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 of 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 form of 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, unstable structures.

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 Rotational Unit 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 ג for 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

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