THE SIMPLE MATH OF TRUE UNITS CONTINUED

NOTE:The first part of this was posted on 2/1/16, and that part is re-posted 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 down-quarks 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 space-time domain of our experience. This family of Diophantine equations is represented mathematically by the expression

Σⁿ_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 9-dimensional reality to our observational domain of 3S-1t, we call this expression the “Conveyance Expression”, and individual equations of the expression ”Conveyance Equations”.

 When n = m = 2, the expression yields the equation

(X₁)² + (X₂)² = Z²

which, when related to areas, describes the addition of two square areas, A₁ and A₂ with sides equal to X₁ and X₂ respectively, to form a third area, A₃, 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 sub-set of solutions with AB = X₁, BC = X₂ and AC = Z equal to integers. Members of this subset, e.g. (3,4,5), (5,12,13), (8,15,17), etc. i.e., (3² + 4² = 5², . 5² + 12² = 13², 8² + 15² = 17², … ) are called “Pythagorean triplets”.

To describe the combination of two three-dimensional particles, we have the Conveyance equation when n = 2 and m = 3:

(X₁)³ + (X₂)³ = Z³.

When we define X₁, X₂ 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 three-dimensional 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 r₁, and if there is a second uniform spinning particle rotating at maximum velocity, with radius r₂, we can describe the combination of the two particles by the expression 4/3π(r₁)³ + 4/3π(r₂)³. If this combination produces a third spinning spherical object we have: 

4/3π(r₁)³ + 4/3π(r₂)³ = 4/3π(r₃)³, 

where r₃ is the radius of the new particle. Dividing through by 4/3π, we have:

(r₁)³ + (r₂)³ = (r₃)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 r₁ and r₂ are integers, r₃ cannot be an integer. This means that the right-hand 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 right-hand side of the equation represents an unstable asymmetric spinning particle. The combined high-velocity 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 Newton-Leibniz world, but we actually exist in a Planck-Einstein world. In the real world, revealed by Planck and Einstein, the most perfectly spherical object in three dimensions is a regular polyhedron. (polyhedron = multi-sided three-dimensional form; regular; all sides are of equal length.) The most easily visualized is the six-sided regular polyhedron, the cube. In the Newton-Leibniz 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:

(X₁)³ + (X₂)³ + (X₃)³= Z³, 

which does have integer solutions. The first one (with the smallest integer values) is: 

3³ + 4³ + 5³= 6³ 

It is important to recognize the implications of Σⁿ_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 3S-1t 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 nine-dimensional 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 3S-1t, 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/c² ≈ 1 x10^-47 kg.) to quarks ranging from the “up” quark at about 2.4 MeV/c², to the “top” quark at about 1.7 x10⁵ MeV/c², to the Hydrogen atom at about 1x10⁹ MeV/c² (1.67 x10^-27kg.), to the heaviest known element, Copernicum (named after Nicolaus Copernicus) at 1.86 x10^-24kg . So the heaviest atom has about 10²³ 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 n-dimensional domain is embedded in an n+1 dimensional domain. This means that all distinctions of extent, from the ninth-dimensional 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 tinker-toy 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 sub-atomic 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 three-dimensional Conveyance Equation, they do not simply stick together - they combine to form a new, dimensionally stable, symmetrically-spinning 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 down-quarks, 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 symmetrically-spinning objects formed by the combining of smaller vortical objects form closed vortices in 3S-1t 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 sub-quark 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 time-like 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 Phoenician-Aramaic-Hebrew 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 sub-quark 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 3S-1t measurement, will be determined by the appropriate Conveyance Equation, as demonstrated below. When n = m = 3, Σⁿ_i=1 (X_n)^m = Z^m yields: 

(X₁)³ + (X₂)³ + (X₃)³= Z³ 

The integer solutions of this Diophantine equation in TRUE units represent the possible combinations of three symmetric vortical distinctions forming a fourth three-dimensional symmetric vortical distinction.

THE PRIMARY LEVEL OF SYMMETRIC STABILITY – QUARKS

With the appropriate integer values of X₁, X₂, X₃, 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 3³ + 4³ + 5³= 6³.

Trial Combination of Two Up-Quarks and One Down-Quark, i.e.

The Proton, With Minimal TRUE Units

Particle	Charge*	Mass/Energy	ג	Total TRUE Units	MREV**
u1	+ 2	4	-1	3	27
u2	+ 2	4	0	4	64
d	- 1	9	-4	5	125
Total	+ 3	17	-5	12	216=63

* 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, 3³ + 4³ + 5³= 6³, to find the appropriate values of ג for the Proton, we obtain negative values for גfor the first up-quark and the down-quark and zero for the second up-quark. 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 up-quark and two down-quarks in a neutron, all of the quarks have negative ג units. See the table below:

Trial Combination of One Up-Quark and Two Down-Quarks in TRUE Units

Particle	Charge	Mass/Energy	ג	Total TRUE Units	MREV
u	+ 2	4	-1	3	27
d1	- 1	9	-5	4	64
d2	- 1	9	-4	5	125
Totals	0	22	-10	12	216=63

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 3S-1t 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 1³ + 6³ + 8³= 9³, 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 6³ + 8³ + 10³= 12³, using this solution we have the electrically and symmetrically stable Proton:

The Proton (P⁺)

Particle*	Charge	Mass/Energy	ג	Total TRUE Units	MREV
u1	+ 2	4	2	6	216
u2	+ 2	4	4	8	512
d1	- 1	9	1	10	1,000
Total	+ 3	17	7	24	1728=123

* u₁ and u₂ have the same number of TRUE units of mass and energy, and therefore will register as up-quarks 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 3S-1t 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: (X₁)³ + (X₂)³ + (X₃)³= Z³. If the number of TRUE units in the proton is equal to the integer X₁, the number of TRUE units in the neutron = X₂, the number of TRUE units in the electron = X₃, then the resulting compound entity, will be stable in the 3S-1T domain of physical observations.

We know that the 24 TRUE-unit 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 X₁ 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: 3³ + 4³ + 5³ = 6³ and 1³ + 6³ + 8³ = 9³. (A primitive Diophantine solution is defined as one without a common factor in all terms.) We have also used 6³ + 8³ + 10³= 12³, 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 (X₁)³ + (X₂)³ + (X₃)³ = Z³ are listed below in ascending order. Primitive solutions are in bold.

3³ + 4³ + 5³ = 6³

1³ + 6³ + 8³ = 9³

6³ + 8³ + 10³ = 12³

2³+ 12³ + 16³ = 18³

3³ + 10³ + 18³ = 19³

7³ + 14³ + 17³ = 20³

12³ + 16³ + 20³ = 24³

4³ + 17³ + 22³ = 25³

3³ + 18³ + 24³ = 27³

18³ + 19³ + 21³ = 28³

11³ + 15³ + 27³ = 29³

15³ + 20³ + 25³ = 30³

4³ + 24³ + 32³ = 36³

18³ + 24³ + 30³ = 36³

2³ + 17³ + 40³ = 41³

6³ + 32³ + 33³ = 41³

16³ + 23³ + 41³ = 44³

5³ + 30³ + 40³ = 45³

3³ + 36³ + 37³ = 46³

27³ + 30³ + 37³ = 46³

24³ + 32³ + 40³ = 48³

8³ + 34³ + 44³ = 50³

29³ + 34³ + 44³ = 53³

12³ + 19³ + 53³ = 54³

36³ + 38³ + 42³ = 56³

15³ + 42³ + 49³ = 58³

21³ + 42³ + 51³ = 60³

30³ + 40³ + 50³ = 60³

7³ + 42³ + 56³ = 63³

22³ + 51³ + 54³ = 67³

36³ + 38³ + 61³ = 69³

7³ + 54³ + 57³ = 70³

14³ + 23³ + 70³ = 71³

34³ + 39³ + 65³ = 72³

38³ + 43³ + 66³ = 75³

31³ + 33³ + 72³ = 76

The numbers appearing in the totals in the tables describing quarks, protons, neutrons and atoms are the smallest possible non-negative 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 three-dimensional 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 sub-atomic particles. The correct unique solution can be found for each triadic sub-atomic 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 6³ + 8³ + 10³= 12³, the first attempt to find the TRUE unit configuration of the neutron is shown below:

Trial Combination of One Up-Quark and Two Down-Quarks in TRUE Units

Particle	Charge	Mass/Energy	ג	Total TRUE Units	MREV
u	+ 2	4	2	6	216
d1	- 1	9	-1	8	512
d2	- 1	9	1	10	1000
Totals	0	22	2	24	1728=123

Since this solution still produces a negative value of ג for d₁, 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 9³ + 12³ + 15³= 18³

Second Trial of Quark Combinations for the Neutron

Particle	Charge	Mass/Energy	ג	Total TRUE Units	MREV
u3	+ 2	4	5	9	729
d2	- 1	9	3	12	1,728
d3	- 1	9	6	15	3,375
Totals	0	22	14	36	5,832=183

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 TRUE-units analysis of Helium as an example, we have:

Attempt to Construct a Helium Atom with P⁺ = 24 and N⁰ = 36

Particle	Charge	Mass/Energy	ג	Total TRUE Units	MREV
2e	- 6	2	78	80*	512,000
2P+	+ 6	34	14	48	110,592
2N0	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 7³ + 14³ + 17³ = 20³.

Third Trial of Quark Combinations for the Neutron

Particle	Charge	Mass/Energy	ג	Total TRUE Units	MREV
u3	+ 2	4	3	7	343
d2	- 1	9	5	14	2,744
d3	- 1	9	8	17	4,913
Totals	0	22	16	38	8,000=203

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 N⁰, the first integer solution of the conveyance equation containing the values X₁ = 24 and X₂ = 38 is obtained by multiplying both sides of the primitive solution 12³ + 19³ + 53³ = 54³ by 2, yielding  the integer solution 24³ + 38³ + 106³ = 108³.

Helium Atom with P⁺ = 24 and N⁰ = 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
2N0	0	44	32	76	438,976
Totals	0	80	256	336	10,077,696=2163

^*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 down-quark 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
N0	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 right-hand column in these tables contains cubed integer amounts representing the Minimum Relative Equivalence Volume (MREV) for each particle making up the combination of sub-atomic particles.

**The TRUE unit values for the elementary particles are uniquely determined by conditions necessary for a stable universe. The values for up- and down-quarks 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 semi-stable 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 3³ + 18³ + 24³ = 27³. 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 co-multiples of 24 and 38 as terms in the left side of the equation, we can use the solution that provided a stable Helium atom: 24³ + 38³ + 106³ = 108³. Using it to represent the Hydrogen atom, we have:

TRUE-Unit 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
Cג*	0	0	38	38	54,872
Totals	0	18	150	168	1,259,712=1083

* 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 3S-1t. This satisfies the conveyance equation and produces a stable Hydrogen atom with a total volume of 108³.

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 3S-1t, 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
N0	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
2N0	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 TRUE-unit 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
2N0	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
4N0	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 multi-dimensional 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 semi-stable vortices, spinning structures consisting of stable triads of protons, neutrons and electrons. A third level of stable and semi-stable 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 semi-stable molecules in mathematically predictable ways, depending on the number of electrons in their outer-most 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 two-electron-capacity 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. 

H₂O, 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
	8N0+2Cג	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⁺ + 2N⁰ is volumetrically symmetric and electron-shell stable, and is, therefore, the form of Helium most often found in nature. Valence is an expression of the atom’s relative electron-shell 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
2N0	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
5N0	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
6N0	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 semi-stable; 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 electron-shell 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
6N0	0	132	96	228	11,852,352
Totals	0	140	768	1,008	272,097,792=6483

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
7N0	0	154	112	266	18,821,096
Totals	0	280	896	1,176	432,081,216 =7563

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
8N0	0	176	128	304	28,094,464
Totals	0	320	1,024	1,344	644,972,544=8643

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
10N0	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
10N0	0	220	160	380	54,872,000
Totals	0	400	1,280	1,680	1,259,712,000=10803

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
12N0	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
12N0	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
14N0	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.