer shell.
H_{2}O, Water, Valence = 2 8 + 10 = 0
Atoms

Particles

Mass/Energy

ג

Total TRUE
Units

Volume

2(H2)+O^{*}

10e

10

1050

1060

1,191,016,000

10P^{+}

170

70

240

13,824,000


8N^{0}+2C_{ג}

176

204

380

54,872,000


Totals

356

1,324

1,680

1,259,712,000=(1,080)^{3}
=(10x108)^{3}

^{*}
See detailed TRUE units analysis for Oxygen listed in order below.
As shown below, Helium with neutrons, 2e + 2P^{+}
+ 2N^{0} is volumetrically symmetric and electronshell stable, and is,
therefore, the form of Helium most often found in nature. Valence is an
expression of the atom’s relative electronshell stability. A symmetric atom
with no valence atoms is very stable.
HELIUM Valence =  2 + 2 = 0 (Inert)
Particle

Charge

Mass/Energy

ג

Total TRUE
Units

Volume

2e

 6

2

210

212^{*}^{}

9,528,128

2P^{+ }

+ 6

34

14

48

110,592

2N^{0}

0

44

32

76

438,976

Totals

0

80

256

336

(2x108)^{3}
^{}

The next natural
element after Lithium is Beryllium. Since it is asymmetric and has two valence
electrons, it is much less stable than Hydrogen (H1) and Helium.
Beryllium, Valence = – 2 + 4 = +2
Particle

Charge

Mass/Energy

ג

Total TRUE
Units

MREV

4e

 12

4

420

424

76,225,024

4P^{+ }

+ 12

68

28

96

884,736

5N^{0}

0

110

80

190

6,859,000

Totals

0

182

528

710

(437.8976…)^{3}

BORON, Valence = – 2 + 5 = +3
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

MREV

5e

 15

5

525

530

148,877,000

5P^{+ }

+ 15

85

35

120

1,728,000

6N^{0}

0

132

96

228

11,852,352

Totals

0

222

656

878

162,457,352=(545.648…)^{3}

We see that Boron is also asymmetric
with valence electrons and is therefore semistable; but the next element,
Carbon, is more stable, being volumetrically symmetric. Carbon and the next two
atoms, Nitrogen and Oxygen are the most stable and abundant elements after
Hydrogen and Helium, and since they are not electronshell stable, they readily
combine with Hydrogen to form natural organic compounds. This establishes
Hydrogen, Carbon, Nitrogen and Oxygen as the primary building blocks of life,
making up between 92% and 96% of living matter ^{ref}.
As we proceed with the TRUE unit
analysis, we will see that the other elements and compounds necessary for life
and the manifestation of consciousness in sentient beings are produced in
abundance by the organizing action of the third
form as ג units, and the conveyance equation.
CARBON, Valence = – 2
+ 6 = +4
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

MREV

6e

 18

6

630

636

257,259,456

6P^{+ }

+ 18

102

42

144

2,985,984

6N^{0}

0

132

96

228

11,852,352

Totals

0

140

768

1,008

272,097,792=648^{3}

NITROGEN, Valence = – 2 + 7 = +5
Particle

Charge

Energy/Mass

ג

Total
TRUE
Units

MREV

7e

 21

7

735

742

408,518,488

7P^{+}

+ 21

119

49

168

4,741,632

7N^{0}

0

154

112

266

18,821,096

Totals

0

280

896

1,176

432,081,216
=756^{3}

OXYGEN, Valence = – 2 + 8 = +6
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

MREV

8e

 24

8

840

848

609,800,192

8P^{+ }

+ 24

136

56

192

7,077,888

8N^{0}

0

176

128

304

28,094,464

Totals

0

320

1,024

1,344

644,972,544=864^{3}

Moving
on to Fluorine, we find it to be volumetrically asymmetric and volatile.
FLUORINE, Valence = – 2 + 9 = +7
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

MREV

9e

 27

9

945

954

868,250,664

9P^{+ }

+ 27

153

63

216

10,077,696

10N^{0}

0

220

160

380

54,872,000

Totals

0

382

1,168

1,550

(977,218…)^{3}

NEON, Valence = – 2 – 8 + 10 = 0 (Inert)
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

Volume

10e

 30

10

1050

1060

1,191,016,000

10P^{+ }

+ 30

170

70

240

13,824,000

10N^{0}

0

220

160

380

54,872,000

Totals

0

400

1,280

1,680

1,259,712,000=1080^{3}

Notice that Hydrogen, Carbon,
Nitrogen, and Oxygen, the basic elements of organic life thanks to the presence of ג in their atomic structure are
volumetrically symmetric and have available valence electrons. Helium and Neon
are also symmetric, but are not among the basic elements of organic life
because they are inert and therefore unable to readily combine with Hydrogen.
All of the other elements analyzed so far, are asymmetric and less abundant in
nature.
It
is no accident that the reactive, volumetrically symmetric elements are
important building blocks of natural organic compounds, and that complex combinations
of them manifest life and consciousness.
SODIUM, Valence = – 10 +11 = +1
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

Volume

11e

 33

11

1,155

1,166

1,585,242,296

11P^{+ }

+ 33

187

77

264

18, 399,744

12N^{0}

0

264

192

456

94,818,816

Totals

0

462

1,424

1,886

(1,193.12…)^{3}

MAGNESIUM, Valence = – 10 +12 = +2
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

Volume

12e

 36

12

1,260

1,272

2,058,075,648

12P^{+}

+ 36

204

84

288

23, 887,872

12N^{0}

0

264

192

456

94,818,816

Totals

0

480

1,536

2,016

(12X108)^{3}

ALUMINIUM^{*}, Valence = – 10 + 13 = +3
Particle

Charge

Mass/Energy

ג

Total
TRUE
Units

Volume

13e

 39

13

1,365

1,378

2,616,662,152

13P^{+}

+ 39

221

91

312

30,371,328

14N^{0}

0

308

224

532

150,568,768

Totals

0

542

1,680

2,222

(1,409.057…)^{3}

*It
is my position that this is the correct spelling, consistent with metal
nomenclature, however, being an American, I tend to pronounce it ‘Aluminum’.
Notice that water and the elements that support life have volumetric equivalence equal to a multiple of 108 TRUE units cubed, and are symmetric and balanced, making them the most abundant elements in the universe. Also note that gimmel had to exist in the first Hydrogen atom coming out of the big bang. This is discussed in greater detail in published technical papers.
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