Like it or not, theories will eventually come down to how you “add up” the Universe, so understanding some basic concepts about numbers and numerical systems is an essential building block. The descriptions following are “common sense” interpretations, based on historical and mythological use of numbers, not the mathematician's entries in encyclopedias.

There are three types of numbers, which are not apparent from their value but only from the context they are used in:

• Cardinal, a number that represents a quantity or magnitude. “I have 5 friends.”

• Ordinal, a number that represents a rank or position within a set (ordering). “Your seat is in row 5.”

• Nominal, a number that is just a name or identity for something, like a phone number.

In the context of the Reciprocal System, motion (a ratio) is composed solely of cardinal numbers, which is why Larson refers to that ratio as a scalar motion. “Scalar” means, “magnitude only.” Scalar motion does not contain ordinal nor nominal numbers.

Larson's coordinate space (or time) is composed of ordinal number he calls absolute locations, since their value is determined by an ordered set starting at zero.

One of the problems identified with both Larson's Reciprocal System and conventional theories is that they lack the proper context for numbers, particularly in mathematical equations. This has led to a great deal of confusion and misinterpretation in both systems. By pointing this out now, it is hoped that confusion can be minimized.

Natural Numbers (Cardinal numbers)

It all starts with simple counting. I have one, he has two, she has three. Larson's Reciprocal System and this reevaluation both start with the ancient Greek concept of Natural Numbers that begin with one and increase by single, whole amounts to a finite¹ maximum. Zero² and infinity³ are not included. Counting is the cardinal, scalar concept of magnitude only, not inferring any direction, geometry or coordinates.

Because the minimum quantity of the natural number system is unity, “one” becomes the reference point for natural ratios, the multiplicative inverse.

Whole Numbers (Ordinal numbers)

Once you can count, you can also “order”—what comes first, second, third… to define a sequence. Sequences imply direction, fewer to greater or vice versa. Once a sequence is established, it can point in two directions… you can always have more, but what about not having any? The ancient Greek argument of “how can nothing be something?” comes into play here, nothing is undefined—either you've got it, or you don't. It isn't until we impose order, a sequence, that the concept of “not having” comes into play and zero is born. The addition of zero to the set of natural numbers gives the concept of whole numbers, making a more complete system of representation, beginning with “I don't have any” through “I have this quantity.” But this is misleading!

Whole numbers are not cardinal quantities; they are ordinal distances from an origin of zero. Using whole numbers, you do not actually have “one” of something, but “one more than nothing.” It may sound like a trivial difference, but it is an important one⁴.

Integer Numbers

Now it's pay-back time! With the rise of trade and mercantilism, simple counting gave way to accounting, the tallying of quantities and the concept of debt (or owing). “You owe me two apples,” and when those apples were returned, the count of 2 went against the debt of 2, resulting in no debt or “nothing”—back to zero. This gave rise to negative quantities, those “owed,” and the integer number system, which included none (zero), the natural, counting numbers, and their opposite—the negative counting numbers, which are again, ordinal in structure.

Rational Numbers (Cardinal ratios)

Trade also gave rise to the concept of fractional parts, as individual items could be subdivided, such as one apple pie into 8 pieces. This resulted in the proportions expressed as the ratio—how many times one thing is contained in another—and the rational number system to represent it. Rational numbers are a ratio of natural numbers, meaning that the minimum quantity of both aspects⁵ is unity and the minimum ratio is 1/1.

Larson's concept of scalar motion is defined by a rational number. By assigning a unit of space to the numerator and of time to the denominator, he creates the concept of scalar speed⁶.

Real Numbers (Ordinal measure)

The use of counting numbers and ratios lead to the general system of measure, where numbers no longer represented quantities but relative amounts. If something is “4 miles distant,” that does not mean you have “4 miles” in a box marked “distant.” Relative measure required a starting point, or a “datum” as used in the Reciprocal System, from which measurement of quantity is made. Note the distinction here: quantity is absolute (cardinal); measurement is relative (ordinal). Given that quantity can have proportion (ratio of one quantity to another), measurement can also have proportion… these proportional measurements became the set of real numbers, which can express any measurement, when given a place to start.

Irrational Numbers (Ordinal ratio)

To this point, the number systems have been primarily linear in use. The ordering systems can be placed in a straight line, the ratios of rational numbers are analogous to slopes of a line—but still a straight line. But that is only one aspect… the other is the curve, when ordering becomes dependent upon two dimensions and is no longer commutative. When a curve interacts with a straight line, such as the ratio of a circumference to its diameter, another type of real number comes into existence, that of the irrational number—a number that cannot be expressed as a ratio of two integers, as it is not a ratio of quantities, but a ratio of measurements. Common examples of irrational numbers are π and the natural log, e. These numbers represent a different type of ordering system based on angles, with different forms of representation—polar sequencing.

Complex Numbers

The basic problem with mixing linear and polar numbering systems together was that they did not play by the same rules. Trigonometry was the first to correlate the two; later on, the concept of the imaginary number came into play and when joined with the linear became the complex number system, allowing both the linear and polar values to be represented on a linear graph known as the complex plane or Argand diagram. This simple concept allowed both linear and polar coordinates to be represented with a single “number,⁷” creating a global system of all the number systems preceding it.

Extended Complex Numbers

The final step is to close the system of complex numbers to include the counterpoint to nothing—that of totality (everything), or infinity⁸. Zero and infinity represent the numeric extremes of nothing to everything and are treated as boundaries—not limitations—in the system of extended complex numbers.

Summary

As explained, number systems have developed from simple, cardinal counting, through ordinal sequencing, measurement, proportions and mixing lines with curves. Though there are many other number systems for specialized branches of science and mathematics, RS2 only requires comprehension up to the extended complex number system to represent its universe of motion.

As can be seen from the number systems, once assumptions are made against the natural, counting numbers there is a change in context as to what a number means, moving from the natural, cardinal count to a constructed, ordinal measure from a specific reference.

Two reference systems are being defined here, the first being the “natural” or cardinal reference that will define scalar motion as a rational number. The second is the projection of the natural reference system onto a screen we call “reality,” which is a normalized snapshot of the natural reference system that our physical senses can make use of to make observations and measurements. This second reference system will be referred to as the “diagrammatic” reference, a term coined by Miles Mathis in his papers¹ concerning, “The Greatest Standing Errors in Physics and Mathematics.”

The Natural Reference System

The natural reference system is characterized by rational numbers, where both aspects have a minimum quantity of unity with a finite² upper bound. No zero or infinity here, as those are ordinal concepts. The minimum dimension in this realm is also unity, since there is no zero:

( [1..a] : [1..b] )^[1..n]

In other words, the smallest quantity that can exist in the natural realm is (1 : 1)¹, which students of the Reciprocal System will recognize as Larson's “unit speed.”³

In order to distinguish natural ratios from diagrammatic ones, natural ratios will use the colon separator: a:b, rather than the slash.

The Diagrammatic Reference System

The diagrammatic reference system consists of relative measurements defined by the ordinal numbers. Since it is a reference system based on measure, the system defines a minimum measure of zero (nothing, the origin) and a maximum measure of infinity (everything). Zero and infinity are required in a diagrammatic reference system because there are certain, geometric conditions that cannot be expressed and the results extend to the limits of expression.

In the Reciprocal System, the diagrammatic reference system is what Larson refers to as “Euclidean geometry” in his Fundamental Postulates, referring to both the conventional reference of 3D space with clock time, and 3D time with clock space. Clock time, clock space, time dilation and space dilation only occur as a result of the normalization of the natural reference system into a diagrammatic reference system. These concepts, as conventionally understood, do not exist outside of the Euclidean stratum of geometric projection.

The Conventional Reference System

The conventional reference system is the one we use in our everyday perception of the world around us. It is the one we observe and measure, a subset of the diagrammatic reference system with the properties of three dimensions of space, clock time, uniform scaling in all dimensions, and Euclidean geometry.

One of the prerequisites for observation and measurement is to know where you are measuring from. Larson's original work contains zero (physical location reference) and unity (scalar motion reference). RS2 defines several datums¹ of measurement based on basic, arithmetic concepts:

• Additive inverse: Zero, the one we are all familiar with, 0 ± n.

• Subtractive inverse: Infinity, the other end of the number line, ∞ ± n.

• Multiplicative inverse: Unity, the natural datum of the Reciprocal System, n/1, 1/n.

• Evolutive inverse: Dimensional, from where powers (evolution) and roots (involution) are measured, n^m/1, n^1/m.

Zero and unity are fairly well understood, as we use them in everyday life. The dimensional datum—why the Universe has 3 dimensions—is there, but goes unnoticed. However, infinity is probably one of the least understood concepts that mankind has ever conceived of. It is commonly defined as “unbounded or without end.”² For the purposes of the Reciprocal System, this is inadequate without some clarification.

Zero and Infinity

For the purpose of this reevaluation, zero and infinity are nothing more than pseudo-locations in the diagrammatic reference system from which we count or measure a finite quantity. I realize that sounds a little strange, but consider them as the conditions of “nothing” (zero) and “everything” (infinity) in the number system. From this perspective, the references can be described as:

Zero: the location or point measure, used when starting with nothing and building up to something.

Infinity: a directional or plane measure, used when starting with everything and excavating down to something, as in a sculptor creating a piece of art from a block of rock.

To clarify, measuring from these reference locations do not measure the same values. When measuring from zero, you are adding “things” to nothing, like counting the number of balls in a bucket. When measuring from infinity, you are subtracting “holes” from everything, analogous to counting bubbles in a bucket of water.

The situation arises in mathematics when using these datums, where an attempt is made to count “holes” by counting “things.” This, of course, does not make much sense. One cannot determine the number of apples in a bucket by counting the gaps between the apples³. But mathematics has devised a way to count “remainders,” a process of starting at a quantity and estimating how much more could be added before “everything” is reached. This technique uses difference equations, discussed later on in this section.

The Natural Datum: Unity

By postulating motion, a rational number, as both the contents and the container of a universe of motion, the natural datum is unity: the minimum quantity for the multiplicative inverse. The maximum quantity is finite—zero and infinity are never actually reached in any situation regarding motion, so the problems associated with 1/0 and 1/∞ are not an issue.⁴

We are accustomed to considering zero as the center of measurement. With a natural datum of unity, one becomes the center—not zero. When taking measurements, we can either measure either from unity or towards unity. Larson calls this a scalar direction and refers to “from unity” as outward and “towards unity” as an inward, scalar direction. These concepts indicate a shift from the projective to affine stratum of geometry used to express the system, discussed in the section on Architectural Patterns.

The Dimensional Datum

The dimensional datum concerns the counting of powers, n¹, n², n³,... which gives us the concept of dimension. In order to determine a dimensional datum, one must count the quantity of operators involved, not the specific magnitudes of the operators.

In a closed group of operators, like [i j k], the result of the combination of any number of the basal elements is also a member of the same group. The result of any such combination can be known only if all the possible binary combinations of the elements are first defined in terms of the basal elements i, j and k themselves. Let there be n basal elements in a group. Then the number of unique binary combinations of these elements, in which no element occurs twice, is n(n-1)/2. We can readily see that a group becomes self-sufficient and finite only if the number of binary combinations of the basal elements is equal to the number of those basal elements themselves, that is:

In the natural reference system, the only solution for n is 3. Therefore, there are three, and only three, scalar dimensions of motion. The natural, dimensional datum is (1:1)³, which Larson refers to as the “progression of the natural reference system,” a unit speed (not velocity) in 3 dimensions.

In the diagrammatic reference system, the solutions are 0, 3 and ∞. When the number of dimensions is less than 3, the system will cascade down to zero. When greater than three, it will cascade up to infinity.

If we regard any coordinate system as a group of orthogonal rotations about a datum, its dimensionality has to be three in order to make it self-sufficient, dimensionally. Three is the point of dimensional stability, the diagrammatic, dimensional datum, and can be considered independent dimensions. There is no restriction to the number of dimensions involved, however anything beyond the stable three dimensions would become dependent upon that structure and hence, dependent dimensions.

The dimensional datum also has a reciprocal structure to it, just as the unit datum does, which is expressed mathematically as powers and roots.

Opposites come in different forms. Basic mathematics represents them as inverses: the additive inverse and the multiplicative inverse. The use of complex numbers add another “opposite,” that of the complex conjugate and the dimensional datum adds an evolutive opposite¹, roots and powers. Opposites require a reference point from which to measure the “opposite” character. This reference is called a datum². A datum can have any value, but zero and unity are the common ones.

Additive Inverse: Polarity from Zero

The most familiar algebraic inverse is the additive inverse, which has a datum of zero and a polarity of positive or negative from zero. Larson uses the additive inverse for displacements³ and the 3D coordinate systems: material extension space or cosmic extension time. Note that the number line includes zero and both positive and negative directions towards infinity.

Subtractive Inverse: Polarity from Infinity

A completely unfamiliar inverse is the subtractive inverse, which has a datum of infinity and a polarity of positive or negative from infinity. Note that the number line includes infinity and both positive and negative directions towards zero. This is possible because infinity is just a reference datum—not a quantity.

Multiplicative Inverse: Dichotomy

The multiplicative inverse has a datum of unity, with a dichotomy⁴ of greater or less than unity. Larson uses the multiplicative inverse for scalar motion (speed and energy). Note that the number line does not include a negative direction towards infinity, as does polarity, but an inverse direction—expanding towards infinity or shrinking to it. 0, 1, ∞.⁵

Complex Opposite: Conjugate

Complex numbers add another form of opposite that can be represented on an Argand diagram (a linear, real axis and an polar, imaginary axis). Both have a datum of zero and the relationship is called the conjugate.⁶ The complex plane contains a single datum at 0+i0, which is the center of the diagram. As a 2-dimensional representation, the complex plane contains four directions, a composite of both polarity and dichotomy.

Some of the characteristic features of the ideas in calculus has to do with the concept of infinitesimals, and the corresponding infinitesimal elements, such as dx, dy and dt. In order to first understand the different applications of calculus, we need to clarify what a differential means, and what it means to integrate them. A geometrical derivation of those concepts are outlined here.

First, consider a 2-dimensional plane as an example. There are two ways of characterizing elements on that surface, which are essentially duals¹ of each other. Duality is a geometric reciprocal where one aspect is a single point at the intersection of two lines and the other is a line drawn through two points. This means that every given curve can be defined in two ways: one which gives points (where we can fill in the lines by “connecting the dots”), and the second as an envelope of lines, where the lines are given, and we “fill in” the dots.

The concept of defining a curve by its envelope is well known, but seldom applied in conventional science, as the point-based system has been developed for plotting. But essentially what we are doing when we move from the envelope determined curve, to the point-wise determined curve, is to shift our reference from the line at infinity² to the point at zero. A similar thing can be envisioned in three dimensions, where a spherical surface, for example, is either constructed out of points on it, or out of a series of circles, while at the same time being an envelope of either a planes coming from all sides, or lines coming from all sides.

So this means we can either “build up” a surface, or even “build down” a surface. Hence there are two ways of accomplishing the same thing, depending upon the reference point selected.

Now let us consider a curve such as the one shown in the figure above and identify what the process of differentiation does to it. There is a point selected on the curve and the differential at the point gives us the tangent to the curve, at that point. In other words, we are getting the slope of the curve at that point.

Hence, if

where 2x is now its slope.

What differentiation has done is to converted to an angular measure; the ratio dy/dx is actually the tangent of the angle of the slope. The geometrical transformation is from a point to a line, where we go from (x, y) to dy/dx, and from two lines (parallel to the axes) defining a point, to two “points” (dy and dx) defining a line. An inversion of geometric duality is accomplished.

The inverse process, integration, is that of building up. It is generally the transition from a line to the area under the curve, or from the area to the volume in three dimensions. The logic is the same in that from the reference line at infinity, we are shifting back to the reference zero point, therefore needing to the increase the dimensionality by one in the process.

Since differentiation is a “building down” approach, it is insensitive to changes in the origin of the coordinate system. Since it is reckoned with respect to the line at infinity, every line is “equidistant” from the line at infinity, and shifting around a given graph anywhere in the coordinate system will not change its differential. In other words, if the function y is replaced with y+c, there would be no change in the differential. This is a direct consequence of the fact that the point at zero is not fixed, unlike the unique line at infinity.

This provides the reason for asymmetry in the behavior of the differential and the integral. Integration always results in the addition of an arbitrary constant, while differentiation always removes the effect of the arbitrary constant.

This also provides a means of determining whether or not the application of differential equations is valid or not in any particular system. From just geometry, we can say that in systems where the point at zero is uniquely defined, we cannot use differential equations. Nor can we use them where the two quantities are not related with the same line at infinity³. We cannot use it for motion in Larson's time (or space) region, where one component progresses while the other is fixed at unity, except for the only function that is unchanged under the differential, the exponential function. It is seen that the application of quantum mechanical differential equations leads to the solutions which are combinations of exponentials, both real and imaginary.

Also note that this means a reversal of the previously accepted pattern of analyzing the problem. Instead of setting up a differential equation and then trying to solve it for physical answers, we instead set up the geometry from the physics considerations and then create the differential equation from it. This had largely been the case before calculus started being applied to mechanics during the time of Newton, when it was purely based on the kinematic aspects of velocity and acceleration. Thereafter, after Leibniz and Newton attributed a reality to differentials, it has been followed by a tendency to “find the solutions to a given differential equation.” It is the exact analogue of finding the behavior of the world under a given set of forces, instead of observing that force is a property of motion.

From the logic developed, it can also be predicted that the appropriate equations that should be utilized in the case of time region phenomena will have the form of difference equations, dealing with finite differences instead of differential equations. Liquids must be treated with rotational differentials, taken with respect to the center of the earth as the absolute frame of reference⁴.

This provides the reason for the difficulty faced for the last 150 years of aligning two major branches of physics into the differential equation mold... it is a question of the right geometry for the right motion, and thereby the right equation, and not the reverse.

Larson's technique for modeling the universe is very similar to today's technique of presenting information on the World-Wide Web, specifically separating out the concepts of a data model (scalar motion), how the data is viewed (coordinate locations) and how the two correlate (Fundamental Postulates). This is analogous to modern software architecture known as MVC, the model, view and controller method of programming computers.

“This theory introduces two new concepts into physical science: the concept of physical location, and the concept of scalar motion.”¹

Larson refers to his model as “scalar motion,” rational numbers (a ratio of natural numbers) that comprise the data model to define any object in the universe. This would be analogous to your checkbook data that is stored in a banking computer. The numbers are stored there as thousands of binary digits, and though useful to a computer, is rather useless to find your account balance. Larson's scalars are the same concept; just numbers, without any “formatted output.”

Next, he defines the concept of “physical location,” the coordinate view of a grid of points, spaced one natural unit apart, forming a framework in which his scalar motion can be observed and measured, his diagrammatic reference system. This would be analogous to a blank check register, either hard copy or on a monitor. The view just defines how the model will be presented—the actual numbers are not yet filled in. Larson refers to these views as extension space or time; the concept being that the scalar motion was extended out into a system of representation.

Lastly, Larson introduces his Fundamental Postulates—the controller² of the system that outlines the interaction between the model and view. This is analogous to formatting the binary data in the computer into readable numbers and placing those numbers on the check register, including the running balance and totals. It connects the unrepresentable data to an observable and measurable system of reference. As a theatrical analogy, the script of a play is the controller, the actors are the model and the stage is the view. This is the classical, scientific view of the Universe.

Having developed the Reciprocal System in the 1950s, Larson had this same, conventional view which is expressed in his postulates. Scalar motion is projected into a coordinate system, where it is observed and measured. However, like modern science, one important consideration is lacking: that of feedback from the coordinate system affecting scalar motion. Using the computer analogy, the concept of “user input,” where something is typed on the keyboard or the mouse is used to select something on the viewing screen. In the context of the Reciprocal System, how kinetic interaction of locations can alter scalar motion³.

The reevaluation separates the controller into two, different sections, analogous to the way Larson split the Universe into material and cosmic sectors. To incorporate this additional controller, the MVC model has been updated to an “MCVC” model:

• Model (database): Scalar Motion.

• Model-View Controller (server programming): The process that projects scalar motion into a coordinate system.

• View (browser on monitor): The 3D coordinate system and clock we experience.

• View-Model Controller (keyboard and mouse): The process that integrates coordinate locations back to scalar motion.

By separating the controllers into two, distinct processes, the architectural pattern now shows the same, inverse relationships that exist throughout the Reciprocal System of theory between Model and View, and the Model-View and View-Model controllers.

With the updated MCVC model, the Model and View components are now reciprocally related, what would be called in philosophy the “objective” and “subjective” perspectives.

The Model Component

The scalar motion that comprises the Model component is objective in the sense that it is independent of any observer. Scalar motion is literally “pure data,” and is nothing more than a ratio of magnitudes, without any interpretation as to orientation, direction nor location.

With respect to the Reciprocal System, it should be noted that scalar motion, being nothing more than magnitude, is considered to be part of the projective stratum of geometry, not the Euclidean, as concepts such as length, direction and location do not exist in the pure data model. Larson recognized this, and it is the reason he has two, separate postulates⁴. It is important for further development not to confuse the Model (scalar) with the View (coordinate).

The View Component

The View component is subjective in the sense that it is dependent on all observers, being by a series of assumptions defined by our physical senses and consciousness. Unlike the fixed ratios of scalar motion, what we observe and measure as “reality” is totally defined by our body, mind and spirit and the rules that we have imposed upon ourselves. The “View” is created by the rules of the observer⁵, not by the organization of the data in the Model. In this architectural pattern, the view deals strictly with the presentation of scalar data in a 3-dimensional, diagrammatic, coordinate reference frame, of which there are two defined in the Reciprocal System:

• The Material Sector: 3-dimensional, coordinate space and 1-dimensional clock time.

• The Cosmic Sector: 3-dimensional, coordinate time and 1-dimensional clock space.

Again, I will emphasize that the View is not the data, it is a projection screen on which scalar motion is observed and measured after it has been transformed by the Model-View controller. The transformation process, described below, will lose information resulting in some apparent contradictions, such as wave-particle dichotomies, but it is only the misinterpretation of this “shadow on the screen” that is causing the contradiction, not the data in the Model, itself.

The Model-View Controller Component: Projective Geometry

The Model-View controller is the formal definition of the functions that we use to interpret scalar motion and bring it in to our consciousness, to make us aware of the surrounding environment. Its character is defined by Larson's First Postulate, which defines one component, motion, existing in three dimensions, in discrete units and with two reciprocal aspects⁶.

“Motion” and “two reciprocal aspects” define the type of information being obtained from the Model, the ratio of scalar magnitudes that Larson labels, “motion.” “Existing in three dimensions” indicates how many of these parameters are being passed to the controller for projection into a coordinate view⁷.

Projective geometry is the study of relationships between geometric entities, divided into a series of strata that have distinct, and well-defined, assumptions. To project a model (such as scalar motion) through these strata is much like projecting a beam of light through a series of glass plates, where each plate has a different property (assumption) associated with it. The final result contains an image that has been filtered and rearranged, based on the properties of the plates the light has traveled through, although some information is lost in the process. This is indeed the case with the Reciprocal System's scalar motion: only a single, scalar dimension can be fully represented in a coordinate view. The other two dimensions just modify that representation. So what is seen as atomic structure is a “dimensional reduction” from the 3 dimensions of scalar motion to the 3D coordinate structure.

Geometric Strata

Geometry is divided into four “strata” or “layers”. Each strata adds a set of assumptions that create certain invariants for that layer. An “invariant” is a property of a configuration of geometric entities that is not altered by any transformation⁸ belonging to the specific strata.

There are normally four strata defined, with the Euclidean stratum being “what you see is what you get,” the realm of our conventional experience.

• 		Stratum	Invariants		Degrees of Freedom
  			Invariant Property	Projective Invariant
  	G E O M E T R Y	Projective	1. cross-ratio	1. incidence 2. collinearity 3. tangency	15	15 scale
  		Affine	2. relative distance along direction 3. parallelism 4. plane at infinity	Lines or planes that intersect at infinity are called “parallel.” Ratio of lengths along a certain direction is a cross-ratio with one point at infinity.	12	9 scale 3 translation
  		Metric	5. relative angles 6. relative lengths 7. absolute conic (circle at infinity defining rotation)	Scale applies equally to all dimensions.	7	1 scale 3 orientation 3 translation
  		Euclidean	8. absolute distance	Everything is scaled to unity, removing that degree of freedom.	6	3 translation 3 rotation

Summary of Geometric Strata

• Projective Stratum	This is the least structured stratum of geometry, dealing with the concept of a “ratio of ratios” — the cross-ratio.
  Affine Stratum	Creates the assumption of a plane at infinity. This creates parallel and orthogonal relationships between geometric entities by placing one of the 4 points of the cross-ratio in the plane at infinity. This “squares up” geometric entities by making lines and planes parallel.
  Metric Stratum	Adds the concept of scaling to the affine stratum.
  Euclidean Stratum	Fixes the scale at unity.

Application in the Reciprocal System of physical theory

It can be readily seen that the Reciprocal System concept of scalar motion is in actuality that of the projective invariant, the cross-ratio. The only difference is that the cross-ratio is a more generic term than scalar motion, with the latter having named aspects of “space” and “time” whereas the former does not label the aspects⁹.

• Projective Stratum cross-ratio	Reciprocal System scalar motion
  Four scalars reciprocally related as a ratio of ratios, with no zero or infinity.	Two scalars reciprocally related. Whereas Larson assumed zero and infinity, his motion is actually a cross-ratio with one value at zero, two forming a ratio, and the final at infinity.
  Unnamed aspects.	Named aspects of space and time (defining speed).
  Has magnitude only.	Has magnitude only.
  Non-zero.	Non-zero.
  Finite.	(Unaddressed, but all motion applied as finite values.)
  Cannot be translated.	Cannot be translated.
  Cannot be rotated.	Cannot be rotated.
  Can be scaled.	Can be scaled.

• Geometric Concept	Application	Results in Reciprocal System:
  Projective Stratum cross-ratio	Name aspects space and time.	Scalar motion.
  Affine Stratum plane at infinity	Makes motion relative to other motion resulting in towards/away (in/out) direction of motion.	Scalar direction; linear vibration.
  Metric Stratum absolute conic	Defines angular relationship between motions.	Rotational base.
  Euclidean Stratum fixed scale	Defines common perspective by fixing scale at unity.	Vectorial motion in absolute Euclidean coordinate system.

The View-Model Controller Component: The Observer Effect

Larson, in his publish works, does not address the Observer Effect¹⁰. In his system, designed by a chemical engineer considering the inanimate realm of atoms and chemistry, there is no feedback into the system. Chemical reactions behave the same way, whether you are looking at them or not. However, with the inclusion of projective geometry as a system to transform scalar motion into coordinate motion, it became apparent that the observer—who defines the way in which things are viewed—could not be overlooked, nor omitted as a source of potential change to the data model.

The View-Model controller component is often not considered as a separate function. In the computer industry where the concept gets its origin, the mouse and keyboard are attached to the computer, right along with the monitor. They are essential to the interaction of the system, as you could not browse to a web site if you could not type in its URL, or be able to point the mouse at a link and click the button. But the architectural pattern being adopted by this reevaluation is more generic than a simple, user interface to a computer, so separating the component and identifying the functions it performs is a required part of the pattern, in order to come “full circle.”

During consideration of this architectural pattern, the question arose as to how (or what) actually provided “feedback” in a physical sense. As Larson describes in Nothing But Motion, all chemical interactions are done at a scalar level (the Model), not in the coordinate system (the View). This, along with information in Larson's final book, Beyond Space and Time, indicates that feedback is of two, distinct types: a natural consequence of the projection in the view, and what could be termed an unnatural consequence initiated by a biological organism.

Feedback from a natural consequence would be anticipated as part of the projection of the Model, such as a burning piece of wood igniting an adjacent piece of wood. The thermal conditions of the atoms define whether or not it is burning, and the projection into a coordinate system determines if the adjacency is sufficiently close to induce thermal motion in the atoms of the non-burning piece, therefore sending feedback to the model that, in turn, increases the thermal motion at the scalar data level. The entire system is 100% predictable, or predestined.

Now enter a young child with a box of matches, and he ignites a piece of wood that would not normally catch fire in the given environment. This is an unnatural consequence from the perspective of the inanimate realm (though it may be predictable from the biological—or parental—realm). As such, the act of the child to alter the thermal state of the scalar motion comprising the atoms in the Model is feedback—an observer effect from the view, changing the model. The system is unpredictable, or an act of free will¹¹.

Knowing this, the View-Model controller component is basically the reciprocal of the Model-View component, taking coordinate information and altering the model from it. Projective Geometry is the tool we are using for the Model-View differentiation, and can also be used—in reverse—to perform a View-Model integration, where information from the view (primarily structural location) is dimensionally expanded through the use of a type of constant of integration, to affect the scalar motion of the Model¹².

In summary, it can be seen that the four components of this architectural pattern work together to make a full circle of interaction, where each can affect the other through a series of well-defined relationships.

Yin-Yang: the Gender of Motion

In the Reciprocal System, motion is composed of two aspects: space and time, which are inseparable—you cannot have space without time, nor time without space. Larson calls the ratio of the two, “motion.”¹

The inseparable pairing of opposites is not an unfamiliar concept. Eastern philosophies have long accepted the idea of the yin-yang, a pairing of the feminine and masculine principles. That which is yin is feminine, curved, rotational, smooth or cold. That which is yang is masculine, straight, linear, rough or hot. These philosophies accept that opposites can only exist in relation to each other. The uniform motions discussed next have similar characteristics, so one can draw a correspondence between yin-yang and uniform motion: linear motion is yang; rotational motion is yin.

However, the yin-yang concept differs in one, important aspect from Larson's space-time: the aspects have gender… yin and yang are “opposites” in the truest sense, embodying the concepts of additive inverses, multiplicative inverses and conjugates. Larson's Reciprocal System was limited to linear motion (the yang aspect). It was Prof. K.V.K. Nehru's work on birotation² that represented the yin aspect of the Reciprocal System.³

Larson's and Nehru's research merge into RS2, as the best of both worlds. This includes the gender distinction by defining a reciprocal relation that Larson never considered: the geometric reciprocal of the linear, Euclidean geometry being polar, Euclidean geometry. Motion, in RS2, has aspects of yin and yang that can manifest as either space or time, depending on the observer perspective. What is meant by this, is that the observer often determines what is yin and what is yang in the field of observation, just as one would determine what is “right” and what is “left” depending on where you are standing.⁴

In the case of our everyday existence, we see the coordinate system of space as yang, extending linearly outwards from us towards infinity. Time, as used in the context of the Reciprocal System (not clock time) is yin, but typically only observable at the atomic level, once we cross the unit boundary out of space and into the time region of atomic rotation. From our conventional reference, space is yang and time is yin.

However, if you were born in the Cosmic sector, the sector of the Universe where time is 3-dimensional with “clock space,” you would argue that point, saying that time is yang and space is yin, based on the way they behave. Time would have the linear, coordinate system extending around the Cosmic observer, while atoms would have a “space region” of rotational, atomic systems.

To avoid this confusion with the terms “space” and “time,” RS2 uses the term “yang” to represent the linear, translational aspects of motion and “yin” to represent the polar, rotational aspects of motion, because that correlation is consistent in all frames of observation.

Motion as a Complex Quantity

For scalar quantities, where one is dealing solely with magnitudes only, the simple ratio of a/b or a:b is fine, as neither aspect has any properties other than magnitude. But to properly represent the yin-yang of motion, both its yin (polar) and yang (linear) properties need to be represented clearly in a single “formula.” Thanks to 16^th century mathematician Gerolamo Cardano, such a formulation exists: the complex number:¹

The question may arise, “Is a ratio (motion) and a complex number the same, basic concept?” Consider that motion is a ratio of space to time and that ratio is simply the slope of a line. Take, for example, a motion of 3/5. That is 3 “space” and 5 “time” in Larson's system and plot it on a graph where “time” is on the vertical axis and “space” is on the horizontal, then compare it to a plot of 3+i5 on the complex plane²:

The only visible difference between the two graphs are the labels. But the mathematical difference is what is crucial: the real axis represents the linear, yang aspect of motion, while the imaginary axis represents the polar, yin aspect of motion.

These are not new Reciprocal System concepts, just a different way of expressing the ratio of motion that introduces a new, intermediate step between Larson's scalar motion and coordinate motion: sector motion. Sector motion adds the yin-yang property to the magnitude-only scalar motion, creating the concepts of linear (yang) and angular (yin) motion. Sector motion is not scalar motion because it possess this yin-yang property; nor is it coordinate motion, as there are no coordinate components—it sits somewhere inbetween, sort of a “missing link” between the scalar and the coordinate systems defining two different systems of measurement.

The two aspects of a ratio can be expressed as a:b or b:a (now many times a contains b, or how many times b contains a). The magnitudes of a and b do not change, but the choice is determined by how the observer wants to express the ratio.

Larson uses the terms space (s) and time (t) to represent the two aspects of motion, so there are two possible forms of sector motion: one where space is yang and time is yin (which Larson calls the Material Sector), and the other where time is yang and space is yin (Larson's Cosmic Sector):

Material Sector Properties

In the conventional frame of reference, space is yang, linear and is measured by the real aspect of sector motion. It is therefore observable and measurable by conventional techniques.¹ Time is yin, polar and measured by the imaginary aspect of sector motion and is unobservable and unmeasurable by conventional techniques. However, the effect that time has on space can be observed and measured as force fields.² The clock³ is considered temporal: clock time.

Cosmic Sector Properties

The cosmic sector is not observable from the conventional frame of reference. However, time is yang, linear and measured by the real sector motion axis and observable and measurable to a Cosmic observer, as coordinate time would be their “conventional” frame of reference. Space is yin, polar and measured by the imaginary aspect of sector motion. The clock is considered spatial: clock space, analogous to the conventional concept of length.

Simulating Yin: Larson's “Rotational Base”

Because Larson's Reciprocal System did not include the yin aspect of motion, he could have “motion without something to move”—providing it was linear motion—but could not have “rotation without something to rotate.” Since the Universe requires⁴ both yang and yin to function, Larson had to come up with a way to simulate the yin aspect of motion, which was lacking in the RS.

To circumvent this lack, he used the linear vibration generated by the “direction reversal” to create a structure that could be rotated, which he termed the rotational base—his version of yin. Even Larson admits the rotational base has no observed physical properties,⁵ which is what you get if you had started with polar geometry to begin with.

Table 1: Comparison of the Derivation of Yin-Yang in RS and RS2

In RS2, neither the direction reversal nor the rotational base are necessary structures, as both concepts are embodied in the fundamental concept of the uniform and dependent motion. All motions identified as photons, particles or and atoms have the same, foundational structure (Larson's photon is not built upon a rotational base) and there is a clear distinction between location and structure. Lex parsimoniae⁶.

Uniform Motion

Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.⁴
—Isaac Newton

Though motion is fundamental to the Reciprocal System of theory, the concept of uniform motion is often overlooked. Uniform motion simply means a constant velocity; the motion does not accelerate or decelerate, which applies to both scalar and coordinate motion. The most obvious example of uniform motion in the RS is the progression of the natural reference system at the speed of light—a scalar motion of 299,792,458 m/s, or 1.0 s/t (in Larson's natural units).

The forceless nature of uniform motion gives things permanence. The progression will never slow down and stop (as in astronomical “Big Crunch” theories) or speed up. The only motions that are permanent are the uniform motions.

Professor KVK Nehru in his paper, “The Law of Conservation of Direction,”⁵ points out that there are actually two forms of uniform motion: linear, the commonly recognized, straight-line motion that is used exclusively by Larson and angular, which is more commonly known as angular velocity. Any object exhibiting a uniform, angular motion will be observed as spinning, from the spin of electrons to the orbits of planets about the sun. This motion is also permanent when no “forces” are acting upon it.

The linear and angular aspects of uniform motion are independent motions, as they do not require any external forces or other motions to maintain their integrity, so they remain in their state of motion permanently.

4 University of Tennessee definition. Newton's first law is: “Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.” (Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.) Modern interpretation excludes the “straight forward” or “in a straight line” clause to account for gyroscopic motion, which would otherwise defy Newton's first law (uniform, angular velocity).

5 K.V.K. Nehru, “The Law of Conservation of Direction”, Reciprocity XVIII, № 3 (Autumn, 1989).

Non-Uniform Motion (Dependent Motion)

When a force⁶ acts upon uniform motion, that motion begins to vary in time (Δv/Δt), resulting in speeding up (acceleration) or slowing down (deceleration). The common term is accelerated motion for both cases—a consistent change in velocity over a period of time. All waves are accelerated motions, including simple, harmonic motion:

A form of periodic motion of a particle, etc, in which the acceleration is always directed towards some equilibrium point and is proportional to the displacement from this point.⁷

Accelerated motions require at least TWO other motions to maintain their behavior, whether periodic or not. For example, if you have a car moving at a constant velocity down the highway (uniform motion), all the other related motions (seats, occupants, etc) are dependent upon the car, but cannot change from that constant velocity unless another motion is introduced, such as stepping on the gas or turning a corner. Therefore:

1. Any motion derived from a single, uniform motion will act in unison with the uniform motion. There can be no change in linear or angular velocity.⁸

2. Any motion derived from a combination of two or more motions can change its velocity, accelerate or vibrate.

These non-uniform motions will be referred to as dependent motions, as they are dependent upon either a uniform motion or another dependent motion to maintain integrity.

The relationship between uniform and dependent motions can be expressed graphically:

The fundamental motions are the uniform motions, which are called the “scalar motions” in Larson's Reciprocal System of theory. However, Larson only considers the linear form of uniform motion in his analysis. RS2, the reevaluation of the Reciprocal System, considers both linear and angular aspects of uniform motion in its structure.

Simple Harmonic Motion (Direction Reversals)

In order to create simple harmonic motion in Larson's Reciprocal System, the outward, scalar expansion of the Universe must reverse itself at regular intervals at specific locations, but not necessarily all locations. Larson gives no reason for this “direction reversal”²³ where, at the end of each “unit” of motion, the progression can just decide to reverse itself and go inward rather than outward—and does it with clock-like regularity. Students have had a problem with this description for decades, citing that a direction reversal would generate a square or triangle wave, not a sine wave, and a truly random function would not be the perfect wave that the photon exhibits. Larson provided no mechanism for this reversal to occur but requires it to build his system of theory.

With the understanding of uniform and dependent motion provided by RS2, we know that simple harmonic motion is an accelerated motion and must therefore be a dependent motion, derived from two or more uniform motions. The RS2 model is based on Nehru's paper, “The Photon as Birotation” that uses two dimensions of uniform, angular motion (rotation) to represent the photon. These uniform dimensions of motion, acting in opposition²⁴, create the simple harmonic motion commonly associated with the sine wave of the photon. Euler's formula²⁵, using complex quantities, can be used to represent this relationship:

Where R₁ and R₂ are the two, uniform angular velocities (rotations) and λ is the resulting dependent motion: simple harmonic motion. Note that the two-dimensional system of rotation is reduced to a single dimension of linear vibration.²⁶ No “direction reversals” required, no question as to the wave shape or randomness of direction and a direct, predictable, and consistent coupling between the scalar dimensions and the resulting wavelike motion.

Simulating Yin: Larson's “Rotational Base”

Because Larson's Reciprocal System did not include the yin aspect of motion, he could have “motion without something to move”—providing it was linear motion—but could not have “rotation without something to rotate.” Since the Universe requires both yang and yin to function, Larson had to come up with a way to simulate the yin aspect of motion, which was lacking in the RS.

To circumvent this lack, he used the linear vibration created by the “direction reversal” to create a structure that could be rotated, which he termed the rotational base—his version of yin. Even Larson admits the rotational base has no observed physical properties,²⁷ which is what you get if you had started with polar geometry to begin with.

Table 1: Comparison of the Derivation of Yin-Yang in RS and RS2

In RS2, neither the direction reversal nor the rotational base are necessary structures, as both concepts are embodied in the fundamental concept of the uniform and dependent motion. Lex parsimoniae.

Anyone who has explored the realm of the science beyond what is taught in the classroom will undoubtedly run across the term scalar, without any consistency of application. Scalar waves, scalar motion, scalar this, scalar that… it appears the term is popular to describe something that the author does not quite understand himself. So, let us start with a clear definition of what the term "scalar" means:

Scalar

"a quantity possessing only magnitude."

Quantity

"an exact or specific amount of measure"

Magnitude

"greatness of size or extent"

From the definitions, a scalar is simply the "specific amount of
greatness." Sounds nebulous, but it is fairly precise and a good definition of
"scalar." First, consider the term amount. It comes from the old trading
days, where people would barter for one "exact or specific amount of measure"
for another. "I'll trade you this sack of sugar for two bags of flour." Amounts
were the counting numbers. There are three attributes of the counting
numbers that make them unique:

1. There is no zero. Suppose I came up to you, and said, "I'll trade you nothing
   for your new Lexus." Sound like a good deal? If so, please contact the author
   ASAP. If not, then you understand why zero is not included in the counting
   numbers. Since they are based in measures, and measures are used in trade, you
   can only trade what you have and if you have zero of it, then it cannot be used
   in trade.
2. There are no fractional parts. "I'll trade you two and a half necklaces for
   three-quarters of your mule." Possible, but pointless.
3. There are no negative amounts. With counting numbers, there must be something
   to count, and there is no such thing has having "-4" cats in the house.

Now that the idea of quantity is understood to be the whole or
counting numbers, consider the term magnitude. How does a magnitude
differ from an amount? In simplest terms, and amount is actually
an amount of something. You can't have just six. You need six
somethings. Amounts qualify other concepts.

But what about magnitude? The magnitude refers to the "greatness of size or
extent", which means that it is the quantity specified in the amount of
measure, the "six" in "six somethings." The "somethings" is not included in the
magnitude, because it doesn't matter what it is, only how many
there are.

And there you have the definition of scalar: "A quantity possessing
only magnitude", which is one of the non-zero, non-negative, non-fractional,
whole counting numbers, without any identification of what they are a quantity
of. The minimum scalar magnitude is therefore one and the maximum is unlimited.

Some people may say that zero and negative amounts are valid, but they are
not part of the counting number system. If the computer at "Cars-R-Us" says they
have "-2" brake pads in stock for you, are you going to walk home with anything?
A promise won't stop your car. Until you have them, for all practical
purposes, "promises" don't exist, and cannot be counted as an item up for trade.

Since we will be dealing totally with the natural systems of reference in
this work, we have to stick to what is "real", not "promises" created by the
inventive mind of man. They don't exist in Nature. Can you have "-1" ocean?

When two scalars are brought into relationship with each other, a ratio is the result.

Ratio

"the relation between two similar magnitudes with respect to the number of times the first contains the second."

As can be seen in the definition of the ratio, the ratio adds the concept of proportion to the concept of magnitude. This gives rise to three possibilities for the proportions of the ratio: one magnitude is either equal, greater or less than the other. This introduces the concept of a scalar orientation.

Orientation

"to adjust with relation to, or bring into relation to surroundings, circumstances, facts, etc."

The three possible scalar orientations for a ratio of scalar magnitudes A and B, as A:B, are: A=B, A>B and A<B. These relations shall be referred to as scalar orientations, since the definition of "motion" requires a place or location and that concept does not yet exist.

Recall that the minimum scalar magnitude is unity, so the ratios of A/B or B/A will never become undefined since neither A nor B can be zero.

Note that in the low and high orientations, the possible combinations of scalar ratios are unlimited. But, where A=B, only one ratio is possible: unity. The scalar orientation structure thus shows a natural separation across a common "scalar boundary" of unity, which can be used as a reference point, a natural datum. This gives a "place" or "location" in which we can begin to define scalar motion, but it has a problem: given any ratio, there is no way to determine if you are observing A/B or B/A… another reference point is needed to determine the orientation of the ratio, itself, with respect to an observer or environment. This can be found in the invariant property of the cross-ratio.

Invariant

"a quantity or expression that is constant throughout a certain range of conditions."

A cross-ratio is literally a "ratio of ratios", and is the only projective invariant in all strata of geometry. In a scalar sense, it relates two scalar orientations through a ratio, and that ratio remains constant—giving a secondary orientation to the ratios and producing scalar motion. One can also think of it as the ratio of slopes between two lines on a graph.

Note: A "projective invariant" simply means that you can't alter it, regardless of what perspective transformation you apply to it. A perspective transformation is the process of introducing assumptions to coordinates, to produce a reference system (such as a plane at infinity, and the "eye cone"). More on this later.

There are now two things to consider as the basis of scalar motion: the cross-ratio and the ratio orientation.

The cross-ratio introduces the concept of association. In a geometric sense, it is like two points joining to form a line, except here you join two ratios to form the cross-ratio. The result is a concept we call "dimension."

Dimension

"a magnitude that, independently or in conjunction with other such magnitudes, serves to define the location of an element within a given set."

There are only two possible ratio orientations, since there are only two scalar magnitudes involved in a specific relationship: A:B or B:A.

Scalar motion is another term that is often used with very little understanding of its meaning. Scalar has already been defined, so let us examine the term motion and its connection with the concept of a scalar:

Motion

"changing place or position."

Motion is a simple enough concept to understand, but when you consider it in the context of "scalar motion", it becomes like "military intelligence"—a contradiction in terms. How is it possible for quantity possessing "magnitude only" to change place or position, when both concepts are totally foreign to the idea of a "magnitude only" scalar? It can't, and there lies the problem with the term "scalar motion."

Exactly what is meant by the term, "motion," when associated with the concept of "magnitude?" The answer is found in how we express the concept of motion as speed—an inverse relation between some "quantity of spatial distance", s, and some "quantity of time," t, as s/t. In other words, speed is just a ratio of space to time and therefore motion, and in a more generic sense is simply a ratio of quantities.

It is important to understand that the concept of motion is a subset of ratio, because ratios deal with magnitudes and motion deals with quantity (magnitudes of something, namely space and time). In essence, we have two similar concepts: that of scalar ratio (generic) and that of scalar motion (specific to space and time).

Scalar Motion is therefore the projectively invariant cross-ratio, with specific aspects of space and time.

In physics, a lepton is a particle with spin-1/2 that does not experience the strong interaction (that is, the strong nuclear force). The leptons form a family of fermions that are distinct from the other known family of fermions, the quarks. There are three known flavors of lepton: the electron, the muon, and the tau.

In chemistry and physics, an atom (Greek ?τομος or átomos meaning "the smallest indivisible particle of matter, i.e. something that cannot be divided") is the smallest particle still characterizing a chemical element.

The atom in the Reciprocal System is different from conventional physics in that it is composed solely of a single, compound motion, not protons and neutrons lumped together into a nucleus and electron cloud.

Atomic properties are not solely due to the compound motion of the atom. The counterspatial fields generated by atomic motion allow the capture of other particles, such as electrons and neutrinos, which account for much of the currently observed behavior of conventional physics (though technically not part of the atom, itself).

The mass of the RS atom is twice its atomic number. Isotopes of that mass occur through the capture of material electron neutrinos, adding 1 AMU per capture.

RS atoms have no electrons as part of their atomic motion. They do have a 1-dimensional rotational component that has "electron" properties. Also, electrons can be captured by the atomic rotations under certain conditions, which result in electron orbitals.

Atomic valences (oxidation states) are just the speeds of the motions making up the atom. These speeds can be 2-dimensional (magnetic) or 1-dimensional (electric).

During the course of examining the difference between the measurements of charged and uncharged electrons in a conductor, something unusual turned up: electrostatic theory. The equations for electrostatic relationships in space-time units are very different from the conventional “electric current” units described by Larson in “Basic Properties of Matter.”

Also, examine the electrostatic unit values, as compared to the natural units. Notice any numerical similarity? (The values for the conventional electrostatic units were taken from Webster’s Unabridged Dictionary, 1966.)

As the table shows, the significant digits between natural and electrostatic units are identical, except for scaling. The measurement for statamps is low by a factor of 10, the measurement for resistance is low by 10⁵ and the measurement for statvolts is also too low by 10⁶. But the scaling relationship is identical... if you increase the statampere measure by 10, and increase the statohm measure by 10⁵, statvolts (according to V=IR) adjusts by 10⁶. We are dealing with the same relationship in conventional electrostatics that we are with the Reciprocal System’s natural units!

But what does it mean, and how does it relate to electric current? From the equations, it looks like we are dealing with an entirely different phenomenon. What is being measured as electrostatic voltage, appears to actually be current! Let’s examine the difference:

The question is: what are we actually looking at, when we look at electrostatic units versus conventional ones? First, it appears that a dimension of “space” has been removed from the equation--in other words, the electric quantity has disappeared. Let’s add it back in, by multiplying the equation V=IR on both sides by (s), which will not change the numerical outcome:

(s) (s/t) = (t/s) (s²/t²) (s)

- reduces to -

(s²/t) = (t/s) (s³/t²)

Low and behold, the typical V = IR equation reappears--but it is inverted! It is actually (1/V) = (1/I) (1/R). When viewed in the inverse sense, the unit measurement returns to non-static conventions, voltage is again a force, current a speed, and resistance a mass per unit time.

I believe legacy science adopted this electrostatic convention for two reasons:

1. When the electron acquired a charge and became electrostatic, the spatial rotation (q) became immeasurable, and was removed from the equations (hence the loss of the “s” term required for force and resistance).
2. Normal current is based on electric quantity, s, and is measured as speed, s/t. But electrostatics measures the associated charge, t/s, instead. Therefore, normal current, “space per unit time” (or speed) suddenly turned into electric charge, “time per unit space” (inverse speed, or energy). In order to keep the equations working, it had to be viewed from the inverse perspective--that of energy, versus speed.

In Larson's astronomy, there is a constant exchange of motion between the two sectors of the Universe. The primary manifestations of this exchange are the cosmic background radiation (CBR, inflowing particles from stellar combustion), and gamma ray bursts (GRBs, inverse-matter ejecta from stellar and galactic explosions moving into the 3-x speed range).

In the simple, scalar picture, it is nothing more than changes in speed, from one extreme to the other. There is no sharing, or composite motion between the two realms. Larson refers to this as "Level 1-inanimate" (also known as "first density").

The Cosmic Doctrine, by Dion Fortune has some interesting concepts in it, that roughly parallel concepts in the Reciprocal System. She does describe a universe based solely on motion, and has many features that are in agreement with Larson. But she carries it a step further with the concept of the Logos, which adds an idea of "intelligence" or "life" to structures such as planets, stars, stellar neighborhoods, galaxies, galactic neighborhoods, supergalaxies and cosmic bubbles.

Nehru wrote:

This has some significance that we need to recognize explicitly. Consider the quantity speed [s/t]. The corresponding pertinent quantity in the counterspace of the Time Region (of the Material Sector) is the INVERSE speed [t/s], which manifests to us as energy, as we already know. Now if we consider the quantity pertaining to the Cosmic Sector, namely, the CONJUGATE speed [t/s], it manifests to us not merely as the material energy, but as PRANA—the self-organizing power of LIFE—apparent in all the living systems.

If one considers Nick Thomas' work on "counterspace linkage" (an integral part of RS2), which is the "speed/energy" relationship of the time region, one must also consider a similar "conjugate linkage" between speed and prana, as Nehru defines it. This has some interesting characteristics.

From a material point-of-view, a cosmic galaxy would not be visible because it is non-local from that perspective. Even though localized in time, would be randomly distributed throughout space. However, what happens if the stars comprising a cosmic galaxy have "conjugate linkage" to the material sector, due to their intermediate and ultra-high-speed motion? In other words, if what the metaphysicians say is true, and structures like stars and galaxies are "living", then they have such a conjugate linkage, because "prana" IS "life."

How does that change things? The points of linkage will be localized in both the material and cosmic sectors, and each sector will respond to that sector's gravity, forming similar, structural relationships. Simply put, cosmic stars and galaxies will be "detectable" in the material sector, and will have localized structure.

I quote "detectable", because they will not necessarily be "visible." The only motions that can move freely between the material and cosmic sectors are the sub-atomic particles, with a net motion of 1 or 2 units of displacement (such that they have no net effect outside of the unit boundary). This limits the possibilities to photons, positrons/electrons, and neutrinos. Positrons and electrons are readily absorbed by matter in the material sector, so they will probably remain undetected. Neutrinos tend to pass through matter, so it would be very difficult to identify their flight paths to get any type of orientation to their source. Photons, however, are detectable and tend to travel very far in space.

But what kind of photons? Remember conjugate relationships. A "visible" cosmic photon would be viewed in the material sector as an X-ray. Radio c-photons would be viewed as Cosmic and Gamma rays. However, photons near the unit speed level, in the infrared according to Larson, would still be detected as infrared—the only difference being that "near" infrared would become "far" infrared, and vice versa, as the unit speed boundary sits between near and far infrared. And this infrared signature, along with X, cosmic and gamma rays, would be localized in the space of the material sector.

Since we are crossing the unit speed boundary, the inter-regional ratio of 154:1 must also apply. Thus, we should observe IR sources that are very massive, in the neighborhood of 150 times larger than your average, supergiant star. Curiously enough, the IRR seems to be a cut-off point for stars, in general:

But we're looking for something very big and red… and that brings us to "infrared protostars". A quick search on the Internet will reveal a large number of papers talking about infrared protostars and their unusual relationship with "intense X-ray emissions"—cosmic "visible light"—something you wouldn't expect for relatively "cool" stellar matter.

Legacy astronomers assume that what they are witnessing is a star in the very early "birth" process. We have now identified the infrared protostar as the "conjugate-prana" localization of a cosmic star. Both situations, however, may be true, because the density gradient of the infrared photons will interact and heat local matter, which may, in time, actually form a material-sector star in the same location.

We also know from "Beyond Space and Time", that this linkage between a material unit and a cosmic unit is what Larson calls a "life unit"—and that when a material star forms at the location of a cosmic star (or vice-versa), it will form a gigantic "life unit" that is known as a "Logos."

On the astronomical scale, these stellar Logoi may be viewed as mere "cells" in a larger organism, a galactic Logos. Therefore, the large-scale structure of the universe should show stars forming into galaxies, galaxies into galactic groups, and groups into superclusters—and all these structures should manifest geometric relationships similar to what we commonly observe as the galactic shapes, defined by age, according to Larson's galactic evolutionary process.

But what of cosmic galaxies, and their infrared protrusions into our local environment? Same rules apply… Inter-regional ratio and structure. However, we should observe, if we can get a telescope large and sensitive enough, super-infrared-galaxies, many times larger than any galaxies we observe in the visible range.

In Reciprocity, Volume 12, No. 3, Summer, 1983, p. 12, Dewey B. Larson writes:

Some of the readers of my latest book, The Neglected Facts of Science, are apparently interpreting the conclusions of this work as indicating that the Reciprocal System of theory leads to a strict mechanistic view of the universe, in which there is no room for religious or other non-material elements. This is not correct. On the contrary, the clarification of the nature of space and time in this theoretical development removes the obstacles that have hitherto prevented science from conceding the existence of anything outside the boundaries of the physical realm.

In conventional science, space and time constitute a framework, or setting, within which the entire universe is contained. On the basis of this viewpoint, everything that exists, in a real sense, exists in space and in time. Scientists believe that the whole of this real universe is now within their field of observation, and they see no indication of anything non-physical. It follows that anyone who accepts the findings of conventional science at their face value cannot accept the claims of religion, or any other non-material system of thought. This is the origin of the long-standing antagonism between science and religion, a conflict which most scientists find it necessary to evade by keeping their religious beliefs separate from their scientific beliefs.

In the Reciprocal System, on the other hand, space and time are contents of the universe, rather than a container in which the universe exists. On this basis, the "universe" of space and time, the physical universe, to which conventional science is restricted, is only one portion of existence as a whole, the real "universe" (a word which means the total of all that exists). This leaves the door wide open for the existence of entities and phenomena outside (that is, independent of) the physical universe, as contended by the various religions and many systems of philosophy.

Inasmuch as the Reciprocal System is a theory of the physical universe only, it arrives at no conclusions as to the validity of the contentions of the various non-scientific schools of thought, but it removes all justification for the assertions that are frequently made to the effect that those contentions are scientifically impossible. Those scientists with strong religious convictions who are now looking askance at the Reciprocal System under the mistaken impression that it envisions a purely materialistic universe should, in fact, welcome it, because it removes the basic conflict between science and their religious beliefs.

Those who are familiar with the study of UFOs are well aware of the fact that “close encounters” with this phenomena often result in the erratic behavior of electrical devices. It is well known that cars stall out and cannot be started—though the engine will often crank—and things like radios, television and kitchen appliances will suddenly turn on and cannot be turned off.

Legacy science has no explanation; and Larson’s RS doesn’t fair too well here, either, but comes a bit closer.

Examine first Larson’s view of electricity, which comes in two varieties:

1. The uncharged electron: being a space displacement, becomes trapped in wire (temporal displacement) and gives us the phenomenon of electric current.
2. The charged electron: still a spatial displacement, but being neutralized half the time by the charge (as a vibration), thus able to jump gaps across the vacuum of space (when charge is effective), and remain trapped in matter (when charge is ineffective). This gives us the phenomenon of static electricity.

In a conductor, the behavior of “current” and “static” electricity is a bit different; electric current, without a charge, will distribute itself evenly over the cross-section of the wire, making the current proportional to the cross-sectional area, which is a known and measured quantity. Static electricity exhibits the “skin effect”, where the “same charge” on the electrons pushes them apart, so they move to the surface of the conductor. In a basic conductor, we will find both forms of electricity, static on the surface, and current in the middle.

In the Law of One material, Ra indicates that the majority of UFOs are actually manifestations from the “4^th density”, our own density being defined as “3^rd”. (4^th density is commonly referred to as “astral” and the 3^rd is “physical”). Therefore, when a UFO enters our realm, it has about it a “field” of 4^th density “energy” (I quote the terms, because they are not technically correct, but relate the concept). Therefore, it is logical to assume that the weird electric effects are a result of this 4^th density “energy field”, which we will now evaluate.

One of the characteristics that the RS2 evaluation brought out concerning electrons is that they can “pair up” or form “triplets”, the triplet being known as the compound motion of the charged electron neutrino, which is also a “rotating unit of space” as Larson describes it (“This charged neutrino is thus, in effect, a rotating unit of space, similar in respect to the uncharged electron, and, as matters now stand, indistinguishable from it.”, Basic Properties of Matter, p. 262). The big difference between the electron and electron neutrino is that the neutrino is a magnetically-rotating (2-dimensional) unit of space, whereas the electron is an electrically-rotating (1-dimensional) unit. But, being a unit of space, it, too, can become trapped in a conductor—and can flow thru matter just like an electron. Larson refers to this as a “magnetic temperature”.

It has been theorized that UFOs use some form of magnetic propulsion. An analysis of the data indicates that the type of “magnetism” it uses is not the common rotational vibration, but the non-local field generated by charged, electron neutrinos moving thru conductors—a “4^th density” magnetic field. This field will, of course, induce a similar movement of electron neutrinos in the matter in the vicinity, and hence cause the electrical problems exhibited. But why do some things shut down, and others turn on?

As indicated above, Larson identified two forms of electricity, charged/static and uncharged/current. The presence of this 4^th density field will try to form charged electron neutrinos by transferring the charges from the static, charged electrons and placing them on existing uncharged neutrinos in the area (since static charges are not confined to conductors, and are free to move). Thus, any device that relies on a “static effect”, such as the spark plugs in a car or a neon sign, will fail! But, devices that rely solely on electric current will continue to work, which is why one can still crank the starter motor on a car during one of these events, but it will not start because it cannot generate a spark.

But what of the charged, electron neutrinos being generated… increasing Larson’s “magnetic temperature”? Since the charged electron neutrino behaves like electric current, everything in the vicinity that requires electric current will suddenly have electric power available, even if it is not plugged in! This effect was noted by Nikola Tesla during his experiments, as well as during UFO close encounters. Radios and TVs turn on, blenders in the kitchen start spinning… usually out of control, digital appliances tend to burn out.

And what other effects does the flow of charged, electron neutrinos have? The charge on a neutrino is inward, not outward, as other magnetic charges. Thus, the location of electron neutrinos in a conductor will be the inverse of it’s static equivalent—they will accumulate at the core of the conductor, not on its surface, forming a thin, intense stream running down the middle of a conductor, which gives us a 3^rd form of electricity, and a 3^rd type of behavior—cold electricity—a type of electric current with no thermal/heat properties, because it’s “thermal” condition is magnetic.

One of the dangers this analysis brings out, however, is summed up by a statement Ra made in Session 63:

“If a third-density entity were, shall we say, electrically aware of fourth-density in full, the third-density electrical fields would fail due to incompatibility.”

“To answer your query about death, these entities will die according to third-density necessities.”

Every advancement has its dangers; fortunately, there are some warning labels.

Distribution of Electron Types in a Conductor

This diagram shows where regions of static (charged) electrons, current (uncharged electrons) and cold electricity (electron neutrinos) exist within a cross-section of a conductor.

I also noted that cold electricity tends to aggregate at the center of gravity of the cross section, and since electron neutrinos are also responsible for isotopic mass, one would expect to find the higher isotopes of an element near its center of mass. From the little I was able to find on the topic on the Internet, this does indeed seem to be the case—right down to the Earth, itself, where the heaviest isotopes occur closest to the center of the planet.

This might change the view of “magnetic temperature” slightly; Larson had attributed it to a field effect (magnetic ionization level), but it may actually be the rotational vibration of the electron neutrinos, concentrating at the center of mass, that causes the increase in magnetic ionization along with the related effects (increased radioactivity, stellar combustion, co-magnetism, thredules, etc).

It is an interesting cascade situation; the presence of neutrinos causes the magnetic ionization level to increase, causing more elements to become radioactive, releasing more neutrinos, increasing the magnetic temperature further, causing more radiation. Sounds like the basis of nuclear fission—another interesting side-effect of “cold electricity”.

See: http://www.subtleenergies.com/ORMUS/

ORME, or monoatomic elements, are metals that exist as a powder rather than a metallic crystal. David Hudson, back in the 1970s, worked with these monoatomic elements discovering some unusual properties, such as their ability to superconduct, aid in healing the body and induce psychic experiences. The monoatomic state of an element, the m-state, is investigated here in light of the RS2 understandings of polar geometry and birotating electron pairs (aka "Cooper Pairing").

Background Information

It is recommended that one read KVK Nehru's article, "Superconductivity: A Time Region Phenomenon?" prior to reading this article, to gain insight into birotating electrons and the mechanism behind superconductivity.

Also recommended is Bruce Peret's article, "Positrons and Electrons" in the RS2 theory website, to understand the dimensional relations involved in creating charged electrons, electron pairs and electron triplets.

Monoatomic Elements

Monoatomic elements are elements that exist as single atoms only, as compared to diatomic elements which exist in pairs. When evaluated in terms of the magnetic and electric rotations of the Reciprocal System, they all fall within a specific range of electric rotations:

• Items listed in green were not on Hudson's list and were added to fill out the table, based on projection.
• First notation after the element name is the metallic crystal structure:
  BC = body centered, Hex = hexagonal, FC = face centered, Rho = rhombic.
• Second notation after the name is the Theosophical "Anu" structure:
  DB = Dumb Bell group, Tb = Tetrahedron group B, Bar = Bar group.
• Whereas the elements in the 4-4 magnetic range do not naturally occur in nature, and were not observed, the crystal structure and Anu structure are unknown.
• Elements below the 3-2 magnetic range have insufficient displacement to have an electric speed greater than 7, which is required for the m-state.

ORME Characteristics

The bulk of the metallic structures are face-centered cubic, followed by hexagonal. Only mercury is rhombic. It is interesting to note that the ONLY "Bar group" element that isn't considered an ORME is iron, which is not on Hudson's list. ALL the remaining Bar group elements are ORME.

There is definitely something about this range of elements that works with biologic structures. The Theosophical Bar group elements are a set of 7 rods piercing a cube, which is virtually the SAME geometric configuration that I cam up with when I plotted the intersection between polar and rectangular geometric of the Cosmic to Material sector linkage, respectively:

Theosophical "Bar Group"	RS2 Diagram of Polar/Rectangular Intersection

Nehru has identified the energy in the Cosmic sector, existing as a conjugate rather than an inverse to the Material sector as prana, the life force. This does appear to indicate that some of the unusual shapes that were seen by the clairvoyant viewing of atomic structure by Theosophists has some validity, as it is the resulting geometry of Material-Cosmic linkage.

Nehru identifies "Cooper pairs" as bi-rotating electrons, which might be exactly how the monoatomic elements work--put simply, the outer "valence" electrons transform from individual units into bi-rotating pairs. Due to the dimensional reduction that occurs when electrons pair (described in Nehru's article), the element will lose its electric valence and act a noble gas element. I notice that ALL the noble gas elements have face centered cubic crystalline structures, the same crystal structure as the bulk of the ORME, so there is some superficial evidence to support this conclusion.

Monoatomic elements are noted for their inability to form a crystal lattice or to bond with anything, existing in a fine powder form only. In the link referenced above, even hydrochloric acid and aqua regia have no effect. The behavior as a noble element would explain this inability to form bonds with other atoms.

The presence of bi-rotating electrons in the outer bands will give the element room-temperature "superconductivity", as described by Nehru's article on superconductivity, being a property of the bi-rotating electrons within the material, rather than a characteristic of the material, itself. This is also exhibited in monoatomic elements when exposed to a magnetic field.

Indirect References to ORME

The monoatomic powder has turned up in other references. During Tesla's radiant energy experiments, his copper wiring would often turn to powder, ruining his experiment. This would indicate that certain frequencies cause pair production in metallic elements making them drop out of the metallic bond state and into a monoatomic form, literally turning the metal into dust.

Notable experimenter John Keely demonstrated resonant devices that were able to reduce solid rock and metals to dust, as an alternative to drilling in mines. Again, this is exhibiting similar properties to Tesla's accidental work by using specific sets of frequencies to dissolve the atomic bonds in metals and crystals.

In a second application, Keely was able to fuse metals together in such a way that no weld could be found--almost as though the metals were blended together as a single piece, without any heat. The transition state between crystalline and ORME could explain this, where the atomic structure was beginning to lose it bond, forming a gel rather than solid powder, thus allowing two pieces of metal to be brought together and fused without heat. Upon removal of the effect causing the transition, the material would return to the crystalline state.

Canadian John Hutchinson reproduced this effect repeatedly during his demonstrations of the "Hutchinson Effect", where objects and metals would exhibit superconducting "anti-gravity" effects, along with objects fusing together without any heat, or one object being pushed through another without any distortion of material. Unfortunately, Hutchinson's work was not easily duplicated.

The RS2 Model of the M-State

The RS2 atomic model is closer to the conventional physics model than Larson's atom. It includes both valence and conduction bands, which are nothing more than speed zones created by the magnetic rotations of the atom. Electrons, being cosmic in RS2, are captured in the speed zone that matches their energy, resulting in zero net motion--captured. They are not an inherent part of the atomic rotation, but due to the extreme number of free electrons in the environment, all atoms will capture their quota in a small amount of time. It should be noted that the number of electrons that can be captured is directly proportional to the net magnetic speed. Therefore, it appears that each atom has the same number of "protons" and "electrons", as conventional science would understand it.

The RS2 explanation proposed to explain monoatomic behavior is that the outer, "valence" electrons are converted into birotating electron pairs (the "Cooper Pairing"). Electrons are cosmic and are therefore adjacent in time, not in space, so when viewed by our technology, the electron pairs will be distributed across the environment, not stuck together.

When electrons form a birotating pair, two things happen: first, the rotational torque associated with a "rotating unit of space" is cancelled out, and the electron becomes a boson, with all the characteristics of a photon. Secondly, as Nehru described in his article, it undergoes "dimensional reduction"; the area that comprises resistance to the movement of the electron is reduced to a 1-dimensional structure, with no resistance and would appear as a HF photon in the category of hard ultraviolet or X-rays.

Electrons, in RS2, only occupy a single, rotational dimension. Birotating electrons, like photons, occupy two dimensions and are still carried by the progression of the natural reference system and thus would move easily from atom to atom. When an electron (1-dimensional) captures a photon (2-dimensional), the resulting structure is 3-dimensional, without any free dimensions to be carried by the progression--the "charged" or "static" electron. However, the birotating electron appears as a photon/boson, a 2-dimensional system, which can be captured as a "charge" by another electron.

Microclusters

When the environmental conditions are set up to form birotating, "Cooper" pairs, the probability is that the electron that captures the "charge" will be part of another birotating system, resulting in a type of covalent bond between them--one electron being shared to make each of the birotating systems appear as a stable, captured triplet, remaining fixed in the valence band.

However, due to the non-locality of the electron pairing, it is unlikely that the other member of a pair will exist within the same atom. Thus, a type of covalent bond will form between atoms. It has been noted in Cooper pairing that the spatial separation is limited; so the neutral, noble-gas like behavior of monoatomic elements will bond together, but as a loose aggregate. The most stable structure of a loose aggregate is a sphere, observed as the microcluster.

Microclusters would have no limit on size, since the bond between atoms is random. However, external influences, such as electric, thermal and magnetic ionization levels, will tend to break up larger clusters since the electron pairs are not as stable as magnetic and electric bonds that one normally finds.

Identification

Identification of monoatomic elements would be difficult, since most test equipment determines the element number based on the number of electrons present (assuming the same number of protons present). Bi-rotating electron pairs appear as PHOTONS, not electrons, and hence would be invisible to that type of testing equipment, resulting in the appearance the the element is of a substantially lower atomic number.

Tests that involve chemical bonding would also be useless, because of the inability of ORME to form chemical bonds.

Spectrophotometers would have a similar difficulty, because the electron triplets forming the aggregate bonds would not be able to absorb and release photons to change energy levels, thus making those spectral lines invisible.

The only way an ORME could be properly identified would be through the rotations of the atom, itself. The magnetic rotations will express themselves through magnetic valence states, but without an ability to directly measure the electric rotation, it is unlikely that the ORME can be identified as their proper elements, unless the element is known prior to conversion.

Reintegration

Hudson, when creating the m-state, said it was like converting an apple to applesauce... once the conversion had been accomplished, it could not be reverted. This may not necessarily be true.

Unlike converting applesauce to an apple, elements do not need to retain their original pattern. Based on Keely's research, sympathetic resonance can be used to create the birotating electrons, so therefore discordant resonance should be able to destroy the pairing, returning the material to a crystalline powder, with all of its original properties.

The “zero point” is the energy flux that occurs within the vacuum of space. Legacy science refers to it as “zero” because it should be motionless; nothing should be there—but there is. It is often described as a “sea of electrons”, which pop into existence then back out again, making the capture of electrons a bit tricky since you never know when they will appear, disappear, or even where the event will happen.

Within the projective geometry model of RS2, the region of the observer appears Euclidean, but the regions beyond any unit boundary appear as the geometric inverse, polar-Euclidean, or commonly called “counterspace”. The space/counterspace interaction creates a set of “duals” between the geometric objects within the region. In space, we see a point, but in counterspace, we observe a plane. A planar construct in space is then viewed as a point in counterspace, BUT a line is a line in both regions. This linear intersection, therefore, is the geometry underlying the “zero point” domain.

It makes sense if you consider that the zero point flux is referred to as a “sea of energy”. We know from Larson that “energy” is inverse speed, namely t/s. Speed in space is translational; speed in counterspace (time) is rotational. The linear intersection of translation and rotation is the projection of rotation upon a line, or the projection of a line upon a rotation—namely, linear vibration.

This sea of energy is, therefore, a sea of vibration… not a “zero point” but a “unit line”.

In RS2, the first manifestation of motion is the m-positron or c-positron (electron). They are both simply ’rotating units’ in counterspace (or countertime), without sufficient magnitude of motion to make any effect outside of their unit time (or unit space) boundaries. Also consider that counterspace is the full geometric inverse of space—with a plane at zero and a point at infinity (the center). This means that the first displacement of speed is physically located ON the unit boundary, extending inward. Any displacement, even those that do not have sufficient magnitude to have an outside effect, can still be seen at the unit boundary.

Given that we are not actually looking at “zero space” but at “unit space”, in a vibrational aspect, one would expect to see a totally stationary system, with the material and cosmic sectors perfectly divided at this sub-atomic level. So how does one account for “flux”?

From observation by experimenters, the zero point is a fluxuating sea of electrons and positrons, winking in and out of existence. But DO they? In RS2, NO. Only the projection upon the measuring instruments is winking in and out; the physical motion, itself, remains constant. The difficulty occurs because of the intersection between space and counterspace—the resultant linear vibration has no orientation. Therefore, at times we view it head-on, and see an electron or positron. Other times, when viewed from the side, it appears as a wave. Other times, when viewed end-on, it disappears entirely. But it is not the motion that disappears! It is the projection of that motion into our coordinate space that disappears.

Based on a new understanding of this “unit line” sea of energy, and the fact that it is the projection that is causing the problems with utilizing it for a source of power—not the motion itself—a new approach can be taken to examine the zero point energy devices, determine exactly how they work, and what can be done to eliminate their problems and increase their efficiency.

The following table defines the aspects and powers of units of space and time commonly associated with mechanical, electrical and other phenomenon.

Mechanical

Electrical

* "q" is considered the same as "Q" (charge) in conventional science

Magnetic

** Magnetic charge is not recognized.

Inside the Unit Boundary

TR=Time Region; SR=Space Region